TABLES AND FORMULAE 
 
 FOR 
 
 THE COMPUTATION 
 
 OF 
 
 LIFE CONTINGENCIES; 
 
 WITH 
 
 COPIOUS EXAMPLES OF ANNUITY, ASSURANCE, AND 
 FRIENDLY SOCIETY CALCULATIONS. 
 
 SECOND ISSUE, WITH AN ADDENDUM, 
 
 COMPRISING A LARGE EXTENSION OF THE PRINCIPAL TABLE. 
 
 BY 
 
 PETEE GEAY, P.B.A.S., F.B.M.S., 
 
 HONORARY MEMBER OP THE INSTITUTE OP ACTUARIES J AUTHOR OF TABLES FOR THE 
 FORMATION OF LOGARITHMS AND ANTILOGARITHMS OP TWELVE PLACES, &C. 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 CHARLES AND EDWIN LAYTON, 
 
 150, FLEET STREET, E.G. 
 1870. 
 
0. & B. LAYTON, PIIINTEHS, 150, FLEET STREET, 
 LONDON. 
 
To AUGUSTUS DE MORGAN, ESQ., 
 
 OK TR15ITY OOLLBGH, GAMBQIDGB, 
 S-.B.A.S., F.C.P.B., &C. 
 
 Dear Sir, 
 
 Accept my grateful thanks for your having so promptly 
 acceded to my request to be allowed again to place your name in the 
 front of my work. 
 
 But for the assistance derived from your writings the work would 
 never have been undertaken ; and to that assistance mainly it owes 
 whatever of utility it has been or may be found to possess. 
 
 I am, Dear Sir, 
 
 With much respect, 
 Your obedient and obliged Servant, 
 
 P. GRAY. 
 
 
EEEATA. 
 
 Page 33, line 13 from bottom, for " six favour the occurrence of black," read " six 
 favour the occurrence of white, three favour the occurrence of 
 black." 
 
 40, 10 from top. The probability here assigned is mis-named. It really 
 designates, (and is so referred to, p. 78,) the probability that the 
 joint existence of (#) and (y) will be desolved in the nth year, an 
 event which may arise by the decease in that year of either (x) 
 or (y\ or both. 
 
 45, In the Elementary Table, log. (p^-l) should be '654119. 
 
 46, 24 from top, for log. (l+r), read % log. (l+r). 
 
 93, 11 from bottom, for '568989, read '586989. 
 
 110, 7 from top, for 1, 2, 6, 10, &c., read 1, 3, 6, 10, <tec. 
 
 122, 9 from top, for A log. (p x+ i 1 ), read A log. (p x +i~ l - l ) 
 
 TABLE I. 
 
 15, Diff. 83. Pro. parts for 76 should be 63. 
 
 19, in T! [0-8991] /or 0798 read 0698. 
 
 35, in Tx [1-6578] for 7226 read 7246.* 
 
 * This is he only additional error which twenty years' use of this table has brought to 
 light. 
 
PREFACE. 
 
 THE subject of the construction of the requisite tables is one of no 
 small importance in connexion with the doctrine of life contingencies. 
 There is nothing so effectual for imparting facility and confidence in 
 the use of tables as some experience in their actual construction, even 
 although the more immediate result should be nothing more than 
 the reproduction and verification of tables already existing: while, 
 moreover, necessity for the formation of new tables frequently arises. 
 It is desirable, therefore, to have the best methods of proceeding 
 pointed out and exemplified ; and as methods of constructing tables 
 in aid of life calculations have been in a great measure overlooked by 
 preceding writers, it is hoped that a work devoted to their considera- 
 tion will not be unacceptable. 
 
 The present work claims to afford greater facilities for the formation 
 of tables of life contingencies than have been hitherto at the command 
 of the computer. What these are will be learned from the following 
 brief enumeration of its contents. 
 
 Chapter I. contains a description of the appended Tables, with the 
 manner of using them, and the necessary exemplifications. The pri- 
 mary object of these tables is to afford the means of determining the 
 logarithm of the sum or the difference of two quantities whose loga- 
 rithms are known, without the preliminary labour of determining the 
 quantities themselves. The idea and the first formation of ta,bles for 
 this purpose, are due to Professor Gauss, of Gottingen. Gauss's 
 Tables were published in 1812, and they have been since repeatedly 
 reprinted in Germany. More extensive tables of the same kind were 
 also published by E. A. Matthiessen in 1817, at Altona.* 
 
 * An account of the various publications referred to will be found in the 
 Supplement to the " Penny Cyclopaedia," articles, " Logarithms, Gauss's," and 
 "Table." The dates to be consulted in the article last named are, 1812, 1817, 
 1821, 1832, 1840, 1845. See also four papers, by the present writer, in con- 
 junction with Mr. Wcollgar, in the " Mechanics' Magazine " for June and July, 
 Captain Shortrede's valuable ami extensive Tables, about to be re-issued, 
 contain the tables of Gauss, with certain modifications. 
 
VI PREFACE. 
 
 In all previous tables of this kind, (with the exception of those of 
 Mr. Sheepshanks, referred to in the preceding note, under date, 1845,) 
 the tabulated functions, corresponding to values of log.<r, by the aid 
 of which the objects in view are to be effected, are log. (1 -fa?), and 
 log.(l + 1-T-a?) ; and these are arranged, along with the argument, log.#, 
 in parallel columns. In the present tables the tabulated functions, 
 corresponding to values of log.#, are log. (1 + #) and log.(l #), which 
 moreover, are formed into separate tables. The advantage of the 
 change is, that we are thereby furnished with a simpler operation for 
 the formation of log. (a b) -, and that of the separation into distinct 
 tables is, that it permits the adoption in each of what may be called 
 the standard arrangement, the importance of which soon comes to be 
 known and felt by all users of tables. 
 
 Both tables have been computed expressly for the present publica- 
 tion. This was obviously necessary in the case of Table II., which, as 
 has been stated, has no prototype. But the necessity is less obvious 
 in the case of Table I. Gausses Tables extend only to five places, but 
 Matthiessen, in one of his columns, has given log.(lH-#) to seven 
 places, for the same values of log.a? that form the argument series in 
 this table ; and it might therefore seem to have been sufficient, for 
 the formation of a six-figure table, to adopt Matthiessen's values, cur- 
 tailed of their last figure. But it is impossible to form a six-figure 
 table which shall be always true to the nearest unit in the last place, 
 from a seven-figure table which is in like manner true to the nearest 
 unit in the last place. For, an increase of a unit in the sixth place 
 being due only when the seventh figure is not less than 5, when we 
 find a seven-figure value ending with 5 we require to know whether 
 the seventh figure is truly 5, or has only been made so in consequence 
 of the eighth being not less than 5. When to this it is added that a 
 good many errors were found in Matthiessen's table, the propriety of 
 reconstruction will be apparent. 
 
 It was intended to have given in this place an account of the methods 
 employed in the construction of these tables. Several theorems of 
 great practical utility in the Calculus of Finite Differences having been 
 deduced, and many contrivances devised, which had the effect of most 
 materially lightening the labour, and of which no previous account 
 has been given, a detail of them here might have been of service to 
 others who may hereafter engage in work of a similar kind. But 
 the prescribed space being already more than exhausted, this must be 
 reserved for another opportunity, and the merest outline must suffice 
 for the present. The following description applies to Table I. 
 
 The method employed was the differential method; and the funda- 
 mental values were formed by the aid of Dodson's "Anti-Logarithmic 
 
PREFACE. "VII 
 
 Canon ."* From that work, which gives the numbers to eleven 
 places corresponding to every logarithm of five places, the numbers 
 corresponding to the logarithms 0-00,0'01,0'02, &c., up to T99, were 
 taken out. Unity was then added to each, and the logarithms of the 
 numbers thus increased were formed, to ten places, by the inverse use 
 of the same table. There were thus obtained 200 values at equal 
 intervals. Interpolation was then employed, three values, by the use 
 of four orders of differences, being inserted in each interval, which was 
 thus divided into four smaller equal intervals, and the whole table 
 consequently into 800 intervals. Interpolation by two orders of diffe- 
 rences then sufficed to fill up the table, a theorem being employed in 
 the formation of the several second differences which gave the same 
 order of accuracy as would have resulted had the second series of 
 intervals been previously bisected. Fifteen places of figures were 
 employed in the first series of interpolations, and twelve in the second. 
 
 By the arrangements that were made for verification at every point, 
 it was rendered all but impossible that any error committed should 
 fail to show itself, if ordinary care were exercised ; and as it is proved r , 
 that at the most unfavourable part of the table the error incident to 
 the process of interpolation is barely sufficient to affect the tenth place, 
 the interpolated values may be considered as equally correct with those 
 formed by the more laborious processes above mentioned.f 
 
 In the methods heretofore laid down for extensive series of interpo- 
 lations, it has been usual to assign the values of the differences of the 
 various orders required, in terms of what may be called the root of the 
 function whose successive values are to be tabulated. J Formulae so 
 expressed always possess more or less of complexity, and are conse- 
 quently laborious and subject to error in their numerical applications. 
 Now, since in the employment of the method of differences, values, at 
 
 * See Note, p. 19. 
 
 t Matthiessen's Table is not properly a seven-figure table ; since, although 
 the tabular values extend to seven places, the argument values have only four, 
 as in the present table ; and, as a consequence, in using it three figures have to 
 be proportioned for, which is a tedious and troublesome operation, and attended 
 with considerable risk of error. A proper seven-figure table would be one in 
 which the tabular values should extend to seven places, and the argument values 
 to five. Such a table, with an argument ranging from OOOOOO to 2-00000, 
 would be ten times the extent of the present table, or nearly twice the extent of 
 Babbage's, and therefore its formation is not likely to be soon undertaken, more 
 especially as six figures are sufficient for all ordinary purposes. But should it 
 be wanted the materials now accumulated would be immediately available for its 
 construction. Interpolation by a single order of differences would suffice for a 
 great part of the table, and a slight modification of the same operation for the 
 remainder. 
 
 J See Airy's Trigonometry, in the " Encyclopaedia Metropolitana," pp. 702 
 708 ; Lardner's " Differential and Integral Calculus," p. 490 ; and Young " On 
 the Computation of Logarithms," pp. 38-48. See also the Introduction to 
 Matthiessen's Tables, passim, and the Introduction to Callet's Tables, pp. 57 69. 
 
Vlll PREFACE. 
 
 stated intervals, belonging to the series to be formed, must be pre- 
 viously known, the inconveniences referred to may be always avoided 
 by deducing expressions for the required differences in terms of these 
 values, or rather of their differences. This was the process here em- 
 ployed. But comparatively simple as the resulting formulae were, 
 they were only had recourse to for the purpose of occasional verifica- 
 tion. The differences and other functions wanted for the second and 
 more extensive set of interpolations, were all formed in series, and by 
 mere addition. 
 
 The methods employed for the formation of the second table did 
 not differ materially from those made use of for the formation of the 
 first. On this point, therefore, it is unnecessary to be more particular. 
 
 In the construction of Table I. the author had no assistance till the 
 whole was prepared for the final additions, by the formation in their 
 places of the entire series of first differences, and the insertion of the 
 values for verification. When this point was reached he gladly availed 
 himself of all the competent assistance he could conveniently command. 
 Accordingly, for 1200 of the final additions he has to express his obli- 
 gations to Mr. J. W. Woollgar, F.R.A.S., a gentleman whose skill in 
 calculation is well known ; for the same number he is indebted to Mr. 
 Alexander Yeats, F.I.A., Actuary of the Aberdeen Mutual Assurance 
 and Friendly Society ; for a like number to Mr. Thomas J. Coles, a 
 relative of his own; for 7200 his acknowledgments are due to Mr. 
 Henry Ambrose Smith, F.I. A., Actuary and Secretary of the Northern 
 Assurance Company in Aberdeen; and the remaining 9200 were 
 performed by himself. 
 
 The only assistance he has to acknowledge in the construction of 
 Table II. is the formation, by Mr. Woollgar, of sixty of the funda- 
 mental values. 
 
 The author is farther indebted to Mr. Yeats for a reading of both 
 tables with the printer's copy. They were subsequently read with the 
 original computations ; but he is satisfied that to Mr. Yeats' s skill and 
 care users of the tables will be indebted for the accuracy by which it is 
 hoped they will be found to be characterized. A twelvemonth's tole- 
 rably extensive use of them, in operations admitting generally of inde- 
 pendent verification, has detected one error, which is elsewhere noticed ; 
 and it turns out to be an error, not of the printer, but of the computer, 
 insufficiently corrected. 
 
 A word now as to the typographical details in these tables. First. 
 It will be observed that Mr. Babbage's mode of indicating the point 
 at which the leading figures of the tabular values change, has been 
 adopted. This was done after mature deliberation and several trials, 
 
PREFACE. IX 
 
 the result of which seemed to prove that it is, on the whole, the most 
 effectual for the end in view. 
 
 Secondly. The leading figures are in all cases separated by a space 
 from the adjoining figures. They are, in consequence, much more 
 easily picked out in the rapid using of the tables. The want of some 
 such separation is much felt in using Babbage's or Mutton's Tables. 
 
 Thirdly. The horizontal lines across the page, after every tenth 
 line of the table, have been restored, notwithstanding the typographi- 
 cal difficulty assigned by Mr. Babbage as his reason for omitting them. 
 They will be found considerably to facilitate the use of the tables. 
 
 Fourthly. The leading figures are re-inserted, when necessary, 
 after the horizontal lines just referred to ; so that the eye has never 
 far to travel in search of the leading figures belonging to any given 
 tabular value. 
 
 The peculiar arrangement of the proportional parts in Table I. de- 
 mands a moment's notice. The arrangement in question was adopted 
 from necessity and not from choice ; and it is a fortunate circumstance 
 that the use of the table is thereby facilitated rather than obstructed. 
 The reason may be briefly explained. 
 
 It is sufficiently well known that in proportioning for two figures 
 by the ordinary tables of proportional parts, the results are not in all 
 cases true to the nearest figure. In point of fact, they do, as often as 
 not, differ from the whole number nearest to the truth by a unit, 
 either in excess or defect. If the probabilities of deviation in both 
 directions be equal, but little inconvenience will arise from this cause, 
 even in continuous operations, where each result depends on that 
 which precedes. But if these probabilities are different, if the cases 
 which give rise to deviations in one direction are more in number 
 than those that give rise to deviations in the opposite direction, then 
 we must expect that in continuous operations a tendency will manifest 
 itself to deviation in that direction for which there exists the greatest 
 probability in the individual results. 
 
 Now the differences that belong to the first half of our Table I., 
 namely, 50 to 90, possess the property, when arranged in tables of 
 proportional parts of the usual form, of giving results which, when 
 they differ from the whole number nearest to the truth, do so as often 
 in excess as in defect. So far as this part of the table is concerned 
 then, the ordinary form of the proportional parts might have been 
 retained. But the case is otherwise with respect to the differences 
 which belong to the latter half of the table. Of these, when arranged 
 in the ordinary way, the first three A 91 to 93, give results whose devi- 
 ations from the whole number nearest to the truth arc oftener in 
 
X PREFACE. 
 
 excess than in defect ; and the remaining differences, 94 to 99, give 
 results whose deviations are always in excess. 
 
 In the first trials of the table, in manuscript, with the usual pro- 
 portional parts, the effects of this were abundantly apparent. The 
 operations were continuous, and while the former part of the table was 
 in use the results were satisfactory; but when the latter part was 
 reached, a tendency to deviation in excess soon began to manifest itself. 
 This became more and more decided as the work proceeded, and at the 
 close the later results were found to be in error as much as 6 or 7 in 
 the last place. Careful investigation showed that the cause of this was 
 that which has just been explained ; and the remedy which suggested 
 itself was the formation and employment of tables of proportional 
 parts which should give their results by a single entry, and in all 
 cases true to the nearest figure. This plan has been successful. The 
 tendency to permanent and increasing deviation in one direction has 
 been completely removed. The occasional deviations from the truth 
 that now occur are sometimes in the one direction and sometimes in 
 the other, and they are certainly neither greater nor more frequent 
 than may be expected to arise from the application to like purposes of 
 the common logarithmic tables of six figures.* 
 
 The first intimation, in our own language, of the existence of Sum 
 and Difference Logarithm Tables, is contained, the author believes, in 
 a paper by Mr. Woollgar, in the " Mechanics' Magazine" for Sept., 
 1830. They have been since noticed, as already stated, in the Sup- 
 plement to the " Penny Cyclopaedia ; " but the only illustrations that 
 have yet been given of their practical applications, and especially of 
 their very remarkable adaptation to annuity computations, are to be 
 found in the series of papers referred to in the note, page v. The 
 numerous additional applications, shown in the present work, are the 
 results of subsequent investigation. 
 
 In Chapter II., the leading results in the Doctrine of Compound 
 Interest are briefly deduced, and compendious methods of forming the 
 various tables for their application to practical purposes are explained 
 
 * Extensive use of Table I. enables us to assign the following statistics of the 
 amount of deviation that may be expected to arise in continuous operations. An 
 error in the 6th place of j: 1 is of not unfrequent occurrence ; (see the example 
 pp. 8, 9, and remark 4 upon it;) + 2 also occurs, but much less frequently; and 
 in rare cases an error of + 3 has been observed. The results of seven-figure 
 logarithms being here taken as the standard of comparison, it is not easy to say 
 how much of the deviations observed is due to the curtailment of the seventh 
 figure. The errors in the natural numbers will not in general be greater than 
 those in the logarithms ; and it may be mentioned that the application of the same 
 standard to the Carlisle 3 per cent, annuities, as given in Milne and Jones, shows 
 that of 62 values consisting of five figures, 31, (exactly one half,) err by a unit in 
 the fifth place. Yet these were formed by seven-figure logarithms, but without 
 interpolation. 
 
PREFACE. XI 
 
 and exemplified. This chapter, it is hoped, will furnish the young 
 computer with some useful hints. The first examples of the practical 
 applications of the tables occur in this chapter; but we defer to 
 another place such observations as may appear to be called for on this 
 subject. 
 
 In Chapter III., the fundamental principles of the Theory of Pro- 
 bability are laid down ; the manner in which that theory is applied to 
 the determination of the probabilities of human life is explained ; and 
 the expressions connected with it, which are required in subsequent 
 parts of the work, are deduced. 
 
 Chapter IV. is devoted to an exposition of the properties of the 
 Table of Elementary Values, and the manner of applying it for the 
 formation of the various auxiliary series that are required. Of the 
 four series of which, with their differences, this table consists, the first 
 two only have been heretofore tabulated in connexion with any rate of 
 mortality. The increased power which the possession of the appended 
 tables confers, and the new formulae of computation to which the 
 acquisition of that power has given birth, render the formation of the 
 two remaining series well worth while, in connexion with any rate of 
 mortality which is to be made the foundation of extensive calculations. 
 
 The principal series in the Elementary Table were formed as follows : 
 The fundamental logarithms are those of l xi 2(^ + ^+1); an( ^ l x ^+i> 
 with the arithmetical complement of the former. The whole of these 
 were first cut down to six places, to secure the requisite correspondence 
 amongst the several values to be deduced from them. Then, 
 since p x = l x+l -*-l x , /. log.^=log./, 
 
 Hence also, colog.j9, c =log./, r +colog./, l . + 1 =ar. co. 
 
 VH 
 
 Again, 
 
 Also, 
 
 By these formulae the individual values were constructed ; and their 
 differences being then taken, the table was completed.* A, fitting 
 addition to the table would be the logarithms of l x l x+ i, with their 
 differences, which would facilitate the operation suggested in (413). 
 
 The methods explained and exemplified in this chapter, for the 
 formation of the various combined series, by means of the differences 
 of their component series, are newf; and it is believed, that by their 
 
 * To ensure accuracy the table was constructed in duplicate, the author's 
 coadjutor in the work being Mr. Henry Ambrose Smith. 
 t Since they were matured the author has found that the principle they in- 
 
Xll PREFACE. 
 
 employment the labour requisite for the construction of those series is 
 reduced to an absolute minimum, while means are afforded of securing 
 their perfect accuracy. 
 
 Chapter V. treats of Mean Durations of Life, (Curtate and Com- 
 plete,) and Probabilities of Survivorship ; and methods of computing 
 these, by continuous processes, are explained and fully exemplified. 
 Curtate mean durations have been frequently ere now formed continu- 
 ously, and by the same formulae as are here employed, but applied by 
 means of the common tables. So also have probabilities of survivor- 
 ship, but by other and much less commodious formulae than the one 
 here made use of.* The complete mean duration of one life is now for 
 the first time computed continuously, by the aid of one of the newly 
 tabulated series. 
 
 In this chapter the typical method of indicating the manner of 
 conducting the various operations is first employed. This method, 
 when the principle of it is understood, will be found to afford all requisite 
 facility for the end in view. If difficulty is felt, let the letters desig- 
 nating the several lines in the type be placed against the corresponding 
 lines in the operation typified ; then, remembering that the terms of 
 the subsidiary series are supposed to have been at the outset inserted 
 in their places, reference to the type will show how the blanks are to 
 be filled up. 
 
 Chapter VI. is occupied with the subject of Benefits on Lives; and 
 the manner of forming continuously tables of the values of the prin- 
 cipal of them is fully explained and exemplified. Independently of 
 the application to this purpose of Tables I. and II., many of these 
 benefits are now for the first time shewn to be susceptible of this mode 
 of formation. This is effected by means of the Lemma in (193), which 
 is a generalization of the famous formula assigning the value of an 
 annuity upon (an) in terms of a like benefit upon (a?+ 1), by the aid of 
 which most, if not all, of the existing tables of annuities have been 
 formed. t The relation upon which this formula is founded is here 
 
 volve was employed by Mr. Barrett in the construction of his joint life commu- 
 tation Tables. See Baily's " Appendix," pp. 41, 42 ; or De Courcy's French 
 Translation of the same, pp. 272, 273. But Mr. Barrett was obviously unaware 
 of the power of the method he thus indicates. The only series to which he 
 applies it is log. l x , and the purpose is to verify a series formed by another 
 method. It is here applied to many series; and recourse to any other method 
 is rendered unnecessary, except for the purpose of occasional verification. 
 
 * For previous formulae see Morgan, pp. 185, 187; Baily, (French Transla- 
 tion), pp. 95, 96; Milne, p. 73; Jones, pp. 151, 152. Mr. Jones's formula is 
 an improvement upon Mr. Milne's, and the present is a farther improvement in 
 the same direction. 
 
 t This formula, as improved by Mr. Milne, (who was the first to tabulate 
 log.^, by which its application is much facilitated,) will be found in (202) ; and 
 its history is somewhat singular. Mr. Milne (Introduction, pp. xv, xvi,) informs 
 
PREFACE. Xlll 
 
 shown to hold almost universally, and to be commensurate in its 
 practical applications with the power of forming, from tabulated values 
 for single lives, the requisite subsidiary series for joint lives. We do 
 not stop to enumerate the cases to which, for the first time, the con- 
 tinuous method of computation is here applied. These will be appa- 
 rent to all acquainted with the subject. But we add a few remarks 
 explanatory of the advantages gained by the application, in all such 
 cases, of the appended tables, more especially of Table I. 
 
 It will be noticed that the principal part of the operation by which 
 we pass from each value to the next, is the formation of log.(l +x) 
 from log.,3?, the latter logarithm being either the value last found, or 
 the difference between that value and a given logarithm. To effect 
 this by the common tables two entries are necessary ; first, an inverse 
 entry, to determine the number x from its logarithm, and then a direct 
 entry, to determine the logarithm of ! + <# from the corresponding 
 number. By the new table a single entry suffices for the purpose in 
 view, since the table in question enables us to pass at once from log.# 
 to log.(l +#) Hence, when oc is a number which it does not concern 
 us to know, a saving of one tabular entry for each value determined is 
 effected by the employment of the new table. 
 
 In the case of one, and only one, of the formations with which we 
 have to do, namely, that of a uniform annuity, a? is a number which 
 it may concern us to know ; so that if the values are wanted in num- 
 bers, we have not in this case the advantage just claimed for the new 
 table. But in all other cases we do not care to know what x is ; and 
 therefore, whether the values be wanted in logarithms or in numbers, 
 the saving of one entry for each value to be determined is effected by 
 the employment of the new table. 
 
 In the annuity operation, however, as in all the others, when the 
 new table is used, we possess the power of stopping short with the 
 formation of the logarithmic values, a power which, in the case in 
 question, the employment of the common tables does not afford. 
 
 us that^we are indebted for it to Thomas Simpson, who first published it in the 
 year 1742; and that Euler, in apparent ignorance of what Mr. Simpson had 
 done, afterwards gave it to the world as original in 1760. Subsequent writers 
 have repeated Mr. Milne's account of the parentage of the formula, some of 
 them however, as Mr. Galloway, (Treatise on Probability, p. 93,) attributing it 
 entirely to Euler. But it has been recently shown by Mr. Farren, in the work 
 cited in (384), that the originator of the formula in question was no other than 
 the illustrious De Moivre, in the first edition of whose Treatise on Annuities, 
 published in 1725, it is distinctly laid down and explained. The singularity of 
 the case does not cease here. Useful, nay indispensable, as modern computers 
 have found this formula, we search in vain in the third (the "last and best") 
 edition of De Moivre's Treatise for anv trace of it ! 
 
XIV PKEFACE. 
 
 Hence, universally, since the logarithms are nearly, if not quite, as 
 useful as the numbers, we may consider that a saving equivalent to 
 that of one tabular entry for each value determined, is effected by the 
 use of the new table . 
 
 But the advantage does not cease here. When the common tables 
 are used, the different kinds of entries requisite direct and inverse 
 are intermingled, whereby the operations are rendered exceedingly 
 irksome and liable to error. This intermingling is entirely avoided 
 by the use of the new table : entries of the same kind are brought 
 together,* and there need be no inverse entries at all if the final 
 taking out of the natural numbers be performed by means of Captain 
 Shortrede's Anti-Logarithmic Table. 
 
 Chapter VII. is occupied with the subject of Commutation Tables. 
 The leading properties of these tables are laid down, and methods for 
 their formation, for both single and joint lives,t are explained and 
 exemplified, which it is believed will be found more certain and efficient 
 than any heretofore proposed. The Postscript contains two additional 
 methods, which suggested themselves since the chapter was completed. 
 
 Chapter VIII. contains some details in reference to the history of 
 the Commutation Method of computation ; and an attempt is made 
 to vindicate Mr. Barrett's claim to the invention of that method, of 
 which it has been recently sought to deprive him. 
 
 We close with one or two general remarks. The Carlisle Table of 
 Mortality, and 3 per cent, interest, have been probably more extensively 
 employed as the bases of calculation than any other existing data, and 
 a proportion ably greater number of results have been tabulated in 
 connexion with them. The facility thereby afforded for comparison 
 and verification of our results, has been a principal reason for the 
 selection of the data in question as the basis of the examples in the 
 following work. But, to guard against misconception, it is necessary 
 to remark, that some of the values with which ours may come to be 
 compared, have been found to be very incorrect. Thus, some of the 
 mean durations given by Mr. Jones, in his Table XL., pp. 948, 
 &c., are altogether different from those for the same ages formed here 
 
 * Of the facility thus afforded an idea may be formed when the author men- 
 tions, that he finds the formation of from 140 to 150 logarithmic annuity values 
 in an hour to be a fair rate of working. This rate has been surpassed by 
 Mr. Orchard, a friend of the author, who has had the tables in use for some 
 time. Mr. O. finds that the formation of 200 such values in an hour is a rate 
 of working that can be maintained. The operations in which a subtraction 
 occurs occupy of course a little more time. By the operation in (255) 120 loga- 
 rithmic survivorship assurance values have been formed in an hour. In all the 
 cases mentioned the terms of the auxiliary series were previously inserted. 
 
 t The illustrations apply to both Davies's and De Morgan's forms of the joint 
 life tables. 
 
PREFACE. XV 
 
 on p. 62. Mr. Jones's adjoining columns afford the means of deter- 
 mining which set of values is the correct one. Again, Mr. Jones's 
 Table XXII., pp. 538, &c., contains the values of assurances by the 
 Carlisle rate of mortality, at eight different rates of interest ; and the 
 result of an examination of the 3 per cent, column shows, that in the 
 first half of that column the values err generally by about 3 or 4 in 
 the fifth place, and in the remaining half usually by 1 or 2.* These 
 are just such deviations as would arise from the employment of the 
 incorrect three-decimal annuity values, (see ante, p. x.,) by which to 
 deduce the assurance values.f In all cases of discrepancy, we appeal 
 from Mr. Jones's results to his data. 
 
 In Mr. Sang's very remarkable work the 3 per cent, assurances are 
 also given. But as they are there, to reduce them to the moment of 
 death, multiplied by the constant factor, Vl+r, his values cannot be 
 directly compared with those formed in our examples, p. 93. 
 
 In the note p. vii., the practicability of forming with facility, from 
 the materials now accumulated, a seven-figure table of log.(l-f x), is 
 mentioned. Such a table would no doubt be very useful, but from 
 its great extent, (200,000 values,) it is not likely to be soon under- 
 taken. A much more practicable undertaking would be an addition 
 to the table here given of half its present extent, making it commence 
 with argument 1 instead of 0. Such an extension would enable us to 
 dispense with the double form of the types of operation, except for 
 one or two values in the case of an annuity on three or more lives. 
 Could the author venture to anticipate the necessity for a reprint of 
 his work the extension in question should certainly be made. 
 
 37, BAKER STREET, LLOYD SQUARE, 
 
 2 1st February, 1849. 
 
 * One value differs much more widely from the truth. Age 25, '36808 should 
 be '36894. 
 
 t A portion of Mr. Jones's column is copied, with all its imperfections, (and 
 one more,) by Mr. Willich, in his Popular Tables," Longman and Co., 1846. 
 
Preparing for Publication, 
 
 BY THE AUTHOR OF THE PRESENT WORK, IN CONJUNCTION WITH 
 HENRY AMBROSE SMITH, F.I. A., AND WILLIAM ORCHARD, A. I. A., 
 
 TABLES OF THE VALUES 
 
 OF 
 
 SURVIVORSHIP ASSURANCES; 
 
 Showing the Single and the Annual Premium for an Assurance on (x] against 
 (y), for every age of both (a?) and (y), according to the Carlisle Rate of Mortality, 
 at 3 per cent. With an Introduction explanatory of the methods employed in 
 the construction of the Tables, and the manner of their application to various 
 important Problems ; and an Appendix of several Subsidiary Tables. 
 
CHAPTER I. 
 
 OF THE TABLES. 
 
 1. IN most of the cases hereafter to be considered, one or other of 
 the subjoined tables comes into use. To avoid interruption, there- 
 fore, in treating those cases, it is expedient to commence with a 
 description of the Tables, their principal properties, and the manner 
 of using them. 
 
 2. All mathematical tables consist of two series of corresponding 
 values, each value in either series having a value corresponding to it 
 in the other. One of these series forms always an arithmetical pro- 
 gression, having, to facilitate interpolation, 1, 10, 100, &c., as the case 
 may be, for its common difference; while the other series forms a 
 progression, varying usually according to other and less obvious laws. 
 In reference to any table, then, we may, for present purposes, designate 
 the two series of which it consists by the terms regular and irregular, 
 respectively. 
 
 3. The manner in which the two series forming a table are arranged, 
 so that the corresponding values shall oe readily ascertained, is 
 familiar to all who are acquainted with the standard tables of Hutton, 
 Babbage, &c. And as none others will seek to use the subjoined 
 tables, which are arranged in the same form as the tables in question, 
 it is unnecessary to be more particular on this point. 
 
 4. The number with which a table is entered, as the phrase is, that 
 is, which is sought in one of the series of which the table is composed, 
 and the number corresponding to which is to be taken out from the 
 other series, is called the argument ; and the number so taken out is 
 called the tabular result* corresponding to that argument. 
 
 5. When the argument is sought in the regular series, and the 
 result consequently taken out from the other series, the table is said 
 to be used directly ; and, conversely, when the argument is sought in 
 
 * This very commodious term was first proposed by Professor De Morgan in 
 his valuable and interesting article, "TABLE," in the "Penny Cyclopaedia," and 
 a continuation of which will be found in the " Supplement " to the same work. 
 
the irregular series, and the result taken out from the regular series, 
 the table is said to be used inversely. 
 
 6. For the more convenient application of logarithmic functional 
 tables, to which class the tables subjoined belong, it is desirable to 
 possess a means of commodiously denoting their results, when used 
 either directly or inversely. For this purpose the following notation 
 is proposed : 
 
 Let a result, arising from the direct use of a table, be denoted by 
 the letter T, with the argument appended, in brackets ; and let a 
 result arising from the inverse use of a table, be denoted by the 
 inverse symbol T" 1 , also with the argument appended. 
 
 Thus, if the given argument for any table be log. m ; then the 
 corresponding tabular result will be denoted by T [log. m] if the 
 table is to be used directly, and by T -1 [log. m\ if the table is to be 
 used inversely. 
 
 7. These symbols being quite general, they may be employed to 
 characterize the results of any table whatsoever ; and hence the 
 functions they will represent in any particular case, will depend 
 altogether on the nature of the table in connexion with which they are 
 used. We now inquire, therefore, what are the functions represented 
 by the foregoing symbols, when used in connexion with the subjoined 
 tables. 
 
 A slight extension of the notation proposed is desirable, before 
 proceeding farther. The tables are two in number, and it will be 
 convenient to possess the means of distinguishing the results of the one 
 from those of the other. This end will be attained very commodiously 
 by employing, in connection with the symbols T and T" 1 , the suffixes 
 1, and 2, thus T } , T 2 ; T^ 1 , T^ 1 , according as the results indicated 
 are those of the first or of the second table. We have then four cases 
 to consider, namely, 
 
 T x [log. m\, the result of the direct use of Table I.; 
 
 Tj- 1 [log. w], inverse I. ; 
 
 T 2 [log. m], direct II. r ; and 
 
 T 2 -1 [log. m], inverse II. 
 
 8. The two series composing Table I., are headed respectively, 
 " Log. x," and " Log. (1 -f x) ;" the former being what we have hitherto 
 called the regular series, and the latter the irregular series. The 
 meaning of this is, that any term in the first series being considered 
 as the logarithm of some value of x, (what that value of x is, we 
 do not need to know,) the corresponding term in the second series 
 will be the logarithm of the same value of x, increased by unity. 
 For example, corresponding to log. a? = 0*245400, we find log. (1+ x) 
 
= 0-440837. The former logarithm happens to be the logarithm of 
 1- 75954, and the latter is consequently the logarithm of 1 + 1-75954, 
 or 2-75954. 
 
 Hence we assign the interpretation of the first symbol ; which is, 
 making log. m the log. x of the table, 
 
 T^log.roJ^log. (l+m). 
 
 9. Corresponding values in Table I., being log. x, and log. (!+<#), 
 let for a moment \ + x~y } then is x=y 1, and the corresponding 
 values may be written log. (y 1) and log. y. Let our log. m= log. y, 
 and we shall have for the interpretation of the second symbol, 
 
 Tj-ipog. w]=log. (m-1). 
 
 10. The two series composing Table II., are headed, respectively, 
 " Log. x" and "Log. (\x}"; which means that, any term in the 
 first series being considered as the logarithm of some value of x, the 
 corresponding term in the other series will be the logarithm of unity 
 diminished by that value of x. Thus, corresponding values are 
 1-520200 and T'825242 : the former of these will be found to be 
 the logarithm of 0-331284, and the latter of 0-668716, which is 
 1-0-331284. 
 
 Hence, if our log. m be made the log. x of this table, we shall have 
 T 2 [log. m] =log. (1-m). 
 
 11. Let, as in (9), 1 x=y ; then is x = \ y, and corresponding 
 values in Table II. may be written log. (1 y} and log. y. 
 
 Hence, if our log. m=log. y, we shall have for the interpretation of 
 the fourth symbol, 
 
 T 2 -i[log.ro]=log. (l-m). 
 
 From this it appears, as might indeed have been inferred without the 
 formality of demonstration, that the series composing Table II., are 
 reciprocally related. The tabular result is the same function of the 
 argument in whichever way the table is used. Hence, if the two 
 series were co-extensive, we need never employ this table inversely. 
 This is not the case, however ; the series log. (1 x} extends consider- 
 ably beyond the other in one direction. The inverse use of this table 
 may therefore sometimes be necessary. 
 
 12. It remains to say something on the subject of interpolation for 
 values intermediate to those contained in the tables. We take the 
 tables separately, as the processes are different. 
 
 In Table I. the proportional parts are arranged in an unusual if not 
 an unprecedented form. x\s ordinarily arranged, the proportional parts 
 corresponding to each tabular difference consist of tenths, from one- 
 tenth up to nine-tenths of that difference ; and when two figures have 
 to be proportioned for, two entries of the table of proportional parts 
 
are requisite. Here the proportional parts of each tabular difference 
 consist of hundredths, from one-hundredth up to ninety-nine hun- 
 dredths, made true to the nearest unit in the last place. In the use 
 of this table, consequently, with an argument of either five or six 
 figures, one entry of the table of proportional parts is always sufficient. 
 With this single variation the direct and inverse modes of using Table I. 
 correspond with the like modes of using the common tables of loga- 
 rithms, with which all are familiar. 
 
 13. In Table II. the proportional parts are arranged in the usual 
 way. So far, therefore, it corresponds, as to the modes of using it, 
 with the like modes of using the common tables. But if, as in using 
 the common tables, the results corresponding to the tabular value next 
 lower than the given argument, be taken out, the proportional parts 
 will be subtractive ; because, of the two series composing the table, an 
 increase in the one corresponds to a decrease in the other. This in- 
 convenience, however, is remedied, as in the case of the complemental 
 functions in the trigonometrical tables, by taking out the result corre- 
 sponding to the tabular argument next higher than the given argu- 
 ment, and adding to it the proportional parts for the excess of the 
 tabular argument over the given argument. 
 
 The foregoing remarks, and precepts for the use of the tables, will 
 find abundant illustration in the examples which will presently be 
 given. 
 
 14. We now proceed to the general problems which may be solved 
 by means of the tables. 
 
 PROBLEM I. 
 
 The logarithms of two numbers are given : it is required to find the 
 logarithm of their sum. 
 
 Let a and b be the two numbers, of which a is the greater. 
 Log. a and log. b are given, and it is required thence to determine 
 log. (a +b). 
 
 Converting into logarithms the identical equation, 
 
 we have, log. (a -f b) = log. b -f log. (T + 1 Y 
 
 But, by (8), log. g+ l^flog.y ^[log. a-log. b] ; 
 
 whence, log. (a -f b) = log. b + T x [log. a log. b] . 
 
 We have, therefore, the following Rule : 
 Subtract the less of the two given logarithms from the greater, and 
 
 CI/YW^ 
 
enter Table I. directly with the difference. The tabular result added 
 to the less logarithm will give the logarithm required. 
 
 15. Here we may remark, that, the argument being log. a log. 6, 
 or log. (a-r-6), the table will apply, for the solution of this problem, 
 to any two numbers, however large or however small those numbers 
 individually may be, provided their ratio does not exceed that of 
 100 to 1 ; since 2, which is the logarithm of 100, is the greatest 
 tabular argument for the direct use of the table. 
 
 16. EXAMPLE 1. Given log. a = 1-543268, and log. 6=0-467568; 
 required log. (a + 6) . 
 
 log. = 1-543268 
 
 log. 6=0-467568 1-075 7,00= log. a-log b. 
 ^[1-0757]= 1-1 10731 
 
 log. (a + b) = 1-578299 
 
 The working here hardly needs explanation. The difference between 
 log. a and log. b is set down at one side, that it may not interfere 
 with the subsequent addition of the latter logarithm to the tabular 
 result. The line between log. a and log. b is to give farther facility 
 by separating the latter logarithm, which has to be included in the 
 addition, from the former. 
 
 Ex. 2. Given log. a = l'623482, and log. 6 = 1-338253 ; required 
 log. (a + 6). 
 
 log. a = 1-623482 
 
 log. 6 = 1-338253 0-2852,29= log. a-log. 6. 
 ^[0-2852]= 0-466632 
 pro. parts for 29= 19 
 
 log. (a -f 6) =1-804904 
 
 In this example the proportional parts come into use. The tabular 
 difference is 66 ; and in the column of proportional parts headed 66, 
 opposite 29, we find the pro. parts corresponding, namely, 19. 
 
 Ex. 3. Given log. = 1-806180, and log. 6=1-556303 ; required 
 log. (a +6). 
 
 log. a = l-806180 
 
 log. 6 = 1-556303 0-2498,77 =log. a-log. 6. 
 Tj [0-2498] =0-443648 
 p. p. for 77= 49 
 
 log. (a -f6) =2-000000 
 
In this example the given logarithms are those of 64 and 36. The 
 sum of these numbers is 100, whose logarithm is 2, as found above. 
 
 Ex. 4. Given log. a =0-362415, and log. 6=1-276342 ; required 
 log. (a + b). 
 
 log. 0=0-362415 
 
 log. 6=1-276342 1-0860,73= log. a-log. b. 
 Tj [1-0860]= 1-120241 
 p. p. for 73= 68 
 
 log. (0 + b) =0-396651 
 
 Ex. 5. The logarithms of two numbers are 1*243689, and 
 2-683426; find the logarithm _of their sum. 
 
 1-243689 
 
 2-683426 0-5602,63 
 T! [0-5602] =0-665811 
 p. p. for 63= 49 
 
 log. required = 1-349286 
 
 Ex. 6. The logarithms of two numbers are 4-328653, and 
 2-437248; required the logarithm of their sum. 
 
 4-328653 
 
 2-437248 1-8914,05 
 Tj [1-8914] =1-896941 
 p. p. for 05= 5 
 
 log. required = 4-334194 
 
 17. Occasionally, in practice, it is not convenient to arrange the 
 logarithms to be operated upon in the manner here exhibited, namely, 
 the less under the greater. The greater will sometimes stand under 
 the less, when it would be exceedingly inconvenient to proceed in the 
 way directed. In this case, therefore, the operation may be modified 
 in the manner following : Subtract the greater logarithm from the 
 less, and set the difference under the greater. Make the arithmetical 
 complement of the difference the tabular argument, and set the result 
 under the difference. Addition of the three lines will give the logarithm 
 required. 
 
 For since a-f b = a(l +-J = a - (r + 1 ), we have, on passing to 
 logarithms, 
 
lo. + = log. a + lo. - 
 
 . . . 
 
 =log. a + (log. b log. aJ+Tj^og. a log. I 
 the last term, TJlog. a log. ti], being the same as 
 
 T^O-aog. i-log. a)], or T^colog. j], 
 
 We repeat the last example, worked as here directed : 
 less log. =2-437248 
 
 greater log. = 4-328653 
 
 diff. =2-108595 1-8914,05 =ar. co. diff. 
 T l [1-8914] = 1-896941 
 
 p. p.= 5 
 
 log. required =4-334194 
 
 This form of the operation requires a line more than the other, and 
 we cannot make provision for this additional line, since it is impossible 
 to know beforehand the cases in which the necessity for it will occur. 
 This inconvenience will be obviated by adding the proportional parts 
 to the tabular result before setting it down. 
 
 18. COROLLARY 1. If the less of the two quantities, the logarithm 
 of whose sum is to be found, be unity ; then, by substituting in the 
 general expression in (14), 1 for b, and for log. b, we have, in this 
 case, 
 
 log. (fl+l)=T 1 [log. a]. 
 
 This is a symbolical representation of the fundamental property of 
 the table, with which we set out (8). We give a few examples of its 
 application. 
 
 19. Ex. 1. Log. a = 1-637400; required log. (a + 1). 
 
 T! [1-6374] =1-647295 =log. (a + 1). 
 Ex. 2. Log. a = l -995635; required log. (a+1). 
 ^[1-9956]= 1-999965 
 p. p. for 35 = 35 
 
 log. (a+1) =2-000000 
 
 The given logarithm here is that of 99. The required logarithm, 
 consequently, is the logarithm of 100, which is 2, as just found. 
 
 Ex. 3. The logarithm of 1 is ; it is required thence, by means 
 of the table, to determine the logarithms of the natural numbers up 
 to 20. 
 
8 
 
 The operation is in the margin ; and as it may 
 be regarded as a type of the continuous opera- 
 tions in which the power and utility of the table 
 are chiefly exhibited, it may be proper to offer a 
 few remarks upon it. 
 
 (1.) The principle of the operation is obvious. 
 Each logarithm, as it is found, is made the argu- 
 ment for finding the next. The logarithms of 
 the natural numbers up to 100 might be found 
 in this way, but no farther, as log. 100 is just 
 beyond the greater limit of the table. 
 
 (2.) When the last two places of the argument 
 are ciphers, a line is still left for the proportional 
 parts, for uniformity, and because it is convenient 
 in these operations to use paper previously ruled. 
 
 (3.) In continuous operations, such as the pre- 
 sent, the indices of the logarithms need never be 
 set down. It is something to save the writing of 
 one figure out of seven, if the omission give rise 
 to no obscurity. And this it cannot do, since, 
 the index of the initial value being known, and 
 the values increasing gradually, an addition of 
 unity to the index is always marked by a large 
 decrease in the mantissa; as here, in passing 
 from log. 9 to log. 10. 
 
 (4.) The result of every tabular conversion, 
 when interpolation is used, and the tabulated 
 values, as in all logarithmic tables, are only ap- 
 proximately and not absolutely true, is subject to 
 some degree of uncertainty. From a careful con- 
 sideration of the causes of this uncertainty, in 
 connexion with the present table, it appears that 
 its results, in detached operations, will be occa- 
 sionally affected by an error of a unit (not more) 
 in the last place; and this either in excess or 
 defect, both kinds of error being about equally 
 probable. In a number of such operations, con- 
 sequently, the errors of the one affection will 
 
 log. i* 
 [0-0000] 
 
 Tjp-3010] 
 p.p. for 30 
 
 T^O-4771] 
 p. p. for 21 
 
 000000 
 
 301030 
 00 
 
 log. 2 301030 
 
 477101 
 20 
 
 log. 3 477121 
 
 602044 
 16 
 
 log. 4 602060 
 
 698922 
 
 48 
 
 5 698970 
 
 778093 
 
 58 
 
 6 778151 
 
 845054 
 44 
 
 7 845098 
 
 903004 
 
 86 
 
 8 903090 
 
 954163 
 79 
 
 9 954242 
 
 999962 
 '3S 
 
 10 000000 
 
 041393 
 00 
 
 11 041393 
 
 079096 
 
 86 
 
 12 079182 
 
 * In this and succeeding examples the natural and the logarithmic numbers are 
 distinguished from each other by being printed in different forms of type. The 
 old type, now again coming into extensive use, is employed for the natural 
 numbers, and the modern type, of uniform size, for the logarithms. 
 
about equal those of the other, so that the average 
 may be regarded as correct. And hence, also, 
 when the operations are dependent, as in the 
 example before us, errors of the one kind will 
 probably be compensated by those of the other, 
 and the total de.viation will, in general, be ex- 
 ceedingly small, certainly quite undeserving of 
 attention in practice. 
 
 To illustrate this, compare the logarithms 
 found in this example with the logarithms, to 
 six places, of the same numbers as given in the 
 standard tables, and it will be found that, up 
 to log.8, inclusive, they are correct. In log.9 
 there is a deviation of 1, which is neutralized 
 at the next step, and logs. 10 and 11 are correct. 
 Logs.12 to 15, inclusive, err by + 1, and logs. 
 16 to 20 are correct.* 
 
 20. COROLLARY 2. If the greater of the two 
 quantities, the logarithm of whose sum is to be 
 found, be unity, then, by substituting in the ex- 
 pression for log. (a+ b) in (14), 1 for a, and for 
 log.a, we have, 
 
 12 079182 
 
 113868 
 76 
 
 i 4 146129 
 
 15 176092 
 
 204034 
 86 
 
 i6 204120 
 
 230430 
 19 
 
 17 230449 
 
 1 8 255273 
 
 278685 
 69 
 
 19 278754 
 
 300979 
 51 
 
 20 301030 
 
 [colog.6], 
 
 where colog.6 denotes the arithmetical comple- 
 ment of log.6. 
 
 21. This case is to be distinguished from that 
 of Corollary 1. There the number whose log- 
 arithm was to be found was, 1 plus a number greater than unity ; 
 here, the number whose logarithm is required is, 1 plus a number 
 less than unity. Between the two cases the table affords a wide 
 range within which the number whereof, when increased by unity, 
 the logarithm is required, may be situated. For 2, the limit of the 
 tabular argument in this application of the table, being the logarithm 
 of 100, and the co-logarithm of -01, the number may have any value 
 within these limits. 
 
 * If it be objected that this is a clumsy way of finding the logarithms of the 
 natural numbers, it is at once admitted that it is so. The example has been 
 chosen solely on account of the facility afforded by existing tables for verifying 
 the results of this table ; and the operation would have been, practically, just the 
 same, had the numbers corresponding to the logarithms found consisted of six 
 places instead of one or two only. In the latter case the example could not hav'e 
 been worked by any existing table so easily as by this. 
 
 C 
 
10 
 
 22. It may be well here to give the rules for the two cases, although 
 that for the first case has been anticipated. 
 
 A logarithm being given, and it is required to find the logarithm of 
 the corresponding natural number, increased by unity : 
 
 CASE 1. When the given logarithm is positive, and its number 
 consequently not less than unity, enter Table I. directly with the 
 given logarithm, and the tabular result will be the logarithm required. 
 
 CASE 2. When the given logarithm is negative, and its corre- 
 sponding number consequently less than unity, enter Table I. directly 
 with the arithmetical complement of that logarithm, and add the 
 tabular result to the given logarithm ; the sum will be the logarithm 
 required. 
 
 23. The following are a few examples illustrative of the second 
 case. 
 
 Ex. 1. Log. = 2-326341; required log. (1 +b). 
 
 log.6 = 2-326341 1 -6736,59 = colog.d 
 T^l-6736^1-682712 
 p. p. for 59= 58 
 
 log.(l + b) =0-009111 
 
 Ex. 2. The logarithm of a certain number is 1-698970 ; required the 
 logarithm of that number increased by unity. 
 
 T-698970 0-3010,30 
 T^O-3010] =0-477101 
 p.p. for 30= 20 
 
 log. required=0-176091 
 
 Here the given logarithm is log.5, and the result is log.1-5. 
 
 24. We now pass on to 
 
 PROBLEM II. 
 
 The logarithms of two numbers being given, it is required to find 
 the logarithm of the difference of those numbers. 
 
 Let, as before, a and b be the two numbers, of which a is the 
 greater. Hence, log. and log. being given, it is required to find 
 log. (a b). 
 
 (1.) From the identical equation, 
 
 we have, on passing to logarithms, 
 
11 
 
 Now, by (9), 
 
 consequently, 
 
 log.(a-^)=lo 
 Hence we have the following 
 
 RULE : To find the logarithm of the difference of two numbers 
 whose logarithms are given: Subtract the less logarithm from the 
 greater, and enter Table I. inversely with the difference. Add the 
 tabular result to the less logarithm, and the sum will be the logarithm 
 required. 
 
 (2.) The relative magnitudes of a and b remaining, a b may be 
 also written thus : 
 
 which, on passing to logarithms, gives, 
 
 log. (a-b)= log.a + log|l - -). 
 But, by (10), 
 
 Whence, log.( b] =log.a + T 2 [log. log.a]. 
 
 RULE : Subtract the greater logarithm from the less, and enter 
 Table II. directly with the difference. Add the tabular result to the 
 greater logarithm, and the sum will be the logarithm required. 
 
 (3.) By (11), 
 
 log.(l -J)=T 2 -'[log. j] =T 2 -> pog.6-log.a]. 
 
 Whence also, log.(a ^)=log.fl4-T 2 ~ 1 [log.6 log.0]. 
 
 The rule here is the same as that last given, with the single change 
 of the word directly into inversely. (11.) 
 
 25. We have thus three modes of solution of Problem II. ; but all 
 are not applicable throughout the same limits. That is to say, in 
 many of the cases that arise in practice, some one or more of the 
 methods may not admit of being applied, in consequence of the argu- 
 ment being beyond the limits of the series in which, in the use of 
 those methods, it has to be sought. The second method applies to 
 all the cases to which the first applies, and also to some in which the 
 third applies ; and it is generally to be preferred, as being at least as 
 correct as the others, and also somewhat easier. The results of the 
 third method, where it alone is applicable, namely, towards the com- 
 mencement of the table, where the ratio of a to b approaches a ratio 
 
12 
 
 of equality, must be used with caution. The deviation from the truth, 
 in the results arising from the inverse use of that part of the table, 
 will often be very considerable. 
 
 26. We now give some examples, illustrative of the application of 
 these different methods of solution. 
 
 Ex. 1. Given log.a = 5 -994605, and log. = 4-011274; required 
 log.a b). 
 
 log.a = 5-9946051 
 
 log. a log. 
 log.6=4-011274| 1-983331 
 [1-983237] = 1-978700 237 
 
 p. p. for 94= 95 
 
 y^ 
 
 -fl =5-990069 
 
 log.6= 4-01 1274 
 
 _log.a = 5-994605 
 T 2 [2-01 7] =1-995460 
 p. p. for 33= 3 
 
 log. (a -) =5-990068 
 
 log. b log. or 
 2-016,669 - 
 
 331 
 
 In this example the given logarithms are those of 987654, and 
 10263. The logarithm required, therefore, is that of 987654 10263 
 =977391, which is 5-990068. 
 
 27. A few words of explanation may be useful. In the first solu 
 tion Table I. is entered inversely with log.a log. = 1-983331, and 
 1*978700, the result corresponding to the next lower tabular value, 
 namely, 1*983237, is taken out. The excess of the argument over 
 the tabular value used, namely 94, is then sought in the Table of 
 Proportional Parts, in the column headed 29, which is the tabular 
 difference at the point of the table in use, and opposite is found 95, 
 the pro. parts corresponding. The sum of the three lines then gives 
 the logarithm required. 
 
 28. In the second solution Table II. is entered directly, with 
 argument log.b log.a, namely 2*016669, and, agreeably to (13), 
 that the pro. parts may be additive, the result corresponding to the 
 next higher tabular value, namely, 2'017, is taken out. We have 
 now to proportion for the excess of 2'017,000 over 2*016,669, and this, 
 it is easy to see, is the same thing as the excess of 1000, the constant 
 difference in the argument series, over 669, the last three figures of 
 the given argument. The property here taken advantage of may be 
 thus simply demonstrated. If t denote any value in the argument 
 series, and d the constant difference in that series, the value next 
 higher than t will be denoted by t + d. Any value, consequently, in- 
 termediate to t and t+ d, will be denoted by t-\-e, where e is less than 
 d. Hence the excess of the greater tabular value over the interme- 
 diate value is plainly (t+ d) (t-\-e)=de. In the first four pages 
 of Table II. d is 1000, and e, consequently, the number composed of 
 
13 
 
 the last three figures of the given argument. In the remainder of 
 the table d is 100, and e } therefore, the number composed of the last 
 two figures of the given argument. The complement of these figures, 
 therefore, to the constant difference may always be at once set down 
 without the formality of a regular subtraction. The last figure of the 
 complement, in this portion of the table, does not affect the pro. parts, 
 and no farther account need be taken of it than, when it is not less 
 than 5, increasing the next figure by unity. 
 
 29. Ex. 2. Given log. = 1-999996, and log.b = 1 -045753 ; re- 
 quired log.( b). 
 
 log.b = 1-045753 
 
 log. = 1-999996 
 
 log.a log.ZO 
 0.954243 > 
 163 
 
 80 
 
 log.fl = 1-045 753 
 [0-954163] =0-9030 
 p. p. for 80= 91 
 
 log. (a -b) = 1-948844 
 
 log. = 1-999996 1-0457,57 
 T 2 [1-0458] =1-948842 
 p. p. for 43= 5 
 
 43 
 
 1-948843 
 
 Here a=99999, and = 11-111; hence - = 88-888, whose 
 logarithm is 1-948843. 
 
 Ex. 3. Log.a =1-823908, and log. = 1-346787; required log. 
 (a-*). 
 
 log. = l-823< 
 
 j.A = 1-34678710.477121 
 101 
 
 [0-477101] =0-3010 
 p.p. for 20= 30 
 
 log.(a-) =1-647817 
 
 log.A = 1-346787 
 
 log.a = 1-823908 1-5228,79 
 T 2 [1-5229] =1-823898 
 p. p. for 21 = 
 
 11 
 
 21 
 
 log.=I-346787 
 
 log.a = 1-823908 
 s - 1 [f-522896] =1-8239 
 p. p. for 17= 8 
 
 1-522879 
 96 
 
 17 
 
 log.(- b) = 1-647816 
 
 Here the three methods of solution apply. The values of a and b 
 are -666666 and -222222. Hence b = '444444, the logarithm of 
 which is 1-647817. 
 
14 
 
 Ex. 4. Log.a= 1-64781 7, and log.6 = 1-522878; required log. 
 
 (a-b). 
 
 . = 1-522878 
 
 log.a = 1-64781 7 
 T 2 [1-8751] =1-397824 
 p. p. for 39= 117 
 
 .a-d =1-045758 
 
 1-875061 
 
 39 
 
 log. = 1-522878 
 
 log.a = T64781 7 1 -875061 
 
 T 2 - 1 [l-875075[ = l-3979 
 p. p. for 14= 43 
 
 14 
 
 log. (a-*) = 1-045760 
 
 Here only the second and third methods apply. The values of a 
 and b are -444444 and -333333 ; a-b therefore is -111111, and its 
 logarithm is 1-045758. 
 
 Ex. 5. Log.a = 1-753327, and log. A = 1 -744727 ; required log. 
 (a-*). 
 
 lo.& = 1-744727 
 
 log.a=JL- 753327 1-991400 
 
 1 [1-991408] =2-292 08 
 
 p. p. for 8= 400 
 
 8 
 
 log.(a-)=0-045727 
 
 In this example the third method alone applies. = 56-6666, and 
 6 = 55-5555 j hence a 6 = 1-1111, the logarithm of which is 0-045753. 
 There is, therefore, here considerable deviation from the truth, in ac- 
 cordance with what was stated in (25). The uncertainty in such cases 
 does not arise from error in the table ; but is a necessary consequence 
 of the relation subsisting between numbers and their logarithms, as 
 might easily be shewn did space permit. The common tables applied 
 to similar cases will give results in no respect more satisfactory. It 
 is but seldom, in practice, that occasions arise in which the inverse 
 use of our second table is requisite. 
 
 30. COROLLARY 1. If the less of the two numbers, the logarithm 
 of whose difference is required, be equal to unity, we have, by substi- 
 tuting in the general expressions in (24), 1 for b, and for log. 6, 
 
 log.(-l) = T r i[log.] (1); 
 
 log.(fl-l)=log.a + T 2 [colog.a] .... (2); 
 log.(a-l)=log.fl + Tj-^colog.a] . .(3). 
 
 We have here, as before, three solutions. We confine ourselves 
 in our examples to the first and second, and chiefly to the latter, 
 
15 
 
 for the reasons already stated (25). It is unnecessary to give rules 
 in words. 
 
 31. Ex.1. Given log.0 = 1-890855: required log. (a 1) 
 
 log.a = 1-890855 
 T 1 - 1 [l-890820]= 1-885200 
 p.p. for 35 = 35 
 
 log.(a-l) = 1-885235 
 
 _log.a = 1-890855 2-110 
 T 2 [2-110] = 1-994369 
 p.p. for 86= 11 
 
 log.(fl-l) =1-885235 
 
 Here a = 77'7777 ; hence a 1 = 76-7777, whose logarithm is 
 1-885235. 
 
 32. The operation by Table I. needs no explanation. In that by 
 Table II., the argument is the arithmetical complement of log. a, 
 namely, 2'] 09145. The next higher tabular argument is 2*110, the 
 result corresponding to which being taken out we should have to add 
 to it the proportional parts corresponding to 1000 145 (28). But 
 this is 855, the last figures of the given argument. We therefore 
 save labour by forming the complement of only the first three or four 
 figures of the given argument, (according to the part of the table in 
 use,) and proportioning for the remaining figures. 
 
 33. Ex. 2. The logarithm of a certain number is 1-698970; 
 required the logarithm of that number diminished by unity. 
 
 log. = 1-698970 
 T 1 - 1 [l'698876] = 1-690100 
 
 p.p. 
 
 for 94 = 
 
 log.(fl-l) =1-690195 
 
 _ log.fl =1_-698970 2-302 
 T 2 [2-302] = T-991206 
 p.p. for 97= 20 
 
 log.(fl-l)= 1-690196 
 
 Here a being 50, a I is 49, the logarithm of which is 1-690196. 
 
 Ex. 3. The logarithm of a certain number is 0-698970 ; find the 
 logarithm of that number diminished by unity. 
 
 Given log. =0-698970 
 
 T 1 ~ 1 [0-698922] =0-602000 
 p. p. for 48 = 60 
 
 log. required =0-602060 
 
 Given log. =0-698970 1-3011 
 T a [l-3011] =1-903072 
 
 p. p. for 70= 18 
 
 log. required = 0-602060 
 
 Here the given logarithm is that of 5, and the required logarithm 
 consequently is log.4 = 0-602060. 
 
16 
 
 34. Ex. 4. The logarithm of 100 is 2; 
 determine the logarithms of 99, 98, 97, &c. 
 
 The operation, by Table II., is in the 
 margin.* Each value is deduced from that 
 last found. In the values from 97 to 90, 
 inclusive, there is a deviation of 1 in the 
 last place ; but 89 is correct. 
 
 Ex. 5. The logarithm of 10 is 1 ; find 
 thence the logarithms of 9, 8, 7, &c. 
 
 The operation is on this and the next 
 page, and all the values are correct. This 
 operation differs from that in the last ex- 
 ample by calling into requisition a part of 
 the table in which the argument is given 
 to four places. 
 
 35. COROLLARY 2. If the greater of the 
 two numbers whose logarithms are given, 
 be equal to unity, on substituting in the 
 expressions in (24), 1 for a, and for log. a, 
 we have 
 
 log.(l-4)=log.6+T 1 -i[colog.6] .. (1); 
 
 log.(l-i)=T 2 [log.i] (2); 
 
 log.(l-fi)=T 2 -'[log.S] (3). 
 
 We give two examples of the application 
 of the second formula. 
 
 Ex. 1. The logarithm of a number is 
 1-647817; required the logarithm of the 
 excess of unity over that number. 
 
 Given log. = 1-6478,17 83 .... (28) 
 
 it is required thence to 
 
 T 2 [l-6479] =1-744661 
 p. p. for 83 = 
 
 log. ioo 
 T 2 [2-000] 
 
 log. 99 
 T 2 [2-005] 
 p. p. for 64 
 
 98 
 
 T 2 [2-009] 
 p. p. for 23 
 
 97 
 
 96 
 
 95 
 
 94 
 
 93 
 
 92 
 
 90 
 
 2-000000 
 
 1-995635 
 
 00 
 
 J-995635 
 1-995584 
 
 7 
 
 J-991226 
 
 1-995543 
 
 2 
 
 986771 
 995491 
 
 8 
 
 9822/0 
 
 995450 
 
 3 
 
 977723 
 995397 
 
 7 
 
 973217 
 
 995354 
 
 1 
 
 968482 
 
 995300 
 
 5 
 
 963787 
 995245 
 
 2-000 
 
 005 
 
 009 
 
 014 
 
 018 
 
 023 
 
 027 
 
 032 
 
 037 
 
 959040 041 
 995201 
 1 
 
 954242 046 
 995145 
 3 
 
 89 949390 
 
 66 
 
 log. required = 1-74472 7 
 
 Jog. 10 !_ -000000 1-0000 
 T 2 [l-0000] 1-954243 
 00 
 
 Here the given logarithm is that of log p -954243 0458 
 
 444444. The required logarithm, con- T 2 [l-0458] 948842 
 
 sequently, is that of -555556, which is P- P- for 43 ^ 
 
 1-744728. log. 8 903090 0970 
 
 * See the remarks and note on the similar operation in (18). 
 
8 
 
 T 2 [l-0970] 
 p. p. for 90 
 
 903090 
 
 941995 
 
 13 
 
 845098 
 
 933037 
 
 16 
 
 778151 
 
 920809 
 
 10 
 
 17 
 
 0970 Ex. 2. Required the logarithm of (1 
 
 log.- 1 2-464284).* 
 
 698970 
 903072 
 
 18 
 
 1550 
 
 2219 
 
 3011 
 
 2-464284 716 
 
 T 2 [2-465 = 1-987141 
 p. p. for 72= 22 
 
 required log. =1-987163 
 
 3980 
 
 The given logarithm here is that of 1 v, 
 (v = 1 1 -03) ; and the logarithm found is, 
 consequently, that of v = -970874. 
 
 36. For convenience of after reference, 
 5229 the formulae we have deduced are here sub- 
 joined in a tabular form. In the formula 
 in which b alone appears, it is, for sym- 
 metry, replaced by a. The formulae will 
 hereafter be referred to by the Roman 
 numerals. The Arabic numerals in the 
 last column denote the paragraphs in which 
 the formulas are deduced or illustrated. Of the formulae those desig- 
 nated by the odd numbers, I., III., V., &c., will be most frequently 
 employed. 
 
 602060 
 
 875041 
 
 20 
 
 477121 
 
 823898 
 11 
 
 301030 
 
 698940 
 
 30 
 
 000000 
 
 6990 
 
 a 7 b 
 
 7l 
 
 log.fl-f T 2 pog.6-log.fl] 
 ;. log.fl] 
 
 TI H 
 
 .a + T 2 - 1 [colog.ff] ---- 
 
 T 2 [log.a] 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XL 
 
 XII. 
 
 * Log. -1 a means the number whose logarithm is a. 
 
CHAPTER II. 
 
 OF INTEREST. 
 
 PROBLEM III. 
 
 37. To construct a table of the amount of 1., in every number of 
 years, from one to one hundred. 
 
 Let the interest of 1. in one year be denoted by r. Then will r 
 be one-hundredth of the rate per cent. ; so that, according as this rate 
 is 3, 3i, 4, 5, &c., per cent., the value of r will be -03/035/04/05, 
 &c., respectively. One pound, by the operation of interest, increases 
 in one year to 1 -\-r pounds. And since any other sum will, in the 
 same time, increase in the same ratio, namely, that of 1 to 1 + r, 1 -fr 
 will in a year become (1 -f r)' 2 ; (1 +r) 2 will in a year become (1 + r) 3 ; 
 (1 +r) 3 will in a year become (1 + r) 4 , and so on. Hence 
 I-i-r is the amount of 1. in one year, 
 
 (1 + r) 2 <!. in two years, 
 
 (1 +r) 3 1. in three years ; 
 
 and generally, 
 
 (I + r) n will be the amount of 1. in n years, 
 n being any whole number whatsoever. 
 
 The table proposed for construction, then, will evidently consist of 
 the first hundred integer powers of the quantity 1 +r; and the object 
 is to effect the construction with the least expenditure of time and 
 labour. 
 
 For this purpose there is no method so easy and effectual as that by 
 actual multiplication. When the rate of interest is an integer, as 2, 
 3, 4, &c., per cent., L + r consists of only two significant figures, one 
 of which is unity. For the raising of each power then, in this case, 
 there needs but a single multiplication by one figure, and an addition 
 of two lines. And when the rate is fractional, as 3|, 4|, &c., per cent., 
 one multiplication is still sufficient, simple aliquot parts of the result of 
 this multiplication, or of the multiplicand, supplying the place of the 
 product by the remaining figures of 1 -f r. 
 
19 
 
 38. As a first example, let the rate of interest be 3 
 is l-fr = r03, and the operation for the raising 
 powers will be as in the margin. The product of 
 each multiplicand by 3 being carried two places to 
 the right, and added to that multiplicand, gives the 
 next multiplicand. 
 
 As a second example, the raising of the first ten 
 powers of T0375 (that is, rate 3| per cent.) is shown. 
 Each power is multiplied by 3, and the product set 
 down two places to the right. Instead of now mul- 
 tiplying by 7 and by 5, one-fourth of the last pro- 
 duct is taken, ('75 being one-fourth of 3,) and the 
 three lines being added, the result is the next higher 
 power. 
 
 39. The number of places in the successive powers 
 will go on increasing if not checked. The limit to be 
 set to the increase will depend upon the number of 
 places that are to be finally retained. When so many 
 as 100 powers are to be formed, about two places be- 
 yond what are wanted for use had better, for security, 
 be employed in the computation. Hence, if a table 
 extending to 7 decimal places is to be formed, it will 
 be requisite to let the increase go on till 9 decimal 
 places are reached. The figures that fall beyond the 
 9th place are then to be omitted, taking due account 
 of the carriage from them, however, as in ordinary 
 contracted multiplication. The extra figures are to 
 be cut off at the close, increasing, as is usual, the 
 last figure retained by a unit, when the first of those 
 cut off is not less than 5. 
 
 40. A verification of the process may be had by 
 finding the last term independently, by logarithms 
 extending to the requisite number of places.* But 
 it is not advisable to wait till the close for a verifica- 
 tion ; since, if error is then found to have been com- 
 mitted, revision of the whole work may be necessary 
 to discover its source. By the following method, 
 
 per cent. Then 
 of the first ten 
 
 The amount of 1. 
 
 I 103 
 
 309 
 
 3 1092727 
 
 3278181 
 
 4 112550881 
 
 33765264 
 
 5 1159274074 
 
 34778222 
 
 6 1194052296 
 
 35821569 
 
 7 1229873865 
 
 36896216 
 
 8 1266770081 
 
 38003102 
 
 39143*95 
 10 I 3 439 l6 378 
 
 The amount of 1. 
 I 10375 
 
 778125 
 
 2 107640625 
 
 32292188 
 8073047 
 
 3 1116771485 
 
 33503144 
 
 8375786 
 
 4 1158650415 
 
 34759512 
 
 8689878 
 
 5 1202099805 
 
 * Such are Vlacq's, (by far the best edition of which is Vega's, Leipsic, 17-94,) 
 and Dodson's "Anti-logarithmic Canon," London, 1742. In the "Mechanics' 
 Magazine" for November 1846, and February 1848, the author of the present 
 work described two easy methods of forming logarithms to 12 places, and gave 
 the tables for their application. These methods will be found to form efficient 
 substitutes for the large and expensive works just named. 
 
20 
 
 verification may be obtained at as many points as 5 1202099805 
 
 may be desired. 36062994 
 
 41. Having ascertained the accuracy of the 2nd . 
 
 and 3rd terms, multiply them together, and the pro- 6 1247178548 
 duct will be the 5th term. The square of the 5th 
 
 term will be the 10th term ; and the 10th term, con- 
 tinually multiplied by itself, will give in succession ^ ^388 18432 
 the 20th, 30th, 40th, &c., up to the 100th term. 9704608 
 All these multiplications may be performed in the 8 g 
 contracted mode ; and the continued multiplication 40274123 
 by the 10th term will be much facilitated by using a 10068531 
 
 small table of the first nine multiples of that term. 9 1392813437 
 
 41784403 
 
 PROBLEM IV. 
 
 10 1445043941 
 
 42. To construct a table of the logarithms of the 
 
 amount of 1. in every number of years, from one 
 to one hundred. 
 
 The terms of this table will be the logarithms of the corresponding 
 terms in the table of the amount of l. And, since log.(l + r) w = 
 M log.(l +r), they will evidently be the first hundred integer multiples 
 oflog.(l-fr). 
 
 A table of multiples is most conveniently formed by addition. The 
 first multiple being set down, and also written at the bottom of a card, 
 the continual addition to the last result of the number on the card, 
 which is moved down as the work proceeds, soon produces as many 
 multiples as may be required. And a verification is obtained in the 
 course of the operation, since, allowing for the position of the decimal 
 point, the 10th value is the same as the first, the 20th as the 2nd, the 
 30th as the 3rd, and so on. 
 
 43. It is necessary in this operation, as in the last, although not for 
 precisely the same reason, to use two places more than are wanted for 
 use. The logarithm with which we set out, although true to the 
 nearest unit in the last place, is not absolutely correct. The error, 
 whatever it may be, is increased in each succeeding multiple, and in 
 the last the increase will be a hundred-fold. In fact, multiplication 
 by 100 being equivalent to removing the decimal point two places to 
 the right, the 100th multiple will always necessarily have its last two 
 places occupied by ciphers, which can only be correct on the supposi- 
 tion (which is probably in no case true,) that the two figures following 
 the last figure used in the first multiple are ciphers. That the values 
 found then may be true to the extent wanted, two places more than 
 are to be retained must be used in the operation ; and when these are 
 
21 
 
 cut off at the close, the usual increase, when necessary, must be given 
 to the last figure. 
 
 44. The operation for the first ten values of a seven-figure table, 
 when the rate of interest is 3 per cent., is given in the margin. The 
 8th and 9th figures have to be cut off, and as it Logarithm of the 
 happens, in every alternate pair of values the 7th amount of l - 
 
 mOQQ7O OP* 
 
 figure has to be increased.* 2 Q256744 50 
 
 45. The trouble of going over the operation for 3 0385116 75 
 the purpose of correcting the last figure will be Q641861 25 
 avoided, if the figure in the value first set down, im- 6 0770233 50 
 mediately following the last figure to be retained, ? 1026978 00 
 (but not in the value on the card,) be increased by 5. 9 1155350 25 
 All the succeeding values receive the same increase ; I0 1283/LL 50 
 and the effect is, that in every case in which the first of the figures 
 cut off is not less than 5, the figure preceding is increased by a unit. 
 
 The example last given, with this change in the mode of operation, 
 is here presented. All that now remains is to strike Logarithm of the 
 off the last two figures from each value. iaaoiuA <**\ 
 
 46. It may be proper now to show how the log- * 0256745 00 
 arithm of 1 + ?*, with which we set out, is to be 3 038511725 
 formed to the requisite number of places. When the Jj Q641861 75 
 rate of interest is an integer, 1 + r is a number of 6 0770234 00 
 three places; its logarithm, therefore, will be found ? ^^ *jj> 
 by inspection in Button's Table of the logarithms of 9 1155350 75 
 numbers from 1 to 1000 to 20 places. When the I0 12 83723 00 
 rate is fractional, 1 + r will overpass the limits of this table ; but it 
 may generally be separated into factors, the logarithms of which are 
 contained in the table ; and by the addition of the logarithms of those 
 factors, the required logarithm will be obtained. If these means fail 
 recourse may be had to those indicated in the note, p. 19. 
 
 PROBLEM V. 
 
 47. To construct a table of the present value of one pound, to be 
 received at the end of every number of years from one to one hundred. 
 
 48. The present value of a sum whose period of payment is other 
 
 * It must be remarked, in regard to this particular example, that it belongs 
 to what may be called the critical case. The last two figures of log.(l-|-r), as 
 written, are 25, just half the least quantity, the cutting off of which occasions an 
 increase of a unit in the preceding figure. Hence, if 25 be greater than the true 
 value of the last two figures of log.(l-fr), the indication of increase in the 2nd, 
 6th, 10th, &c., multiples will be fallacious. This is actually the case here. The 
 real value of the figures written 25 is only 24' 7 ; and this value must be used, that 
 is, we must here employ log.(l-j-r) to 10 places, if strict accuracy be required. 
 
22 
 
 than the present time, whether past or future, is that sum whose pay- 
 ment now is equivalent to the payment of the given sum at the speci- 
 fied time. Hence, since by Problem III., (l + r) M denotes the present 
 value of 1. paid (or received) n years ago, the present value of 1. 
 to be paid (or received) n years hence, will, by the laws of algebraical 
 interpretation and extension, be denoted by (l-\-r)~ n , or ln-(l-f /*)". 
 Otherwise, since 1. amounts in n years to (l+r) n pounds, 1. is 
 the present value of (l-\-r) n pounds to be received n years hence; 
 from which we find, by proportion, that the present value of 1. to be 
 received n years hence is l-r-(l -\-r} n , as before. 
 
 49. It thus appears, that the present value of a sum due at an 
 after period, is that sum which, invested now, and improved during 
 the interval, will, at the period of payment, just amount to the sum 
 due. 
 
 50. It is usual to denote !-=-(! + r) by v. Hence, making n = 
 1, 2, 3, ... n, successively, v, v 2 , v*, . . . v n , will denote the present 
 value of 1. to be received at the end of 1, 2, 3 > . . . n years re- 
 spectively. The table proposed for construction, therefore, will consist 
 of the first hundred integer powers of the quantity v, obtained by 
 dividing unity by 1 -f-r ; and the several terms will be the reciprocals 
 of the corresponding terms in the table of the amount of 1 . 
 
 Three modes of construction present themselves. The first is, by 
 the continual multiplication of v by itself; the second is, by the 
 continual division of v by 1 + r ; and the third is, commencing with 
 the last term, to compute backwards, by the continual multiplication 
 of that term by 1 +r. Having already seen the facility with which 
 multiplication by 1 -f r can be effected, there can be no doubt that 
 the third of these methods is that which ought to be preferred. 
 
 51. The operation may be conducted in the manner following : 
 The last term, v 100 , being the reciprocal of (1 -fr) 100 , will be determined 
 either by logarithms, or by actual division. Multiply this term con- 
 tinually by (1 + r) 10 , and the results will be v 90 , v 80 , v? , and so on, 
 down to v, or 1. Now, multiply v wo continually by 1+r, in the 
 manner of Problem III., and the results will be the values required, a 
 verification being had at every 10th value, by means of the values 
 previously found. 
 
 52. The operation for the last ten terms of this The p T^i. value 
 table, (or the first ten terms as they will ultimately 10 7440939149 
 be arranged,) when the rate of interest is 3 per 
 
 cent., is in the margin. The 10th term, that is v 10 , 9 7664167323 
 
 has been found by logarithms, and the succeeding 
 
 values are found by continually multiplying by 1*03. 8 7894092343 
 
23 
 
 7894092343 
 
 236822770 
 
 8130915113 
 243927453 
 
 8374842566 
 251245277 
 
 8626087843 
 258782635 
 
 8884870478 
 266546114 
 
 9151416592 
 
 274542498 
 
 942595909 
 
 282778773 
 
 9708737863 
 291262136 
 
 9999999999 
 
 The value of v should be unity. There is, there- 
 fore, in our operation a deviation of a unit in the 
 10th place. 
 
 PROBLEM VI. 
 
 53. To construct a table of the logarithms of the 
 present value of 1., to be received at the end of 
 every number of years, from one to one hundred. 
 
 The series of logarithms here to be formed, are the 
 logarithms of the numbers in the table whose con- 
 struction was last shown. Being the logarithms of 
 
 v, v* 1 , v 3 v 100 , they will be the first 100 integer 
 
 multiples of log.v, (42) ; and the table may be 
 formed in the same way as the table described in 
 Problem IV. 
 
 54. The formation of the first ten values, when 
 the rate is 3 per cent., is here given. The logarithm 
 of v is the arithmetical complement of log.(l+r) ; 
 and it has been increased in the 8th place by 5, that 
 the 7th place may be correct throughout (45). The 
 index, so far as we have g one in the computation, is 
 1. But it need not be set down, as the point at 
 which a change takes place is easily seen. 
 
 55. Take, as a second example, the formation of 
 the first ten values when the rate is 4 per cent. On 
 comparing the 9th and 10th values here found with 
 the corresponding values in Mr. Jones's work, p. 104, 
 they will be seen to differ by a unit; and if the 
 table be completed the difference in the latter values 
 will be no less than 6 in the 7th place. This seems 
 to have arisen from the log.v used in the computation, 
 of Mr. Jones's table, having been taken as 982966667, 
 instead of 982966661 a likely enough error, since 
 the true logarithm to 10 places is 9829666607. 
 There are discrepancies also in the 3 per cent, column, 
 but they do not amount to so much as in the case 
 adduced.* 
 
 PROBLEM VII. 
 
 56. To construct a table of the amount of an annuity of l. in 
 every number of years, from one to one hundred. 
 
 * Mr. Jones's table is understood to have been copied from an uncompleted 
 work on Assurances, by Mr. Griffith Davies, to which there will be occasion 
 hereafter to refer. 
 
 Logarithm of the 
 present value of 1. 
 
 1 9871628 25 
 
 2 9743256 00 
 
 3 961488375 
 
 4 9486511 50 
 
 5 9358139 25 
 
 6 922976700 
 
 7 9101394 75 
 
 8 897302250 
 8844650 25 
 8716278 00 
 
 9 
 10 
 
 Logarithm of the 
 present value of 1. 
 
 9 
 10 
 
 9829667 11 
 9659333 72 
 9489000 33 
 9318666 94 
 9148333 55 
 8978000 16 
 8807666 77 
 8637333 38 
 8466999 99 
 8296666 60 
 
24 
 
 57. By the amount of an annuity in n years is meant the aggregate 
 amount, at the instant the nth payment is made, of all the payments, 
 supposing each to have been improved at interest from the time of its 
 becoming due. Now, the first payment having been made n 1 years 
 ago, the second n 2 years ago, and so on, the amounts of these pay- 
 ments will be (37), of the first, (1 -f r) n ~ l , of the second, (1 +r) w ~ 2 , of 
 the third, (l-fr) tt ~ 3 , and so on to the nth, which, having just been 
 made, its amount will be 1. Hence, taking the payments in reverse 
 order, the amount of the annuity will be, 
 
 Amount of an Annuity 
 ofl. 
 
 Consequently, whatever whole number be denoted by n, the sum of 
 n terms of the foregoing series will be the amount of an annuity of 
 1. for n years. But the terms of the series, after the first term, are 
 the values in the table of the amount of 1. Hence, when a table of 
 the amount of 1. has been previously formed, a table of the amount 
 of an annuity of 1. may be readily constructed. 
 
 58. The formation of the first ten terms of this 
 table (rate 3 per cent.) is here given. Unity, which 
 is the amount for one year, is first set down, and on 
 the next and alternate following lines the successive 
 values in the table of the amount of jl. Continual 
 addition of these gives, in succession, the required 
 amounts for 2 years, 3 years, &c. 
 
 59. This method requires a table of the amount 
 ofl. to have been previously calculated. When 
 this has not been done, the mode of proceeding will 
 be different. Denote by A( n _ 1 ) the amount of an an- 
 nuity for n 1 years, and by A (w) the amount of an 
 annuity for n years. Now the amount of the latter 
 annuity manifestly consists of the amount of an an- 
 nuity for n 1 years improved for one year, and 
 increased by <!., the amount of the nth payment. 
 Hence, 
 
 30909 
 1092727 
 
 112550881 
 
 530913581 
 1159274074 
 
 6468409884 
 1194052296 
 
 7662462180 
 1229873865 
 
 8892336045 
 1266770081 
 
 And by this formula it will be easy to construct the 
 table required. We repeat the formation of the 
 first ten values of the table by this method. We set 
 out, as before, with unity, the amount for one year. 
 It is there multiplied by l-fr=l'03, in the man- 
 ner before shown (38), a unit being added, along 
 with the product by 3. This operation is repeated 
 till the required number of values is found. There 
 
 9 10159106126 
 1304773183 
 
 10 11463879309 
 
 Amount of an Annuity 
 ofl. 
 
 30909 
 
25 
 
 is a difference of 2 in the eleventh place between the 3 30909 
 last values in the two operations. But this, of course, 1092727 - 
 
 will disappear when the last two figures are cut off at 4 4183627 
 
 112550881 
 the close. _ 
 
 60. It cannot fail to be observed, that when ciphers 5 539 I 35 81 
 shall have been supplied in the three spaces now 
 
 blank, in the second operation, the figures appearing 6 6468409884 
 in both operations are absolutely identical. The re- 
 sults being the same, the differences of those results 7 7662462181 
 are necessarily also the same. But the differences in 1229873865 
 
 the first operation are, by the mode of construction, g 8892336046 
 the successive amounts of 1. Hence it follows, 1266770081 
 
 that in forming a table of the amounts of annuities, 9 10159106127 
 we form one also of the amounts of 1. A demon- I 34773 l8 4 
 
 stration of this will be immediately given ; but, in I0 11463879311 
 the mean time it is worth while to remark, that, at 
 least when the rate of interest is an integer, instead of forming a table 
 of the amounts of 1. separately, it is better at once to form a table 
 of the amounts of an annuity of 1., since the labour of forming this 
 table is no greater than that of forming the other, and in forming 
 this, the other likewise is formed. 
 
 61. Verification may be obtained in the following manner. The 
 expression for the amount of an annuity in n years, 
 
 forms a series in geometrical progression, summation of which, by the 
 common rule, gives, 
 
 A (l + 'O'-l* 
 
 A ()= -- - -- 
 
 By this formula, therefore, when (1 +r) w is given, for all values of n, 
 that is, when a table of the amounts of 1. has been previously formed, 
 values for the verification of the table in hand, at as many points as 
 are judged requisite, may be readily obtained. If the second mode 
 of operation be used, and the table of the amounts of 1. has not been 
 
 * When the table of the amounts of I . is given, it is not easy to conceive a 
 simpler or shorter operation for the construction of a table of the amounts of 
 annuities, than the first of the methods described in the text. Nevertheless, the 
 late Mr. Morgan, in a note in the Appendix to his Treatise on Assurances, 
 (2nd edition, p. 316,) says, that a "shorter process," in these circumstances, will 
 be, to find the values separately, by the formula of verification given above. It 
 is certain that Mr. Morgan never tried the two methods. Besides that a division, 
 even when the divisor is but a single digit, is less easy than an addition of two 
 lines, the values formed by the method Mr. M. prefers would require a separate 
 verification for each ; whereas, when the values are formed in series, verification at 
 a few points ensures the accuracy of the whole. 
 
26 
 
 previously formed, the requisite powers of 1 + r may be formed in the 
 manner described in (41.) 
 
 62. Resuming the foregoing formula, and solving the equation for 
 (l+r) M , we have, 
 
 which proves the property noticed in (60), in regard to the second 
 mode of construction. 
 
 PROBLEM VIII. 
 
 63. To construct, by logarithms, a table of the amount of an annuity 
 of 1. in every number of years, from one to one hundred. 
 
 For this purpose we make use of the formula deduced in (59), 
 namely, 
 
 which, on passing to logarithms, becomes 
 
 and this, on replacing a, in Formula V., by (1 -fr)A (w _ l) , is reduced to 
 log.A w =T X [log.(l + r) + log.A (n _ l} ] . 
 
 64. The computation of the first ten values 
 by this formula is here given. The working 
 stands in need of but little explanation. The 
 value with which we set out is 1, the amount 
 in one year. The logarithm of this is 0, which 
 is first set down. Under it, and on every 5th 
 succeeding line, is set down the logarithm of 
 (1+r), namely, 012837. The error in this 
 logarithm, when reduced to 6 places, is about 
 a fourth in the last place in defect. It is 
 proper, therefore, to increase the sixth place by 
 a unit, about once in every four times.* It is 
 accordingly twice so increased in the example. 
 The first two lines are now added, and Table I. 
 being entered with the sum, the result is log.A (2 ). 
 This result and the next line are then added, 
 and the table being again entered with the sum, 
 the result is log A (3) , and so on. The taking 
 out of the natural numbers corresponding to 
 these results is reserved till the close of the 
 operation. 
 
 65. We cannot complete the series by this 
 mode of operation, on account of the limited 
 
 Amount of an Annuity of 1 . 
 
 1 000000 r ooooo 
 012837 
 
 012837 
 
 307477 
 19 
 
 2 307496 2-03000 
 012837 
 
 320333 
 
 490062 
 22 
 
 3 490084 3-09089 
 012837 
 
 502921 
 
 621536 
 16 
 
 4 621552 4-18362 
 012838 
 
 634390 
 
 724951 
 73 
 
 5 725024 5-30914 
 
 * The correction for the error in the last place of a logarithm, which frequently 
 occurs in a continuous operation, ought always to be made, when, as in the present 
 case, the tables employed are used to their full extent. 
 
27 
 
 extent of Table I. We shall be brought to a 
 stand at the first term whose logarithm is not 
 less than 2 ; in this case the 47th. We there- 
 fore have recourse to Table II,, which is applied 
 as follows : 
 
 Resuming the formula, 
 
 and transposing, and dividing by 1 -\-r, we have 
 
 A 1 
 
 
 
 i+F=(Aw-i), ( 5 ); 
 
 which, on passing to logarithms, becomes, 
 
 or, finally, replacing , in Formula VII., by A (w) , 
 log. A (n _ l) = log.v + log. A (M) -f T 2 [colog. A fw )] . 
 
 By this formula then, commencing at the 
 other end of the series, we can compute down- 
 wards till we reach the point where the pre- 
 ceding formula fails. The operation for the 
 last eleven terms is here given, and we add a 
 few words of explanation. 
 
 66. The 100th term being found by the for- 
 mula of verification, (or by reference to existing 
 tables,) its logarithm, 2*783395, is set down, 
 and under it, and on every 4th following line, 
 the logarithm of v = 1-987163*, which, being 
 about one-fourth of a unit in the last place in 
 excess, must, at every fourth occurrence, be 
 diminished by a unit in that place. The table 
 is now entered, as described in (30), with the 
 logarithm of the 100th term, and the result 
 set down under log.v. Addition of the three 
 logarithms gives the logarithm of the 99th 
 term. This process is repeated till the log- 
 arithms of all the terms are found, the taking 
 out of the natural numbers corresponding being, 
 as in the former case, reserved till the close. 
 
 67. No great importance is attached to the 
 two operations last given. It is not often that 
 
 9 006856 
 012837 
 
 019693 
 
 059247 
 85 
 
 6-46840 
 
 725024 
 012837 
 
 737861 
 
 810746 
 51 
 
 810797 
 012837 
 
 823634 
 
 884338 
 30 
 
 7 884368 7*66245 
 012837 
 
 897205 
 
 8 949015 8-89232 
 012838 
 
 961853 
 
 006808 
 
 48 
 
 10-1591 
 
 10 059332 11-4639 
 
 * * * 
 
 Amount of an Annuity of 1. 
 
 ioo 2-783395 3-217 
 
 987163 607-288 
 999284 
 
 
 99 769842 231 
 
 987163 588-630 
 999260 
 2 
 
 98 756267 244 
 
 987163 570-515 
 999238 
 
 
 97 742668 258 
 
 * No indices after the first, which^directs to the part of the table to be em- 
 ployed, need be set down. 
 
28 
 
 729044 271 
 987163 535-851 
 999189 
 
 
 the logarithms of the amounts of annuities are 97 742668 258 
 wanted, and when they are they can be readily 999213 ^ ' 
 
 derived from the natural numbers, tables of 1 
 
 which, as we have seen, are of very easy for- 
 mation. The operations remarked upon, how- 
 ever, serve to illustrate the mode of applying 
 the tables to practical purposes, and it was 
 chiefly with this view that they were given. 
 
 PROBLEM IX. 
 
 68. To construct a table of the present value 
 of an annuity of <!., for every number of years, 
 from one to one hundred. 
 
 The annuity being supposed to be now con- 
 stituted, it will make its first payment a year 
 hence, its second two years hence, its third 
 three years hence, and so on to its nth or last, 
 which will be made n years hence. Now the 
 present value of the annuity being evidently 
 equal to the sum of the present values of its 
 several payments, and these being v, # 2 , v 3 , . . 
 . . v n , respectively, if we denote the present 
 value required by c (w) , we shall have, 
 
 a (n} =v-rv*+ v 3 +.. .+v n . 
 
 From this it appears, that when a table of 
 the present values of 1. is given, a table of the present values of an 
 annuity of <!. may be formed by continued addition, in direct order, 
 of the terms of that table. We give the formation of the first ten 
 terms, rate 3 per cent, as usual, employing the values 
 deduced in Problem V. The powers of v are set 
 down in their places at the outset, before commencing 
 the addition, which then goes on without interruption. 
 
 69. If the table of the present values of 1. has 
 not been previously formed, the table of the present 
 values of annuities may be constructed as follows : 
 
 96 
 
 95 
 
 94 
 
 93 
 
 92 
 
 91 
 
 715396 
 987163 
 999162 
 
 1 
 
 701722 
 987163 
 999135 
 
 1 
 
 688021 
 
 987162 
 
 999108 
 
 
 
 674291 
 
 987163 
 
 999079 
 
 1 
 
 660534 
 
 987163 
 
 999049 
 
 1 
 
 90 646747 
 
 285 
 519-274 
 
 299 
 503-178 
 
 312 
 
 326 
 472-380 
 
 340 
 457^50 
 
 443*360 
 
 Present value of an 
 Annuity of 1. 
 
 1 9708737863 
 9425959090. 
 
 2 19134696953 
 
 9151416592 
 
 But the expression in parentheses is evidently equal 
 to I+OGI_I)> (68). Hence, 
 
 28286113545 
 
 8884870478 
 
 37170984023 
 8626087843 
 
 8374842566 
 6 54171914432 
 
29 
 
 By this formula, commencing with the value for one 
 year, the table might be constructed. But a multi- 
 plication by v for each value is involved, to avoid the 
 labour of which the formula is subjected to a slight 
 transformation. 
 
 Dividing both sides of the equation by v, we have, 
 
 } , or, (50), (1 + r)( n ) = 1 -f a (li _ l} ; 
 
 whence, a (n _ l} = (1 + r) a (n} - 1 . 
 
 70. We give a re-calculation, by this formula, of 
 the terms already found. They are here deduced in 
 reverse order, and, in practice, a commencement 
 would be made at the 100th term, which would be 
 found, by the application of logarithms to the for- 
 mula of verification, to be immediately given. Here 
 a commencement is made at the 10th term, and the 
 values to the first are found in succession. The 
 operation is to multiply continually by 1*03 (38), 
 abating a unit from the integer part at each addition. 
 
 71. Comparison of this operation with the last 
 shows that the mixed quantity, here added at each 
 operation, is equivalent to the subtraction of a power 
 of v ; that, in fact, the subtraction from unity of 
 the several positive addends, will give the successive 
 powers of v. Here, therefore, in strict analogy with 
 what was observed in (60), previous construction of 
 a table of the powers of v is seen to be superfluous, 
 as they may be readily deduced from the materials 
 of this equally simple operation. 
 
 72. A formula of verification is deduced as follows 
 
 a (n )=v+v 2 + v 3 + . . . . + v n , 
 
 summing this geometrical series in the usual way, we have 
 
 l-v n 
 
 8130915113 
 
 7 62302829545 
 
 7894092343 
 
 8 70196921888 
 
 7664167323 
 
 9 77861089211 
 
 7440939149 
 
 10 85302028360 
 
 Present value of an 
 Annuity of 1. 
 
 10 85302028360 
 
 2559060851 
 
 9 77861089211 
 2335832676 
 
 8 70196921887 
 2105907657 
 
 7 62302829544 
 1869084886 
 
 6 54171914430 
 1625157433 
 
 5 45797071863 
 1373912156 
 
 4 37 T 70984019 
 1115129521 
 
 3 28286113540 
 848583406 
 
 2 19134696946 
 574040908 
 
 9708737863 
 
 : Since 
 
 Or, it may be deduced thus : An annuity of 1. in n years amounts 
 to , (61). The value of that annuity, therefore, at the time 
 
 of its being entered upon, is the present value of 
 
 , to be 
 
30 
 
 received in n years. Hence, multiplying this expression by v n , or 
 l-i-(l-t-r) 71 , we obtain 
 
 as before. 
 
 By this formula, calling in the aid of logarithms if necessary, as 
 many values as may be judged requisite for the purpose of verification 
 may be obtained. 
 
 73. Reduction of the previous equation gives ra (w ) = l v n , which 
 establishes the property above noticed (71), in virtue of which the 
 separate construction of a table of the powers of v is rendered un- 
 necessary. 
 
 PROBLEM X. 
 
 74. To construct, by logarithms, a table of the present value of an 
 annuity of <!., for every number of years, from one to one hundred. 
 
 Taking the formula deduced in (69), namely, 
 
 and passing to logarithms, we have, 
 
 Hence we have, for working formula? : 
 when ( a ( n -i) *- -^ 
 and when ( 7l _ 1 > 7 1, 
 
 In practice, the use of the first of these 
 formula? is requisite only for a single value, 
 since the present value of an annuity for two 
 years will always be greater than 1. 
 
 75. The calculation of the first ten terms is 
 in the margin. The value for one year is v. 
 Log.v, therefore, is first set down, and imme- 
 diately under it, and on every 4th following 
 line, the same logarithm is repeated, since, as 
 the formula? show, this quantity forms part of 
 the addend in each partial operation. The 
 table is then applied, as indicated by the for- 
 mula? ; and when all the logarithms have been 
 found the natural numbers may be taken out. 
 
 76. SCHOLIUM. A treatise on Interest is 
 not here contemplated. A few results in that 
 theory, for which we may hereafter have occa- 
 sion, are therefore introduced in this place. 
 
 IXJ 
 V. 
 
 Preaent value of an Annuity 
 ofl. 
 
 i 1-987163 -970874 
 987163 0-012837 
 307477 
 
 987163 
 
 464396 
 
 15 
 
 3 451574 2-82861 
 
 987162 
 
 582987 
 
 55 
 
 4 570204 
 
 987163 
 
 673672 
 
 3 
 
 5 660838 4-5797I 
 
31 
 
 r)' 
 
 =, 
 
 l+r=-,and(l+r)-=-. 
 
 Also, r 
 
 v 
 l-v 
 
 v y 
 
 v n 
 
 and 1 v=vr. 
 
 The interest of 1. for a year being r, it 
 follows that 1. is the present value of a per- 
 petual annnuity of r ; whence we find, by 
 proportion, for the present value of a perpetual 
 annuity of 1., 1-r-r. This is usually denoted 
 by P. 
 
 5 660838 4-5797' 
 
 987163 
 
 746580 
 
 32 
 
 6 733775 5-41720 
 
 98/163 
 
 807282 
 64 
 
 7 794509 6-23030 
 
 987162 
 859149 
 
 8 
 
 TO 
 
 * 
 
 8 846319 7-01970 
 
 987163 
 904142 
 17 
 
 9 891322 7-78613 
 
 987163 
 
 943778 
 
 20 
 
 930961 
 
 * 
 
CHAPTER III. 
 
 OF PROBABILITY. 
 
 77. IT is not intended to enter here into a disquisition on the 
 Doctrine or Theory of Probability. It is necessary, however, for the 
 better understanding of what follows, that we should have before us 
 its fundamental principles. 
 
 78. The probability of a future event is measured by a factor, by 
 which, the value of any advantage which the occurrence of that event 
 is to bring us being multiplied, the product is the value of our expect- 
 ation before the event is determined. 
 
 From this it follows 
 
 1st. That the value of this factor can never exceed unity. For if it 
 could, in that case the value of the expectation would exceed the value 
 of the advantage expected ; which is absurd. 
 
 2nd. That the measure of certainty, which is the highest degree of 
 probability, is unity. For if the event, the occurrence of which is to 
 bring us an advantage, is certain to happen, the value of our expecta- 
 tion is equal to the value of the advantage, and unity is the only factor 
 which, employed as above, will allow this equality to subsist. 
 
 3rd. That the measure of impossibility of occurrence, considered as 
 the lowest degree of probability, is zero. For, if the event cannot pos- 
 sibly occur, our expectation in regard to it is worth nothing, and zero 
 is the only factor which gives for product. 
 
 4th. Hence it farther follows, that all other degrees of probability 
 are measured by fractions less than unity, which may of course be 
 exhibited in either the vulgar or the decimal form. 
 
 79. When, out of a number of events, one or other of which must 
 happen, to the exclusion of the rest, we have no reason for expecting 
 the occurrence of any one in preference to any of the others, we say 
 that all are equally probable ; and the probability of each is measured 
 by the fraction which has unity for its numerator, and the whole 
 number of events for its denominator. Thus, if an urn contain ten 
 balls, one of which is to be drawn by a person who does not see the 
 balls, all the ten events that may arise are equally probable, since, 
 
33 
 
 before the drawing we can assign no reason for the occurrence of any 
 one ball in particular, a similar and equally cogent reason to which 
 may not also be assigned for the occurrence of any of the others. And 
 
 the measure of the probability of the occurrence of each is . For, 
 
 let the balls be marked 1 to 10, respectively, and let it be a condition that 
 an advantage is to be received by either A, B, C, &c., according as the 
 ball drawn is No. 1, No. 2, No. 3, &c. Now, the expectations of A, 
 B, C, &c., are of equal value, and being ten in number, each of 
 them is worth one-tenth of the whole. But the value of the whole is 
 just the value of the advantage to be received ; for, since he who 
 should buy the whole would be certain to receive the advantage, he 
 
 ought in equity to give that value in return. The fraction , there- 
 fore, the multiplication by which of the value of the advantage produces 
 the value of the expectation in respect of each of the events, is the 
 measure of the probability of any one of those events. (78.) 
 
 80. From this it readily follows, that the probability of an event 
 which may be produced by the occurrence of any one out of two or 
 more of the simple events circumstanced as described, is measured by 
 the fraction whose numerator is the number of favourable events, as 
 they are called, and denominator the whole number of events. For 
 example, if, in the case supposed, the balls are of different colours, six 
 being white, three black, and one red, then will the several probabilities 
 of the ball drawn being white, black, or red, be denoted by the fractions 
 
 63 1 
 
 , , and , respectively. For the whole number of simple events, 
 
 all equally probable, is, as before, ten ; and of these, six favour the 
 occurrence of Welt,*and one favours the occurrence of red. 3 /**>*** 
 
 81. It is apparent in each of the cases supposed, that the sum of 
 the probabilities of the different events to which that case can give rise, 
 
 is equal to unity. Thus, in the first case, , (the probability of each 
 
 event,) x 10, (the number of events,) = 1 ; and in the sedond, 
 
 631 
 
 TQ+ TQ-fTpj = l. The same thing will be found to subsist in every 
 
 case in which the probabilities of the different events have been cor- 
 rectly determined; and a little consideration shows that it ought to 
 do so. Hence if, of the events that may arise out of a given case, the 
 probabilities of all except one are known to us, the probability of that 
 one will be determined by subtracting the sum of the known proba- 
 bilities from unity. And, in like manner, if the probability of only 
 
34 
 
 one of the events is known, the sum of the probabilities of all the 
 others will be determined by subtracting the known probability from 
 unity. But the sum of the probabilities of all the others, being the 
 probability that some other event than that specified will be the event 
 that happens, will evidently be the probability that the event in ques- 
 tion will not happen, or in other words, that it will fail. Hence, if p 
 denote the probability that a specified event will happen, the probability 
 that it will not happen will always be denoted by 1 p. Or, if we 
 denote the probability of its failing by q, the following equation will 
 always subsist, p + q = 1 . Two events whose probabilities are so related 
 are called contrary events. 
 
 82. If an event consist in the concurrence of two or more indepen- 
 dent events, that is, events the happening or failing of any of which 
 has no influence on the happening or failing of any of the others, the 
 probability of that event is equal to the product of the probabilities of 
 the events by whose concurrence it is produced. For example, if the 
 event be the throwing of aces with one die twice running, or with two 
 
 dice each thrown once, its probability is ^. For g being the proba- 
 bility of an ace from one throw of each of the two dice, or each throw 
 of the single die, we have ^ x ~ = ^- 
 
 O O OO 
 
 83. That this is consonant with the principle already established is 
 easily shown. Take the case of the single die, which is to be thrown 
 twice. Any one of the six faces may turn up at the first throw, and 
 any one also at the second throw. In the two throws, therefore, we 
 are presented with six times six, or thirty-six different events, all equally 
 probable ; and of these one only favours the occurrence of the event in 
 question, namely, that in which ace turns up at the first throw, and 
 also at the second throw. The probability of this event, therefore, is 
 
 , (79,) as already shown, 
 oo 
 
 So, also, if there are to be three throws, the probability of an ace 
 three times running will be x ~ x ^ = ?n~A* 
 
 O O O .C'lO 
 
 83. When an event can happen in more ways than one, its probability 
 is equal to the sum of its probabilities in respect of the different ways. 
 
 This principle is identical with the first when the probabilities of the 
 different ways of occurrence are the same. For, returning to the case 
 of the urn with ten balls, let the event in question be the occurrence 
 of the colour white as the result of a drawing. Then, as there are six 
 white balls, the event can happen in six different ways, for it will be 
 
35 
 
 produced by the occurrence of any one of the white balls. And the 
 probability of each of these being , the probability of the event in 
 
 in question is, by this principle, Y^X 6 = r-r, as was already found. 
 
 We have, therefore, now to show the application of this principle to 
 cases in which the probabilities of the different favourable and unfavour- 
 able events are unequal. 
 
 Let there be two urns, the first containing four balls, of which three 
 are white and one black, and the second five balls, of which two are 
 white and three black. Then, if a ball is to be drawn from some one 
 of the urns by a person ignorant of their contents, the probability that 
 the ball so drawn will be a white ball will, by the principle just enunci- 
 ated, be determined as follows. The probability in respect of the 
 
 O T O 
 
 first urn is TX o o- For ^ tne fi rst urn i tne one selected, the 
 
 4< A o 
 
 3 
 
 probability of a white ball is ; and that this urn will be selected the 
 
 probability is . The product of these probabilities, therefore, namely, 
 
 o 
 
 , is the probability of the concurrence of these two events, namely, 
 o 
 
 the selection of the first urn, and the drawing of a white ball from that 
 urn. In like manner the probability in respect of the second urn is 
 
 211 
 
 found to be - x o =F- Therefore the probability of the occurrence of 
 
 o -I oo 
 
 a white ball in the circumstances supposed is o+H"=T7v Now, en- 
 
 o O 4U 
 
 quire if this value accords with that indicated by our original principle. 
 
 84. The probability of a specified event, as the drawing of a ball of 
 
 a given colour from an urn, is evidently not affected if the numbers of 
 
 balls of the different colours in the urn be varied in the same ratio. 
 
 o o 
 
 Thus - and - are the same probability, the first arising from two white 
 o y , 
 
 balls to four black, and the second from three white balls to six black. 
 We may therefore suppose the two urns in the foregoing case to be 
 replaced by others, containing twenty (4 x 5) balls each, the first hawng 
 fifteen white and five black, and the second eight white and twelve 
 black. The probabilities in respect of the different urns are the same 
 
 as before, that in respect of the first being x -==-, and that in 
 
 &0 & 4U o 
 
 8181 
 respect of the second ^Q x ^= =-. We may now suppose the balls 
 
36 
 
 placed together in one urn, for the probability of each of the parcels 
 of twenty being the one selected from, will evidently be the same after 
 the mingling of the balls as before. The case is therefore reduced to 
 the drawing of a white ball from an urn containing 15 + 8 = 23 white 
 balls, and 5 + 12 = 17 black, and consequently forty balls in all. And 
 
 23 
 
 this probability is ^, (79,) as already found. 
 
 By similar reasoning the principle may be shown to hold when the 
 number of ways in which an event may happen, differing anyhow in 
 regard to their relative probabilities, is more than two ; and hence it 
 may be considered as universally established. 
 
 85. We shall now, in the way of illustration of what has been said, 
 exhibit in a tabular form the expressions for the probabilities of the 
 various contingencies arising out of the occurrences of three indepen- 
 dent events. Call the events A, B, and C, and denote their probabilities 
 by a, bj and c, respectively. 
 
 That of the three events A, B, C, 
 
 the probability is, 
 
 all will happen 
 
 all will not happen, that is, that") 
 one or more will fail . , .) 
 
 abc. 
 labc. 
 
 all will fail 
 
 all will not fail, that is, that one") 
 or more will happen ...... ) 
 
 A will happen and B and C fail 
 B A and C 
 C A and B 
 A and B will happen and C 
 AandC B 
 
 B and C A 
 
 b(la)(lc) 
 c(la)(I b) 
 ab(l c) 
 
 =bba 
 
 =ab 
 =ac 
 =bc 
 
 abc. 
 acb. 
 bca. 
 
 86. Having thus laid down the fundamental principles of the Theory 
 of Probability, we have now to show how that theory is applied to 
 questions having reference to the duration of human life. 
 
 The application of which we now speak is founded upon two as- 
 sumptions. Of these the first is, that the experience of the future 
 will accord with that of the past ; that is to say, that what ratio soever 
 may have been found to subsist in time past between the number sur- 
 viving and the number dying in a given time, out of a specified number 
 
37 
 
 of individuals observed upon, the same ratio will be found to subsist 
 in time to come between the number living and the number dying 
 in the same space of time, out of any number of individuals similarly 
 circumstanced with the others. And the second assumption is, that 
 the individuals, the probability of whose living or dying, in any speci- 
 fied time, may be in question, have all the same prospects of longevity 
 with each other. 
 
 87. These assumptions, which are absolutely necessary for the pur- 
 pose in view, need little to be said in their support. As regards the 
 first, there is no fact in statistical science better established than that 
 the ratios of the numbers surviving to the numbers dying, from year 
 to year at the different ages, approach fixed ratios ; and we therefore 
 but assume, that what has been observed constantly to happen will 
 continue to happen. As regards the second assumption, it may or 
 may not be true in any particular case. If, however, we assume, as 
 we are warranted in doing, that deviations from the truth in both di- 
 rections are equally probable, it will follow, that if we are to disregard 
 these, and so proceed upon a fixed and uniform principle, the mode 
 proposed is the one most likely to conduct us generally to the truth. 
 
 88. The records of the past, which in these investigations we take 
 for our guidance, are arranged in the form of tables of mortality ; 
 that is, tables showing, out of a number supposed to have entered upon 
 life together, the numbers who completed each year of existence, and 
 consequently the numbers who died in each year. These tables are 
 familiar to all who have any acquaintance with this subject, and there- 
 fore need no farther description here. The Carlisle Table, constructed 
 by Mr. Joshua Milne, is probably the one best known, and is certainly, 
 as shown by the striking confirmations it has received, as much en- 
 titled to confidence as any other. From it, therefore, our numerical 
 illustrations will be taken. Our general investigations, however, will 
 be equally applicable to any table of the same form, as they will be 
 conducted by means of symbols. 
 
 89. Denote by the letter /, with the age attached as a suffix, the 
 number represented by the table as attaining each age. So shall the 
 numbers in the table, opposite the ages, 0, 1,2, ... a?, . . . x + n, be 
 denoted by 1 0) l lf / 2 , . . . l x , ... l x+H , respectively. Then will the 
 probability that a life aged x will survive n years, be denoted by 
 l x+n ~r-l x ' For, if the representation of the table were that there was 
 to be but one survivor, the probability that any of the l x persons would 
 be that survivor, would, in accordance with what has been laid down, 
 be I-H/^. But the number of survivors is to be l f+n . There are, 
 therefore, l f+n ways in which the survivorship in question may take 
 
38 
 
 place. The probability of that survivorship consequently is the sum 
 of l x+n probabilities, each of which is equal to IH-/^, or a+n -r-l x * (83.) 
 
 90. This function is usually denoted by p jen , the first suffix being 
 the age of the life, and the second the number of years, the survivor- 
 ship of that life at the end of which is in question. When the number 
 of years is unity, the second suffix is usually omitted, so that the 
 probability that a life aged x years will survive one year, is denoted by 
 p x> and the like for a life aged y years by p y . 
 
 91. The function^ occupies a most important part in the doctrine 
 of life contingencies. We here direct attention to one of its applica- 
 tions. Many others will be hereafter shown. If a register of a stationary 
 population, comprising lives of all ages, be carefully formed, distinguish- 
 ing the ages, and if the deaths that occur amongst them, in one year, 
 at each age, be noted, the number of survivors at the different ages 
 will be known, and consequently the value of the function p x for each 
 age. Now, since 
 
 P*= l ,+i-*-l*> *A=kn- 
 
 Hence if, commencing at the youngest age, 0, we assume any number 
 
 for the radix, as it is called, and multiply this number by p , we obtain 
 the number of survivors at the end of one year, according to the 
 observed rate of mortality. Multiplying the number thus found by/? p 
 we obtain the number of survivors at the end of two years, and so on. 
 These numbers being tabulated in connexion with the different ages, 
 a table is formed, representing, in a systematic manner, the mortality 
 which has prevailed, during the period of our observations, among the 
 population observed upon. If the values of p x for each age are given 
 in logarithms, it is obvious that the formation of a mortality table will 
 be thereby much facilitated. For, to find the logarithms of the num- 
 bers surviving at each age, out of any assumed radix, we shall have 
 merely to add continuously, commencing with the logarithm of the 
 radix, the logarithms of the successive values of p x . 
 
 92. It thus appears that, / being the radix, 
 
 but /, = l .f = l Q .p 0je . Hence, by substitution, and dividing by / ; 
 
 or, more generally, changing into oc, and x into x + n, 
 
 P*-P*+rP*+f ' 'P X+ n-l=P X .n' 
 
 That is, the probability that a life aged x years will survive n years is 
 equal to the product of the probabilities that n lives, aged respectively 
 x, x+ 1, # + 2, . . . as+n 1, will survive one year. 
 
 93. This may be otherwise shown. Adopt, for brevity, the symbol 
 
(x) instead of the phrase, " a life aged x years." The survivorship of 
 (x) for, say two years, will arise from the concurrence * of these two 
 events, first, his surviving one year at the age x, and secondly his sur- 
 viving one year at the age a?+ 1. The probability of the first event is 
 p x , and of the second p x+r Hence the probability that both will 
 happen, that is, that (x) will survive two years, is p x -p x+l ', that is, 
 Pz'Px+iPxv And by similar reasoning the like may be shown for 
 any other number of years than two. 
 
 94. Since out of l x individuals aged x years, the number who survive 
 n years is l x+n , it follows that the number who die therein is l x l x+n - 
 Hence, in accordance with what has been shown, the probability that 
 
 (x) will die in n years is -^^ = l--^ = l-^, n . This follows 
 
 * l x 
 
 also from a somewhat different consideration. At the end of n years 
 (x} must be either alive or dead. These, therefore, are contrary events, 
 and the probability of one of them being p xn , that of the other is 
 
 95. The number who, out of l x individuals, die in the nth year from 
 the present time, is obviously l x+n _i l x+H * Hence the probability 
 that (x) will die in that year is, (80,) 
 
 ( l x+n-\ l*+n]-*-lx=Px.n-l Px.n' 
 
 That is, the probability of dying in the nth year is equal to the excess 
 of the probability of living n 1 years, that is, of entering upon the 
 nth year, over the probability of surviving that year. 
 
 96. It is easy now, from what has been established, to deduce the 
 probabilities of the various contingencies, as regards survivorships, 
 that may arise out of combinations of two or more lives. We give 
 here the expressions for the cases in which two lives only are concerned. 
 
 The probability that two lives, (x) and (y} } will jointly survive n 
 years, the event in question being the concurrence of the two events 
 which consist in their singly surviving the term, is p xn -p yn , which we 
 shall hereafter write P( xy ) n . 
 
 The probability that both will not survive n years, that is, that 
 either or both will be dead at the end of the term, is 1 P( xy - }n . 
 
 The probability that both will be dead, that is, that neither will 
 survive the term, is, 
 
 (1 -flrJ (1 -P 9M ) = 1 -P XM -P y 
 
 * Observe that by the concurrence of two or more events their simultaneous 
 happening is not implied. Any portion of time may intervene between the 
 occurrences of the different events. The probability of two aces from a single 
 throw of two dice is the same whether both are thrown now or a hundred years 
 hence, or whether one is thrown now and the other a hundred years hence, 
 
40 
 
 The probability that both will not be dead at the end of the term, 
 that is, that either or both will be then alive, is, 
 
 1 - (1 ~P X . J ( l ~P y .n) =P*.n +P,.n ~P( X ,}n' 
 
 The probability that (x) will be alive and (y) dead, is, 
 
 P X .nV-P v .n}=P X .n-P( X . y }n> 
 
 and, in like manner, the probability that (y) will be alive and (x) dead, is, 
 
 Py.n-P( x Jn> 
 
 The probability that both (x) and (y) will die in the wth year from 
 the present time, is shown, as in the case of a single life, to be 
 
 97. If with any of these contingencies be combined the condition, 
 that the deaths or survivorships shall take place in a specified order, 
 the probability of the proposed contingency is materially different from 
 that of the contingency on which it is founded. For the treatment of 
 cases involving the condition in question, a farther assumption with 
 regard to the mortality table is necessary. The assumption now 
 requisite is, that the deaths which take place in each year of age are 
 uniformly distributed over the year in which they take place ; so that 
 if any year of existence be supposed to be divided into m equal parts, 
 (m being any whole number,) the probability, before the year is en- 
 tered upon, of a specified life failing in any one of those parts is the 
 same as that of its failing in any other. From this by no means 
 unreasonable assumption, it at once follows, first, that of any number 
 of lives which fail in a year, the probability of any one of them occu- 
 pying an assigned place in the order of failure, is the same as that of 
 any other occupying an assigned place, the differences of age amongst 
 the proposed lives being any whatsover ; and secondly, that the pro- 
 bability of all the lives failing in any assigned order in one year, is the 
 same as that of their failing in any other order in the same time.* 
 
 98. Hence, if we denote by q the probability that any number of 
 specified lives will fail in a given year, the probability that any one 
 of them will occupy an assigned place in the order of failure will be 
 denoted by q divided by the number of places, that is by the number 
 of lives; and the probability of the failure of the whole taking place 
 in an assigned order, will be denoted by q divided by the number of 
 orders, that is by the number of permutations that can be formed out 
 of the number of given lives taken all together. If the lives are two 
 in number, (#) and (y} } these probabilities are identical, being in 
 that case, if the nth year is in question, each equal to 
 
 * See an elaborate demonstration of this theorem in Mr. Milne's Treatise, 
 pp. 63 to 68, or Mr. Jones's, pp. 148 to 150. 
 
41 
 
 99. Let it now be required to determine the probability that of two 
 lives, (x) and (y) } one of them in particular, as (x), shall die in the 
 nth year, and the other, (/), be alive at the moment of (x)'s death. 
 This event can happen in two ways, first by (#) dying in the nth year, 
 and (y) living over that year; or, secondly, by both lives failing in 
 the nth year, (x) the first of the two. The probability of the first way 
 of happening is, (82 and 95), 
 
 (Px.n-l~~Pxn)Py.n> aT1C ^ ^at ^ tne secon ^ &S 
 jUSt Shown, is \(P x .n-l-Px.n}(Py.n-l-Py.,) '> 
 
 hence, (83), \ (P x . n -i~P^ (P y . n -i+P y .n)> tne sum of these two, 
 
 is the probability required. 
 
 It is unnecessary for present purposes to carry these investigations 
 f-.rther. 
 
 
 
CHAPTER IV. 
 
 OF THE TABLE OF ELEMENTARY VALUES. 
 
 100. THE functions tabulated in the following pages come into use, 
 both singly and variously combined with each other and with constant 
 quantities, in the constructions with which we are hereafter to be oc- 
 cupied. They consist of the following, for each age, according to the 
 Carlisle Table of Mortality: Log.^, Colog.^, Log.-i^" 1 ^ 1), and 
 Log.^- 1 1). The differences between the successive terms of the 
 several series are also given, the object being to facilitate the formation 
 of the combined series for which we have occasion. The manner of 
 thus using the differences is now to be shown. 
 
 101. (1.) To form the series \og-vp x for each age. This function 
 being equal to \og.v + \og.p x , the series to be formed will be the tabu- 
 lated series log.p x , with each of its terms increased by the constant 
 quantity log.v. Now two series so related have obviously the same 
 
 series of differences between their successive terms. 
 Hence, as the tabulated series will be reproduced by 
 the continual addition to its first term of the corre- 
 sponding series of differences, so also the required 
 series will be formed by the continual addition to Us 
 first term of the same series of differences. The ex- 
 ample in the margin shows the formation of the first 
 few terms of the proposed series, when the rate of 
 interest is 3 per cent., and log.t? consequently 987163. 
 10.2. This mode of formation possesses various ad- 
 vantages. First, the terms being dependent on each 
 other, sufficient verification is obtained by occasional 
 recurrence to the tabulated series, as shown in the 
 proof appended to the example. A discrepancy will 
 establish that error has been committed, which must 
 be traced to its source and removed. Secondly, to 
 form this scries in the most obvious way, all the 
 terms of the tabulated series will have to be written 
 down, and underneath each the constant logarithm 
 
 Log.V/7*. 
 
 987163 
 522879 
 
 103 
 A 
 
 102 
 
 101 
 
 TOO 
 
 99 
 
 510042 
 255272 
 
 7(5531-1 
 0/5721 
 
 841035 
 036983 
 
 878017 
 021995 
 
 900013 
 982415 
 
 98 882428 
 
 Proof. 
 
 log.?? 987163 
 8952G5 
 
 log.p, 882428 
 
FIG. ]. 
 
 to be added to it. By the method now proposed half this labour is 
 saved, the series of differences only requiring to be written down before 
 proceeding to the additions. But, thirdly, even this labour may be 
 spared, if the series of differences be written, once for all, on a set of 
 perforated cards of the form shown 
 in the margin. These being pro- 
 perly placed will admit of the ad- 
 ditions being as readily performed 
 as if proceeded with in the ordi- 
 nary way. 
 
 103. If one series only has to 
 be formed, it will obviously not be 
 worth while to prepare a set of 
 cards. But if series involving 
 different rates of interest, and es- 
 pecially in which two or more lives 
 are concerned, have to be formed, 
 the trouble of preparing the cards 
 will be amply compensated by the 
 saving effected in their application 
 to these purposes. If the cards 
 be made of a length sufficient to 
 admit of thirteen values on a side, 
 then, as both sides may be used, 
 four cards will be sufficient for the 
 Carlisle Table, or any other at 
 present in use.* 
 
 104. In these series it is seldom 
 requisite formally to set down the 
 indices of the several terms. In 
 most of them it is usually, (in the 
 series now before us it is always,) 
 1. Where doubt exists, and it is 
 requisite to ascertain the index of 
 
 a particular term, this can be done by reference to the tabulated 
 series ; arid the determination of one index is usually sufficient, ex- 
 cept, perhaps, in one or two instances near the beginning of some 
 of the series, to determine the whole. 
 
 105. The constant addend, log.#, is of course true to the nearest 
 unit in the last place. But it is not absolutely true, and it is desirable, 
 
 
 
 A 
 
 lo 
 
 cr ri 
 O'/V 
 
 103 I 
 
 5 
 
 1 
 
 
 
 
 
 4 
 
 2 
 
 1 
 
 
 2 
 
 5 
 
 5 
 
 2 
 
 7 
 
 2 
 
 
 102 1 
 
 
 
 
 
 
 
 1 
 
 
 
 
 7 
 
 5 
 
 7 
 
 o 
 
 1 
 
 
 "i r i 
 
 
 
 
 3 
 
 6 
 
 9 
 
 8 
 
 3 
 
 
 100 1 
 
 
 
 
 
 
 
 1 
 
 
 
 
 2 
 
 1 
 
 9 
 
 9 
 
 5 
 
 
 99 
 
 
 
 
 
 
 
 1 
 
 
 9 
 
 8 
 
 2 
 
 4 
 
 1 
 
 5 
 
 
 98 r 
 
 
 
 
 
 
 
 1 
 
 
 9 
 
 9 
 
 5 
 
 5 
 
 9 
 
 
 
 
 97 r 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 2 
 
 6 
 
 9 
 
 
 
 
 96 1 i 
 
 
 9 
 
 9 
 
 1 
 
 
 
 6 
 
 2 
 
 
 95 r i 
 
 
 9 
 
 9 
 
 
 
 4 
 
 5 
 
 4 
 
 
 94) 
 
 
 
 
 
 
 
 1 
 
 
 9 
 
 9 
 
 4 
 
 6 
 
 
 
 5 
 
 
 93 1 
 
 
 
 
 
 
 
 1 
 
 
 9 
 
 8 
 
 7 
 
 G 
 
 6 
 
 7 
 
 
 * The best way of making the cards is, when formed into shape, to tie them 
 together, and cut out the perforations with a chisel. 
 
44 
 
 TABLE OF ELEMENTARY VALUES. 
 
 CARLISLE RATE OF MORTALITY. 
 
 X 
 
 L g^* 
 
 A 
 
 Colog.^ 
 
 A 
 
 Log. 
 
 tfPT'+l) 
 
 A 
 
 Log. 
 taT 1 -!) 
 
 A 
 
 a? 
 
 103 
 
 1-522879 
 
 2552/2 
 
 0-477121 
 
 744728 
 
 0-301030 
 
 823909 
 
 0-301030 
 
 522879 
 
 103 
 
 102 
 
 778151 
 
 075721 
 
 221849 
 
 924279 
 
 124939 
 
 954242 
 
 1-823909 
 
 778151 
 
 102 
 
 IOI 
 
 853872 
 
 036983 
 
 146128 
 
 963017 
 
 079181 
 
 978811 
 
 602060 
 
 853872 
 
 IOI 
 
 100 
 
 890855 
 
 021995 
 
 109145 
 
 978005 
 
 057992 
 
 987765 
 
 455932 
 
 890855 
 
 TOO 
 
 99 
 
 912850 
 
 982415 
 
 087150 
 
 017585 
 
 045757 
 
 009760 
 
 346787 
 
 088941 
 
 99 
 
 98 
 
 895265 
 
 995590 
 
 104735 
 
 004410 
 
 055517 
 
 002475 
 
 435728 
 
 020204 
 
 98 
 
 97 
 
 890855 
 
 002690 
 
 109145 
 
 997310 
 
 05/992 
 
 998489 
 
 455932 
 
 987765 
 
 97 
 
 96 
 
 893545 
 
 991062 
 
 106455 
 
 008938 
 
 056481 
 
 005037 
 
 443697 
 
 039673 
 
 96 
 
 95 
 
 884607 
 
 990454 
 
 115393 
 
 009546 
 
 061518 
 
 005429 
 
 483370 
 
 039509 
 
 95 
 
 94 
 
 875061 
 
 994605 
 
 124939 
 
 005395 
 
 066947 
 
 003091 
 
 522879 
 
 021189 
 
 94 
 
 93 
 
 869666 
 
 987667 
 
 130334 
 
 012333 
 
 070038 
 
 007128 
 
 544068 
 
 045757 
 
 93 
 
 92 
 
 857333 
 
 996539 
 
 142667 
 
 003461 
 
 077166 
 
 002015 
 
 589825 
 
 012235 
 
 02 
 
 9 1 
 
 853872 
 
 015029 
 
 146128 
 
 984971 
 
 079181 
 
 991297 
 
 602060 
 
 944953 
 
 y 
 
 91 
 
 90 
 
 868901 
 
 025708 
 
 131099 
 
 974292 
 
 070478 
 
 985406 
 
 547013 
 
 891764 
 
 9 
 
 89 
 
 894609 
 
 997582 
 
 105391 
 
 002418 
 
 055884 
 
 001358 
 
 438777 
 
 011114 
 
 89 
 
 88 
 
 892191 
 
 002005 
 
 107809 
 
 997995 
 
 057242 
 
 998874 
 
 449891 
 
 990801 
 
 88 
 
 87 
 
 894196 
 
 012430 
 
 105804 
 
 987570 
 
 056116 
 
 993076 
 
 440692 
 
 939274 
 
 87 
 
 86 
 
 906626 
 
 009680 
 
 093374 
 
 990320 
 
 049192 
 
 994668 
 
 379966 
 
 947463 
 
 86 
 
 85 
 
 916306 
 
 008598 
 
 083694 
 
 991402 
 
 043860 
 
 995309 
 
 32/429 
 
 948490 
 
 85 
 
 8 4 
 
 924904 
 
 004064 
 
 075096 
 
 995936 
 
 039169 
 
 997798 
 
 275919 
 
 973753 
 
 84 
 
 83 
 
 928968 
 
 005182 
 
 071032 
 
 994818 
 
 036967 
 
 997205 
 
 249672 
 
 964440 
 
 83 
 
 82 
 
 934150 
 
 003463 
 
 065850 
 
 996537 
 
 034172 
 
 998141 
 
 214112 
 
 974768 
 
 82 
 
 81 
 
 937613 
 
 006019 
 
 062387 
 
 993981 
 
 032313 
 
 996785 
 
 188880 
 
 952853 
 
 81 
 
 80 
 
 943632 
 
 001635 
 
 -056368 
 
 998365 
 
 029098 
 
 999130 
 
 141733 
 
 986384 
 
 80 
 
 79 
 
 945267 
 
 004698 
 
 054733 
 
 995302 
 
 028228 
 
 997510 
 
 12811/ 
 
 958631 
 
 79 
 
 78 
 
 949965 
 
 000677 
 
 050035 
 
 999323 
 
 025738 
 
 999642 
 
 086748 
 
 993744 
 
 78 
 
 77 
 
 950642 
 
 002164 
 
 049358 
 
 997836 
 
 025380 
 
 998857 
 
 080492 
 
 979414 
 
 I 
 
 77 
 
 76 
 
 952806 
 
 003592 
 
 047194 
 
 996408 
 
 024237 
 
 998111 
 
 059906 
 
 963801 
 
 76 
 
 75 
 
 956398 
 
 002563 
 
 043602 
 
 99/437 
 
 022348 
 
 998656 
 
 023707 
 
 972386 
 
 75 
 
 74 
 
 958961 
 
 005715 
 
 041039 
 
 994285 
 
 021004 
 
 997017 
 
 2-996093 
 
 9319/8 
 
 74 
 
 73 
 
 9646/6 
 
 004680 
 
 035324 
 
 995320 
 
 018021 
 
 997571 
 
 928071 
 
 935904 
 
 73 
 
 72 
 
 969356 
 
 004303 
 
 030644 
 
 995697 
 
 015592 
 
 997778 
 
 863975 
 
 932108 
 
 72 
 
 7 1 
 
 973659 
 
 003312 
 
 026341 
 
 996688 
 
 013370 
 
 998297 
 
 796083 
 
 939976 
 
 7 1 
 
 7 
 
 9/6971 
 
 001160 
 
 023029 
 
 998840 
 
 011667 
 
 999405 
 
 736059 
 
 976971 
 
 7 
 
 69 
 
 9/8131 
 
 001212 
 
 021869 
 
 998788 
 
 011072 
 
 999379 
 
 713030 
 
 974614 
 
 69 
 
 68 
 
 979343 
 
 000938 
 
 020657 
 
 999062 
 
 010451 
 
 999520 
 
 687644 
 
 979343 
 
 68 
 
 67 
 
 980281 
 
 000857 
 
 019719 
 
 999143 
 
 0099/1 
 
 999562 
 
 666987 
 
 980281 
 
 67 
 
 66 
 
 981138 
 
 000642 
 
 018862 
 
 999358 
 
 009533 
 
 999673 
 
 647268 
 
 984655 
 
 66 
 
 65 
 
 981780 
 
 000595 
 
 018220 
 
 999405 
 
 009206 
 
 999696 
 
 631923 
 
 985268 
 
 65 
 
 64 
 
 982375 
 
 00068/ 
 
 017625 
 
 999313 
 
 008902 
 
 999649 
 
 617191 
 
 982375 
 
 64 
 
 63 
 
 983062 
 
 000380 
 
 016938 
 
 999620 
 
 008551 
 
 999807 
 
 599566 
 
 989956 
 
 63 
 
 62 
 
 983442 
 
 000732 
 
 016558 
 
 999268 
 
 008358 
 
 999627 
 
 589522 
 
 980009 
 
 62 
 
 61 
 
 984174 
 
 001033 
 
 015826 
 
 998967 
 
 007985 
 
 999475 
 
 569531 
 
 970163 
 
 61 
 
 60 
 
 985207 
 
 002337 
 
 014793 
 
 997663 
 
 007460 
 
 998813 
 
 539694 
 
 924153 
 
 60 
 
 59 
 
 98/544 
 
 001814 
 
 012456 
 
 998186 
 
 006273 
 
 999081 
 
 463847 
 
 930721 
 
 5 9 
 
 58 
 
 989358 
 
 001470 
 
 010642 
 
 998530 
 
 005354 
 
 999256 
 
 394568 
 
 934689 
 
 58 
 
 57 
 
 990828 
 
 000841 
 
 009172 
 
 999159 
 
 004610 
 
 999576 
 
 329257 
 
 957828 
 
 57 
 
 56 
 
 991669 
 
 000477 
 
 008331 
 
 999523 
 
 004186 
 
 999759 
 
 28/085 
 
 974178 
 
 56 
 
 55 
 
 992146 
 
 000453 
 
 007854 
 
 999547 
 
 003945 
 
 999771 
 
 261263 
 
 973921 
 
 55 
 
 54 
 
 992599 
 
 000331 
 
 007401 
 
 999669 
 
 003716 
 
 999834 
 
 235184 
 
 980010 
 
 54 
 
 53 
 
 992930 
 
 000417 
 
 00/070 
 
 999583 
 
 003550 
 
 999789 
 
 215149 
 
 973334 
 
 53 
 
 52 
 
 993347 
 
 000401 
 
 006653 
 
 999599 
 
 003339 
 
 999798 
 
 188528 
 
 972826 
 
 5 2 
 
45 
 
 TABLE OF ELEMENTARY VALUES. 
 
 (Continued.) 
 
 X 
 
 Log.^ 
 
 A 
 
 Colog.j^ 
 
 A 
 
 L - + l 
 
 A 
 
 Log.^ 
 
 A 
 
 X 
 
 5 1 
 
 T-993748 
 
 000386 
 
 0-006252 
 
 999614 
 
 0-003137 
 
 999806 
 
 2-161354 
 
 972208 
 
 5 1 
 
 5 
 
 994134 
 
 999882 
 
 005866 
 
 000118 
 
 002943 
 
 000059 
 
 133562 
 
 008612 
 
 5 
 
 49 
 
 994016 
 
 999889 
 
 005984 
 
 000111 
 
 003002 
 
 000056 
 
 142174 
 
 008027 
 
 49 
 
 
 993905 
 
 999707 
 
 006095 
 
 000293 
 
 003058 
 
 000148 
 
 150201 
 
 020639 
 
 48 
 
 47 
 
 993612 
 
 999905 
 
 006388 
 
 000095 
 
 003206 
 
 000048 
 
 170840 
 
 006386 
 
 47 
 
 4 6 
 
 993517 
 
 000003 
 
 006483 
 
 999997 
 
 003254 
 
 999998 
 
 177226 
 
 999766 
 
 46 
 
 45 
 
 993520 
 
 000006 
 
 006480 
 
 999994 
 
 003252 
 
 999997 
 
 176992 
 
 999680 
 
 45 
 
 44 
 
 993526 
 
 000094 
 
 006474 
 
 999906 
 
 003249 
 
 999953 
 
 176672 
 
 993526 
 
 44 
 
 43 
 
 993620 
 
 000093 
 
 006380 
 
 999907 
 
 003202 
 
 999953 
 
 170198 
 
 993620 
 
 43 
 
 42 
 
 993713 
 
 000263 
 
 006287 
 
 999737 
 
 003155 
 
 999868 
 
 163818 
 
 981304 
 
 42 
 
 41 
 
 993976 
 
 000339 
 
 006024 
 
 999661 
 
 003023 
 
 999829 
 
 145122 
 
 974671 
 
 4i 
 
 40 
 
 994315 
 
 000496 
 
 005685 
 
 999504 
 
 002852 
 
 999750 
 
 119793 
 
 960101 
 
 40 
 
 39 
 
 994811 
 
 000312 
 
 005189 
 
 999688 
 
 002602 
 
 999843 
 
 0/9894 
 
 972909 
 
 39 
 
 38 
 
 995123 
 
 000137 
 
 004877 
 
 999863 
 
 002445 
 
 999932 
 
 052803 
 
 987570 
 
 38 
 
 37 
 
 995260 
 
 000133 
 
 004740 
 
 999867 
 
 002377 
 
 999933 
 
 040373 
 
 987573 
 
 37 
 
 36 
 
 995393 
 
 000129 
 
 004607 
 
 999871 
 
 002310 
 
 999935 
 
 027946 
 
 987568 
 
 36 
 
 35 
 
 995522 
 
 000046 
 
 004478 
 
 999954 
 
 002245 I 999977 
 
 015514 
 
 995522 
 
 35 
 
 34 
 
 995568 
 
 000045 
 
 004432 
 
 999955 
 
 002222 999977 
 
 011036 
 
 995568 
 
 34 
 
 33 
 
 995613 
 
 999965 
 
 004387 
 
 000035 
 
 0021991000018 
 
 006604 
 
 003438 
 
 33 
 
 32 
 
 995578 
 
 999967 
 
 004422 
 
 000033 
 
 002217 
 
 000016 
 
 010042 
 
 003265 
 
 32 
 
 3 r 
 
 995545 
 
 000045 
 
 004455 
 
 999955 
 
 002233 
 
 999978 
 
 013307 
 
 995545 
 
 31 
 
 3 
 
 995590 
 
 000121 
 
 004410 
 
 999879 
 
 002211 
 
 999939 
 
 008852 
 
 987903 
 
 30 
 
 29 
 
 995711 
 
 000494 
 
 004289 
 
 999506 
 
 002150 
 
 999751 
 
 3-996755 
 
 946493 
 
 29 
 
 28 
 
 996205 
 
 000408 
 
 003795 
 
 999592 
 
 001901 
 
 999796 
 
 943248 
 
 950448 
 
 28 
 
 27 
 
 996613 
 
 000176 
 
 003387 
 
 999824 
 
 001697 
 
 999912 
 
 893696 
 
 976868 
 
 27 
 
 26 
 
 996789 
 
 000023 
 
 003211 
 
 999977 
 
 001609 
 
 999988 
 
 870564 
 
 996789 
 
 26 
 
 25 
 
 996812 
 
 000096 
 
 003188 
 
 999904 
 
 001597 
 
 999952 
 
 867353 
 
 986593 
 
 25 
 
 24 
 
 996908 
 
 000022 
 
 003092 
 
 999978 
 
 001549 
 
 999989 
 
 853946 
 
 996908 
 
 24 
 
 23 
 
 22 
 
 996930 
 996952 
 
 000022 
 000021 
 
 003070 
 003048 
 
 999978 
 999979 
 
 001538 
 001527 
 
 999989 
 999989 
 
 850854 
 
 847784 
 
 996930 
 996952 
 
 2 3 
 
 22 
 
 21 
 
 996973 
 
 999950 
 
 003027 
 
 000050 
 
 001516 
 
 000025 
 
 844736 
 
 007192 
 
 21 
 
 20 
 
 996923 
 
 000021 
 
 003077 
 
 999979 
 
 001541 
 
 999990 
 
 851928 
 
 996923 
 
 20 
 
 jO 
 
 996944 
 
 000022 
 
 003056 
 
 999978 
 
 001531 
 
 999989 
 
 848851 
 
 996944 
 
 19 
 
 18 
 
 996966 
 
 000020 
 
 003034 
 
 999980 
 
 001520 
 
 999989 
 
 845795 
 
 996966 
 
 lo 
 
 17 
 
 996986 
 
 000091 
 
 003014 
 
 999909 
 
 001509 
 
 999955 
 
 842761 
 
 986767 
 
 J 7 
 
 16 
 
 997077 
 
 000226 
 
 002923 
 
 999774 
 
 001464 
 
 999887 
 
 829528 
 
 964893 
 
 16 
 
 *5 
 
 997303 
 
 000291 
 
 002697 
 
 999709 
 
 001351 
 
 999854 
 
 794421 
 
 950306 
 
 *5 
 
 M 
 
 997594 
 
 000150 
 
 002406 
 
 999850 
 
 001205 
 
 999925 
 
 744727 
 
 972040 
 
 14 
 
 
 997744 
 
 000079 
 
 002256 
 
 999921 
 
 001130 
 
 999960 
 
 716767 
 
 984380 
 
 13 
 
 12 
 
 997823 
 
 000078 
 
 002177 
 
 999922 
 
 001090 
 
 999961 
 
 701147 
 
 984035 
 
 12 
 
 II 
 
 997901 
 
 000145 
 
 002099 
 
 999855 
 
 001051 
 
 999927 
 
 685182 
 
 968937 
 
 II 
 
 10 
 
 998046 
 
 999742 
 
 001954 
 
 000258 
 
 000978 
 
 000130 
 
 644119 
 
 054162 
 
 10 
 
 9 
 
 997788 
 
 999345 
 
 002212 
 
 000655 
 
 001108 
 
 000328 
 
 708281 
 
 112742 
 
 9 
 
 8 
 
 997133 
 
 999030 
 
 002867 
 
 000970 
 
 001436 
 
 000487 
 
 821023 
 
 127093 
 
 8 
 
 7 
 
 996163 
 
 998470 
 
 003837 
 
 001530 
 
 001923 
 
 000769 
 
 948116 
 
 146549 
 
 7 
 
 6 
 
 994633 
 
 997566 
 
 005367 
 
 002434 
 
 002692 
 
 001226 
 
 2-094665 
 
 163604 
 
 6 
 
 5 
 
 992199 
 
 995144 
 
 007801 
 
 004856 
 
 003918 
 
 002456 
 
 258269 
 
 212610 
 
 5 
 
 4 
 
 987343 
 
 995858 
 
 012657 
 
 004142 
 
 006374 
 
 002107 
 
 470879 
 
 125056 
 
 4 
 
 3 
 
 983201 
 
 987648 
 
 016799 
 
 012352 
 
 008481 
 
 006339 
 
 595935 
 
 245583 
 
 3 
 
 2 
 
 970849 
 
 992653 
 
 029151 
 
 007347 
 
 014820 
 
 003812 
 
 841518 
 
 101342 
 
 2 
 
 I 
 
 963502 
 
 963920 
 
 036498 
 
 036080 
 
 018632 
 
 0191/2 
 
 942860 
 
 316957 
 
 I 
 
 
 
 927422 
 
 
 0725/8 
 
 
 03/804 
 
 
 T-259817 
 
 
 
 
46 
 
 for the sake of accuracy, to take account of its deviation from the 
 truth. This is done thus : log.v to 8 places, is 98716278 ; in using 
 987163, therefore, we commit each time an error of '22 in the 6th 
 place in excess. This amounts to about one-fourth in the place in 
 question, and the error will consequently be neutralized, (so far at least 
 as this is practicable,) by abating a unit in the 6th place from about 
 every fourth value.* The portions of the various series we shall em- 
 ploy in our examples will always have been subjected to this correction. 
 
 106. (2.) To construct the series co\og.vp x . 
 
 The terms of this series are obviously those of the tabulated series, 
 colog.j^, increased by the constant quantity, colog.v, or log.(l -f r),(76). 
 The construction will be effected, therefore, by the continued addition 
 to the first term of the series in question, increased by log.(l+r) = 
 012837, of the differences of that series. The correction to be made 
 for the inaccuracy in the last place of log.(l + r) will be, an increase 
 of a unit in the last place of about every fourth value. It is unneces- 
 sary to give an example. 
 
 107. (3.) To form the series colog.Vv.j^. 
 
 This series will be the tabulated series, colog.jt?^, with each of its 
 terms increased by the constant quantity colog.N/tf = log.Vl -fr = 
 ilog.fl -fr)= 006419; and it will be constructed by adding continu- 
 ally to the first term of the tabulated series increased by the quantity 
 in question, the differences of this series. The correction will be, 
 since log.(l + r) to eight places is 00641861, a diminution of the 6th 
 figure by a unit 39 times in a hundred, which is a little more than 
 once in three times. Here, too, an example is unnecessary. 
 
 108. (4.) To form in series all the different values of log.j 
 
 Values of 
 
 x-y 
 
 
 
 i 
 
 2 
 
 3 
 
 4 
 * 
 * 
 
 y 
 103 
 
 102 
 101 
 100 
 
 99 
 * 
 
 * 
 
 ^=103 
 522879 
 
 522879 
 
 y 
 
 102 
 
 101 
 100 
 
 99 
 
 98 
 * 
 * 
 
 X=I02 
 
 778151 
 778151 
 
 y 
 
 101 
 100 
 
 99 
 98 
 
 97 
 * 
 
 * 
 
 a?=ioi 
 
 853872 
 8538/2 
 
 y 
 
 TOO 
 
 99 
 98 
 
 97 
 
 96 
 * 
 * 
 
 X=IOO 
 
 890855 
 890855 
 
 y 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 # 
 
 * 
 
 #=99 
 912850 
 912850 
 
 y 
 9 8 
 
 97 
 
 96 
 
 95 
 
 94 
 * 
 * 
 
 #=98 
 895265 
 895265 
 
 
 045758 
 
 2552/2 
 
 556302 
 075721 
 
 707744 
 036983; 
 
 781710 
 021995 
 
 825700 
 982415 
 
 790530 
 995590 
 
 ,301030 
 075721 
 
 632023 
 036983 
 
 744727 
 021995 
 
 803705 
 982415 
 
 808115 
 
 995590 
 
 786120 
 002690 
 
 788810 
 991062 
 
 376751 
 036983 
 
 669006 
 021995 
 
 7667221 
 982415 
 
 786120 
 9.95590 
 
 803705 
 002690 
 
 413734 
 
 021995 
 
 691001 
 982415 
 
 749137 
 995690 
 
 781/10 
 002690 
 
 806395 
 991062 
 
 779S/2 
 990454 
 
 435729 
 # 
 * 
 
 673416 
 * 
 * 
 
 7447271 
 
 * 
 * 
 
 784400 
 * 
 * 
 
 797457 
 * 
 
 * 
 
 770326 
 * 
 * 
 
 * The correct abatement, in this case, would be 22 units in one hundred terms. 
 
47 
 
 98 
 
 99 
 
 IOO 
 IOI 
 102 
 I0 3 
 
 5 
 4 
 3 
 
 2 
 
 i 
 
 
 
 515078 
 995144 
 
 4 
 
 3 
 
 2 
 
 I 
 
 
 765494 
 995858 
 
 3 
 
 2 
 
 I 
 
 
 
 837073 
 987649 
 
 2 
 I 
 
 
 861705 
 992652 
 
 i 8763521 
 963920 
 
 510222 
 
 .995858 
 
 761352 
 
 987649 
 
 824722 
 992652 
 
 854357 
 963920 
 
 o 840272 
 
 506080 
 987649 
 
 749001 
 992652 
 
 817374 
 963920 
 
 818277 
 
 493729 
 992652 
 
 741653 
 963920 
 
 781294 
 
 486381 
 963920 
 
 705573 
 
 450301 
 
 o |822687ji 
 
 Above is a type of the operation here requisite, and the necessary 
 explanation follows. 
 
 109. The initial value in each column is log./?,,, corresponding to 
 that value of x which stands at the top of the column, and the first 
 addend is log.jt? y corresponding to the same value of y. The succeed- 
 ing addends are the differences of the series log.jt? 7 , commencing with 
 that difference which belongs to the term of the series just set down, 
 and the successive sums are the terms of the required series which 
 correspond to the value of x at the top of the column in which they 
 are found, and to the values of y immediately adjoining them. The 
 number of columns will obviously be 104, (one fewer than the number 
 of ages in the mortality table,) and the type, when completed as indi- 
 cated, will contain all the different values of log.jt?^), as required. 
 For, log.^ and log.j^, of which log.p^ y ) is the sum, being terms of 
 the same series, the function is symmetrical with respect to x and y, 
 and all its different values are obtained by combining, as is here done, 
 each of the values of one of these quantities with all the values of the 
 other that do not exceed it. 
 
 110. It will be noticed, also, that the several horizontal lines of the 
 type consist of those values of the function log.p^^, ranged in order, 
 in which x and y have the same difference. That is to say, opposite 
 each value in the column headed x y, are to be found all those values 
 of the function Iog.j9 (j ij} in which x y has the value indicated, and so 
 arranged that in passing from any one value of the function to the next 
 both x and y decrease by a unit. These values, in order, form the series 
 required for use, and by reason of this commodious arrangement, they 
 admit of being abstracted with facility, unattended by risk of error. 
 
 111. We see that, taking any term of the combined series, and 
 adding to it continually the differences of log.p, , from and after the 
 term with which we set out, other terms of the combined series are 
 successively produced, in which x remains constant while y decreases 
 by a unit in each succeeding term. But this property is obviously not 
 confined to the differences of log. ; it belongs as well to the differences 
 
48 
 
 of log. p x . If, therefore, to any term of the combined series we add 
 continually the differences of log.jt?.,., commencing with the difference 
 belonging to that term of log.^ included in the term of the series 
 with which we set out, the terms of the .series produced will be those 
 in which y remains constant, and x decreases by a unit in each suc- 
 ceeding term. This suggests a method of verifying the operation 
 typified above, which may be applied as follows : Let the first horizon- 
 tal line, (x y=0,) be completed, and also the first column, (x = 103,) 
 which last may be verified as in (102). From this 
 column select three or four values at suitable intervals, <0 frP(*rl 
 and make them the initial terms of as many series in ^ 255272 
 
 which x varies and v remains constant, verifying each 
 
 j- j mi j i -ii t 102-99 691001 
 
 series as just directed. The values produced will be ^^ 075721 
 
 found to be the terms of the required series, which be- 
 
 long to compartments in the type ascending diagonally ^^ 036983 
 
 from the place of the initial term of the several series 
 
 thus formed. They may therefore be inserted in their * 
 
 places, and they will thus serve, as they are successively 
 
 reached, to verify the additions in the several columns. ^ 103 255272 
 
 An example showing the formation of portions of two 
 
 of these auxiliary series is here given, comparison of ^ 102 075721 
 
 which with the type will render farther explanation 
 
 Jr loro 781294 
 
 unnecessary. Am 036983 
 
 112. All the foregoing operations the formation of > 
 
 the auxiliary series, as well as of those occupying the 
 
 columns of the type may obviously be performed by means of the 
 set of cards already described, for they are the terms of the same series 
 of differences, and in the same order, although commencing usually at 
 different points of the series, that form the addends in each case. 
 Hence the labour of construction is comparatively small. Each value 
 is produced by an addition of two lines, and the only writing, after the 
 formation of the first line, (x y = 0,) is the setting down of the 
 results. 
 
 113. It will be observed that, according to the mode of construction 
 now described, the successive terms of the several series in which 
 x y is constant, are found in successive columns; so that to form 
 one of these series in which x y has a given value, all the columns 
 in which terms of that series are contained, must be completed down 
 to the terms in question. Hence, if only a few of those series be 
 wanted, those for example in which x y = 10, and multiples of 10, 
 successively, in forming them by the method described, a great many 
 values of the general function which are not required for the purpose 
 
49 
 
 in view, are also formed. And although each of these is formed in 
 this way with less labour than is required for its formation in any other 
 way, yet it may very well happen, if the values of x y in the series 
 wanted are far apart,, that the labour is greater than needs be en- 
 countered in the circumstances. We therefore proceed to show how 
 any one of these series may be formed independently of the others. 
 A few examples for different values of x y follow. 
 
 103 
 103 
 
 103-103 
 
 A 103 
 
 A 103 
 IO2'I02 
 
 Al02 
 Al02 
 
 lorioi 
 
 A 101 
 Aioi 
 
 IOO* IOO 
 
 100 
 100 
 
 lOO'IOO 
 
 x y=o 
 522879 
 522879 
 
 103 
 
 102 
 I03*I02 
 
 Al03 
 A 102 
 102*101 
 
 A 102 
 
 Aioi 
 lorioo 
 A 101 
 
 A 100 
 
 100-99 
 
 IOO 
 
 99 
 100-99 
 
 xy=i 
 522879 
 778151 
 
 103 
 
 101 
 
 103-101 
 
 A 103 
 
 Aioi 
 
 102*100 
 
 A 102 
 A 100 
 
 101-99 
 
 A 101 
 A 99 
 100*98 
 
 cation. 
 
 IOO 
 
 98 
 
 100*98 
 
 xy=2 
 522879 
 
 853872 
 
 103 
 3 
 iQ3'3 
 
 A 103 
 
 A 3 
 
 IO2'2 
 
 A 102 
 
 A 2 
 
 lOI'I 
 
 A 101 
 Ai 
 
 lOO'O 
 
 IOO 
 
 
 lOO'O 
 
 x y=iOQ 
 522879 
 983201 
 
 045758 
 2552/2 
 255272 
 
 301030 
 2552/2 
 075721 
 
 376751 | 
 255272 
 
 036983 
 
 506080 
 255272 
 987648 
 
 556302 
 075721 
 075721 
 
 632023 
 075721 
 036983 
 
 669006 | 
 075721 
 021995 
 
 749000 
 075721 
 992653 
 
 707744 
 036983 
 036983 
 
 744727 
 036983 
 021995 
 
 766722 
 036983 
 982415 
 
 817374 
 036983 
 963920 
 
 781710 
 
 890855 ! 
 890855 
 
 781710 ' 
 
 803705 
 
 Verifi 
 
 890855 
 912850 
 
 786120 i 
 
 890855 
 895265 
 
 818277 
 
 890855 
 927422 
 
 803705 
 
 786120 
 
 818277 
 
 114. Little explanation of the foregoing seems to be requisite after 
 what has been already shown. All the values here formed will be 
 found in the type previously given. The successive values here as 
 there being mutually dependent, the same sort of verification applies ; 
 and it will be well to form a sufficient number of values for this pur- 
 pose first of all, and to insert them in their places at once. 
 
 115. This operation, as well as that previously described, may be 
 performed by means of cards, which being once prepared, the subse- 
 quent labour of writing is reduced to that of setting down the results. 
 The following figures show the form of the cards here requisite, and 
 the mode of applying them for any given value of xy. 
 
 116. Two sets of cards are requisite, and figs. 2 and 3 show the 
 first, or portions of the first, card of each set. Fig. 4 shows the cards 
 arranged for the formation of the series in which x y = Q, and fig. 5 
 shows them arranged for the formation of that in which x y = l. 
 The arrangement for any given value of x y is sufficiently obvious. 
 The cards have to be so placed that each pair of ages, having between 
 them the given difference, shall be brought together. The value cor- 
 
50 
 
 o* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 II 
 
 
 
 GO 
 
 CM 
 
 (M 
 
 01 
 
 i-H 
 
 I 
 
 * 
 
 CO 
 
 CO 
 
 
 to 
 
 tO 
 
 
 to 
 
 tO 
 
 c 
 
 ?5} 
 
 H 
 
 
 tO 
 
 t^. 
 
 (^ 
 
 
 
 OI 
 
 01 
 
 Tf 
 
 00 
 
 GO 
 
 
 Oi 
 
 Oi 
 
 
 1 1 
 
 ,_, 
 
 
 
 1 
 
 *$ 
 
 ^o 
 
 
 
 
 CM 
 
 OI 
 
 co 
 
 tO 
 
 tO 
 
 t - 
 
 Oi 
 (O 
 
 Oi 
 
 
 Oi 
 
 Oi 
 ( 1 
 
 
 ^ 
 
 OI 
 
 ur 
 
 Lf 
 
 ^ 
 
 < 
 
 
 ^ 
 
 to 
 
 ^0 
 
 to 
 
 t^ 
 
 t^ 
 
 o 
 
 CO 
 
 CO 
 
 
 CN 
 
 01 
 
 
 GO 
 
 GO 
 
 
 
 M 
 
 
 
 O 
 
 OI 
 
 CM 
 
 to 
 
 
 
 
 
 * 
 
 
 
 o 
 
 
 
 
 o 
 
 
 Oi 
 
 Oi 
 
 
 
 
 ro c^ M o ON oo 
 O O O O ON ON 
 
 
 O o O o ON ON 
 
 H M M M 
 
 CO 
 
 d 
 
 to 
 
 
 
 OI 
 
 
 
 
 CO 
 
 
 to 
 
 
 to 
 
 
 
 
 
 
 
 
 
 
 - H 
 
 
 t^ 
 
 
 OI 
 
 
 CO 
 
 
 Oi 
 
 
 1 1 
 
 
 Oi 
 
 ^ 
 
 
 OI 
 
 
 t->. 
 
 
 Oi 
 
 
 Oi 
 
 
 TP 
 
 
 to 
 
 
 
 
 to 
 
 
 to 
 
 
 to 
 
 
 _H 
 
 
 CM 
 
 
 tO 
 
 ^ 
 
 
 to 
 
 
 t^ 
 
 
 CO 
 
 
 CN 
 
 
 CO 
 
 
 Oi 
 
 
 
 CN 
 
 
 o 
 
 
 
 
 
 
 
 
 Oi 
 
 
 Oi 
 
 co N M O ON OO 
 O O O O ON ON 
 
 M M M M 
 
51 
 
 FIG. 5. x = 
 
 responding to the first 
 of these pairs, namely, 
 that in which a; = 103, 
 being then inserted in 
 its place, the addition 
 proceeds as in the previ- 
 ous example. Compare 
 the first two columns 
 of the example referred 
 to with figs. 4 and 5. 
 
 117. (5.) Let it now 
 be required to construct 
 all the different values 
 of Iog.j9( ). 
 
 The type of this ope- 
 ration is on the fol- 
 lowing page, and it 
 will readily appear that 
 when completed as indi- 
 cated, it will contain all 
 the required values. It 
 will consist of 104 com- 
 partments, of which the 
 first will contain 104 
 columns and 104 rows, 
 the second 103 columns 
 and 103 rows, and so 
 on, each succeeding 
 compartment contain- 
 ing both one column 
 and one row fewer than 
 the preceding. The ini- 
 tial values in the first 
 compartment are the terms of the series log.jt?^) for x 2/ = 0, and 
 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 98 
 
 
 Alog.fl,. 
 
 
 
 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 98 
 
 Alog-p, 
 
 
 255272 
 
 301030 
 
 255272 
 
 075721 
 
 632023 
 
 075721 
 
 036983 
 
 744727 
 
 036983 
 
 021995 
 
 
 021995 
 
 982415 
 
 
 982415 
 
 
 
 
 995590 
 
 the first addend in each column is that log.^, in which z is equal to 
 the common value of x and y in that column. The remaining ad- 
 dends are the successive terms of the series of differences of log.p,. 
 The initial values in each of the other compartments are taken from 
 the compartments that precede them respectively, as will appear by 
 reference to the type, and the addends consist of the series of differences 
 of log.j^. 
 
 118. The series of values of this function wanted for use are those 
 
52 
 
 Values 
 
 .y.z) 
 
 yz 
 
 
 
 i 
 
 2 
 
 3 
 4 
 
 2 
 103 
 
 IO2 
 101 
 100 
 
 99 
 
 #=103 
 
 y=I03 
 
 045758 
 522879 
 
 z 
 
 102 
 IOI 
 100 
 
 99 
 98 
 
 y=iO2 
 #=102 
 556302 
 778151 
 
 z 
 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 
 x y- 
 a?=ioi 
 y=ioi 
 707744 
 
 853872 
 
 =o. 
 
 z 
 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 X= IOO 
 
 y=ioo 
 781710 
 
 890855 
 
 z 
 
 99 
 98 
 
 97 
 96 
 
 95 
 
 z 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 #=99 
 #=99 
 825700 
 912850 
 
 t 
 
 98 
 
 97 
 96 
 
 95 
 9 1 
 
 ; 
 
 97 
 96 
 
 95 
 94 
 93 
 
 #=98 
 y= 9 8 
 790530 
 895265 
 
 568637 
 255272 
 
 334453 
 075721 
 
 561616 
 036983 
 
 672565 
 021995 
 
 738550 
 982415 
 
 685795 
 995590 
 
 823909 
 075721 
 
 410174 
 036983 
 
 598599 
 021995 
 
 694560 
 982415 
 
 720965 
 995590 
 
 681385 
 002690 
 
 899630 
 036983 
 
 447157 
 021995 
 
 620594 
 982415 
 
 676975 
 995590 
 
 716555 
 002690 
 
 684075 
 991062 
 
 936613 
 021995 
 
 469152 
 982415 
 
 603009 
 995590 
 
 672565 
 002690 
 
 719245 
 991062 
 
 675137 
 990454 
 
 958608 
 
 451567 
 
 598599 
 
 675255 
 
 710307 
 
 * 
 
 #=99 
 y= 9 8 
 720965 
 982415 
 
 665591 
 
 * 
 
 #=98 
 
 y=9i 
 
 681385 
 995590 
 
 yz 
 
 o 
 i 
 
 2 
 
 3 
 
 4 
 
 f 
 
 102 
 101 
 100 
 
 99 
 98 
 
 #=103 
 
 y=IO2 
 
 823909 
 255272 
 
 z 
 
 IOI 
 IOO 
 
 99 
 
 98 
 
 97 
 
 #=102 
 y=ioi 
 410174 
 075721 
 
 z 
 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 xy- 
 
 #=IOI 
 
 y=ioo 
 598599 
 036983 
 
 = i. 
 
 z 
 
 99 
 98 
 
 97 
 96 
 
 95 
 
 #=100 
 
 y=99 
 
 694560 
 021995 
 
 079181 
 075721 
 
 485895 
 036983 
 
 635582 
 021995 
 
 657577 
 982415 
 
 716555 
 982415 
 
 703380 
 995590 
 
 676975 
 002690 
 
 154902 
 036983 
 
 522878 
 021995 
 
 698970 
 995590 
 
 698970 
 002690 
 
 679665 
 991062 
 
 191885 
 021995 
 
 544873 
 982415 
 
 639992 
 995590 
 
 694560 
 002690 
 
 701660 
 991062 
 
 670727 
 990454 
 
 213880 
 982415 
 
 527288 
 995590 
 
 635582 
 002690 
 
 697250 
 991062 
 
 692722 
 990454 
 
 661181 
 994605 
 
 196295 
 
 522878 
 
 638272 
 
 688312 
 
 683176 
 
 655786 
 
 y * 
 
 o 
 I 
 
 2 
 
 3 
 4 
 
 z 
 
 IOI 
 100 
 
 99 
 
 98 
 
 97 
 
 #=103 
 y=ioi 
 154902 
 075721 
 
 z 
 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 #=IO2 
 
 y=ioo 
 522878 
 036983 
 
 z 
 
 99 
 98 
 
 97 
 96 
 
 95 
 
 xy- 
 #=101 
 
 y=99 
 
 657577 
 021995 
 
 = 2. 
 
 Z 
 
 9 8 
 
 97 
 96 
 
 95 
 94 
 
 #=100 
 y= 9 8 
 698970 
 982415 
 
 z 
 
 97 
 96 
 
 95 
 94 
 93 
 
 #=99 
 
 y=97 
 
 698970 
 995590 
 
 z 
 
 96 
 95 
 94 
 
 93 
 92 
 
 #=98 
 y= 9 6 
 679665 
 002690 
 
 230623 
 036983 
 
 559861 
 021995 
 
 679572 
 982415 
 
 681385 
 995590 
 
 694560 
 002690 
 
 682355 
 991062 
 
 267606 
 021995 
 
 581856 
 982415 
 
 661987 
 995590 
 
 676975 
 002690 
 
 697250 
 991062 
 
 673417 
 990454 
 
 289601 
 982415 
 
 564271 
 995590 
 
 657577 
 002690 
 
 679665 
 991062 
 
 688312 
 990454 
 
 663871 
 994605 
 
 272016 
 995590 
 
 559861 
 002690 
 
 660267 
 991062 
 
 670727 
 990454 
 
 678766 
 994605 
 
 658476 
 987667 
 
 267606 
 
 562551 
 
 651329 
 
 661181 
 
 673371 
 
 646143 
 
 y z 
 
 
 
 i 
 
 2 
 
 3 
 4 
 
 2 
 
 too 
 
 99 
 
 98 
 
 97 
 96 
 
 #=103 
 y=ioo 
 267606 
 036983 
 
 z 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 
 #=102 
 
 y=99 
 
 581856 
 021995 
 
 z 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 x-y 
 
 #=IOI 
 
 y= 9 8 
 661987 
 982415 
 
 =3- 
 
 z 
 
 97 
 96 
 
 95 
 H 
 93 
 
 #=ioo 
 
 y=9l 
 676975 
 995590 
 
 z 
 
 96 
 
 95 
 94 
 93 
 92 
 
 #=99 
 y= 9 6 
 697250 
 002690 
 
 z 
 
 95 
 94 
 93 
 92 
 
 9i 
 
 #=98 
 
 y=95 
 
 673417 
 991062 
 
 304589 
 021995 
 
 603851 
 982415 
 
 644402 
 995590 
 
 672565 
 002690 
 
 699940 
 991062 
 
 664479 
 990454 
 
 326584 
 982415 
 
 586266 
 995590 
 
 639992 
 002690 
 
 642682 
 991062 
 
 675255 
 991062 
 
 691002 
 990454 
 
 654933 
 994605 
 
 308999 
 995590 
 
 581856 
 002690 
 
 666317 
 990454 
 
 681456 
 994605 
 
 649538 
 987667 
 
 304589 
 002690 
 
 584546 
 991062 
 
 633744 
 990454 
 
 656771 
 994605 
 
 676061 
 987667 
 
 637205 
 996539 
 
 307279 
 
 575608 
 
 624198 
 
 651376 
 
 663728 
 
 633744 
 
53 
 
 in which both x y and y z are constant. Hence, the former being 
 constant throughout the whole of each compartment, and the latter in 
 each horizontal line, any particular series required will be found in 
 the compartments headed with the given value of x y, and in the 
 line opposite the given value of y z. 
 
 Values for verification may be found precisely as 
 in (111), as shown in the accompanying example 
 from the portion of the first compartment above 
 given. Only here, as both x and y vary in the suc- 
 cessive columns, two differences have to be included 
 in each addition. 
 
 119. The whole of the operations now described 
 may obviously be performed by means of the cards ; 
 the principal operation by the single card, fig. 1, and 
 the subsidiary operation, for the verification of the 
 other, by means of the two cards, figs. 2 and 3 ; so 
 
 that here also the writing is reduced to the setting down of the 
 results. 
 
 120. Any one of the series in which x y andy z are constant, 
 and have given values, may also be formed independently of any of 
 the others; and although the labour of finding each value by the 
 method now to be exemplified is greater than by the method just 
 described, yet, since by this method we form only values that are im- 
 mediately wanted, it may be advisable, in some cases, to resort to it, 
 After what has been already shown > it is sufficient to give a few 
 examples of the method referred to, which examples accordingly are 
 hereto appended. 
 
 103 r 
 
 103 ^ 936613 
 
 100 L 2552/2 
 102] 255272 
 
 102 1447157 
 ioo J 075721 
 TOT n 075721 
 
 101 1598599 
 looj 036983 
 I00 , 036983 
 
 ioo V 672565 
 
 lOOj 
 
 x y=o 
 y z=o 
 522879 
 522879 
 jo.,-] 522879 
 
 x y=o 
 y-z=i 
 522879 
 522879 
 loal 778151 
 
 x-y=z 
 yz=2 
 522879 
 890855 
 103 ] 895265 
 
 x y=2o 
 y-z=io 
 522879 
 928968 
 ioq"| 964676 
 
 103 \ 568637 
 103! 255272 
 255272 
 I02 -| 255272 
 
 103 > 823909 
 102 J 255272 
 255272 
 102 1 075721 
 
 ioo I 308999 
 98 J 255272 
 021995 
 I02 -j 995590 
 
 83 \ 416523 
 73 J 255272 
 005182 
 102 ] 004680 
 
 102 V 334453 
 102 J 075721 
 075721 
 101] 075721 
 
 102 I 410174 
 ioij 075721 
 075721 
 
 joi-i 036983 
 
 99 r 581856 
 97 J 075721 
 982415 
 IOI -, 002690 
 
 82 I 681657 
 72 J 075721 
 003463 
 ioi ] 004303 
 
 ioi \ 561616 
 ioij 036983 
 036983 
 KXO 036983 
 
 ioi [ 598599 
 iooj 036983 
 036983 
 I00 -j 021995 
 
 98 [ 642682 
 69 J 036983 
 995590 
 I00 -j 991062 
 
 81 I 765144 
 71 J 036983 
 006020 
 I00 , 003312 
 
 ioo I 672565 
 
 IOO J 
 
 ioo I 694560 
 99 J 
 
 97 > 666317 
 95J 
 
 80 I 811459 
 70 J 
 
ioo 890855 
 
 8o 943633 
 
 7 ?76_971 
 
 100-80-70 811459 
 
 54 
 
 1.21. All the values formed in the first three of 
 these examples will be formed in the type already 
 given. Each series may be verified at any point, 
 as shown in the margin, in regard to the last of 
 them. 
 
 122. This mode of construction likewise admits of the application 
 of cards. Three will obviously be necessary, similar to those repre- 
 sented in figs. 2 and 3, but with the perforations larger or farther 
 apart. 
 
 123. It will obviously be easy to apply the principles laid down to 
 the formation of the values of functions in which four or any greater 
 number of lives are concerned ; but it is unnecessary to go farther into 
 the subject here. When so many as four lives are in question the 
 number of different values of the functions will be so great as probably 
 in all cases to prove a bar to attempting the formation of the whole. 
 Any particular series wanted, therefore, will be found by the method 
 last exemplified in the case of three lives, and either with or without 
 the aid of perforated cards. 
 
 124. We have now to exemplify the construction of the same series 
 when each of their terms is increased by the constant quantity log.v. 
 
 (6.) To form all the different values of log.Vjp (xy y 
 
 Values of 
 
 xy 
 
 o 
 
 i 
 
 2 
 
 3 
 
 4 
 * 
 
 * 
 
 9 
 
 103 
 
 IO2 
 IOI 
 TOO 
 
 99 
 * 
 
 * 
 
 #=103 
 
 510042 
 522879 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 98 
 * 
 * 
 
 #=IO2 
 
 765314 
 778151 
 
 IOI 
 IOO 
 
 99 
 
 98 
 
 97 
 
 * 
 
 * 
 
 X=IOl 
 
 841035 
 
 853872 
 
 IOO 
 
 99 
 98 
 97 
 
 96 
 
 * 
 * 
 
 X=IOO 
 
 878017 
 890855 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 * 
 * 
 
 #=99 
 900013 
 912850 
 
 98 
 
 97 
 96 
 
 95 
 
 94 
 * 
 * 
 
 #=98 
 882428 
 895265 
 
 * 
 * 
 * 
 
 * 
 
 * 
 
 * 
 
 032921 
 255272 
 
 543465 
 075721 
 
 694907 
 036983 
 
 768872 
 021995 
 
 812863 
 982415 
 
 777693 
 995590 
 
 288193 
 075721 
 
 619186 
 036983 
 
 731890 
 021995 
 
 790867 
 982415 
 
 795278 
 995590 
 
 773283 
 002690 
 
 363914 
 036983 
 
 656169 
 021995 
 
 753885 
 982415 
 
 773282 
 995590 
 
 790868 
 002690 
 
 775973 
 991062 
 
 400897 
 021995 
 
 678164 
 982415 
 
 736300 
 995590 
 
 768872 
 002690 
 
 793558 
 991062 
 
 767035 
 990454 
 
 422892 
 
 * 
 * 
 
 660579 
 * 
 * 
 
 731890 
 * 
 * 
 
 771562 
 * 
 
 * 
 
 784620 
 
 * 
 * 
 
 757489 
 * 
 
 * 
 
 The only respect in which the operation here typified differs from 
 that typified in (108), is that here the initial values in the several 
 columns are the successive terms of the series* log.vp x , the formation 
 of which is explained in (101). The addends are in every case the 
 
 * This series being here used as corrected for the inaccuracy in the last figure 
 of log.t? (105), no farther attention on this score is necessary. 
 
55 
 
 same, so that the same set of cards may be employed 
 in the construction; and the method of forming x y 
 
 values for verification is also the same, of which, log.t? 
 therefore, it is unnecessary to give an example. 
 
 125. The mode of forming independently a series 103-102 
 of values in which x y is constant, is likewise here 
 
 in strict analogy with that described in (113) and 
 (114). The only difference is, that in the formation 
 now in question log.t? is included in the first addition. 
 An example is given in the margin. This operation 
 may be conducted by means of the cards, figs. 2 and 3. 
 
 126. (7.) To form all the different values of log.vp^ K} . 
 
 Values of 
 
 102-101 
 
 987163 
 522879 
 778151 
 
 288193 
 255272 
 075721 
 
 619186 
 075721 
 036983 
 
 731890 
 036983 
 021995 
 
 790868 
 
 x y=o 
 
 y-Z 
 
 O 
 I 
 
 2 
 
 3 
 
 4 
 
 * 
 * 
 
 z 
 103 
 
 102 
 101 
 100 
 
 99 
 * 
 * 
 
 #=103 
 
 y=i3 
 
 032921 
 522879 
 
 z 
 
 102 
 IOI 
 IOO 
 
 99 
 
 98 
 
 * 
 * 
 
 #=IO2 
 
 y=i02 
 543465 
 778151 
 
 z 
 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 
 * 
 
 * 
 
 #=IOI 
 
 y=ioi 
 694907 
 
 853872 
 
 z 
 
 IOO 
 
 99 
 
 f 
 
 97 
 
 96 
 
 * 
 
 * 
 
 #=ioo 
 y=ioo 
 
 768872 
 890855 
 
 z 
 
 99 
 98 
 
 97 
 96 
 
 95 
 
 * 
 
 * 
 
 *= 99 
 
 y=99 
 812863 
 912850 
 
 z 
 
 98 
 
 97 
 96 
 95 
 
 94 
 
 * 
 * 
 
 37=98 
 y= 9 8 
 777693 
 895265 
 
 * 
 
 * 
 
 * 
 * 
 
 # 
 * 
 
 555800 
 255272 
 
 321616 
 075721 
 
 548779 
 036983 
 
 659727 
 021995 
 
 725713 
 982415 
 
 672958 
 995590 
 
 611072 
 075721 
 
 397337 
 036983 
 
 585762 
 021995 
 
 681722 
 982415 
 
 708128 
 995590 
 
 668548 
 002690 
 
 886793 
 036983 
 
 434320 
 021995 
 
 607757 
 982415 
 
 664137 
 995590 
 
 703718 
 002690 
 
 671238 
 991062 
 
 923776 
 021995 
 
 456315 
 982415 
 
 590172 
 995590 
 
 6597271 
 002690 
 
 706408 
 991062 
 
 662300 
 990454 
 
 945771 
 
 * 
 * 
 
 438730 
 
 * 
 * 
 
 585762 
 
 * 
 * 
 
 662417! 
 
 * 
 * 
 
 697470 
 
 * 
 
 * 
 
 652754 
 
 * 
 * 
 
 x y=i 
 
 ,-. 
 
 O 
 
 I 
 
 2 
 
 3 
 
 4 
 * 
 * 
 
 z 
 1 02 
 
 IOI 
 
 IOO 
 
 99 
 
 98 
 
 * 
 * 
 
 #=103 
 y=i02 
 811072 
 255272 
 
 z 
 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 
 * 
 * 
 
 #=102 
 y=ioi 
 397337 
 075721 
 
 z 
 
 IOO 
 
 99 
 
 98 
 
 97 
 
 96 
 
 * 
 * 
 
 #=IOT 
 
 y=ioo 
 585762 
 036983 
 
 z 
 
 99 
 98 
 
 97 
 96 
 
 95 
 
 * 
 * 
 
 a?=ioo 
 
 y= 99 
 
 681722 
 021995 
 
 z 
 
 98 
 
 97 
 96 
 
 95 
 
 94 
 
 * 
 * 
 
 *=99 
 y= 9 8 
 708128 
 982415 
 
 97 
 96 
 
 95 
 94 
 
 93 
 * 
 * 
 
 #=98 
 
 y=97 
 
 668548 
 995590 
 
 * 
 * 
 * 
 * 
 
 * 
 * 
 * 
 
 066344 
 075721 
 
 473058 
 036983 
 
 622745 
 021995 
 
 703717 
 982415 
 
 690543 
 995590 
 
 664138 
 002690 
 
 142065 
 036983 
 
 510041 
 021995 
 
 644740 
 982415 
 
 686132 
 995590 
 
 686133 
 002690 
 
 666828 
 991062 
 
 179048 
 021995 
 
 532036 
 982415 
 
 627155 
 995590 
 
 681722 
 002690 
 
 684412 
 991062 
 
 688823 
 991062 
 
 657890 
 990454 
 
 201043 
 982415 
 
 514451 
 995590 
 
 622745 
 002690 
 
 679885 
 990454 
 
 648344 
 994605 
 
 183458 
 * 
 * 
 
 510041 
 
 * 
 * 
 
 625435 
 
 * 
 * 
 
 675474 
 
 * 
 * 
 
 670339 
 * 
 # 
 
 642949 
 
 * 
 * 
 
y z 
 
 o 
 I 
 
 2 
 
 3 
 4 
 
 z 
 
 101 
 IOO 
 
 99 
 
 98 
 
 97 
 
 #=103 
 y=ioi 
 142065 
 075721 
 
 z 
 
 TOO 
 
 99 
 98 
 
 97 
 96 
 
 #=IO2 
 
 y=ioo 
 510041 
 036983 
 
 z 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 
 56 
 
 xy- 
 
 X=IOI 
 
 y=99 
 
 644740 
 021995 
 
 -2 
 
 Z 
 
 9 8 
 
 97 
 96 
 
 95 
 94 
 
 r=ioo 
 y= 9 8 
 686132 
 982415 
 
 z 
 
 97 
 96 
 
 95 
 94 
 93 
 
 #=99 
 
 y=97 
 
 686133 
 995590 
 
 z 
 
 96 
 
 95 
 94 
 93 
 92 
 
 #=98 
 y- 9 6l 
 666828 
 002690 
 
 217786 
 036983 
 
 547024 
 021995 
 
 666735 
 982415 
 
 668547 
 995590 
 
 681723 
 002690 
 
 669518 
 991062 
 
 254769 
 021995 
 
 569019 
 982415 
 
 649150 
 995590 
 
 664137 
 002690 
 
 684413 
 991062 
 
 660580 
 990454 
 
 2/6764 
 982415 
 
 551434 
 995590 
 
 644740 
 002690 
 
 666827 
 991062 
 
 675475 
 990454 
 
 651034 
 994605 
 
 259179 
 995590 
 
 547024 
 002690 
 
 647430 
 991062 
 
 657886 
 990454 
 
 665929 
 994605 
 
 645639 
 987667 
 
 254769 
 
 549714 
 
 638492 
 
 648343 
 
 660534 
 
 633306 
 
 
 This formation corresponds in every respect with that described and 
 typified in (117), except that here the initial values in the several 
 columns of the first compartment are the terms of the series log.vp^ ), 
 in which x y = 0. The addends are the same, so that the cards, 
 fig. 1, may be employed ; and the initial terms of the several columns 
 in the successive compartments are found here, as in the former case, 
 in each preceding compartment. 
 
 127. The mode of forming independently any series of these terms 
 in which x y and y z are given and constant, is likewise the same 
 as that described in (120) for the formation in like circumstances of a 
 series of terms of log.p (x y ^ log.v being here included in the first 
 addition. A few examples are given below. 
 
 987163 
 522879 
 522879 
 
 x y=o 
 
 987163 
 522879 
 522879 
 ioO 778151 
 
 y 2=2 
 987163 
 522879 
 890855 
 I03 -| 895265 
 
 y z=io 
 987163 
 522879 
 928968 
 Tft o 964676 
 
 103 f 555800 
 IQ3J 255272 
 255272 
 I02 -| 255272 
 
 103 > 811072 
 iQ2j 255272 
 255272 
 102 1 075721 
 
 ioo )> 296162 
 98J 255272 
 021995 
 102"] 995590 
 
 83 ^> 403686 
 73 J 255272 
 005182 
 102! 004680 
 
 102 J> 321616 
 
 I02j 075721 
 
 075721 
 
 I0 n 075721 
 
 102 !> 397337 
 ioij 075721 
 075721 
 I0 ri 036983 
 
 99 > 569019 
 
 97; 075721 
 
 982415 
 IOI -| 002690 
 
 82 S 668820 
 72j 075721 
 003463 
 IO P, 004303 
 
 ioi > 548779 
 loij 036983 
 036983 
 -, 036983 
 
 ioi }> 585762 
 iooj 036983 
 036983 
 irvr 021995 
 
 98 )> 629845 
 9^J 036983 
 995590 
 Tftrr) 991062 
 
 81 Y 752307 
 7i J 036983 
 006020 
 T - 003312 
 
 100 !> 659728 
 
 IOOJ 
 
 100 )> 681723 
 99J 
 
 97 }^ 653480 
 95J 
 
 80 *> 798622 
 7oJ 
 
57 
 
 128. These values, like the others, admit of verifi- 
 cation at any point of the series, as is here shown. 
 The series formed in this manner require the usual 
 correction for the inaccuracy in the last place of \og.v. 100- 
 
 129. (8.) To form all the values of log.ifav" 1 !)CP 
 
 Values of Log.-IO^- 1 -!)^- 1 ^- 1). 
 x-y 
 
 v. 987163 
 
 loo 890855 
 
 80 943633 
 
 70 976971 
 
 80-70 798622 
 
 -I_L ^\ 
 
 y 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 * 
 
 #=103 
 301030 
 301030 
 
 y 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 * 
 
 823909 
 301030 
 
 y 
 
 io3 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 a?=ioi 
 602060 
 301030 
 
 y 
 103 
 
 IO2 
 IOI 
 100 
 
 99 
 
 * 
 
 a?=ioo 
 455932 
 301030 
 
 y 
 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 #=99 
 
 346787 
 301030 
 
 y 
 
 103 
 1 02 
 
 IOI 
 IOO 
 
 99 
 
 * 
 
 #=98 
 435728 
 301030 
 
 602060 
 823909 
 
 124939 
 
 823909 
 
 903090 
 823909 
 
 756962 
 823909 
 
 647817 
 823909 
 
 736758 
 823909 
 
 425969 
 954242 
 
 948848 
 954242 
 
 726999 
 954242 
 
 580871 
 954242 
 
 471726 
 954242 
 
 560667 
 954242 
 
 380211 
 978811 
 
 903090 
 978811 
 
 681241 
 978811 
 
 535113 
 
 978811 
 
 425968 
 978811 
 
 514909 
 978811 
 
 359022 
 987765 
 
 881901 
 987765 
 
 660052 
 987765 
 
 513924 
 987765 
 
 404779 
 987765 
 
 493720 
 
 987765 
 
 346787 
 * 
 
 869666 
 
 * 
 
 647817 
 
 501689 
 
 * 
 
 392544 
 
 481485 
 
 * 
 
 The foregoing type shows the mode of formation here. It differs, 
 it will be observed, from those in (108) and (124), inasmuch as in 
 the present case the two series, the terms of which have to be com- 
 bined, being essentially distinct, each value of x has to be united 
 with each value of y t and the terms in which x y is constant are 
 found here ranged, not in horizontal lines, but in diagonally descend- 
 ing lines. The initial values in the several columns are the terms of 
 the series log.^/" 1 -1) ; the first addend in each column is log. 
 iCP -1 + 1) for the greatest value of y for which this function sub- 
 sists; and the remaining addends are the terms of the series Alog.J 
 (p y ~ l + 1). These terms being put upon a set of cards similar to that 
 represented in fig. 1, the operation may obviously be performed by 
 means of them, in the manner already described, (1 12.) 
 
 130. Values for verification may be formed as in i_ 
 the margin. Taking any value in the first column 2 
 
 for an initial term, and adding to it continually the 
 terms of the series Alog.^- 1 1), commencing 
 with that in which x is the same as in the initial 
 term, we produce in. succession the terms extending 
 horizontally across the columns. This, too, can be 
 performed by means of a card, having upon it A log. 
 (p x ~ l 1), and it may be verified from time to time, 
 as shown. 
 
 131. The values of the function now under con- 
 
 a? 
 103 
 
 IO2 
 IOI 
 IOO 
 
 346787 
 522879 
 
 869666 
 778151 
 
 647817 
 
 853872 
 
 501689 
 
 Verification, 
 loo 455932 
 
 99 
 
 045757 
 501689 
 
 sideration may also be formed in a somewhat different I00 '99 
 manner. Examples of the mode of formation now referred to follow. 
 
58 
 
 Values of 
 
 I. x7 y 
 
 x-y 
 
 y 
 103 
 
 102 
 101 
 IOO 
 
 I 9 
 
 #=103 
 301030 
 301030 
 
 y 
 
 IO2 
 101 
 IOO 
 
 99 
 
 2 s 
 
 X=IO2 
 
 823909 
 124939 
 
 " 
 
 101 
 
 IOO 
 
 99 
 
 98 
 
 y 
 
 EIOI 
 1060 
 w/JlSl 
 
 y 
 
 IOO 
 
 99 
 98 
 
 97 
 
 2 s 
 
 a?=ioo 
 455932 
 057992 
 
 y 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 
 #=99 
 346787 
 045757 
 
 y 
 
 9 8 
 
 97 
 96 
 
 95 
 94 
 
 #=98 
 435728 
 055517 
 
 602060 
 82390.9 
 
 948848 
 954242 
 
 681241 
 9/8811 
 
 513924 
 987765 
 
 392544 
 009760 
 
 491245 
 002475 
 
 425969 
 954242 
 
 903090 
 
 978811 
 
 660052 
 987765 
 
 501689 
 009760 
 
 402304 
 002475 
 
 493720 
 
 998489 
 
 380211 
 
 978811 
 
 881901 
 987765 
 
 647817 
 009760 
 
 511449 
 002475 
 
 404779 
 998489 
 
 492209 
 005037 
 
 359022 
 987765 
 
 869666 
 009760 
 
 657577 
 002475 
 
 513924 
 998489 
 
 403268 
 005037 
 
 497246 
 005429 
 
 346/87 
 * 
 
 879426 
 * 
 
 660052 
 
 * 
 
 512413 
 
 408305 
 
 502675 
 
 II. x y. 
 
 yX 
 
 
 I 
 2 
 
 3 
 
 4 
 
 * 
 
 X 
 
 103 
 
 102 
 IOI 
 
 99 
 
 ^ 
 
 y=I03 
 
 602060 
 522879 
 
 X 
 102 
 
 IOI 
 IOO 
 
 99 
 
 f 
 
 y=iO2 
 948848 
 778151 
 
 X 
 IOI 
 
 IOO 
 
 99 
 
 98 
 
 * 97 
 
 y=ioi 
 681241 
 
 853872 
 
 X 
 IOO 
 
 99 
 98 
 
 97 
 
 ? 6 
 
 y=ioo 
 513924 
 890855 
 
 X 
 
 99 
 
 98 
 
 97 
 96 
 
 % 
 
 y=99 
 
 392544 
 
 088941 
 
 X 
 
 98 
 
 97 
 96 
 
 95 
 
 9 * 4 
 
 y=9%\ 
 
 491245: 
 020204 
 
 124939 
 778151 
 
 726999 
 
 853872 
 
 535113 
 890855 
 
 404779 
 088941 
 
 481485 
 020204 
 
 511449 
 
 987765: 
 
 903090 
 
 853872 
 
 580871 
 890855 
 
 425968 
 088941 
 
 493720 
 020204 
 
 501689 
 987765 
 
 499214 1 
 039673 
 
 756962 
 890855 
 
 471726 
 088941 
 
 514909 
 020204 
 
 513924 
 987765 
 
 489454 
 039673 
 
 538887 
 039509 
 
 647817 
 * 
 
 560667 
 
 * 
 
 535113 
 
 * 
 
 501689 
 * 
 
 529127 
 
 # 
 
 578396 
 * 
 
 132. Few remarks on this mode of formation seem to be required. 
 The values are formed in two compartments, the first comprising those 
 values in which x is equal to or greater than y, and the second those 
 in which x is equal to or less than y. The initial values in the 
 second compartment are the values in the first corresponding to 
 x y ^j and the addends are, in the first compartment, the terms of 
 the series Alog.K^" 1 + 1), and in the second the terms of the series 
 A log. (/?,,." * 1). The terms in which x y is constant are now- 
 found ranged in horizontal lines. The method of forming values for 
 verification will be sufficiently apparent, without Log. 
 example, after what has been already shown. fm 
 
 133. Any series in which x y is constant may 
 also be formed independently. A single example 
 is given in the margin. The two addends for the 
 formation of each succeeding value are the diffe- 
 rences of the terms of the individual series which 
 compose the preceding value. The values here 
 formed may be compared with the corresponding 
 values in the type of (131). 
 
 Having now shown the mode of formation of 
 the various subsidiary series for which we shall 100-96 512413 
 have occasion, we proceed in the following chapters to show their 
 applications and uses. 
 
 103 
 
 99 
 103-99 
 
 102*98 
 101-97 
 
 301030 
 045757 
 
 346787 
 522879 
 009760 
 
 8/9426 
 778151 
 002475 
 
 660052 
 
 85387-2 
 998489 
 
CHAPTER V. 
 
 OF MEAN DURATIONS OF LIFE, AND PROBABILITIES OF 
 SURVIVORSHIP. 
 
 PROBLEM XL 
 
 134. To construct a table of the curtate mean duration of a single 
 life, or of a combination of two or more joint lives, at each age. 
 
 135. By the mean duration of life at a specified age and according to 
 a given table of mortality, is implied, the average number of years 
 that, in the case of a single life, will be enjoyed by each individual of 
 the specified age ; and, in the case of a combination of lives, by each 
 combination, the ages of the lives composing which are the same as 
 the ages of those composing the combination in question, on the sup- 
 position that the lives fail as indicated in the given table. In other 
 words, the mean duration is the term that would result from dividing 
 equally among a number of persons of the same age, or of combina- 
 tions similarly circumstanced as to the ages of the persons composing 
 them, the sum of the whole future life-times of those persons or com- 
 binations respectively. 
 
 136. It is the complete mean duration that has just been defined. 
 The curtate mean duration is the duration which arises when we neg- 
 lect the portion of time which each life, or combination of lives, enjoys 
 in the year in which its failure takes place ; that is to say, it is the 
 mean duration which would result if the deaths of each year, instead 
 of being distributed over it at intervals, were to take place at the 
 beginning of that year. Hence the reason of the function being 
 called curtate. 
 
 137. Of l x individuals, the tabular number alive at age x, which 
 suppose the present age,- l x+n survive n years, having been reduced to 
 their then present number a year before ; l x+n consequently is the 
 number of years of existence that will be enjoyed in the nth year from 
 
60 
 
 the present time, by the survivors of the l x individuals now alive ; and 
 hence the share of each, in respect of this year, will be ~=p xn . 
 
 X 
 
 Making n, therefore, in this expression equal to 1, 2, 3, &c., suc- 
 cessively, and adding the results, we have, denoting the curtate mean 
 duration of (x) by e l x , 
 
 I 1 +/ _i_j + 
 
 l z+I ' g+2 ' x+3 ' 
 
 e x == 7~ > 
 
 x 
 
 or 
 
 138. Again, if a combination of two lives, of the respective ages of 
 x and y, be in question, l x and l y , the tabular numbers alive at those 
 ages, afford l xy of such combinations, since each of the l x may be com- 
 bined with each of the / to form one. Of these combinations l x+H1+n 
 subsist at the end of n years, having enjoyed amongst them, during 
 the nth year, l x+ny+n years. Hence the share of existence due to each 
 of the l xy combinations, in respect of the nth year, will be 
 
 ^+n.y+n_ 
 
 ~l -P(x.y)n' 
 
 V x.y 
 
 Consequently, making n successively equal to 1, 2, 3, &c., and adding 
 the results, we have for the curtate mean duration of a pair of lives, 
 (x} and (y\ 
 
 _L 
 
 x -y~ 
 
 or 
 
 139. And in like manner we should find for the curtate mean dura- 
 tion of three lives, (x), (y), and (z). 
 
 e \ L 
 
 K x.y.z- 
 
 or 
 
 and so on, for a combination consisting of any number of lives what- 
 soever. 
 
 By the aid of these expressions the curtate mean duration of any 
 single life, or any combination of joint lives, may be determined. But 
 they are unsuited to our present purpose, which is to show how, by a 
 continuous process, a table may be formed of the curtate mean dura- 
 tion of a single life at each age, or of every combination of joint lives 
 in which the differences of the ages of the lives composing it are the 
 same. 
 
 140. By the hypothesis on which we now proceed, the l x who attain 
 
61 
 
 age x are immediately thereafter reduced by death to l x+lt which num- 
 ber attain the age x+I. Each has then a curtate mean duration of 
 e l x+l , and they consequently have amongst them a total duration of 
 l x+l e l x+l , which, therefore, is the quantity of existence due to the l x 
 individuals in respect of all the years after the first. And the quantity 
 in respect of the first year being l x+l , (137,) we have for the whole 
 quantity of existence to be enjoyed by the l x individuals from and after 
 age x, l x+l + l x+l e l x+l , or, 1,+ ^l + e 1 ^). Hence, dividing by l x , we 
 have finally for the share of each, 
 
 141. In a similar manner we should deduce for a combination of 
 two lives, (x) and (y) } 
 
 and for a combination of three lives (x), (y], and (z), 
 
 142. By these formulae it is obvious that, if the curtate mean dura- 
 tion at any age, whether of a single life, or of a combination of any 
 number of joint lives, be known, the durations at all the younger ages 
 of single lives, or of combinations of the same number of joint lives, 
 may be successively deduced ; by the term younger age, in the case of 
 a combination of lives, it being understood, not only that the lives 
 composing that combination are respectively younger than those com- 
 posing the combination with which it is compared, but that they have 
 amongst themselves the same arithmetical relation as to difference. 
 
 143. Passing to logarithms the formulae just deduced become 
 
 log.e l x =\og.p +\og.(l +e l x+l ) ; 
 
 The applicability of Table I. to these formula is obvious. 
 
 144. Initial values are found as follows : In the case of a single 
 life we have obviously, using the Carlisle Table, ^ 1 104| : =0. Let there- 
 fore x 103. Hence, 
 
 145. And in like manner, in the case of a combination of any num- 
 ber of joint lives, when one of those lives is of the oldest tabular age, 
 the curtate mean duration for that combination is equal to 0, and for 
 the next younger combination it is equal to the probability that all the 
 lives composing that combination will survive a year. 
 
 146. We now introduce the following examples : 
 
One Life. 
 
 62 
 
 Curtate Mean Duration. 
 
 Two Lives. 
 
 io 3 522879 -33333 
 778151 0-477121 
 602044 
 16 
 
 x y=o. 
 
 I03 J045758 -mil 
 03 -J 556302 0-954242 
 999962 
 
 38 
 
 a? y=i. 
 
 * 3 } 301030 -20000 
 32 J 632023 0-698970 
 778093 
 
 58 
 
 102 903090 -Soooo 
 853872 096910 
 352177 
 5 
 
 ioi 109144 1-28571 
 890855 
 358997 
 25 
 
 100 249877 1-77778 
 912850 
 443648 
 49 
 
 l 2 \ 602060 -40000 
 102 J 707744 397940 
 544039 
 29 
 
 [-853872 -71429 
 J 781710 146128 
 380195 
 16 
 
 ^71 1204 -51429 
 101 J 744727 288796 
 468940 
 63 
 
 101 } 924934 -84127 
 100 J 803705 075066 
 340147 
 36 
 
 J^} 015793 1-03703 
 
 825700 
 308951 
 47 
 99 [134698 1-36363 
 
 '}068822 1-17172 
 
 808115 
 336791 
 12 
 
 99 1 144918 1-39610 
 
 99 356547 2-27273 
 
 895265 
 514877 
 32 
 
 790530 
 373524 
 56 
 
 786120 
 379495 
 11 
 
 9 [165626 1*46429 
 
 98 410174 2-57142 
 
 9g} 164110 1-45918 
 
 Three Lives. 
 
 X y Q , 
 
 y-z=o. 
 I0 3] 
 103 [668637 '03704 
 103] 334453 1-431363 
 447097 
 102 1 60 
 
 x y=i. 
 
 y *=i. 
 
 I0 3~l 
 ml 154908 -14286 
 ioi J 522878 0-845098 
 903004 
 
 102 1 86 
 
 * jr=3. 
 
 y 2=2. 
 
 T 3l 
 100 ^308999 -20370 
 98 J 581856 0-691001 
 771520 
 
 102 "I 1 
 
 102 }> 350247 -22400 
 102 J 561616 0-649753 
 737491 
 TOT -I 43 
 
 ioi 1680870 -3809(5 
 100 J 657577 419130 
 559287 
 
 Tnr-| 22 
 
 99 j. 662376 -45960 
 97 J 642682 337624 
 501840 
 
 idil 
 
 101 U49397 '44606 
 IOT J 672565 350603 
 510788 
 
 100] 
 
 100 ^797756 -62771 
 99 J 698970 202244 
 413793 
 100 1 27 
 
 98 1806915 -64108 
 96] 66631 7 193085 
 408164 
 
 100 1 
 
 i oo^ 832752 -68038 
 100 J 738550 167248 
 392627 
 
 991 29 
 
 99 \ 910546 -81385 
 98 J 698970 089454 
 348026 
 
 991 3 
 
 97 !> 881448 -76111 
 95] 681456 118552 
 364309 
 
 991 3 
 
 99 1963958 -92036 
 99 J 685795 036042 
 319403 
 9 81 _ 22 
 
 98}* 957572 -90693 
 97 J 679665 042428 
 322747 
 98-1 15 
 
 96 1927243 -84575 
 94 J 649538 072757 
 338899 
 
 98 1 31 
 
 98V969178 -93149 
 98 J 
 
 97 1969999 -91201 
 
 95 [916711 -82359 
 93 J 
 
63 
 
 147. The operation is in all the examples precisely the same, and 
 the mode of conducting it will be best explained by reference to the 
 following type : 
 
 D! negative. 
 
 Dj positive. 
 
 i 
 
 lS-^ + i 
 
 e \ 
 
 K x + l 
 
 D, 
 
 tosA + , 
 
 el *+i 
 
 A 
 
 1()2*.W 
 
 ar.co.D, 
 
 
 , 
 
 
 
 o * x 
 
 
 
 102T-2? 
 
 
 B 
 C 
 
 JT^ar.co.DJ 
 
 
 B 
 C 
 
 KM 
 
 
 D 
 
 (=cr +c 
 
 ^ 
 
 D 
 
 rA+B + C 
 
 ... 
 
 The type exhibits two modes of operation, the first, in accordance with 
 Formula IX., for use while e l x+l continues less than unity, and its loga- 
 rithm consequently negative, (which will never be for more than a few 
 terms ;) and the second, in accordance with Formula V., for use when 
 e l x+ 1 becomes greater than unity, and its logarithm consequently posi- 
 tive. In the examples in which the operation changes from the one 
 formula to the other, the point where the change takes place is indi- 
 cated by a dark line. The values of log.^ that come into use in the 
 first example are taken from the Elementary Table, and those of log. 
 P(xy)> anc ^ ^&P(kfjd ^ n ^ ne ther examples, will be found in the types 
 of formation in (108) and (117). 
 
 148. The operation for a single life may be verified at any point by 
 the aid of the expression, (137,) 
 
 e i __+ g . 
 
 to apply which to age 98, we have 
 
 I _ 
 K ~ 
 
 + ^109+ I 
 
 102 103 
 
 11 + 9+7 + 5 + 3 + 1 
 14 
 
 log.!8 = l-255273 
 7=0-845098 
 
 36 
 14 
 
 18 
 7* 
 
 0-410175, 
 
 which agrees with the value already found. The analogous mode of 
 verifying the operations for joint lives is too laborious for use. 
 
 PROBLEM XII. 
 
 149. To construct a table of the complete mean duration of a 
 single life at each age. 
 
64 
 
 150. The mean duration of life at a given age, called complete when 
 necessary to distinguish it from curtate, is the average duration of all the 
 lives at that age, according to the mortality table, on the supposition 
 that the deaths of each year are uniformly distributed over that year. 
 
 151. A little consideration suffices to show that the complete mean 
 duration of a single life at any age will exceed the curtate mean duration 
 at the same age by half a year ; since, the deaths being here supposed 
 uniformly distributed over each year, each life will in consequence 
 enjoy half a year of existence, on an average, in the year in which it 
 fails. When, therefore, the curtate mean duration at any age is known, 
 the complete mean duration at that age will be obtained by adding '5 
 to the former function. But the complete mean duration for all ages 
 may also be computed independently, and by a continuous process, 
 and the two operations may be made mutually to confirm each other. 
 
 152. We saw that, on the hypothesis of the last problem, the por- 
 tion of existence due to l x individuals now alive, in respect of the rath 
 year, is l x+n years, l x+H being the number, out of the l x now alive, who 
 live over that year. But, on the hypothesis of the present problem, 
 l x+n _ l l x + n > the number who die in the nth year, also enjoy in that year 
 half a year each, or on the whole, i(( r+tt _ 1 l x+n ) years. Adding this to 
 the former therefore, we have | (( e+B _ l + l x+n ) for the portion of existence 
 due to the /,,. individuals in respect of the nth year, on the present 
 hypothesis ; and division of this by l x) gives for the share of each, 
 
 *(*+H-I + *+J t Finally, making n here equal to 1, 2, 3, &c., suc- 
 cessively, and adding the results, we have, if e x denote the complete 
 mean duration at age #, 
 
 or 
 
 153. The working formula is deduced as follows: Of l x now alive 
 l x+l survive one year, and have then a mean duration of e x+l , and 
 therefore a total duration of l x+ i^ x+r This consequently is the 
 portion of existence due to the l x in respect of all the years after the 
 first. And the portion due to them in respect of the first year being 
 
 duration \(l x + l x+l ) + ^+1^+1 J an( l dividing by l x , we have finally, for 
 the share of each, 
 
 4(*. + ln)+*.H.A+. 
 
65 
 
 Passing to logarithms this becomes, 
 
 which is the formula required. 
 
 154. The initial value for the construction of the table will be, 
 obviously, Iog.e 104 = log. J =1-698970; and \og.(p f -i + I) for each 
 age being given in the Elementary Table, the operation will be as here 
 typified : 
 
 A 7 E r 
 
 AzE r 
 
 A 
 B 
 C 
 D 
 
 A) 
 
 T^ar.co.^-A)] 
 
 E 
 
 A+B + C + 
 
 E 
 
 The type here, like that in last problem, consists of two portions, of 
 which the first, (that on the left,) shows the operation when the value 
 in the line A exceeds that in the line Ej ; and the other shows the ope- 
 ration when this relation of the values in A 
 and E L is reversed. The operations change Complete Mean Duration. 
 
 104 
 
 One Life. 
 698970 
 
 from the first to the second after a very few 
 values have been formed. 
 
 155. An example in illustration is here 
 given, and the values formed may be com- 
 pared with those for the same ages in the 
 first example of last problem. 
 
 156. If it were assumed in the case of com- 
 binations of joint lives that the failures of 
 each year are uniformly distributed over that 
 year, so as to take place at equal intervals 
 therein, then it is evident that, as in the case 
 of single lives, the complete mean duration of 
 each such combination would be obtained by 
 
 adding A to its curtate mean duration. But I01 251811 r 7 8 57 2 
 
 301030 
 522879 
 397940 
 698970 
 103 920819 
 
 124939 
 778151 
 795880 
 414973 
 
 113943 
 
 079181 
 853872 
 318726 
 32 
 
 102 
 
 500000 
 0-602060 
 
 833333 
 204120 
 
 1-30000 
 034762 
 
66 
 
 ioi 251811 
 
 057992 
 
 890S55 
 
 408652 
 
 12 
 
 100 357511 
 
 045757 
 
 912850 
 
 484264 
 
 36 
 
 99 442907 
 
 055517 
 
 895265 
 
 536495 
 
 64 
 
 98 487341 
 * * 
 
 1-78572 
 193819 
 
 2-27778 
 311754 
 
 2-77273 
 387390 
 
 such an assumption is inadmissible. It is 
 incompatible with that more rational assump- 
 tion of a uniform distribution of the indivi- 
 dual deaths of each year. This will readily 
 appear when it is considered that the number 
 of pairs, for example, dissolved by the death 
 of one individual of either of the ages which 
 enter into those pairs, is equal to the number 
 of survivors of the other age. And hence, 
 the survivors being more numerous at the 
 beginning of the year than at the end, a death 
 which takes place towards the commencement 
 of the year dissolves more pairs than one 
 that takes place later in the year. 
 
 It is proposed here to investigate an expression for the complete 
 mean duration of two lives, (a?) and (y). 
 
 157. The numbers now alive of the specified ages are denoted, as 
 usual, by l x and l y . But, for convenience, we here employ a and a 1 
 to denote the numbers who out of these survive the nth year, and d 
 and d l the numbers who die therein, respectively. Now let the nth 
 year be divided into t equal parts ; then, the deaths being uniformly 
 distributed, those of the two classes that take place in each tth part 
 
 will be and , respectively. Also, let the deaths of each tth part 
 
 V t 
 
 take place at the beginning of that part ; then will the quantity of 
 existence, in tth parts of a year, due to the pairs of lives of the speci- 
 fied ages, now subsisting, in respect of any one of the tth parts, be the 
 same as the number of pairs who survive that part. Take the pth in 
 order of the tth parts. The survivors at the end of it will obviously 
 
 t ~~ J) t 2? 
 
 be, of the one class, a-\ -- -d, and of the other, a l + d l ; and 
 
 t/ b 
 
 these give, of pairs then surviving, 
 
 
 Hence, dividing by t, we have, 
 
 for the quantity of existence, in years, due in respect of the pth in 
 order of the t parts into which the ^th year is divided. 
 
67 
 
 Making /?=!, 2, 3, . . . t, successively, we have, 
 
 = 2, 
 
 p =t-i afl i + 
 
 ir 
 
 -\_ 
 
 o o 2 
 
 aa l + -(ad l +a l d) + - 
 
 And addition of these gives, 
 ( since 
 
 or 
 
 for the quantity of existence due in respect of the whole of the parts 
 into which the nth year is divided, when the deaths take place in the 
 manner supposed, that is, the deaths of each part at the commence- 
 ment of that part. But to assimilate this hypothesis to that of a 
 perfectly uniform and equable distribution of the deaths, it is requisite 
 that the parts be indefinitely small, or, which is the same thing, that 
 their number be indefinitely great. Increasing t without limit then, 
 we have, 
 
 - -- 1 ) , 
 
 -=J 5 
 
 and hence the expression, which will now denote the quantity of 
 existence due in respect of the nth year, becomes 
 
 aa l + \(ad l + 
 Substituting now in this expression, 
 
 multiplying out, collecting the terms, and dividing the whole by l xy , 
 the number of pairs now subsisting, we have for the share of existence 
 due to each of those pairs, in respect of the nth year, 
 
 3 x+n-l.y + nl *"$ x+n-l.y+n~* Ti'x+n.y + u- 
 
 
68 
 
 By making n in this expression equal to 1, 2, 3, ... successively, and 
 adding the results, we shall obtain the complete mean duration of a 
 pair of lives, whose ages are x and y } respectively. 
 
 
 
 l x.y-\ Py-l 
 
 Hence the required sum is 
 
 or 
 
 158. It thus appears that if tables of the curtate mean duration 
 of pairs of lives of all ages have been formed, the complete mean 
 duration of any pair that may be proposed can be assigned ; but that 
 the excess of the complete over the curtate mean duration is not here, 
 as in the case of single lives of all ages, a constant quantity. 
 
 Neither can a formula be devised for the commodious computation, 
 by a continuous process, of the values of this function for pairs of 
 ages having the same common difference. Any value wanted must 
 therefore be computed by the formula just deduced. 
 
 159. SCHOLIUM I. The preceding investigation, seemingly some- 
 what out of place in such a work as the present, has been given here 
 because it is not generally accessible. The only writer who has 
 attended to the subject is Mr. Griffith Davies, who in a yet uncompleted 
 work on Assurances*, solves not only the present problem, but also the 
 general problem, in which the number of lives in each combination is 
 any whatsoever. Mr. Davies' solutions however, although correct 
 in their results, are rendered obscure by the employment of the symbol 
 
 x 
 " of a year " to denote, not mih parts, in number x, but the #th in 
 
 Vfl 
 
 order of those parts. They are likewise otherwise objectionable, in 
 
 * The sheets of Mr. Davies' work, so far as printed, being obtainable at a fixed 
 price, it must, although incomplete, be held as published. Otherwise the direct- 
 ing of attention to it, as is done in the text, would perhaps be hardly legitimate. 
 
69 
 
 that he assumes a definite relation as subsisting between two quantities, 
 whereof one is finite and the other infinite. Mr. D's. general method 
 of investigation has been here followed. 
 
 160. Mr. Milne appears to have fairly evaded this subject. At 
 p. 71 of his work he employs (f> to denote the excess of the true over 
 the curtate mean duration of a pair of lives ; but he nowhere seeks to 
 assign the value of <. Nor, as has just been stated, does any other 
 writer, with the exception of Mr. Davies. 
 
 161. SCHOLIUM II. The function with the consideration of which 
 we have been occupied, is that which usually receives the appellation 
 of the expectation of life. This appellation it seems to have derived 
 from its being the sum of the probabilities of the life or combination 
 of lives that may be in question, living over one year, two years, three 
 years, &c., to the greatest tabular duration of that life or combination 
 of lives (137,138). Viewed in this light as the sum of a series of 
 probabilities the function may, perhaps, with some show of propriety 
 receive the appellation to which reference has just been made. But 
 the sense in which the term expectation is frequently received, and the 
 manner in which the function it designates is spoken of, and even 
 defined, by writers on life contingencies, receive no countenance from 
 the nature or the mode of derivation of that function. 
 
 162. Of the prevalent misconceptions on this subject one is, that 
 the " expectation " at a specified age, considered as a term of years, is 
 more likely than any other term, to be the actual duration of that life ; 
 and another is, that it denotes a term of years which the life in ques- 
 tion is as likely to survive as not. Neither of these notions is correct. 
 The first would imply that the year in which a life is most likely to 
 fail is that in which its " expectation " terminates. But the year in 
 which this event is most likely to happen is obviously that succeeding 
 the attainment of the age opposite which the mortality table shows 
 the greatest number of deaths. Thus, by the Carlisle Table, after the 
 4th year, more deaths take place in the 75th year than in any other. 
 In this year, therefore, a life of any age, from 4 to 74 inclusive, is, by 
 this table, more likely to fail than in any other single year that can be 
 named ; while the actual expectations of but few of those intermediate 
 ages will be found to terminate in their 75th year. 
 
 163. The second notion designates, not the "expectation," but the 
 term, (called by French writers la vie probable,) during which the 
 number of lives of the given age is reduced one half. This term does 
 not usually differ much from the true " expectation," and on De 
 Moivre's hypothesis, of a uniform distribution of the deaths, from 
 amongst the number living at a given age, over the whole period of 
 
70 
 
 future existence, it actually coincides with it. Still, as this hypothesis 
 is in accordance with no table exhibiting the results of actual observa- 
 tions, deductions from it cannot be received as true exponents of the 
 law of mortality. 
 
 164. These and other notions equally unfounded, which are enter- 
 tained by many in reference to the mean duration of life, have no 
 doubt their origin in the name, " expectation of life," by which this 
 function is usually designated, and in the manner in which it is spoken 
 of by writers on the subject. We propose briefly to advert to a few 
 of the statements here referred to. 
 
 165. Thomas Simpson, in his " Supplement to the Doctrine of 
 Annuities," &c., 1791, 8vo., pp. 7 and 19, defines the mean duration 
 as (e the number of years of life which a person of a given age may, 
 upon an equality of chance, expect to enjoy ;" but, on the page last 
 mentioned, he subjoins a note, warning his readers against taking 
 this definition in its obvious and grammatical meaning, because its 
 real meaning is something else ! Surely a preferable course had been to 
 avoid the employment of language confessedly suggestive of erroneous 
 ideas. 
 
 166. Mr. Milne, whose general accuracy and precision are worthy 
 of all praise and imitation, is far from being faultless in this matter. 
 He says of the mean duration at a specified age, p. 57, that it is "the 
 whole time which each of [those alive at that age] may at present 
 reasonably expect to survive" It is certainly true of the mean dura- 
 tion that the probability of surviving it,and consequently the reason- 
 ableness of expecting to do so, is greater than that of surviving a 
 longer term ; but as the same thing can be predicated of any term of 
 years whatsoever, Mr. Milne ought to have assigned the grounds on 
 which he claims for this particular term the appellation of the term 
 of reasonable expectation. It is assuredly more reasonable to expect 
 to survive any shorter term. 
 
 167. If the appellation now animadverted on has any definite sig- 
 nification, it must mean that it is not reasonable to expect to enjoy 
 any portion of existence after the expiry of the term to which it is 
 applied. Mr. Milne, nevertheless, speaks of another term of years, 
 which he calls the deferred expectation of life, pp. 57, 58, and de- 
 scribes as " the duration of life which [a person of a specified age] has 
 at present the expectation that he shall enjoy after the expiration of t 
 years;" which term in every case extends beyond that of the mean 
 duration, and in some, according to the value given to t, does not even 
 commence till the mean duration the term of reasonable expectation 
 has expired ! 
 
71 
 
 168. It is surely unnecessary to add much more to what has already 
 been said on this subject. To make the matter, however, if possible, 
 still more plain, we take a single illustration. Suppose that an urn 
 contain n balls, marked with the numbers, 1, 2, 3, .... n, respec- 
 tively, and that the balls are to be apportioned, by a person blind- 
 folded, amongst n individuals, one ball to each. Now it is plain that, 
 before the drawing commences, the expectations of all the n indivi- 
 duals in regard to the result of the drawing are, or ought to be, the 
 same. It would be absurd, therefore, to fix upon any one ball, and 
 to say of the number upon that ball, that it is the number which not 
 merely one of the n individuals, but all of them, may reasonably ex- 
 pect to fall to their lot. It matters nothing that the number thus 
 
 n + \ 
 fixed upon is the average number, , since, by hypothes is, the 
 
 numbers being all equally likely, any other number is just as likely, 
 and therefore may as reasonably be expected, as that number. 
 
 169. The case supposed is parallel to that presented by the mor- 
 tality table; and it is just as absurd to say that the average' duration 
 of lives aged x, is the term which each of those lives may reasonably 
 expect to enjoy, as to say that the average value of the numbers on 
 the balls is the number which each of the supposed individuals may 
 reasonably expect will fall to his lot. Nay, it is more absurd, for 
 while it can be said of the n individuals that each is as likely to ex- 
 ceed as to fall short of the average number, , the like cannot be 
 
 A 
 
 said of the lives aged x, in regard to the average duration of life. 
 The term in regard to which this can be said, is not the mean duration, 
 but the vie probable. 
 
 170. A more recent writer than Mr. Milne defines the mean dura- 
 tion of life as " the number of years which lives of any given age, 
 taken one with another, have an equal chance of attaining." (For 
 " attaining " read " surviving." Lives attain, not years, but ages.) 
 The writer's meaning is not clear. One interpretation of his words is, 
 that the mean duration at a given age is a number of years which all 
 the lives of that age have the same probability of surviving. This, of 
 course, does not describe the mean duration. It applies to any num- 
 ber of years whatsoever, the principle upon which the mortality table 
 is subjected to calculation being, that all lives of the same age are simi- 
 larly circumstanced in regard to probability of longevity. It is likely 
 therefore, (and the likelihood is increased by the employment of the 
 
72 
 
 ambiguous term, "equal chance/'*) that the author wished to say, 
 that the mean duration of lives of a given age is the term which each 
 of those lives is as likely to survive as not. This also, however, is in- 
 correct. It describes the vie probable, and not the mean duration. 
 
 171. Another recent writer says, that " the expectation of life is 
 often confounded with the chance of living an equivalent number of 
 years." It seems probable that the meaning here intended to be con- 
 veyed is, that the mean duration, at a specified age, is often confounded 
 with the term which lives of that age are as likely to survive as not 
 (162). For it cannot be supposed that any would be so absurd as to con- 
 found a term of ten, twenty, or thirty years, as the case may be, with a 
 chance a probability an abstract number, whose greater limit is unity. 
 
 172. The same writer elsewhere says : " At the earlier ages there 
 is always a chance of outliving the period of years represented by the 
 expectation of life ; but the converse is the case for the older ages." 
 Here too it is obvious that the writer cannot mean what he says, since 
 a life of any age whatsoever has a definite and assignable chance of 
 attaining even the oldest tabular age, and therefore of outliving the 
 term of its mean duration. The idea intended to be conveyed is ap- 
 parently this, that lives of the younger ages are more likely to exceed 
 than to fall short of their mean duration, while the reverse is the 
 case with regard to lives of the older ages. These are examples of 
 that vagueness and lack of precision which, in scientific works especi- 
 ally, often prove serious stumbling-blocks to the learner, and which 
 therefore ought to be studiously avoided. 
 
 173. See, on the subject of this scholium, Price on Reversionary 
 Payments, vol. ii., pp. 4 7, 6th or 7th edition. Also, Encyclo- 
 paedia Metropolitana, art. THEORY OF PROBABILITIES, sec. 101, p. 
 445 ; and Sixth Report of the Registrar- General, Appendix, pp. 536, 
 554, 557, 8vo. edition, and pp. 301, 302, 307, 314, folio edition. 
 
 PROBLEM XIII. 
 
 174. To construct a table of the probabilities of survivorship be- 
 tween two lives, for every combination of ages. 
 
 175. By the probabilities of survivorship of any number of lives are 
 meant, the probabilities of the different orders of survivorship that can 
 take place amongst those lives. When two lives only are in question, 
 the number of orders of survivorship is obviously also two. There are 
 
 * The propriety of the use of this term in any particular case may generally 
 be tested by examining whether the connection furnishes an answer to the query, 
 Equal to what ? Try this upon the definition now under consideration. 
 
73 
 
 hence two values connected with each combination of two lives, the 
 one indicating the probability of the dissolution of the combination by 
 the failure of the older life, and the other the probability of the same 
 by the failure of the younger life. The sum of these corresponding 
 probabilities is in every case equal to unity, since the combination must 
 be dissolved by the failure of one or other of the lives. Hence if one 
 of these probabilities be determined the other will be found by sub- 
 tracting the first from unity (81). In forming a table, however, it 
 may be desirable to compute both probabilities independently of each 
 other, to serve for mutual verification. 
 
 176. Let (SB) and (y} be any two lives, and let the probability to be 
 determined be, that of these two (x) will be the first that fails ; that 
 is, that the combination (x.y} will be dissolved by the death of (x). 
 This probability in regard to a single year, the nth, has been already 
 determined (99) ; and by making n, in the expression for that proba- 
 bility, successively equal to 1, 2, 3, &c., to the greatest tabular joint 
 duration of (x.y}, and adding the results, we should have an expression 
 denoting the probability required. We prefer, however, first deducing 
 a working formula, that is, a formula expressing the value of this pro- 
 bability in respect of (x.y}, in terms of the like probability in respect 
 of (ar + l.y+1). 
 
 177. Let, then, these probabilities be denoted by S^, and 
 Sg+i.y+i' respectively. The combination (x.y) may be dissolved by 
 the death of (%}, either in the first year or after that year ; and the 
 sum of the probabilities in respect of these portions of time will 
 obviously be the total probability required (83). That this disso- 
 lution will take place after the first year depends on the concurrence 
 of these two events ; first, that no dissolution take place during the 
 first year, that is, that both lives survive that year, the probability 
 of which isp^y) ; and, secondly, that a combination of the ages attained 
 at the end of the first year, namely, (x + \.y+ 1), will be dissolved by 
 the death of (x+l), the probability of which is S g * ly+1 . Hence 
 (82), the probability required in respect of the period after the first 
 year is fe. y )S g | Ly+1 And the probability in respect of the rctli year 
 being, (99,) 
 
 if we make n \ we get for that in respect of the first year, 
 
 4(i-jg(i+ig. 
 
 Hence the total probability required is, 
 
 or, in logarithms, 
 
 log.SL.=log. Pr , 
 
74 
 
 178. This is the formula required, of which it may be remarked 
 generally, that the formation of log.i(/y~*~~ \)(p y ~ l -fl) for each 
 combination of ages having been already shown, (129 . . . 131), and 
 log.S, r+1;+1 being in each case the result of the preceding operation, 
 the formation of the logarithm to be added to log./^ j (also known) is 
 effected by means of Table I. This will be distinctly seen in the type 
 of the operation to be immediately given. 
 
 179. To find initial values, take the expression in its first algebraic 
 form : 
 
 1st. Let #=y. Then, when (a?) and (y) are of the oldest tabular 
 age, p x , p y) and jo^) obviously vanish, and the expression reduces to 
 
 the logarithm of which, 1*698970, is in this case the initial value. 
 
 2nd. Let x 7 y. Then, when x is the oldest tabular age, p x and 
 P(xy) vanish, so that the expression in this case becomes,, 
 
 or, in logarithms, 
 
 which will readily be obtained from the Elementary Table. 
 
 3rd. Let x L y. Then, when y is the oldest tabular age, p y and 
 p (xy ) vanish, and the expression is reduced to 
 
 or, in logarithms, 
 
 which also may be formed by the aid of the Elementary Table. 
 
 180. When an entire table of the probabilities of survivorship of 
 two lives is to be constructed, the initial values for the different com- 
 partments may be formed as follows : 
 
 Probability that (x 
 Initial 
 
 522879 
 301030 
 
 823909 
 255272 
 
 823909 
 
 903090 
 075721 
 954242 
 
 933053 
 036983 
 978811 
 
 104-100 948847 
 
 I04T03 
 
 104*102 
 
 io4'ioi 
 
 ) will die before (y) 
 Values. 
 
 zy. 
 
 log.* 
 log^oa 
 
 698970 
 522879 
 301030- 
 
 103-104 
 
 102*104 
 
 101-104 
 
 ioo'i04 
 
 5228/9 
 255272 
 522879 
 
 301030 
 075721 
 778151 
 
 154902 
 036983 
 
 853872 
 
 045757 
 
75 
 
 It will be observed that the differences by the addition of which 
 each value is here successively formed, are those of the terms whose 
 sum is the immediately preceding value. 
 
 181. We now give a type explanatory of the operations. 
 
 E, 
 
 A 
 B 
 C 
 D 
 
 E 
 
 T 1 [ar.co.(E 1 -A)] 
 
 Sri- 
 
 A) 
 
 E l7 A. 
 
 E, 
 
 A 
 B 
 C 
 D 
 
 E 
 
 rA+B + C + D 
 
 E,-A 
 
 fe 
 
 And further illustration will be afforded by the following examples. 
 
 Probability that (x) will die before (y). 
 I. x=or 7 y. 
 
 x-y 
 698970 
 602060 
 045758 
 096910 
 954242 
 
 698970 
 
 948848 
 556302 
 750122 
 443698 
 
 698970 
 
 =o. 
 
 0-903090 
 
 2498/8 
 
 103 
 
 x y=i. 
 823909 -666667 
 
 425969 
 301030 
 397940 
 698970 
 
 0-602060 
 
 666667 
 079181 
 
 628571 
 
 823909 
 
 903090 
 632023 
 920819 
 342423 
 
 798355 
 
 x y=: 
 903090 
 
 380211 
 376751 
 522879 
 602060 
 
 800000 
 0-477121 
 
 761905 
 000000 
 
 881901 
 669006 
 301030 
 
 
 851937 '711110 
 
76 
 
 681241 
 707744 
 309970 
 15 
 
 017729 
 185046 
 306425 
 
 \l} 798355 
 
 660052 
 744727 
 3/5662 
 2 
 
 628571 
 138303 
 
 603174 
 
 278754 
 
 585860 
 365489 
 
 538960 
 
 200000 
 
 0-602060 
 
 238095 
 204120 
 
 288888 
 034762 
 
 324674 
 017728 
 
 388888 
 088134 
 
 449494 
 
 102 } 851937 
 
 204120 
 
 675323 
 318063 
 
 6m 10 
 381340 
 
 550504 
 
 142857 
 0-602066 
 
 185185 
 204120 
 
 236364 
 141329 
 
 316327 
 013789 
 
 388888 
 100370 
 
 446639 
 
 647817 
 766722 
 414961 
 12 
 
 Jj:} 698970 
 
 100 } 780443 
 
 } 829512 
 
 513924 
 781710 
 403307 
 
 28 
 
 } 698969 
 
 501689 
 803705 
 462363 
 36 
 
 " J 511449 
 786120 
 
 488508 
 42 
 
 ^} 767793 
 
 }786119 
 
 392544 
 825700 
 480709 
 17 
 
 402304 
 808115 
 521077 
 61 
 
 404779 
 803705 
 532248 
 
 28 
 
 99 | 740760 
 J"} 164902 
 
 99 1 698970 
 
 y x=l. 
 3 L 522879 "333333 
 124939 0-602060 
 301030 
 397940 
 698970 
 
 99 1 731557 
 II. XA 
 
 2 \ 301030 
 
 903090 
 376751 
 397940 
 698970 
 
 ' 756962 
 413/34 
 397940 
 698970 
 
 ;~}267606 
 
 471726 
 691001 
 795880 
 414973 
 
 102 1 522879 
 
 "333333 
 204120 
 
 371428 
 034762 
 
 396824 
 193819 
 
 414141 
 135663 
 
 461039 
 
 J*} 376751 
 
 103 ^726999 
 632023 
 
 795880 
 414973 
 
 580871 
 669006 
 795880 
 414973 
 
 101 1 569875 
 
 1021 535113 
 744727 
 318726 
 32 
 
 *j 460730 
 
 425968 
 766722 
 318726 
 32 
 
 ,99)373580 
 
 J 514909 
 749137 
 858671 
 377418 
 
 1598598 
 
 T 99 1 51 1448 
 
 9 8 1500135 
 
 404779 
 803705 
 408652 
 12 
 
 493720 
 786120 
 309970 
 14 
 
 513924 
 781710 
 986211 
 307979 
 
 ,} 617148 
 
 9%\ 589824 
 
 971 589824 
 
 489454 
 806395 
 354069 
 39 
 
 481485 
 808115 
 374101 
 
 501690 
 803705 
 347310 
 19 
 
 9 8 } 663738 
 
 97J652724 
 
 96} 649957 
 
 182. When x=y we see that we have, in every case, (within 
 narrow limits,) 698970, which is the logarithm of J, for the required 
 value. This we obviously ought to have, and it is satisfactory to find 
 that the operation gives it. With regard to the cases in which x and 
 y are unequal, it has been already remarked (175), that the sum of 
 
77 
 
 every two corresponding values, that is, values in which x and y are 
 interchanged, ought to be equal to unity. This too will be seen to 
 hold. Thus :- 
 
 Diff. i. 
 666667 
 '333333 
 roooooo '999999 
 
 Diff. 2. 
 
 104*103 
 103-104 
 
 IO2-IOI 
 101*102 
 
 628571 
 371428 
 
 538960 
 461039 
 
 roooooo 
 
 104*102 
 102*104 
 
 800000 
 
 '200000 
 
 roooooo 
 
 101-99 
 99'ioi 
 
 675323 
 324675 
 
 99*97 
 97'99 
 
 550504 
 449494 
 
 999998 
 
 999998 
 
 And in this way the operations, if carried on simultaneously, may be 
 verified from time to time. 
 
 183. This function may also be expressed in terms of the curtate 
 mean durations of pairs of lives. If, in the expression in (99), which 
 denotes the probability of survivorship in respect of the nth year, n be 
 made equal to 1, 2, 3, &c., successively, and the results added in the 
 manner of (157), we obtain for the total probability, 
 
 '6* e> 
 
 Px-i Py-i 
 
 As a formula of verification this expression, with existing tables, is 
 almost useless. The mean duration is never given to more than two 
 decimal places. The consequence is that, e l x _ ll and e l xi _ l being 
 necessarily nearly equal, their difference will seldom consist of more 
 than two significant figures, and this therefore will be the utmost 
 extent to which a probability of survivorship, deduced by this formula, 
 can be relied on, as matters now stand. 
 
CHAPTER VI. 
 
 OF ANNUITIES AND ASSURANCES ON LIVES. 
 
 184. BY a Contingent Benefit (or, simply, a Benefit,) is meant a 
 payment to be made, either periodically or once for all, in accordance 
 with previously stipulated conditions, as to the subsistence or non- 
 subsistence, at the period or periods of payment, of the life or lives on 
 which the payment depends. In accordance with this definition it is 
 obvious, that the present value of a benefit, in respect of any one year 
 of existence, is equal to the present value of the sum that may be re- 
 ceived in that year, multiplied by the probability of its being received, 
 (78) ; and hence the whole present value is equal to the sum of those 
 partial values in respect of each year of the possible future existence 
 of the life or lives concerned. 
 
 185. Thus, an annuity of 1. upon (x) being a payment of 1. to 
 be made at the end of each year at which the individual so designated 
 shall be alive, its present value is, 
 
 and in like manner the present value of an annuity upon two lives, 
 (#) and (y} } which is payable at the end of each year at which both 
 lives are in existence, is, 
 
 So also, an assurance of 1. upon (a?), being a payment of }.. to be 
 made at the end of the year in which the life of (a?) fails, its present 
 value is, (95), 
 
 and if the assurance be upon the joint duration of the two lives (x) 
 and (y), its present value will be, (96), 
 
 and so on. 
 
 186. These series will consist of as many terms as there arc years 
 in which a payment of the respective benefits may be received ; so 
 that, if these are to last the whole of life, the annuity series will con- 
 sist, in the case of a single life, of as many terms as there are years in 
 
79 
 
 the tabular remainder of lives of the same age as that life, and in the 
 case of two or more joint lives, of as many terms as there are years in 
 the tabular remainder of lives of the same age as the oldest of the 
 specified lives ; and the assurance series will consist of one term more 
 than these respectively, since the failure, which entitles to the receipt 
 of the benefit, may not take place till the year after the attainment by 
 the single life, or the oldest of the joint lives, of the greatest tabular 
 age. 
 
 187. If the number of years that the benefits are to last be less 
 than the greatest tabular duration of the life or lives on which they 
 are constituted, say for n years, the series expressing their present 
 values will, in all cases, consist of n terms only, the terms beyond the 
 nth vanishing, in consequence of the vanishing of v for those years to 
 which they respectively belong. 
 
 188. And in short, to whatsoever years, during the possible dura- 
 tion of the life or lives on which it depends, the duration of a benefit 
 may be restricted, the present value of that benefit will be the sum of 
 those terms of the series expressing its value for the whole of life, 
 which belong to the years to which its duration is restricted. 
 
 189. It thus appears that to compute directly the present value of 
 a benefit, the numerical values of as many terms as there are years in 
 the duration of the benefit must be computed ; and if these are nu- 
 merous, the labour of the computation will obviously be considerable. 
 And if a table had to be formed in this way, exhibiting the value of 
 even one benefit for but a single life of each age, not to speak of com- 
 binations of lives, the labour would be altogether intolerable. This is 
 avoided, however, by adopting the continuous method of computation, 
 of which examples have already been given in previous problems. The 
 application of this method is facilitated by the Lemma which will 
 presently be given ; but, in the mean time, the principle of the notation 
 to be employed must be briefly explained. 
 
 190. A distinct letter will be appropriated to denote each kind of 
 benefit, the age of the life or lives on which it depends being attached 
 as a suffix; and the portion of existence to which its duration is 
 restricted being indicated by symbols occupying the place of exponents 
 on both the right and the left of the general symbol. The exponent 
 on the left will indicate the number of years that are to elapse before 
 the benefit is entered upon ; and that on the right the number of years 
 that the benefit is to last after it is entered upon. The absence of an 
 exponent on the left will denote that the benefit is to be entered upon 
 immediately ; and the absence of an exponent on the right will denote 
 that the benefit, when entered upon, is to last for the whole remainder 
 
80 
 
 of life. Hence the absence of exponents on both the right and the 
 left of the general symbol will indicate that the duration of the benefit 
 is to be commensurate with that of the life or lives on which it de- 
 pends ; that is, that it is to be entered upon immediately, and to last 
 for the whole remainder of life. 
 
 Orders of failure among the lives on which a benefit depends, when 
 we have occasion to indicate them, will be denoted by numerals placed 
 over the symbols designating the ages. 
 
 191. Thus, in accordance with these conventions, a being the letter 
 chosen to designate the class of annuity benefits generally, 
 
 a x will denote the present value of an annuity of 1. for the whole 
 
 life of (a?) : this is called simply a life annuity. 
 a n x will denote the same for the first n years : this is called a tempo- 
 
 rary annuity. 
 n a x will denote the same for what remains of the life of (x) after n 
 
 years : this is called a deferred annuity. 
 n a t x will denote the same for t years of what remains of the life of (x) 
 
 after n years : this is called an intercepted annuity. 
 And if in these symbols x.y be written for x, they will denote cor- 
 responding benefits on the joint duration of the two lives (x] and (y} ; 
 and so also for any greater number -of lives. 
 
 192. And in like manner, A x being chosen to denote the present 
 value of an assurance of 1. on (#), and A^ the present value of a 
 survivorship assurance of 1. on (x) against (y), by modifying these 
 symbols in the manner just shown, we acquire the means of indicating 
 the values of the several benefits under the variety of circumstances 
 to which they are respectively subject. 
 
 193. We now proceed to the 
 
 LEMMA. 
 
 If B denote the present value of a benefit of l . upon a given life or 
 combination of lives, and such that, in the case of a combination of 
 lives the risk is determined by the failure of any one of them ; and if 
 B L denote the present value of a similar benefit on a life or a combi- 
 nation of lives respectively one year older than those on which B de- 
 depends ; if, moreover, TT denote the probability of a payment of B 
 being received in the first year, and IT the probability of the single 
 life, or of all the lives, on which that benefit depends, surviving a 
 year ; then will the following equation always subsist : 
 
 For, if the benefit make its payments at the end of the year in which 
 
81 
 
 they respectively become due,* the value in respect of the first yeair 
 is obviously VTT (78. 184). And the value in respect of the years fol- 
 lowing the first is vIIB l -, for B x is the value, at the time it is entered 
 upon, of the remaining portion of the benefit ; II is the probability of 
 its being entered upon ; and v is the ratio in which the value is dimi- 
 nished on account of the time that has to elapse before it is realized. 
 Hence, a whole being equal to the sum of its parts, 
 
 or, B=vJT(- 
 
 194. On the foregoing Lemma and the resulting formula we offer 
 the following remarks. 
 
 (1.) The benefits to which the demonstration applies, and in regard 
 to which consequently the equation deduced subsists, must fulfil these 
 two conditions : First, that the payment in respect of any one year in 
 which it may become due shall be always l. An exception may be 
 made in favour of the last year, as will be hereafter seen ; but the ful- 
 filment of this condition obviously precludes the application of the 
 formula, in its present state, to increasing or decreasing benefits. 
 Secondly, the benefit must be such that, in the case of two or more 
 lives the risk will be determined by the first failure that takes place 
 from amongst those lives. The fulfilment of this condition excludes 
 from the application of the formula benefits depending upon specified 
 orders of survivorship amongst three or more lives. The values of such 
 benefits must therefore continue to be found as heretofore from other 
 tabulated values. 
 
 195. (2.) The adaptation of Table I. to the foregoing formula 
 is sufficiently apparent. Throwing it into the logarithmic form it 
 becomes, 
 
 log.B =log.tfH -f log.^+ B,). 
 
 Log.B is thus the sum of two logarithms, whereof the first, log.^II, 
 is always a term of one of the subsidiary series whose formation has 
 been shown; and the second is the logarithm of the sum of two 
 quantities whose logarithms are known. For, the logarithm of the 
 first, TT-r-IT, will be found to be generally a term of one of the subsi*. 
 diary series, and the logarithm of the other, B p is the preceding result. 
 Hence the requisite operations for finding the successive values of B, 
 may, by reference to Formula I., be typified as follows : 
 
 * As is always the case unless it be otherwise expressly stated. 
 
 M 
 
82 
 
 T zA. 
 
 E l7 A. 
 
 E, 
 
 log.B, 
 
 B, 
 
 E, 
 
 log.B 1 
 
 B, 
 
 A 
 
 log.(Tr-JT) 
 
 ar.co.(Ej A) 
 
 A 
 
 log.(7T~n) 
 
 E X ~A 
 
 B 
 
 log.vIT 
 
 
 B 
 
 log.vIT 
 
 
 C 
 
 E,-A 
 
 
 C 
 
 I T rE 
 
 
 I) 
 
 Tjfar.co^Bj-A)] 
 
 
 D 
 
 ) ! 1 
 
 
 E 
 
 1 =log.B 
 
 B 
 
 E 
 
 rA + B + C + D 
 
 l^log.B 
 
 B 
 
 196. The double form of the type is occasioned by there being no 
 negative arguments in Table I. These arise by the subtraction of the 
 line A from the line E p when of the two the former is the greater. 
 The mode in which this difficulty, such as it is, (for it seldom has 
 place in more than two or three values at the extremity of life,) is sur- 
 mounted, is shown in the form typified on the left, and it will be 
 clearly understood by reference to (17). In the examples to be here- 
 after given of the applications of the formula, the point where the 
 operation changes from the one typified form to the other, will be 
 marked by a strong dark line. This mode of indication is employed 
 in some of the preceding examples. 
 
 197. (3.) The benefit B p it is stated in the Lemma, must be 
 " similar " to the benefit B. The only restriction here implied, beyond 
 those of its being of the same amount, and subject to like conditions 
 in respect of the life or lives on which it depends, as Bj, is, that its 
 duration shall extend to the same period of life ; in other words, that 
 both benefits shall cease at the same age. It is no matter what that 
 age is, whether the limiting age of the table or any less tabular age ; 
 the formula will still apply, provided only the age of cessation in respect 
 of both benefits is the same. The formula consequently holds in the 
 case of temporary as well as whole life benefits. 
 
 198. (4.) But farther, the formula holds also in the case of deferred 
 benefits. To adapt it to such it is only necessary to make 7r = for 
 those years in which no payment is to be made. This reduces the 
 formula to 
 
 B=^ITB 1 , or log.B=log.i;IT+log.B 1 ; 
 
 so that, log.Bj being found for the age immediately beyond that at 
 which the blank years begin, log.B for each younger age will be found, 
 
without any tabular operation, by the continuous addition to the former 
 of the successive terms of the series log.vll. 
 
 199. It remains now to be shown how the initial values, to serve as 
 the grounds of the computations, are to be found. 
 
 For this purpose the following precepts will be available. First, if 
 the benefit be either temporary or for the whole of life, then, for the 
 age at which all payments cease, B manifestly equals 0. Hence, for 
 the next younger age, Bj vanishes, and the formula, for this age, is 
 reduced to 
 
 B = vHjf or V7T ' 
 
 from one or other of which forms the initial value may be readily 
 deduced, by the aid of the Elementary Table, if it be not found as a 
 term of one or other of the subsidiary series. 
 
 200. Secondly, if the benefit be deferred, in which class, for our 
 present purpose, intercepted benefits may be included, then, for the 
 age at which it assumes its distinctive character, B is, in the case of a 
 simple deferred benefit, equal to the value of the corresponding whole 
 life benefit ; and in the case of an intercepted benefit it is equal to the 
 value of the corresponding temporary benefit. Hence, as these values 
 are those of B T in the formula for the next younger age, the values of 
 B for all the younger ages may be successively deduced. 
 
 We now proceed to the applications of the foregoing Lemma. 
 
 PROBLEM XIV. 
 
 201. To construct a table of the present value of an annuity of \ y 
 on a single life, or on the joint duration of any number of lives. 
 
 This problem comprises two cases, according as the annuity is to 
 cease at, or not to commence till, a specified age. 
 CASE I. The annuity to cease at a specified age. 
 
 202. First, let the annuity be upon one life, (x), and to last for 
 the whole duration of that life. It will then be denoted by a x . Hence, 
 recurring to the formula in (193), we have here, 
 
 B = fl r ; B 1 =a a , + 1 ; also II=p x ', and since the condition of 1. 
 being received at the end of the first year, is that (x) be then alive, 
 rjr= P X ) an d '. 7r-r-IT = l. Hence, 
 
 or, log.o x = lo&vp, + log. 
 
 203. Secondly, if the annuity, still upon one life, (a?), be to cease on 
 the attainment by that life of the age x + n, it will be denoted by a n x ; 
 and we should find, in like manner, 
 
84 
 
 or, log.a" =log.vp s + log. (1 
 
 204. So also, if instead of one life the foregoing annuities are de- 
 pendent on the joint continuance of two lives, (a?) and (y), their values 
 will be respectively denoted by , and #* y ; and we should find, 
 
 It is needless to give the formulae for three or any greater number of 
 lives, as the respects in which they would differ from the foregoing 
 are sufficiently obvious. 
 
 205. The tabular operation in the case of all these formulae is mani- 
 festly the same. It will be enough, therefore, to give a type corre- 
 sponding to one of them ; and for this purpose the first formula is 
 selected that for the value of an annuity upon (a?), for the whole of 
 life. The type referred to follows : 
 
 D x negative. 
 
 D l positive. 
 
 A' 
 
 log.tfc, 
 
 ar.co.Dj 
 
 DI 
 
 lg., + 1 
 
 w, 
 
 A 
 
 log.v^ 
 
 C 
 
 JT^ar.co.DJ 
 
 
 B 
 C 
 
 }*iPJ 
 
 
 D 
 
 {=il +B+c 
 
 a. 
 
 D 
 
 rA+B + C 
 
 l=log.a x 
 
 . 
 
 206. Now to find the initial values: Agreeably to (199), Bj 
 vanishes when the life, or the oldest life in the combination, on which 
 the annuity depends, is of the next younger age to that at which the 
 benefit is to cease. For that age then we have, 
 a x =vp x ; a n x = vp x -, 
 
 And these, which are the initial values, are all terms of the subsidiary 
 series, log.vp x and log.i^^^. 
 
 207. We now give some examples of the application of the foregoing 
 formulae, which will be fully understood by comparison with the type 
 of operation. 
 
85 
 
 Present Values of Annuities. I. For the Whole of Life. 
 
 One Life. 
 
 103 510042 -32363 
 765314 0-489958 
 611679 
 44 
 
 Two . 
 x y=o. 
 
 ' 3 1 032921 -10788 
 103 J 543465 0-967079 
 011498 
 72 
 
 Lives. 
 
 J3 1 288193 -19418 
 102 J 619186 0-711807 
 
 788869 
 6 
 
 102 887079 -77104 
 
 841035 112921 
 361138 
 12 
 
 ^} 587956 -38722 
 102 J 694907 412044 
 554157 
 31 
 
 J 02 } 696254 -49688 
 101 J 731890 303746 
 478903 
 31 
 
 101 089264 1-22818 
 
 878017 
 347916 
 35 
 
 ll} 837051 "68715 
 101 J 768872 162949 
 390073 
 29 
 
 ioo J 790867 092922 
 349959 
 12 
 
 100 225968 1*68255 
 
 *| 996025 -99089 
 100 J 812863 003975 
 302984 
 38 
 
 100 } 047916 rn665 
 
 99 795278 
 325640 
 
 8 
 
 900013 
 428505 
 43 
 
 99 328561 2-13089 
 
 99} 1H910 1-29392 
 
 777693 
 360574 
 6 
 
 99} 120926 1-32107 
 
 773283 
 365674 
 15 
 
 882428 
 495626 
 41 
 
 98 378095 2-38833 
 
 * * * 
 
 * * * 
 
 60 020833 10-4914 
 
 9^} 138273 1-37491 
 
 * * 
 * * * 
 
 f\ 863039 7-29523 
 
 9*J138972 1-37712 
 * * * 
 * * * 
 
 6 \ 871945 7-44638 
 
 974707 
 060342 
 30 
 
 J 962251 
 918794 
 34 
 
 964065 
 926631 
 40 
 
 59} 890736 7-77564 
 
 59 035079 10-8413 
 
 976521 
 073325 
 72 
 
 58 049918 11-2181 
 
 59} 881079 7-60465 
 
 9 965878 
 934663 
 70 
 
 J 96/349 
 943247 
 32 
 
 5*1.900611 7-95447 
 
 f\ 910628 8-14007 
 
 977991 
 086986 
 17 
 
 968819 
 952030 
 10 
 
 969660 
 960924 
 25 
 
 57 064994 11-6143 
 978831 
 100777 
 
 87 
 
 571 920859 8-33410 
 
 57} 930609 8-52332 
 
 57 970501 
 970020 
 53 
 
 J 970977 
 978781 
 t, 8 
 
 56 0/9695 12-0142 
 
 5jj 19405 74 8-72116 
 
 5"} 949766 8-90770 
 
 979309 
 114330 
 
 87 
 
 5 971455 
 987651 
 67 
 
 971908 
 995914 
 59 
 
 55 093726 12-4087 
 
 551959173 9-10276 
 
 55 972360 
 004374 
 66 
 
 54} 976800 9-47982 
 * * * 
 
 5 5 } 967881 9-28712 
 
 972692 
 012221 
 73 
 
 979762 
 127362 
 24 
 
 54 107148 12-7982 
 * * * 
 
 *} 984986 9-66020 
 53 J 
 
 * # * 
 
x 2/=o. 
 y Z=Q. 
 103 1 
 103^666800 -03596 
 103 J 321616 1-444200 
 459542 
 
 1021 
 
 86 
 
 Three Lives. 
 
 x-y=i. 
 y-z=2. 
 
 I0 3l 
 102 1 1 79048 -15102 
 TOO J 532036 0-820952 
 881991 
 TOSH 45 
 
 x y=2. 
 y-z=4- 
 I0 3l 
 101 ^254769 -17979 
 97 J 549714 0-745231 
 817010 
 
 T02-I 26 
 
 102 [^336958 -21728 
 102 J 548779 663042 
 748387 
 TOT-, 34 
 
 101 [> 593120 -39185 
 99 J 62/155 406880 
 550415 
 
 101 1 
 
 100 [ 621519 -41833 
 96] 638492 378481 
 530201 
 
 IOI "I 
 
 101 j> 634158 -43068 
 101 J 659727 365842 
 521356 
 
 100 1 29 
 
 100 } 770748 -58986 
 98 J 681 722 229252 
 430578 
 
 TOO I 33 
 
 99 |> 790270 -61698 
 95] 648343 209730 
 418416 
 
 100] 19 
 
 iooUl5270 -65354 
 100 J 725713 184730 
 403126 
 
 991 18 
 
 99 }> 883081 -76398 
 97] 688823 116919 
 363401 
 
 991 U 
 
 98 1 857048 -71953 
 94 J 660534 142952 
 378331 
 
 991 3 
 
 99 |> 944127 -87928 
 99 J 672958 055873 
 329826 
 
 98] 39 
 
 98 |>935316 -96162 
 96] 657890 064684 
 334530 
 
 91 45 
 
 97 |> 895943 -78694 
 93 J 633306 104057 
 356136 
 
 9 1 32 
 
 08 !> 946950 -88*01 
 
 98J * 
 * * * 
 
 60-1 
 60 1 749120 5-61203 
 
 97^927781 -84680 
 
 ^ : : 
 60 1 
 
 59^779356 6-01667 
 
 96 \- 885417 -76810 
 92 J * * 
 * * * 
 
 60 1 
 
 58 ^ 806905 6-41070 
 
 60 J 949795 
 820318 
 - , 17 
 
 57 J 955734 
 846083 
 
 591 48 
 
 54 J 958465 
 869855 
 
 591 4 
 
 50 1 770130 5-89020 
 
 58U01865 6-33673 
 
 57 1 828324 6-73479 
 
 59 J 955237 
 838206 
 
 r8l 26 
 
 5J 959495 
 865446 
 58-1 57 
 
 53 J 961538 
 888428 
 5 81 _ 21 
 
 48 1 793469 6-21540 
 
 57^824998 6*68341 
 
 56^849987 7-07925 
 
 58 J 959647 
 858201 
 _, 59 
 U 817907 6-57*17 
 
 55 J 962259 
 885469 
 57 -, 5 
 56 1847813 7'04390 
 
 5 2 -l 963885 
 907295 
 
 571 - 
 55^871256 7-43457 
 
 57] 962169 
 879386 
 , fi , 6 
 
 54 J 963907 
 905455 
 
 5*1 " 
 
 5 1 J 965563 
 926014 
 ^j 49 
 
 \ 6 841561 6 . 0432I 
 
 55 1869373 7-40240 
 
 54 j.891626 7-79158 
 
 56 J 963601 
 899943 
 54 
 
 53 J 965256 
 924339 
 
 551 65 
 
 50 J 966255 
 944044 
 
 551 23 
 
 ^1 863598 7-30463 
 
 5^ 889660 7-75640 
 
 53 1 910322 8-13433 
 
 55 J 964960 
 919234 
 
 541 86 
 
 5 2 J 966440 
 942272 
 
 541 53 
 
 49-1 967016 
 960657 
 
 541 2 
 
 54 !> 884280 7-66090 
 
 54J * 
 
 * * * 
 
 53 1908765 8-10622 
 
 5i J * 
 * * * 
 
 52 1927693 8-46628 
 
 4 8J * 
 * * * 
 
87 
 
 57 
 
 55 
 
 54 
 
 Present Va 
 
 One Life. 
 To cease at 60. 
 
 974707 -94342 
 976521 0-025293 
 313813 
 
 47 
 
 ues of Annuities. II. r 
 
 Two Lives. 
 To cease at 60.59. 
 
 5?} 964065 -92059 
 50 J 967349 0-035935 
 319351 
 18 
 
 Pemporary. 
 
 Three Lives. 
 To cease at 60.58.54. 
 
 591 
 57^958465 -90879 
 53 j 961538 0-041535 
 322276 
 
 265088 
 977991 i'84ii4 
 453436 
 57 
 
 5?} 250783 1-78149 
 969660 
 444224 
 53 
 
 56 242297 r74?02 
 
 52 J 963885 
 438800 
 
 431484 2-70075 
 
 57} 413937 2-59380 
 
 jH 402746 R9 
 
 978831 
 568228 
 61 
 
 970977 
 555528 
 
 27 
 
 51 J 965563 
 547473 
 
 547120 3-52468 
 
 5} 526532 3-36149 
 
 gtwaow 3 . 25 888 
 
 50 J 966255 
 629243 
 53 
 
 979309 
 655573 
 15 
 
 "" 971908 
 639610 
 25 
 
 634897 4*31417 
 
 5 5 1 611543 4-08830 
 
 glwm ,.^0 
 
 979762 
 725356 
 
 80 
 
 972692 
 706538 
 35 
 
 49 J 967016 ' 
 693730 
 
 511 41 
 
 705198 5-07222 
 * * 
 
 54} 679265 4'7782i 
 * * * 
 
 52 } 660787 4-579I7 
 48 J 
 * * * 
 
 208. COROLLARY I. If in the expression, 
 
 we suppose = 1, which, since v = l-r-(l + r), is equivalent to sup- 
 posing r=Q, or, in other words, that money bears no interest, the 
 expression is converted into (137.140.185), 
 
 which is the formula for the curtate mean duration of (a?), and the 
 same hypothesis effects a like transformation in the expression for the 
 value of an annuity on the joint duration of two or more lives. Whence 
 it follows, that the number of years in the curtate mean duration of 
 any life or combination of lives, is the same as the number of pounds 
 in the value of an annuity on the same life or combination of lives, 
 when money bears no interest ; and also, that it is the same as the 
 average number of payments that each annuitant, of the same speci- 
 fied age, (whether an individual or a combination,) will receive. 
 
 209. COROLLARY 2. As in each single operation of those just 
 exemplified, we form the logarithm of an annuity, so in that which 
 follows we form the logarithm of the same annuity increased by unity. 
 This is evident by reference to the formulae and the type of operation. 
 In fact, the line D being equal to log.a, when D is negative, 
 
88 
 
 log. (1+0) is equal to the sum of D and the 
 
 succeeding B and C ; and when D is positive 12 
 
 log.(l + ) is equal to the sum of the succeed- Iog.(l+a 102 ) 0*248229 
 
 ing B and C alone. This will be understood 495626 
 
 from the examples in the margin, which are 
 
 . i c T^p log.(l+ 99 ) 0-495667 
 
 taken from the foregoing operations. 
 
 210. Now log.(l + ) is a function that 302984 
 
 comes frequently into use, as will hereafter be 
 
 more fully shown. In the meantime it will lg-(l+ fl ioo.ioo) 0*299047 
 suffice to explain a single application of it. 
 Suppose that a benefit, whose present value is 
 B, is to be purchased by an annual premium, 
 which makes its first payment now, and its 
 succeeding payments at the end of every year, Ig-(l~l~ a 56) 0-655588 
 either to the extremity of life, or till the attainment of a specified age. 
 Now employing a as a general symbol, to denote the present value 
 of an annuity payable till the attainment of the specified age, the 
 present value of the premium, if its amount were 1., would evidently 
 be 1 -f a ; or, if its unknown amount be called m, its present value 
 will bem(l + a). Now, that the arrangement may be an equitable 
 one, we must have, 
 
 T> 
 
 whence, m=~ -- ; or, log.m=log.B log.(l-fa). 
 
 Hence, log.(l + a) for all ages and combinations being known, we have 
 the means of readily determining the amount of the uniform annual 
 premium equivalent to any benefit whose present value is known. 
 
 211. CASE 2. The annuity not to commence till the attainment of 
 a specified age. 
 
 First, let the annuity be upon one life (#), to be entered upon at 
 age (x + ri), and to continue for the remainder of life. Then, in the 
 general formula of (193) we have, 
 
 B=X; B 1 =-x+i; T=O; n= Px . 
 
 Hence, * a *=vp*"~ la x+i> 
 
 or, log."^ = hg.vp x + log."- l a x+ 1 . 
 
 212. Secondly, if the annuity, when entered upon, is to last only 
 t years thereafter, the expression just deduced takes the following 
 form : 
 
 or, og- = o g.v/ 
 
 And if it depend on two or more lives the requisite changes will 
 
89 
 
 be made in the formulae, by writing x.y, x.y.z, &c., for x in the 
 suffixes. 
 
 213. It thus appears that, according to the number of lives on 
 whose continuance the several benefits depend, a table of the present 
 values of any one of them may be formed by the continual addition to 
 the logarithmic initial value, of the successive terms of the proper one 
 of the series log.vp x , log.vp^), &c. 
 
 214. The initial values are formed by making n in the formulas 
 = 1, and x, x.y, &c., consequently the age next to that at which the 
 benefit is to commence. The formula then become, 
 
 log. l a x =log.vp x +log.a x+l ; 
 and, log. V =log.^+ log.< +1 ; 
 
 and similarly for two or more lives. The initial values then are log. 
 a x+1 , log.fl^ +1 , &c., which will have been found by the first case of the 
 present problem. 
 
 215. The most common case of an intercepted annuity is that in 
 which t = I t and the annuity consequently is to make but a single pay- 
 ment. In this case it is called an endowment, to be received, not at 
 the period at which the corresponding annuity would be entered upon, 
 but a year later, namely, at the period at which the first and only 
 payment is to be made. Thus, an endowment of 1., to be received 
 at 60, is the same as an intercepted annuity of 1. for one year, to be 
 entered upon at 59. 
 
 216. We now give a few examples of the formation of the values of 
 the benefits under consideration. 
 
 Present Values of Annuities. III. Deferred. 
 
 One Life. 
 
 To be entered upon at 
 60. 
 
 60 020833 10.4914 
 
 974707 
 59 995540 9-89781 
 
 9/6521 
 
 58 972061 9-37694 
 
 977991 
 57 950052 8-91358 
 
 978831 
 
 56 928883 8-48952 
 979309 
 
 55 908192 8-09454 
 
 979762 
 
 54 887954 7-72598 
 
 Two Lives. 
 
 To be entered upon at 
 60.59. 
 
 60.59 871945 7-44638 
 964065 
 
 59.58 836010 6-8^04 
 967349 
 
 58.57 803359 6-35856 
 969660 
 
 87*6 ggg S . 9295I 
 
 56.55 743996 5-54620 
 971908 
 
 55.54 715904 5-19881 
 972692 
 
 54.53 688596 4-88198 
 
 * * * 
 
 Three Lives. 
 
 To be entered upon at 
 60.58.54. 
 
 60.58.54 806905 6-41070 
 958465 
 
 59-57-53 765370 5-82600 
 961538 
 
 58.56.52 726908 5-33222 
 963885 
 
 57.55.51 690793 4-90674 
 965563 
 
 56.54.50 656356 4-53269 
 966255 
 
 55-53-49 622611 4*19383 
 967016 
 
 54.52.48 589627 3-88711 
 * * * 
 
90 
 
 Present Values of Annuities. IV. 
 
 One Life. 
 One Payment. (Endowment.) Ten Payments. 
 
 To be received at 60. 
 
 To be entered upon at 60. 
 
 60 
 
 000000 
 
 I -000000 
 
 60 
 
 844978 
 
 6-99807 
 
 
 974707 
 
 
 
 974/07 
 
 
 59 
 
 974707 
 
 943424 
 
 59 
 
 819685 
 
 6-60214 
 
 
 976521 
 
 
 
 976521 
 
 
 58 
 
 951228 
 
 893774 
 
 58 
 
 796206 
 
 6-25470 
 
 
 977991 
 
 
 
 977991 
 
 
 57 
 
 929219 
 
 849608 
 
 57 
 
 774197 
 
 5-9456i 
 
 
 978831 
 
 
 
 978831 
 
 
 56 
 
 908050 
 
 809188 
 
 56 
 
 753028 
 
 5-66276 
 
 
 979309 
 
 
 
 979309 
 
 
 55 
 
 887359 
 
 '77 I 54i 
 
 55 
 
 732337 
 
 5'3993 
 
 
 979762 
 
 
 
 979762 
 
 
 54 
 
 867121 
 
 736411 
 
 54 
 
 712099 
 
 5-I5346 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 Intercepted. 
 
 Two Lives. 
 
 Ten Payments. 
 
 To be entered upon at 60.59. 
 
 60.59 770031 5-88884 
 964065 
 
 59.58 734096 5*42121 
 967349 
 
 58.57 701445 5-02858 
 969660 
 
 57.56 671105 4-68927 
 970977 
 
 56.55 642082 4-38613 
 971908 
 
 55.54 613990 4-11140 
 972692 
 
 54.53 586682 3-86084 
 
 217. An annuity to last n years, and an annuity to commence in n 
 years and continue for the remainder of life, are obviously together 
 equal to an annuity for the whole of life. That is, in symbols, 
 
 : + X=*- 
 
 Advantage is taken of this relation as a means of verifying the values 
 determined as above. For, if a x , a n x) and n a x have been formed, for the 
 same values of x and n, it is obvious that the subsistence amongst them 
 of the foregoing relation will be a sufficient presumption that all of 
 them have been correctly determined, while its non-subsistence will be 
 an undeniable proof that error has been committed. An example is 
 given in the margin. 
 
 218. But it is seldom that we shall be provided 
 with such a complete array of values for verification 
 as is here supposed. It will hardly ever be thought 
 necessary, in addition to a x , to compute both a* and 
 *a x . The foregoing relation can still, nevertheless, 
 be rendered available. 
 
 0^=5-07233 
 
 4 = 12- 
 
 054*2.48 
 6 54.52.48 
 
 054.52.48 8*46628 
 
 The value of a benefit to be entered upon at 
 
 an after period is obviously equal to its value at the time of its being 
 entered upon, multiplied by the probability of its being entered upon, 
 and by the present value of 1. to be received at the same time, (78.) 
 
 that is, since 
 
 to,=*-!4 
 
 or, 
 
 ./,+ log.fl +n . 
 
91 
 
 We have thus the means, if log.a x for all ages be known, of readily 
 determining both \og. H a x and logaj ; and hence, if the series of values 
 of either of these quantities has been constructed, of verifying that 
 series at as many points as we please. 
 
 219. The same properties hold, mutatis mutandis, in regard to 
 benefits depending on the joint continuance of any number of lives, 
 and may be applied in the same way for like purposes. 
 
 220. It will be observed that no means have yet been explained for 
 verifying the series log.a^. This will be done hereafter, (335.) In 
 the mean time some examples are given of the application of the 
 methods of verification just explained. 
 
 Endowment on (54). 
 
 To be entered upon at 6C 
 
 Log.4o=3-561459 
 colog./54=4;382685 
 = 1-9229/7 
 
 1-867121 
 
 Annuity on (54). 
 Deferred 6 years. 
 
 Log.fl6o= 1-020833 
 ^0=3.561459 
 
 colog.Z 54 =4-382685 
 log.v 6 =1-922977 
 
 0-887954 
 
 Annuity on (54.53). 
 Deferred 6 years. 
 
 Z 60 =3-561459 
 / 59 =3-573915 
 colog.Z 5 4=4;382685 
 43=4-375615 
 log.z; 6 =1-922977 
 
 0-688596 
 
 PROBLEM XV. 
 
 221. To construct a table of the present value of an assurance of 
 1. upon a single life, or the joint continuance of any number of 
 lives. 
 
 This problem, like the last, comprises two cases, according as the 
 benefit is to cease upon, or not to commence till, the attainment of a 
 specified age. 
 
 222. CASE 1. The assurance to cease at a specified age. 
 
 First, let the assurance be upon a single life, (%}, and to last till 
 the end of life. Hence, in the formula of (193), we have, 
 B=A X ; B 1 =A, C+1 , TT, the probability of 1. being received in the 
 first year, = 1-^; andIT=^, .-. 7rH-IT=(l pj+p^p,- 1 !. 
 
 or, oS- 
 
 223. Secondly, if the assurance be to cease on the attainment of 
 age x+n, being then denoted by A^, the formula will be, 
 
 or, 
 
92 
 
 224. These formulae, like those of last problem, will be adapted to 
 two or more lives by making the requisite changes in the suffixes. 
 Thus, for a whole life assurance on two lives, (x) and (y} } we should 
 have, 
 
 1S- A,, =log.t?Pc,,) + log. [(p (xjj - l - 1) + A x+ 1 y+ J . 
 The expression, log.fjt^." 1 1), which occurs in the formulae for a 
 single life, is tabulated for all ages amongst the elementary values, 
 and hence those formulae admit of immediate application. But it is 
 otherwise with the analogous expression in the formulae for two or 
 more lives. Log^/^)" 1 1) is not tabulated, nor can it be formed by 
 combination of any of the tabulated values. For two or more lives, 
 therefore, we have recourse to a different formula, the method of 
 deducing which will be presently shown. 
 
 225. The foregoing formulae give the following type of operation : 
 
 E l7 A. 
 
 rX+B + C + D 
 UtofcA. 
 
 226. The initial values are determined as follows. When the 
 assurance is to last to the end of life, then if x be the oldest tabular 
 age, A,,. + 1 = 0; and since p x also in that case equals 0, the formula 
 of (222) reduces to &*=*>'> 
 
 that is, log.A, c =log.v. 
 
 227. If the assurance is temporary, then when a is the age next 
 younger than that at which the benefit is to cease, that is when w = l, 
 A-J=0, and the formula of (223) gives for the initial value, 
 
 or 
 
 which may be constructed from the Elementary Table. 
 
 228. The following examples show the application of the foregoing 
 formula). 
 
93 
 
 Present Values of Assurances. First Method. 
 
 I. For the \\ 
 
 104 987163 '970874 
 
 One Life. 
 r hole of Life. 
 
 60 823017 -665298 
 
 II. Temporary. 
 To cease at 60. 
 59 438554 -027451 
 
 301030 0-313867 
 510042 
 686133 
 485721 
 
 463847 1-359170 
 974707 
 377693 
 67 
 
 394568 0-043986 
 976521 
 323534 
 46 
 
 103 982926 -961448 
 
 823909 159017 
 765314 
 387766 
 10 
 
 102 976999 '948415 
 
 602060 374939 
 841035 
 527737 
 
 27 
 
 59 816314 -655110 
 
 58 694669 '049507 
 
 394568 421746 
 976521 
 437843 
 44 
 
 58 808976 -644133 
 
 329257 365412 
 977991 
 521077 
 
 8 
 
 57 828333 '067349 
 
 329257 479719 
 977991 
 493857 
 18 
 
 57 801123 -632591 
 
 287085 541248 
 978831 
 650983 
 37 
 
 ioi 970859 '935101 
 
 455932 514927 
 
 878017 
 630698 
 21 
 
 56 916936 -082592 
 
 287085 -514038 
 978831 
 527098 
 37 
 
 261263 655673 
 979309 
 742317 
 60 
 
 100 964668 -921866 
 
 56 793051 -620941 
 
 55 982949 -096150 
 
 235184 747765 
 979762 
 819130 
 55 
 
 346787 617881 
 900013 
 711607 
 66 
 
 99 958473 -908810 
 
 261263 531788 
 979309 
 544283 
 
 85 
 
 55 784940 -609453 
 
 54 034131 -108176 
 
 435728 522745 
 882428 
 636685 
 35 
 
 235184 549756 
 979762 
 561779 
 54 
 
 215194 818937 
 980093 
 880254 
 32 
 
 98 9548/6 -901313 
 
 54 776779 -598106 
 
 53 075573 '119007 
 
 455932 498934 
 878018 
 618496 
 26 
 
 215194 561583 
 980093 
 573260 
 
 82 
 
 53 768629 -568989 
 * * * 
 
 188528 887045 
 980510 
 939970 
 40 
 
 97 952472 '896339 
 * * * 
 
 52 109048 -128543 
 
 * * * 
 
 229. We have now to deduce a formula adapted to the computation 
 of cases involving two or more lives. 
 The formula of (222) is, 
 
 Performing the multiplication by p x this becomes, 
 
 * 
 
 This formula, how many soever may be the lives on whose joint 
 continuance the benefit depends, admits of the application of Table II., 
 agreeably to the following type : 
 
94 
 
 H l 
 
 log.A, +1 
 
 A,,, 
 
 A 
 
 kg*. 
 
 
 B 
 
 
 
 C 
 
 } T a [HJ 
 
 
 D 
 
 llog.^(l-A x+1 ) 
 
 
 E 
 
 log.v 
 
 
 F 
 
 
 
 G 
 
 } T 2 [D] 
 
 
 H 
 
 e B+F+G 
 I =log.A, 
 
 A, 
 
 230. There is no argument for the direct use of Table II. and the 
 inverse use of it is not recommended > (25) greater than TOOOOOO. 
 Hence, when log.A x+1 exceeds this limit, the formula is nugatory. 
 The ages thus excluded are of course the older ages, for which, as it 
 happens, the value of this benefit is least frequently wanted. The 
 table, however, can be completed by other means. There exist, as is 
 hereafter demonstrated (239, &c.), certain relations between the values 
 of an annuity and an assurance on the same life, or combination of 
 joint lives, which enable us, when the value of either is known, readily 
 to pass to that of the other. We can therefore in this way supply the 
 values which the preceding formula fails to give, and also form initial 
 values for its use at those parts of the mortality table where it can be 
 applied. 
 
 231. The relation we here make use of for the latter purpose is, 
 
 Av(l r^Jj 
 
 which, as usual, is adapted to two or more lives by changing the 
 suffixes into x.y, x.y.z, &c. ; and the initial values for the succeeding 
 examples are determined as follows : 
 
 Iog.fl6o=l;020833 
 log.r=2-477121 
 
 log.m 60 =T-497954 
 
 log.z?=T-987163 
 -835833 
 
 21 
 
 log.A,=T-823017 
 
 Iog.a 60 .59=0-871945 
 log.r=2-477121 
 
 log.ra 60.59=1' 349066 
 
 log.A co .59=T-877365 
 
 log.a co .58.54=_0-806905 
 log.r=2'477121 
 
 log.ra 6 o.58.54=T'284026 
 
95 
 
 Present Val 
 
 One Life. 
 60 823017 '665298 
 
 nes of Assurances. Sec( 
 For the Whole of Life. 
 Two Lives. 
 
 *} 877365 -753988 
 
 md Method. 
 
 Three Lives. 
 
 661 
 
 5 8l894402 -784155 
 
 987544 
 524492 
 165 
 
 512201 
 
 976902 
 390847 
 107 
 
 367856 
 
 54 J 971302 
 333785 
 357 
 305444 
 
 774717 
 764684 
 '754337 
 
 987163 
 829103 
 
 47 
 
 987163 
 884630 
 13 
 
 987163 
 901965 
 
 591 15 
 
 59 816313 '655109 
 
 59 1 871806 -744400 
 
 989358 
 537518 
 165 
 
 57 \ 889143 
 
 980186 
 407288 
 274 
 
 53 J 974375 
 352533 
 196 
 
 327104 
 
 987163 
 896293 
 
 sai 16 
 
 56 1883482 
 5 2 J 976722 
 371593 
 59 
 
 527041 
 
 387748 
 
 987163 
 821783 
 30 
 
 987163 
 878389 
 18 
 
 58 808976 '644133 
 
 5?} 865570 -733787 
 
 990828 
 551243 
 43 
 
 J 982497 
 425147 
 83 
 
 542114 
 
 987163 
 813916 
 45 
 
 407727 
 987163 
 871724 
 26 
 
 348374 
 
 987163 
 890394 
 
 571 8 
 
 57 801124 -632593 
 * * * 
 
 57 1 858913 -722625 
 * * * 
 
 55^877565 
 5i J 
 
 * * 
 
 232. These operations are tedious. Two tabular entries are requi- 
 site for each assurance value, and one or other of them is always in 
 that part of the table whose results are but little to be relied on (29) . 
 The consequence is, that in a succession of values deduced in this way 
 some of them will be found to differ materially from the truth. In 
 extensive trials we have found deviations of so much as 6 and 7 in the 
 6th decimal place. It is obviously unnecessary to have recourse to 
 this method in the case of single lives, since we have already another 
 method applicable to such, which is not only much simpler, but which 
 also, like all the operations making use of Table L, gives results as 
 correct as any succession of logarithmic results can possibly be. 
 
 233. COROLLARY. If, in the case of an assurance to cease at a spe- 
 cified age, less than the oldest tabular age, the sum assured is to be 
 paid on the attainment of that age, if it shall not previously have 
 become due, that is, if the life or lives on which it depends shall be 
 
96 
 
 then in being, the benefit bears a close analogy to a whole life assu- 
 rance. In both cases the payment depends, for a specified time, on 
 the same conditions, and in both also, at the expiry of that time, it 
 becomes certainly payable if it have not previously done so. If there- 
 fore, for the oldest tabular age in the one benefit, we substitute in the 
 other the age immediately preceding that at which the sum assured is 
 certainly to become payable, the treatment of both will be in all respects 
 similar. In the one case as in the other the initial value will be log.v, 
 and the subsequent operations are identical in form, although, of course, 
 the results will differ in value. 
 An example is here given : 
 
 Present Values of Assurances. 
 To be received at death, or at 60. 
 
 59 987163 -970874 
 
 394568 1-592595 
 976521 
 603460 
 92 
 
 58 974641 -943280 
 
 329257 645384 
 
 9/7991 
 
 655019 
 
 82 
 
 57 962349 -916958 
 
 287085 675264 
 978831 
 684279 
 63 
 
 56 950258 -891780 
 
 261263 688995 
 979309 
 697700 
 93 
 
 55 938365 -867690 
 * * * 
 
 234. This benefit is one for which provision is frequently made in 
 the regulations of Friendly Societies. Its present value is obviously 
 that of a short term assurance, together with an endowment payable 
 at the end of the term, so that by adding together the values of these 
 two benefits, separately obtained, we are furnished with a ready method 
 of verifying the computation. This is here shown : 
 
 Assurance on (55) to cease at 60 '096156 (228) 
 
 Endowment on (55) to be received at 60 .. -771541 (216) 
 
 867691 
 
97 
 
 235. CASE 2. The assurance not to commence till the attainment 
 of a specified age. 
 
 The operations here, in accordance with (198), the requisite change 
 being made in the initial values, are identical with those of (216) for 
 the same ages and number of lives. 
 
 Present Values of Assu 
 
 One Life. 
 To commence ot 60. 
 
 60 823017 '665299 
 974707 
 
 ranees. III. Deferred. 
 
 Two Lives. 
 To commence at 60.59. 
 
 6o -59 877365 -753988 
 964065 
 
 59 
 58 
 57 
 56 
 
 55 
 
 # 
 
 797724 
 976521 
 
 774245 
 977991 
 
 627660 
 594627 
 565244 
 
 538353 
 
 '5 I 33o6 
 
 * 
 
 59.58 
 58.57 
 57-5 6 
 5 6 -55 
 
 55-54 
 
 * 
 
 841430 
 967349 
 
 808779 
 969660 
 
 694113 
 643841 
 600398 
 561586 
 
 526140 
 * 
 
 752236 
 978831 
 
 778439 
 970977 
 
 731067 
 979309 
 
 749416 
 971908 
 
 710376 
 * 
 
 721324 
 * 
 
 236. The series of values thus formed admit of verification after 
 the manner of (220). Thus : 
 
 log.A6o=l-823017 
 
 log.4o=3-561459 
 
 colog./55= 4-390086 
 
 lo.^ 5 = 1-935814 
 
 log. 5 A55= 1-710376 
 
 237. Intercepted assurances belong to this case ; but it does not 
 seem worth while to give an example of its application to these. 
 
 238. The present values of an annuity and an assurance upon the 
 same life or combination of lives are intimately connected, and in 
 fact mutually dependent upon each other. In virtue of this relation 
 we can readily pass from the one to the other, and hence obtain, from 
 point to point, verification of the results of both the last and the pre- 
 sent problem. 
 
 239. The nature of the relation thus subsisting may be shown in 
 various ways. 
 
 1st. By (185), 
 
 =0 
 (!+<,)-, 
 
98 
 
 
 240. By subjecting this expression to a few obvious transformations 
 it assumes various forms, some of which are, in particular circum- 
 stances, better adapted than others to numerical computation. Thus : 
 
 v(l+a x ) a x =v a x + va x =v (I v)a x .. (2) 
 Adding to this the neutral quantity, 1 1, we have, 
 
 v-(l-v}a x = \-l + v-(\-v}a x = l-(l-v)(l + a x }..(3) 
 Also, since 1 v=vr, (76), 
 
 v (lv)a x =vvra x =v(I ra x ) .. .. (4) 
 
 Again, since = T-T- , and -=P, (76), 
 
 Of these five expressions for the same function, A x , (1), in the form 
 
 l+a x 
 
 , x a x , and (5), are given by Mr. Morgan; (2) and (3) we owe to 
 
 "1 ^^ y/j 
 
 Mr. Milne j and (4), in the form - -, to Mr. Baily. 
 
 241 . 2nd. A second method of deducing the value of an assurance 
 from that of an annuity is as follows. Suppose that (#) is in posses- 
 sion of a perpetuity of jl. per annum. What remains of this after 
 the death of (a?) is evidently now worth Y a x . But at the end of 
 the year in which (x) dies its value will be 1 -f P, for his representa- 
 tives will be put into possession of 1., the rent for that year, and the 
 remainder of the perpetuity, which being itself a perpetuity, its value 
 will then be P. Hence P a x is the present value of an assurance of 
 1-fP upon the life of (a?). That is, 
 
 p-a,= 
 
 242. 3rd. Again. Since r is the interest of 1. for a year, 
 invested now will produce r per annum during the life of (#), and at 
 the end of the year in which that life fails, the produce will be r, the 
 interest for that year, together with a return of the principal. Con- 
 sequently 1. is the present value of an annuity of r and an assurance 
 of 1+r upon the life of (a?). That is, 
 
 
 "1 ' Vfl 
 
 Whence, A,=-y-^=i;(l -ra x \ 
 
 243. 4th. Once more. The present value of an annuity of 1. upon 
 (a?), which makes its first payment immediately, is l+a x . Suppose 
 
99 
 
 now that each payment is deferred for a year, so as not to be made 
 till a year after (#)'s title to receive it has been determined : in this 
 case the value will obviously be v(l +a x ). But in these circumstances 
 the annuity will be the same as an ordinary annuity making an addi- 
 tional payment at the end of the year of death. In other words, it 
 will be the same as a life annuity of l. together with an assurance of 
 1. Hence, 
 
 244. It is easy now, from any of the foregoing expressions for A z 
 in terms of a x , to deduce a value of the latter quantity in terms of the 
 
 former. Thus : 
 
 & x =v-(l-v)a x . 
 
 P.JL 
 .-. (1 v)a x =v k x , and a x = y^ 
 
 245. A more convenient form for tabular computation is, 
 
 which arises by dividing both numerator and denominator of the other 
 by v. 
 
 246. We have thus the means, within certain limits, if the series 
 log.c^ and log.A,,. have been formed by continuous operations, of 
 proving the accuracy of both, by passing from the one series to the 
 other at a few points, by means of the foregoing formula?. Or, if one 
 of them only, as log.a,,., has been formed, and otherwise proved, (as 
 by the operation of Problem XXII.,) we have evidently the means of 
 readily finding any term of the other. The formulae most convenient 
 for this purpose, and which admit of the application of Table II, are, 
 
 i - 
 
 and = - 
 
 , 
 
 which give the following types of operation : 
 
 * Of the four methods just given for deducing the value of an assurance from 
 that of an annuity on the same life or lives, the first consists of an obvious alge- 
 braical transformation; the second is adapted from Milne, pp. 167, 168, and the 
 " Penny Cyclopaedia," article REVERSIONS; the third, which is substantially the 
 same as the second, is taken from a " Report on the Bengal Military Fund," by 
 Mr. Griffith Davies ; and for the fourth, which is now for the first time published, 
 the author is happy to express his obligations to an esteemed friend, Mr. Peter 
 Hardy, F.R.S., Actuary to the Mutual Life Assurance Society, and to the Equi- 
 table Reversionary Interest Society, and one of the Vice-Presidents of the Insti- 
 tute of Actuaries. 
 
 t The method of using another of the formulae of (240) has been shown 
 in (231). 
 
100 
 
 A 
 B 
 
 C 
 
 D 
 
 E 
 
 F 
 
 log. (1+0,) 
 log.(l-t>) 
 
 A, 
 
 rA+B= 
 
 llog.(l-tO(l + <0 
 
 }T 2 [C] 
 
 rD + E = 
 
 llog.A, 
 
 A 
 B 
 
 C 
 
 D 
 E 
 F 
 
 G 
 
 log.A. 
 
 log.(l-fr) 
 
 fl , 
 
 |A+B = 
 
 llog.(l + r)A, 
 
 colog.r 
 
 }T 2 [C] 
 
 cD + E + F 
 
 l=log.a x 
 
 247. The value in the line C will occasionally be found to exceed 
 the greater limit of Table II., in which case the operations may be 
 completed either in numbers, or by the use of the common tables. 
 Log.(l + a x ), which occurs in the first of these formulse is formed, as 
 already noticed (209), in the operation for the construction of 
 
 248. We now give a few examples in illustration. 
 
 Assurances deduced from Annuities. 
 
 103 
 
 60 
 
 One Life. 
 0-121765 
 2-464284 
 
 Two Lives. 
 0-044491 
 2-464284 
 
 Three Lives. 
 0-015342 
 2-464284 
 
 2-586049 
 
 2-508775 
 
 2-479626 
 
 F982888 
 38 
 
 f-982926 -961448 
 
 1-060372 
 2-464284 
 1-524656 
 
 1-985747 
 8 
 
 1-986682 
 
 103 ] i? 
 103 M '986694 -969826 
 
 "3 1 1-985755 .0677*2 
 
 i3 J 
 
 0-918828 
 2-464284 
 
 103] 
 0-820335 
 
 2-464284 
 
 1-284619 
 
 1-383112 
 
 f-822995 
 22 
 
 T-907078 
 * -, 19 
 
 ^Q T , 
 
 1-879865 
 29 
 
 T823017 '665298 
 
 60 1 1-907097 -80741* 
 
 |r-879894 -758392 
 
 ties deduced from Assui 
 Two Lives. 
 T879894 
 0-012837 
 
 Annui 
 One Life. 
 1-823017 
 0-012837 
 
 60 J 
 
 Dances. 
 Three Lives. 
 T-816543 
 0012837 
 1-829380 
 1-522879 
 1-511683 
 
 53] 42 
 
 T-835854 
 
 1-892731 
 
 1-522879 
 1-497855 
 100 
 
 1-522879 
 1-339913 
 
 60 1 246 
 
 1-020834 10-4914 
 
 60 1-0-863038 7 . 20WI 
 
 35 ^1-034604 10*8294 
 
 F-603419 
 0-012837 
 f-616256 
 1-522879 
 1768392 
 31 
 
 T855292 
 0-012837 
 
 7J 
 1-830393 
 0-012837 
 
 1-868129 
 1-522879 
 1-417896 
 200 
 
 1-843230 
 
 1-481288 
 531 161 
 
 1-291302 19-5570 
 
 }0-940975 ., aoaa 
 
 37 }> 1-004328 10-1002 
 
 ^r* \ ' 
 
101 
 
 PROBLEM XVI. 
 
 249. To construct a table of the present value of a survivorship 
 assurance of l. on (#) against (y) ; that is, an assurance of 1. to be 
 paid at the end of the year in which (#) dies, provided he shall have 
 been survived by (y). 
 
 This is the benefit whose present value is denoted by A~, (192). 
 
 The general formula of (193) is, 
 
 and hereB=Ai 5 ; B 
 n=p (t - l ; .-.7r-i-II= 
 
 -=i(l -p f )(I +^,(99.177); and 
 l + l), and the formula becomes, 
 
 or, in logarithms, 
 
 250. This is the working formula, the series log. vp (x ^ and 
 
 tog-idV 1 iX-P," 1 * 1 ) havin g been alrea <ty formed, (124.129); 
 and it gives the following type of operation : 
 
 A 
 B 
 C 
 D 
 
 E 
 
 A-T 
 
 A) 
 
 A 
 B 
 
 C 
 D 
 
 E 
 
102 
 
 251. It will readily appear on comparison of this type with that of 
 (181), that with the single distinction of log.vp (xy ) here taking the 
 place of log.p^y) in the other, the two are identical. The reason of 
 this is obvious. Of the functions to be computed the one is the ab- 
 stract value of a probability, and the other is the money value of the 
 same probability. Hence the introduction in the expression for the 
 latter of v and its powers. 
 
 252. The properties of this function, analogous to those of the other 
 mentioned in (182), are, 1st, that when #=y, the resulting present 
 value is equal to iA fty , (Milne, p. 184) ; and, 2nd, that when x and y 
 are unequal, the sum of any two values in which x and y are inter- 
 changed is equal to A ry . (Milne, p. 185). These properties expressed 
 in symbols are as follows, 
 
 ^~x = 2 A *., > A ^ + A ^ = ^x.y > 
 
 and the first is obviously a particular case of the second. We are 
 thus in possession of a ready means of testing the accuracy of our 
 computations. 
 
 253. The initial values here are deduced in precisely the same way 
 as in (179). The expression in (249) becomes, on multiplying out 
 
 A 1 - = v[\ (1 -ft 
 
 1st. Let x=y. Then if (x) and (y} are of the oldest tabular age, 
 P*> P y > au &P(*y) vanish, and the expression is reduced to 
 A ^=i" ' log.A^=log.i+log.t>, 
 = (when interest is 3 per cent.), 698970+987163 = 686133. 
 2nd. Let x 7 y. Then if x be the oldest tabular age, p f and p^ y ) 
 vanish, and we have, 
 
 or log-A=log.^4-log.4(^ y ~ 1 + 1), 
 
 the mode of forming which is obvious. 
 
 3rd. Let x L y. Then if y be the oldest tabular age, p y and 
 vanish, and the expression becomes, 
 
 or, log.A^=log.^ + Iog4 -f log.Qtv 1 1), 
 
 which also may be readily formed. 
 
 254. When a whole table is to be computed the best way of form- 
 ing the initial values for the several compartments will be in two 
 series, as in (180). Indeed the operations will only differ in that 
 here the first term of each series will include log. v. The formation 
 of a few terms is given : 
 
103 
 
 Survivorship Assura 
 
 Initial 
 x7 y 
 
 log.t^os 510042 
 ^Ioa+1) 301030 
 
 nee. (x) 
 
 Values. 
 
 104.103 
 104.102 
 104.101 
 104.100 
 
 811072 
 255272 
 ' 823909 
 
 890253 
 075721 
 954242 
 
 920216 
 036983 
 978811 
 
 936010 
 
 against 
 
 103.104 
 
 102.104 
 
 IOI.IO4 
 
 100.104 
 
 x L y 
 
 698970 
 510042 
 301030 
 510042 
 255272 
 522879 
 
 288193 
 075721 
 778151 
 
 142065 
 036983 
 
 853872 
 
 032920 
 
 Verification. 
 
 log.vpm 878018 log 
 log.iCpuxf^ + l) 057992 
 
 .vp m 878018 
 log.-J 698970 
 i_l) 455932 
 
 104.100 936010 log.^uxT 
 
 3.104 032920 
 
 255. We now introc 
 formula. 
 Survivorshi 
 
 x=.y. 
 ^4} 686133 -485437 
 
 10 
 uce a few examples of 
 
 p Assurances. On (x) 
 I. x= or 7 y. 
 x y=i. 
 104 \ 81 1072 '647250 
 
 the application of the 
 against (y). 
 
 x y = 2. 
 JJ} 890253 -776700 
 
 J 602060 0-915927 
 032921 
 084073 
 965672 
 
 103 425969 0-614897 
 288193 
 385103 
 709270 
 
 380211 O489958 
 363914 
 510042 
 611723 
 
 ioi } 684 726 -483867 
 
 Jo?} 808535 '643480 
 
 ioi} 865890 '734328 
 
 948848 264122 
 543465 
 
 735878 
 452867 
 
 903090 094555 
 619186 
 905445 
 350876 
 
 ** 1 778597 -600616 
 660052 118545 
 731890 
 364309 
 26 
 
 " 881901 016011 
 656169 
 983989 
 309110 
 
 102 >- 681058 '470708 
 
 ^g| 831 169 '677905 
 
 647817 183352 
 753885 
 402280 
 31 
 
 101 } 804013 -636814 
 
 511449 ' 292564 
 
 773282 
 471453 
 42 
 
 ' 681241 000183 
 694907 
 999817 
 301122 
 
 J^} 677087 '475430 
 513924 163163 
 768872 
 390192 
 37 
 
 ** j 756277 -570529 
 
 501689 254588 
 790867 
 446662 
 56 
 
 ;} 673025 -471004 
 
 ^j 739274 -548623 
 
 J | 756226 -570461 
 
 392544 280481 
 812863 
 463477 
 53 
 
 402304 336970 
 795278 
 501361 
 
 48 
 
 99 j 698991 -500024 
 
 * * * 
 
 404779 351447 
 790868 
 511342 
 32 
 
 99 j 668937 '466592 
 
 * * * 
 
 99 1 707021 -509356 
 
 * * * 
 
104 
 
 J3 1 5 io042 -323625 
 
 II. xty. 
 y #=a. 
 H 288193 -194175 
 
 * x j 142065 -138696 
 
 " 124939 0-614897 
 288193 
 385103 
 709270 
 
 903090 0-614897 
 363914 
 385103 
 709270 
 
 756962 0-614897 
 400897 
 385103 
 709270 
 
 * j- 507505 -321740 
 
 \l l \ 361377 -229814 
 
 J| 252232 -178744 
 
 726999 219494 
 619186 
 780506 
 424498 
 
 580871 219494 
 656169 
 780506 
 424498 
 
 471726 219494 
 678164 
 780506 
 424498 
 
 * ] 551 189 -355786 
 
 *j 442044 -276722 
 
 102 } 354894 '226409 
 
 535113 016076 
 731890 
 309104 
 39 
 
 425968 016076 
 753885 
 309104 
 39 
 
 514909 160015 
 736300 
 839985 
 388366 
 
 Jj 576146 -376831 
 
 t ^} 488996 -308316 
 
 ioi} 4 ^ 9560 '3 l6 89 
 
 404779 171367 
 790867 
 395072 
 40 
 
 493720 004725 
 773282 
 995275 
 303399 
 
 513924 034364 
 768872 
 965636 
 318552 
 
 t 99J. 590758 -389725 
 
 !^} 565676 -367854 
 
 j^H 566984 -368964 
 
 481485 109273 
 795278 
 359053 
 
 81 41 
 
 501689 063987 
 790868 
 334154 
 47 
 
 489454 077530 
 793558 
 341506 
 
 Sg} 635857 -432371 
 
 Sg | 626758 -423407 
 * * >n 
 
 9 6 | 624534 -421244 
 * * * 
 
 256. It maybe noted, in regard to the example y a? = 2, that the 
 operation, after changing from the first typified form to the second, 
 reverts, for a single value, to the first of these forms. This may occa- 
 sionally happen in regard to almost any of the formulae we have given, 
 in the use of the Carlisle Table. It will hardly happen, however, 
 when any more regularly graduated table is employed. But it is well 
 to be prepared for its occasional occurrence. 
 
 257. Verification may be obtained as follows, agreeably to (252), if 
 the values of assurances on the joint lives have been computed, either 
 independently, or from the annuities. 
 
 103.103 
 103.103 
 
 483867 
 483867 
 
 967736 
 
 IOI.IOO 
 IOO.IOI 
 
 570529 
 376831 
 
 AIOI.IOO '947360 
 
 99-97 -509356 
 97.99 -423407 
 
 A 99 .97 '932763 
 
 258. The value of a survivorship assurance, in terms of annuity 
 and assurance values, is expressed by the following formula : 
 
 + a ^ L - a ^> (Milne, p. 184) ; 
 
 P,-\ Py-l 
 
105 
 
 and this ought also to serve as a formula of verification. But with ex- 
 isting tables it is all but useless for this purpose. Those tables extend 
 only to three decimal places, and from the manner in which the 
 annuity values enter the foregoing expression it is easy to see that the 
 assurance value deduced by means of it cannot be true to more, and 
 hardly to so many. Mr. Milne (pp. 355, 356,) labours to prove that 
 the error will " seldom amount to '0005." But he is mistaken, of 
 which Mr. De Morgan gives a notable example, (" Companion to the 
 Almanac," 184.2, p. 7.) And others can easily be adduced. Thus : 
 
 p. 352, A\jj 7 
 
 According to Milne . . '08656 
 True Value '08284 
 
 Error '00372 
 
 Per cent. . . 3*96 
 
 52556 
 
 00226 
 
 o*43 
 
 P- 354, AJ_ 
 
 11287 
 "11062 
 
 It is obvious then that till better annuity tables are formed this me- 
 thod cannot be employed to verify results for which we claim a greater 
 degree of accuracy in the sixth place than the results of thejnethod 
 in question possess in the third. 
 
 259. The operation of this problem may be applied to the compu- 
 tation, in series, of the values of deferred and temporary survivorship 
 assurances, in the manner shown in previous problems. But it does 
 not seem necessary to say more on the subject here. When a table of 
 the values of the whole life benefit has been formed, for every combi- 
 nation of ages, it is easy thence to deduce the value of the benefit 
 when it is either deferred or temporary. Thus we have for the deferred 
 benefit, "A- =v n p (r , .A , 
 
 x,y r^x.yin. x j cn.y + n > 
 
 and for the temporary benefit, 
 
 260. SCHOLIUM. This method of deducing the values of deferred 
 and temporary benefits from the value of the same benefit for the 
 whole life, holds in every case, whatever be the number of lives on 
 which the benefit depends, when the risk is determined by the first 
 failure that takes place from amongst those lives. That it holds in 
 the present case seems, however, to have been overlooked by Mr. 
 Milne, whose formulae for the same purpose, on p. 184, although not 
 greatly more complex in appearance than those just given, wall be 
 found to involve a great deal more labour in their numerical applica- 
 tions. 
 
106 
 
 PROBLEM XVII. 
 
 261. To construct a table of the present values of increasing 
 annuities. 
 
 262. 1, 1, 1, 1, 1, 1, 1, (1) 
 
 1, 2, 3, 4, 5, 6, 7, (2) 
 
 1, 3, 6, 10, 15, 21, 28, (3) 
 
 1, 4, 10, 20, 35, 56, 84, (4) 
 
 The foregoing series are formed as follows : (1) is a series of units ; 
 each term in (2) is the sum of the corresponding term and all the 
 preceding terms in (1) ; in like manner, each term in (3) is the sum 
 of the corresponding term and all the preceding terms in (2) ; and so 
 on for the succeeding series, which obviously may be continued at 
 pleasure, both horizontally and vertically. Now it is a property of 
 series so formed, which follows obviously enough from the mode of 
 their construction, that any one of them will be produced by adding 
 its terms, carried one place to the right, to the terms of the series of 
 the next lower order. Thus, to produce (2), we have, 
 
 
 
 i, 
 
 1, 
 
 1, 
 
 1, 
 
 2, 
 
 1, 
 
 3, 
 
 1, .... 
 4, .... 
 
 
 Sum, 
 
 i, 
 
 2, 
 
 3, 
 
 4, 
 
 5, .... 
 
 To produce 
 
 (3), we 
 
 ha) 
 1, 
 
 re, 
 
 i! 
 
 3, 
 3, 
 
 4, 
 6, 
 
 5, .... 
 10, .... 
 
 
 Sum, 
 
 1 
 
 3 
 
 6 
 
 10 
 
 15, .... 
 
 
 
 
 
 
 
 
 To produce 
 
 (4), 
 
 1 
 
 3 
 
 6 
 
 10 
 
 15, .... 
 
 
 
 
 1, 
 
 4, 
 
 10, 
 
 20, .... 
 
 
 Sum, 
 
 1, 
 
 4, 
 
 10, 
 
 20, 
 
 35, .... and so on. 
 
 263. Now let the terms of any one of these series be the number 
 of pounds in the successive payments of an annuity which is to com- 
 mence immediately ; then it is obvious from what has just been shown, 
 that that annuity will be equivalent to two annuities, the one com- 
 mencing immediately and the other a year hence, and of which the 
 payments of the first are the terms of the next inferior series, while 
 the payments of the second are the same as those of the complete 
 annuity under consideration. The value of this annuity, therefore, will 
 be the sum of the values of its two component annuities. 
 
 264. To render this more plain, if the annuity is one whose suc- 
 cessive payments are to be 1, 2, 3, 4, &c.> pounds, it may be decom- 
 
107 
 
 posed into the two annuities whose successive and simultaneous 
 payments are as follows : 
 
 1st year. 2nd year. 3rd year. 4th year, &c. 
 
 1st annuity 1111 &c. 
 
 2nd 1 2 3 &c. 
 
 And in like manner an annuity whose payments are to be 1, 3, 6, 10, 
 &c., pounds, is equivalent to the two whose payments are as follows : 
 1st year. 2nd year. 3rd year. 4th year, &c. 
 
 1st annuity 1 2 3 4 &c. 
 
 2nd ' 1 3 6 &c. 
 
 and so on. 
 
 265. To form the value, therefore, of such an annuity upon any 
 specified life or combination of lives, two elements are requisite ; first, 
 the value of an annuity on the same life or combination of lives, the 
 payments of which are the terms of the series of the next inferior 
 order ; and, secondly, the value of an annuity on a life or a combina- 
 tion of lives a year older than the life or combination in question, and 
 whose payments are the same as those of the annuity whose value is 
 sought. For when this last is known we can easily, in virtue of a 
 principle we have frequently employed, determine thence the value in 
 respect of the life or combination with which we may be occupied. 
 Now the first of these elements, in regard to any of the series, is 
 evidently attainable. For we know the values in respect of one of 
 them the lowest the payments of which are 1, 1, 1, &c. (201, &c.); 
 and from these we can ascend, so far as this element is concerned, to 
 as many as we please of the higher series in succession. It is obvious 
 also, in regard to the second element, that if, commencing with the 
 oldest tabular age or combination of ages, we can assign a single value 
 of the annuity whose values are to be tabulated, we shall be able to 
 deduce in succession the values corresponding to all the younger ages 
 or combinations of ages. 
 
 266. To apply these principles : Let, for facility, the annuities be 
 on a single life, (at), and let their present values be denoted, 
 
 when the payments are the terms of the 1st series in (262), by a x ; 
 j) 2nd i x ', 
 
 3rd ' *"> 
 
 that is, the (m + 2)th series being the figurate numbers of the mth 
 order, when the payments of the annuity are the same as the numbers 
 
108 
 
 of this order, its present value will be denoted by i x (m \ Now, agree- 
 ably to what has been shown in (263), &c., the present value of this 
 annuity will be equal to the sum of the values of an annuity of the 
 (m l)th order, to commence now, and an annuity of the wth order 
 deferred a year. Hence, in symbols, 
 
 i x to> = ij m -u + vpji f+l , (193); 
 and by this formula, which on passing to logarithms becomes, 
 
 the annuities corresponding to each series may be successively formed. 
 267. The formula deduced gives the following type of operation : 
 
 B^C. 
 
 B 7 C. 
 
 B-C 
 
 ar.co.(B-C) 
 
 * 
 
 JT^B-C] 
 
 D+E 
 
 n 
 
 B-C 
 
 268. Since the first payment of all the annuities is 1., the initial 
 value will in every case be simply vp x , corresponding to #= the next 
 to the oldest tabular age. For i x+1 (m} for this age vanishes, and i x (m ~ l) 
 is then also 1. to be received in a year, subject to the failure of the 
 life of (x). 
 
 269. We now give the following examples : 
 
 Present Values of Increasing Annuities. 
 
 Annuity of i, 4, 10, 20, 
 &c., pounds. 
 
 103 510042 -323625 
 765314 
 
 275356 
 
 059976 0-784620 
 
 215380 
 
 850648 
 
 102 126004 1-33661 
 
 Annuity of i, 2 
 pound 
 
 103 510042 
 765314 
 
 , 3, 4, &c., 
 
 s. 
 
 323625 
 0-611723 
 959572 
 
 Annuity of 1,3, 6, 10, &c., 
 pounds. 
 
 103 510042 -323625 
 765314 
 
 275356 
 
 2/5356 
 
 887079 
 388277 
 706722 
 
 982078 0-706722 
 293278 
 784620 
 
 102 982078 
 
 102 059976 1-14819 
 
109 
 
 102 
 101 
 IOO 
 
 99 
 
 98 
 
 
 
 982078 
 841035 
 
 823113 
 
 959572 
 266151 
 1*89363 
 
 070656 
 3-11247 
 
 064557 
 4-60329 
 167401 
 
 5-89986 
 
 * 
 
 102 059976 
 841035 
 
 901011 
 
 089264 
 733849 
 454182 
 
 277295 
 623716 
 528711 
 
 277295 
 
 878017 
 
 101 429722 
 
 878017 
 
 155312 
 
 307739 
 
 225968 
 929344 
 337793 
 
 493105 
 814634 
 403529 
 
 493105 
 900013 
 
 loo 7H268 
 900013 
 
 393118 
 
 328561 
 334476 
 31 
 
 663068 
 
 882428 
 
 611281 
 
 663068 
 948213 
 327695 
 
 99 938976 
 
 882428 
 
 545496 
 
 821404 
 
 770842 
 327014 
 33 
 
 378095 
 392746 
 
 1 
 
 770842 
 * 
 
 98 097889 
 
 * * 
 
 3/6284 
 
 2-68981 
 
 185366 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 98 
 * 
 
 126004 
 841035 
 
 1-33661 
 
 462683 
 3-61672 
 
 274936 
 7-87466 
 
 142731 
 14-9444 
 
 040983 
 
 23-9283 
 * 
 
 967039 
 
 429722 
 537317 
 591276 
 
 558315 
 
 878017 
 
 436332 
 
 711268 
 725064 
 459900 
 
 896232 
 900013 
 
 796245 
 
 938976 
 857269 
 378233 
 
 174478 
 
 882428 
 
 056906 
 
 097889 
 959017 
 322005 
 
 378911 
 
 * 
 
 051787 
 
 8-68912 
 
 050562 
 
 12-5282 
 
 270. In these examples it will be observed, first, that in all, the 
 lines corresponding to that marked A in the type are the same, being 
 the successive terms of the series log.i;^ ; secondly, that in example 1 
 the line corresponding to C is occupied by the successive values of 
 log.o^,, being the results of the first example in (207) ; and thirdly, 
 that in each of the remaining examples the same line C is occupied 
 by the results of the preceding example. 
 
 PROBLEM XVIII. 
 
 271. To construct a table of the present values of increasing assu- 
 rances. 
 
 272. The solution of the present problem, when the annual varia- 
 tion is as in the last problem, follows so obviously from the principles 
 there employed, that to enter at length into it would be but to repeat 
 previous reasoning. The formula here also is entirely analogous to 
 that last obtained. Using h x to denote the present value of an assu- 
 rance upon (a?), which is to be !., ,2., 3., &c., according as the 
 failure of the life of (a?) shall take place in the 1st, 2nd, 3rd, &c., 
 year from the present time, we have, 
 
110 
 
 which formula, differing from that of (266) only in the substitution of 
 h for i y gives a type of operation differing in only the same respect 
 from the type in (267). It is therefore unnecessary to repeat it. 
 
 273. The following are examples of the operation in question, the 
 initial value being in every case log.v. 
 
 Present 
 
 Assurance of i, 2, 3, 4, 
 &c., pounds. 
 
 104 987163 '970874 
 51Q042 
 
 497205 
 
 Values of Increasing As 
 
 Assurance of i, 2, 6, 10, 
 &c., pounds. 
 
 104 987163 -970874 
 510042 
 
 surances. 
 
 Assurance of i, 4, 10, 20, 
 &c., pounds. 
 
 104 987163 -970874 
 510042 
 
 497205 
 
 497205 
 
 982926 0-485721 
 514279 
 608526 
 
 105731 0-608526 
 391474 
 704151 
 
 201356 0704151 
 295849 
 782473 
 
 103 105731 1-27565 
 765314 
 
 103 201356 1-58985 
 765314 
 
 103 279678 1-90405 
 765314 
 
 871045 
 
 966670 
 
 044992 
 
 976999 105954 
 894046 
 357230 
 
 228275 261605 
 738395 
 451239 
 
 417909 372917 
 627083 
 526343 
 
 102 228275 1-69151 
 841035 
 
 102 417909 2-61764 
 841035 
 
 102 571335 3-72679 
 841035 
 
 412370 
 
 069310 
 
 970859 098451 
 353011 
 29 
 
 258944 
 
 323899 064955 
 935045 
 334/21 
 
 593665 181295 
 818705 
 401070 
 
 101 323899 2-10814 
 
 878017 
 
 IOT 593665 3*92342 
 
 878017 
 
 101 813440 6-50789 
 
 878017 
 
 201916 
 
 964668 237248 
 435627 
 30 
 
 100 400325 2-51377 
 900013 
 
 471682 
 
 400325 071357 
 338142 
 31 
 
 691457 
 
 738498 047041 
 952959 
 325187 
 
 100 738498 5-47643 
 900013 
 
 100 016644 10-3907 
 900013 
 
 300338 
 
 638511 
 
 916657 
 860688 055969 
 329879 
 37 
 
 958473 341865 
 504722 
 45 
 
 463240 175271 
 397405 
 43 
 
 99 463240 2-90663 
 * * * 
 
 99 860688 7*25585 
 * * * 
 
 99 190604 15-5097 
 * * * 
 
 274. Any of the results of this problem may be verified by the aid 
 of the following relation, which subsists between each of them and the 
 corresponding result of last problem : 
 
 () 
 
 P*~ 
 
 _ j (TO) 
 
 275. When the assurance is uniform this expression becomes, 
 
 A.=^ -a , 
 
 ' " 
 
Ill 
 
 which is given by Mr. Milne, p. 169, for the value of an assurance in 
 terms of two adjoining annuities. 
 
 276. The remarks in (270) on the examples illustrative of Problem 
 XVII., are equally applicable to those just given. It is sufficient, 
 therefore, here to direct attention to them. 
 
 277. Some general remarks on the variable benefits discussed in 
 the last two problems have been reserved for this place. 
 
 278. (1.) The formulae deduced are equally applicable when the 
 benefits depend on the joint continuance of two or more lives. The 
 only change they require is that of x into x.y, &c. 
 
 279. (2.) If either of the benefits is not to be entered upon till 
 after the lapse of n years, its present value will be found, as in the 
 case of uniform benefits, by multiplying the value of the whole life 
 benefit, at the age then attained, by v n p xn (218.) But a temporary 
 increasing benefit requires different treatment from that of a temporary 
 uniform benefit. And not only so, but the treatment will vary with 
 every variation in the law of increase. We shall confine our remarks 
 to the case of a uniformly increasing annuity, that is, one whose suc- 
 cessive payments are to be 1., 2. } 3., &c. Suppose that this 
 annuity is to cease after n payments have been made. If we deduct 
 from its value for the whole life, the value of a similar annuity deferred 
 n years, it is obvious that the annuity we leave is one of 1, 2, 3, .... 
 n, n, n y &c., to the end of life. We thus cut off, not all payments 
 after n years, but only the increase at and after the expiry of that 
 term, and there remains, therefore, to be deducted the value of a 
 uniform annuity of n pounds deferred n years. Hence we have for 
 the value of the temporary annuity in this case, 
 
 that of the similar annuity whose increase only is arrested being, 
 
 VrirftJU* 
 
 280. (3.) By combining, in the way of addition and subtraction, the 
 values of the various benefits with which we have been occupied, we 
 obtain the values of a great variety of benefits, both increasing and 
 decreasing. A very few examples of the simplest kind may suffice for 
 illustration. The present value of an annuity of 5., increasing by 
 10 shillings per annum, will be 4*\a x + \i x ; that of an annuity of 100, 
 105, 110, &c., pounds, will be 950^+5^; that of an annuity of 100, 
 95, 90, &c., pounds, to be extinguished in 20 years, will be 
 
 a t+w ) -5 P,-*>p rfo (*, +!10 + 20 I+ao )] (279) ; 
 
112 
 
 which after a few obvious transformations is changed into, 
 
 that of an annuity of I 2 , 2 2 , 3 2 , &c., pounds, will be i x 
 and so on. 
 
 281. SCHOLIUM. If we denote by P^ the present value of an 
 annuity upon (x), whose successive payments are to be the terms of 
 the polygonal numbers, of the denomination m, then will the following 
 equation always subsist, 
 
 from which, by a continuous process, the values of P^ for each value 
 of x may be successively deduced. And from these, by a process 
 similar to those of Problems XVII. & XVIII., the values of an annuity 
 for each age, whose payments are to be the pyramidal numbers of the 
 denomination m, may likewise be derived. But such speculations 
 being matters of curiosity rather than utility, we do not farther insist 
 upon them. 
 
 PROBLEM XIX. 
 
 282. To construct a table of the present value of a sick allowance 
 of 1. per week, to cease at a specified age, and commencing at each 
 younger age. 
 
 283. The data requisite for the solution of this problem are, the 
 average quantity of sickness experienced during each year of existence 
 by each of the lives that enter upon that year, and, as in previous 
 problems, the probability of a life at each age surviving a year. De- 
 note the first, expressed in weeks, for age #, by s f ; and the second, 
 as usual, by p x . Denote also the present value of the benefit by S x . 
 
 284. If we suppose that the sickness of each year is uniformly 
 distributed over that year, then, interest also accruing uniformly be- 
 tween the periods of its conversion into principal, it will follow that 
 the value, at the commencement of the year, of the payments to be 
 made on account of sickness in the course of it, will be the same as if 
 the whole were made at the middle of the year. Hence, the present 
 value of one pound to be received half a year hence being 1 -=- (1 + \r), 
 (49.50), which call w, the value of the benefit in respect of the first 
 year will be ws x ; and the value in respect of the subsequent years 
 being vp x S x+v (193), we have for the whole value of the benefit, 
 
 s, 
 
 or, in logarithms, 
 
113 
 
 285. This formula gives the following type of operation : 
 
 B^C.* 
 
 B 7 C. 
 
 F, 
 
 log.S, + l 
 lg-% 
 
 S, + i 
 
 / 
 
 ar.co.(B C) 
 
 s r 
 
 5 
 
 log.S, + 1 
 log.v/^ 
 
 s~i 
 
 B-C 
 *. 
 
 A 
 B 
 
 C 
 D 
 E 
 
 F 
 
 A 
 B 
 
 C 
 D 
 E 
 
 F 
 
 |F l+ A 
 l=log.^ r S, + 1 
 
 |F 1 + A 
 
 t=log.qp,S, + 1 
 
 }og.ws x 
 B-C 
 
 T^ar.co^B-C)] 
 
 \og.ws x 
 
 JT, [B-c] 
 
 |C + D+E 
 t=log.S, 
 
 rC + D + E 
 l=log.S, 
 
 286. For the initial value, S x+l vanishes when x is the age next 
 younger than that at which the benefit is to cease ; so that for this 
 
 age we have, 
 
 log.S,=tog.tp*,. 
 
 287. It will be observed, that for the application of this formula 
 two subsidiary series are requisite, namely log.ws^ and log.t^ r . The 
 latter series for the Carlisle Table, (interest 3 per cent.,) we have al- 
 ready formed, and have had occasion very frequently to use, but the 
 former has not hitherto come under our notice. Friendly Society 
 data usually comprise both a sickness and a mortality table, and the 
 latter is in no case identical with the Carlisle Table. If we select 
 existing data then, we shall have to form both the foregoing series 
 before we can apply our formula to use. This is a simple matter. 
 
 288. We shall employ for our examples the Useful Knowledge 
 Society's data, as contained in Mr. AnselFs " Treatise on Friendly 
 Societies." The values of s x for each age are found at p. 70, and of 
 p x , also for each age, at p. 118, of the work cited. The series wanted 
 then, are the logarithms of s x) each increased by the constant logarithm 
 993534=log.(l-r-l'015)=log,^, and the logarithms of p xt each in- 
 creased by the constant logarithm 987163 =log.#. 
 
 289. We are now prepared for the examples which follow : 
 
 * In computing the values of this benefit it can hardly ever be requisite to 
 use the form of operation typified for the case B ^ C. Such a relation of the 
 quantities occupying the lines so marked, can rarely have place. It has been 
 thought well, nevertheless, not to depart from the customary form of the type. 
 
114 
 
 Present Values of Sick Allowances. 
 
 1. per wee 
 
 69 997253 9-93695 
 961305 
 
 c till age 70. 
 
 64 532979 
 967772 
 
 i. per week till age 60. 
 
 59 473685 2-97636 
 971311 
 
 958558 
 
 500/51 
 
 444996 
 
 931002 0-027556 
 314998 
 29 
 
 641699 859052 
 915278 
 46 
 
 438827 0-006169 
 304091 
 34 
 
 68 246029 17-6200 
 962894 
 
 63 557023 36-0597 
 968/58 
 
 58 742952 5-53289 
 972082 
 
 208923 
 
 525781 
 
 715034 
 
 866913 342010 
 504859 
 
 7 
 
 594616 931165 
 979228 
 59 
 
 406163 -308871 
 482317 
 
 48 
 
 67 S 23 ' 5385 
 
 62 573903 37-4889 
 969648 
 
 57 888528 7-73620 
 9/2828 
 
 336079 
 
 543551 
 
 861356 
 
 805511 530568 
 642696 
 53 
 
 551162 992389 
 034398 
 81 
 
 3/5551 485805 
 608585 
 4 
 
 66 448260 28-0711 
 965552 
 
 61 585641 38-5160 
 970521 
 
 56 984140 9-64140 
 973548 
 
 413812 
 
 556162 
 
 957688 
 
 747270 666542 
 751264 
 35 
 
 510994 045168 
 082581 
 63 
 
 346873 610815 
 705976 
 12 
 
 65 498569 31-5188 
 966673 
 
 60 593638 39-2318 
 971311 
 
 55 052861 11-2944 
 974237 
 
 465242 
 
 564949 
 
 027098 
 
 692591 772651 
 840344 
 44 
 
 473685 091264 
 125049 
 60 
 
 319870 707228 
 785020 
 24 
 
 64 532979 34*1176 
 
 59 598794 39'73 
 
 * * * 
 
 54 104914 12-7325 
 
 * * * 
 
 290. The values of the first of these benefits, for each age, are given 
 by Mr. Ansell, at p. 133, and may be compared with those here 
 determined. 
 
 291. The formula of (284) by a slight transformation becomes, 
 
 (w s 
 --* 
 
 and this gives another mode of construction, which may be used either 
 alone, or to verify the results of that just exemplified. 
 
 292. The type corresponding to this formula is as follows : 
 
115 
 
 EI ZA 
 
 E l7 A. 
 
 E i 
 
 lS-S, + i 
 
 s *+i 
 
 E i 
 
 lo S- s , + i 
 
 s ,+l 
 
 A 
 
 log.. 
 v p x 
 
 /Tj^ A \ 
 
 A 
 
 l s %'p 
 
 Ej-A 
 
 B 
 
 \og.vp s 
 
 
 B 
 
 log.vp x 
 
 
 C 
 
 Ej A 
 
 
 C 
 
 N 
 
 
 D 
 
 T 1 [ar.co.(B 1 A)] 
 
 
 D 
 
 jTaLEi-A] 
 
 
 E 
 
 l=log.S, 
 
 g. 
 
 E 
 
 rA+B-l-C + D 
 
 Ulog.S, 
 
 8. 
 
 fl O 
 
 An additional series here required is log.. - for each age. This is 
 
 w 
 equal to log.^-fcolog./^+log.-, the last logarithm being constant, 
 
 and equal to log. w log.v= '006371. This series will therefore be 
 easily formed. 
 
 293. We give an example of the application of this formula : 
 
 Present Value of Sick Allowance. 
 i. per week till age 70. 
 69 997253 9-93695 
 
 68 
 67 
 66 
 
 65 
 
 64 
 
 * 
 
 969697 
 961305 
 314998 
 39 
 
 0-027556 
 
 17*6209 
 342010 
 
 23'5385 
 530568 
 
 28-0711 
 666542 
 
 31-5188 
 772652 
 
 34-1176 
 
 * 
 
 246029 
 
 904019 
 962894 
 504859 
 7 
 371779 
 
 841211 
 964300 
 642696 
 53 
 
 448260 
 
 781718 
 965552 
 751264 
 35 
 
 498569 
 
 725917 
 966673 
 840344 
 45 
 
 532979 
 
 * 
 
116 
 
 294. It will appear, on comparison of this operation with the first 
 example of (289), that the tabular arguments in both are the same. 
 The reason of this is obvious. In the one case the argument is 
 
 nj o 
 
 log.vp g S x+l log.ws x , and in the other log.S, r+1 log.- . 
 
 =log.S, +1 - 
 
 295. This operation, when the requisite subsidiary series have been 
 formed, is on the whole rather simpler than the other ; and being 
 more assimilated to previous operations it ought, perhaps, on that 
 account to be preferred. Neither operation possesses any analytical 
 advantage over the other. 
 
 296. SCHOLIUM. Existing tables of the values of sick allowances 
 have no doubt been computed by one or other of the formulae here 
 deduced for the purpose, applied by means of the common logarithmic 
 tables. But the only source of distinct information on the subject 
 with which the author is acquainted, is a paper by Mr. Woollgar, in 
 the concluding number of the " Savings Bank Circular," (September, 
 1847,) which contains a detailed example of the actual computation. 
 Mr. Woollgar there remarks, in a note, " Mr. AnselFs is the only 
 work that professes to give the method of calculation ; and, although 
 that author devotes six pages to introductory considerations, he breaks 
 off just at the point where the actual process ought to be explained." 
 
CHAPTER VIL 
 
 OF COMMUTATION TABLES. 
 
 296. TABLES of the kind called Commutation Tables differ in a most 
 important particular from those with the construction of which we 
 have hitherto been occupied. They afford, by inspection, no informa- 
 tion whatsoever as to the value of any one benefit. They nevertheless 
 furnish us with ample and most readily available means of determining 
 the value of any benefit that may be named. We defer for the pre- 
 sent any general remarks we may have to offer on the subject of these 
 tables, and proceed at once to describe their form, give a summary of 
 their leading properties, and explain the most commodious methods of 
 constructing them. 
 
 297. A single life commutation table consists of two sets of columns, 
 called respectively, from the purposes to which they are applied, an- 
 nuity and assurance columns. The columns are designated by letters, 
 arbitrarily chosen, placed at the top, and there is in each a value cor- 
 responding to each year of age. The letters designating the annuity 
 columns are D, N, S, and those designating the assurance columns are 
 C, M, R. The value in any column corresponding to a specified age, 
 is denoted by the letter which designates the column, with the age 
 suffixed. Thus, D 20 , N 20 , &c., denote respectively the values in co- 
 lumns D, N, &c., corresponding and placed opposite to, age 20 ; and 
 D x , N,, &c., denote the same for age x. 
 
 298. Columns D and C form the foundations of the table. From 
 these the remaining columns, in virtue of one of the simplest conceiv- 
 able relations, are readily derived. The composition of those funda- 
 mental columns is as follows : 
 
 299. If l x denote, as before, the tabular number who attain x years 
 of age, the number who die in the following year will be denoted by 
 Jje ^t+i ; and if v* denote the present value of 1. to be received in 
 x years (50), we have, 
 
 .=/,; andC.=+>(/.-/, +1 ). 
 
118 
 
 300. The following relations then subsist : 
 
 the series in each case extending to the end of life. 
 301. From these the following are obvious: 
 
 B.=M,+E^ I 
 
 302. The following also are easily deduced : 
 
 ST\ 
 ~ == u 
 
 303. Let N-N- 
 
 304. Then are 
 
 M.-M. + .=M - . 
 
 1+ >,+._, 
 
 305. Moreover, 
 
 306. By means of the last three formulae the assurance columns 
 may be dispensed with, or their place may be supplied when they are 
 not tabulated, as is sometimes the case, especially with respect to 
 joint lives. The expressions also serve as formulae of verification in 
 the construction of the tables. 
 
 307. An intimate acquaintance with the foregoing formulae is highly 
 advantageous for the simplification of expressions that occur in the 
 treatment of complex problems. A larger collection will be found in 
 the "Companion to the Almanac" for 1840 and 1842. 
 
 308. The meanings above given to S x|n and R^ differ somewhat 
 from those assigned to the same symbols by Professor De Morgan, the 
 author of the papers just cited. The meanings here adopted are 
 equally convenient with those of the learned professor, and we venture 
 to think rather more in accordance with analogy. Mr. Farr has 
 made the same departure from previous usage, as is here done, in his 
 Appendix to the Sixth Report of the Registrar-General, p. 329, folio 
 edition, and p. 595^ octavo edition. 
 
119 
 
 309. It has been already shown (185, 218) that the present value 
 of an endowment of 1. on (x), that is, of 1. to be received in n years 
 if (#) be then alive, is v n .l x+n -r-l x ', and that the present value of an 
 endowment assurance of \. on (#), that is, of 1. to be received at 
 the end of n years, if (x) shall have died in the preceding year, is 
 vn (l x +n-\~lx+n)~*~L' Multiplying numerator and denominator of 
 each of these expressions by v x , we have, 
 
 Present value of endowment 
 
 endowment "I 
 assurance J 
 
 =~j)> ( 2 ") 
 
 4--l~(r+J V> 
 
 D 
 
 310. From these two expressions, that for the present value of any 
 benefit that may be proposed is very easily derived, by means of the 
 relations in (300) . . . (305). We content ourselves, however, with 
 here giving, in a tabular form, the expressions for what may be called 
 the simple benefits ; premising that, as the denominator for a present 
 value is in every case D^, it is sufficient to tabulate the numerators. 
 
 311. 
 
 The present value of an 
 
 Endowment of 1. on (x) 
 Endowment Assurance of) 
 
 1. on(ff) ) 
 
 Annuity of 1. on (x) . . 
 Assurance of 1. on (x) 
 Increasing Annuity of 
 
 Increasing Assurance of 
 
 To be re- 
 ceived in n 
 years, is 
 
 For 
 
 For the first 
 n years, is 
 
 After the 
 
 first 
 years, is 
 
 For n years after 
 k years, is 
 
 Uniform 
 after n 
 years, is 
 
 PROBLEM XX. 
 
 312. To construct column D for single lives. 
 
 313. Since, (299), T> x =v*l x , 
 and D =* 
 
 whence, 
 
 D,=D, +1 . 
 
 
 or, in logarithms, log. D, t =log.D, c+1 4-colog.^, 1 .. 
 
 314. From this expression the mode of construction is obvious. 
 The value for the oldest tabular age being first formed, the others are 
 
120 
 
 successively produced by the continual addition of the terms of the 
 series colog. vp x , the formation of which was explained in (106). 
 
 315. Or, a commencement may be made at the youngest age, and 
 the logarithmic values successively formed by the continual addition 
 of the terms of the series log.vp x ; for 
 
 or 
 
 316. Examples of both these modes of formation are here given. 
 Formation of Column D. One Life. 
 
 First M< 
 
 log.Z 104 000000 
 log.0 lw 664929 
 
 Jthod. 
 
 ii 667069 4645-89 
 014791 
 
 10 681860 4806-84 
 015050 
 
 Second 
 
 o 000000 
 914585 
 
 I 914585 
 9506(55 
 
 Method. 
 
 94 395361 
 862224 
 
 104 664929 -046231 
 489958 
 
 95 257585 
 871769 
 
 103 154887 '142852 
 234686 
 
 9 696910 4976-34 
 015704 
 
 2 865250 
 958011 
 
 96 129354 
 880708 
 
 102 389573 '245230 
 158965 
 
 8 712614 5I59'57 
 016674 
 
 3 823261 
 970364 
 
 4 793625 
 974506 
 
 5 768131 
 979362 
 
 97 010062 
 
 8/8018 
 
 98 888080 
 882428 
 
 101 548538 -353621 
 121983 
 
 7 729288 5361-53 
 018205 
 
 100 670521 -468297 
 099987 
 
 6 747493 559i'05 
 020638 
 
 5 768131 5863-15 
 025494 
 
 99 770508 
 900013 
 
 99 770508 -589533 
 117572 
 
 6 747493 
 981/95 
 
 100 670521 
 
 878017 
 101 548538 
 841035 
 
 98 888080 -772823 
 121982 
 
 4 793625 6217-63 
 029636 
 
 7 729288 
 983326 
 
 97 010062 1-02344 
 119292 
 
 3 823261 6656-73 
 041989 
 
 8 712614 
 984296 
 
 102 389573 
 765314 
 
 96 129354 1-34696 
 128231 
 
 2 865250 7332-46 
 049335 
 
 9 696910 
 984950 
 
 103 154887 
 510042 
 
 95 257585 1-80961 
 137776 
 
 i 914585 8214-57 
 085415 
 
 10 681860 
 985209 
 
 104 664929 
 
 94 395361 2-48520 
 * * * 
 
 o 000000 i oooo-o 
 
 * * * 
 
 ii 667069 
 * * 
 
 317. Verification is obtained by means of the fundamental formula 
 D .=&L. 
 
 log./ 1M =954243 
 log.0 100 =716278 
 
 log.D 10 o=670521 
 
 log.Z 94 =602060 
 log.?? 94 =793301 
 
 log.D 94 =39536l 
 
 log.fc =832317 
 log.t? 5 =935814 
 
 log.D 3 =768131 
 
 318. The column may also be constructed without the previous 
 formation of any other series than \og.p x -, thus, log.D a<+1 =log.D a .+ 
 l&Px + log- 17 - And log.0 being constant, it will be sufficient to write 
 it once for all at the bottom of a card, to be held in the hand over the 
 other two logarithms to be added, and moved down for each new 
 addition. If this method of construction be employed, it will be well 
 to use log.v to two places more than are to be finally retained, that is, 
 
121 
 
 Formation of Column D. 
 Third Method. 
 
 to two more than there are in log. p x , 
 annexing 5 also, in the first of those 
 places, to the initial logarithm, as in 
 (45). The two additional places being 
 struck off at the close, the values 
 formed will then be true to the nearest 
 figure in the last place retained, with- 
 out any alteration. An example of 
 the mode of formation now described 
 is given in the margin. The results, 
 when two figures are cut off, may be 
 compared with those already found. 
 
 319. For the method of verifying 
 these values, that described in (317) 
 having respect to the logarithms only, 
 reference is made to (330). 
 
 320. COROLLARY. When the natural numbers corresponding to 
 the foregoing logarithmic results have been taken out, the formation, 
 in numbers, of Columns N and S is easy. The operation, by means 
 of the formulae of (300), is as follows : 
 
 000000 5 
 927422 
 987162 78 
 
 5 
 
 6 
 
 7 
 8 
 
 9 
 * 
 
 768131 40 
 992199 
 987162 78 
 
 914585 28 
 963502 
 987162 78 
 
 747493 18 
 994633 
 987162 78 
 
 865250 06 
 970849 
 987162 78 
 
 729288 96 
 996163 
 987162 78 
 
 823261 84 
 983201 
 987162 78 
 
 712614 74 
 997133 
 987162 78 
 
 793625 62 
 987343 
 987162 78 
 
 696910 52 
 * 
 
 768131 40 
 
 Formation of Column 
 N in numbers. 
 
 103 -046231 
 142852 
 
 102 '189083 
 245230 
 
 Formation of Column 
 S in numbers. 
 
 103 -046231 
 
 189083 
 
 102 
 101 
 IOO 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 * 
 
 2 353i4 
 '4343^3 
 669627 
 787934 
 
 101 
 100 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 * 
 
 434313 
 353621 
 
 787934 
 '468297 
 
 i'45756i 
 1-256231 
 
 1-256231 
 589533 
 
 2-713792 
 1-845764 
 
 1-845764 
 
 772823 
 
 2-618587 
 1-02344 
 
 4'559556 
 
 2-618587 
 
 7-178143 
 3-64203 
 
 3-64203 
 1-34696 
 
 10*82017 
 4-98899 
 
 6-79860 
 22-60776 
 9-28380 
 31-89156 
 * 
 
 4-98899 
 1*80961 
 
 6-79860 
 2-48520 
 
 9-28380 
 
 * 
 
122 
 
 PROBLEM XXI. 
 
 321. To construct Column C. 
 
 322. Since, (299,) 
 
 and 
 
 Hence, 
 
 or, in logarithms, 
 
 log.C, = log.C, + 1 + colog.tfj^ j + A log. (p x+ j 1 ) . 
 
 323. This is the formula by which the required construction is to 
 be effected. The series colog. vp x occurred in one of the formulae of 
 last problem, and A log. (p x ~ l l) is one of the series contained in the 
 Elementary Table. 
 
 324. The initial value, which is that wherein x is the oldest tabular 
 age, is found from the formula G x =v x+l l x , to which the general for- 
 mula for C, is reduced by the vanishing in this case of l x+r There is 
 also a peculiarity in the logarithmic formula above given, when applied 
 to the next younger age. Here co\og.p x+ 1 = OC, and log. ( p x+ l ~ 1 l) 
 = OC. Hence for this age we have, 
 
 log.C.,. = log.C ^ j -f colog.v -f log. (p x ~ l 1) . 
 
 325. From this point the computation goes on in accordance with 
 the formula first deduced. An example is here given : 
 
 Formation of Column C. 
 
 One Life. 
 000000 
 
 log.0 105 
 
 104 
 colog.0 
 
 103 
 
 1 02 
 
 652091 
 652091 
 012837 
 301030 
 
 965958 
 489958 
 522879 
 
 978795 
 234686 
 778151 
 
 044884 
 
 092461 
 
 095235 
 
 101 991632 -098092 
 158965 
 
 853872 
 
 100 004469 -101034 
 
 100 004469 -101034 
 121983 
 
 890855 
 
 99 017307 -104066 
 099987 
 088941 
 
 98 206235 -160781 
 117572 
 020204 
 
 97 34401 1 -220806 
 121982 
 98/765 
 
 96 453758 -284287 
 
123 
 
 326. Verification is procured by reference to the fundamental ex- 
 pression for C, (299). Thus : 
 
 log.(/ 96 -/ 97 )=log.(23-18) =698970 
 log. v Q7 = 754789 
 log.C 96 = 453759 
 
 327. Comparison may also be made from time to time with Column 
 D, by means of the proper one of the formulae in (305). Thus : 
 
 Log.D 96 (316) = 0-129354 
 log.v = 1-987163 
 
 dD 96 = 1-30773 0-116517 
 097= 1-02344 
 
 C 96 = 0-28429 
 
 328. If none of the subsidiary series have been formed, each value 
 in this column must be constructed independently, although with 
 much greater risk of error, as no verification can then be had, except 
 by separate comparison of each value, till Column M is formed. 
 
 329. COROLLARY. From Column C, in numbers, Columns M and 
 R are formed agreeably to the formulae in (300). An example of the 
 formation of the first few terms in each of these columns is here given. 
 
 Formation of Columns M and R in numbers. 
 
 104 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 * 
 
 M. 
 
 044884 
 092461 
 
 ' J 37345 
 095235 
 
 232580 
 098092 
 
 330672 
 101034 
 
 431706 
 104066 
 
 535772 
 160781 
 
 696553 
 220806 
 
 '9 T 7359 
 
 284287 
 
 1*201646 
 409943 
 1-611589 
 603203 
 
 2-214792 
 869818 
 
 3-084610 
 * 
 
 104 
 
 103 
 
 IO2 
 
 IOI 
 
 IOO 
 
 99 
 
 R. 
 
 044884 
 137345 
 182229 
 232580 
 
 414809 
 330672 
 745481 
 
 '43 I 7 6 
 
 2-4095I2 
 917359 
 3-326871 
 1*201646 
 
 '439508 
 
124 
 
 330. These values may be verified by the formulae in (305), and to 
 avoid unnecessary trouble it will be well to make the requisite verifi- 
 cations before completion of the various columns. Thus, the numbers 
 to be added being set down in their places, they may be summed in 
 groups of ten or twenty, according to convenience ; and these sums, 
 which will of course be terms of the series in hand, being found to 
 stand the test of the formula of verification, sufficient presumption is 
 thereby afforded for the accuracy of the intermediate terms, if the 
 final additions shall be correctly made, which the terms already ascer- 
 tained furnish the means of readily determining. 
 
 331. The formula? of verification are applied as follows : 
 
 ^=9-28380 log.0-967726 
 \-v , 2-464284 
 
 (1 r)N 93 = '27040 1-432010 
 094=2-48520 
 
 893=31-8916 
 1 v 
 
 (1-0)893= '92888 
 ^=9*28380 
 
 1194=8-35492 
 
 log.l;503676 
 2-464284 
 
 , f-967960 
 
 PROBLEM XXII. 
 
 332. To construct Column N in logarithms. 
 
 333. Since, (301), N,=N, 
 
 Hence, (I), 
 
 334. This gives the following type of operation 
 
 D,>A. 
 
 TJar.co.^-A)] 
 
 rA+B + C 
 l=log.N, 
 
 D, 
 
 A) 
 
 log.N, 
 
 335. It will be noticed that in this operation, besides log.N^, we 
 form also log.a x . This, for age a?+l, being equal to log.N >r+1 
 
 * These two D's must be distinguished from each other. The one designates 
 the value occupying the line D in the type of operation, and the other a value in 
 the Column D of the Commutation Table. 
 
125 
 
 1 , it will be found, when the form first typified is used, (which 
 will never be requisite for more than two or three values,) in the line 
 B, and when the second form is used, opposite the line A, in the 
 adjoining column, being in the latter case the tabular argument, and 
 in the former its arithmetical complement. This operation, therefore, 
 and that of (207), will serve most effectually to check and verify each 
 other. Log.(l + J is also formed, agreeably to (209). 
 
 336. Since N z =D, s+1 + N,p 41 , it follows that, when x is the next to 
 the oldest tabular age, N. r = D, i;+1 , and hence, log.D corresponding to 
 the oldest tabular age is the initial value. 
 
 337. An example of this operation is here given : 
 
 Formation of Column N in logarithms. 
 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 664929 
 
 046231 
 
 0-489958 
 
 189083 
 112921 
 
 434312 
 089264 
 
 787932 
 225968 
 
 1-25623 
 
 99 099069 
 
 1-25623 
 328561 
 
 1-84576 
 378095 
 
 2-61858 
 408004 
 
 3-64202 
 431988 
 
 4-98897 
 
 * 
 
 154887 
 510042 
 611723 
 
 2/6652 
 
 389573 
 887079 
 361150 
 
 770508 
 495626 
 41 
 
 98 266175 
 
 888080 
 529919 
 67 
 
 637802 
 
 548538 
 347916 
 35 
 
 97 418066 
 010062 
 551277 
 3 
 
 896489 
 
 96 561342 
 
 670521 
 428505 
 43 
 
 129354 
 568593 
 64 
 
 099069 
 
 95 698011 
 
 * * 
 
 338. COROLLARY. From the logarithms of Column N determined 
 as above, those of Column S may by a like process be deduced. The 
 relation being S X =S^ +1 + N X , (301), we have, 
 
 log.S^log.N^T^log.S.^-log.NJ, (I.) ; 
 
 and for the initial value, when x is the next to the oldest tabular 
 age, log.S. r =log.N, B . It is unnecessary to give an example of this 
 operation. 
 
 PROBLEM XXIII. 
 
 339. To construct Column M in logarithms. 
 
 340. Since, (301), M,=M, +1 + C X 
 
 .-, log.M^log.C.+T^og.M.^-log.CJ ; 
 which gives the following type of operation : 
 
126 
 
 D,>A. 
 
 DI <A. 
 
 , 
 
 log.M, +1 
 
 ar.co.(Dj A) 
 
 D> 
 
 log-M I+1 
 
 M. + , 
 
 A 
 
 log-C. 
 
 A 
 
 log.C, 
 
 B 
 
 Dj A 
 
 
 B 
 
 1 
 
 
 C 
 
 T^ar.co.^-A)] 
 
 
 C 
 
 JT^D^A] 
 
 
 I) 
 
 rA-fB + C 
 l^logjf. 
 
 M, 
 
 D 
 
 rA + B + C 
 
 M, 
 
 341. For the initial value, we have, when x is the oldest tabular 
 age, log.M x =log.Cv 
 
 342. An example of this operation is subjoined : 
 
 Formation of Column M in logarithms. 
 104 
 
 103 
 
 IO2 
 
 IOI 
 
 ioo 635188 -431706 
 
 343. Verification may be obtained by one of the formulae in (305) 
 as follows : 
 
 log.N 96 =0-561342 
 log. (l-t;)= 2-464284 
 1-025626 
 log.D 97 =0-010062 1-015564 
 
 (1-952474 
 T 2 [l -015564] = { 5 
 
 652091 
 
 044884 
 0-313867 
 
 137345 
 159017 
 
 232579 
 374939 
 
 330671 
 514927 
 
 ioo 635188 
 
 431706 
 617880 
 
 535772 
 522744 
 
 696554 
 498944 
 
 917360 
 
 * 
 
 965958 
 686133 
 485721 
 
 017308 
 711607 
 65 
 
 137812 
 
 978795 
 387766 
 10 
 
 99 728980 
 
 206236 
 636685 
 34 
 
 366571 
 
 98 842955 
 
 991632 
 527737 
 27 
 
 344011 
 618496 
 33 
 
 519396 
 
 004469 
 630698 
 21 
 
 97 962540 
 * * 
 
127 
 
 345. We now enter on the subject of Commutation Tables for two 
 joint lives. 
 
 In these tables the pairs of lives subsisting at the successive ages, 
 take the place of the single lives so subsisting in the tables which 
 formed the subject of the last four problems. And as these last named 
 tables answer all the questions that can be proposed in regard to a 
 single life, so the tables for the joint lives answer all the questions that 
 can arise in regard to a pair of lives. The formulae of (300. ... 311) 
 will be adapted to a pair of lives, (x) and (y), by simply writing in them 
 x.y for x. Other formulae for joint lives will be found in Professor De 
 Morgan's paper on the subject, in the " Companion to the Almanac " 
 for 1842, and we may advert to one or two of them presently. 
 
 346. In the single life Commutation Table D x is v x l x . But it might 
 be v k l x> where k is any function of x which increases by a unit, when x 
 increases by a unit. Such a function is x+t, where t is any number 
 independent of x. And the sole reason for selecting from all the 
 suitable values of A: that in which /=0, and consequently k=x, is that 
 thereby a degree of symmetry is imparted to the value of D, which it 
 would not otherwise possess. These remarks apply likewise, to a 
 certain extent, to a joint life Commutation Table. That is to say, all 
 the questions that concern two lives may be answered by means of 
 tables in which the exponent of the power of v which appears in D xy 
 is any function of either x or y or both, which increases by a unit 
 when both x and y increase by a unit. But here certain advantages, 
 by no means unimportant, are gained by a judicious selection of the 
 function to be employed. If we use a function of either x or y only, a 
 consequence is that in the case of benefits depending upon an assigned 
 order of failure or survivorship amongst two lives, two formulae become 
 necessary, the one for use when the failing life, and the other for use when 
 the surviving life, is the elder of the two given lives. This needless and 
 troublesome multiplication of formulae is avoided by choosing for the 
 exponent of v a function in which us and y are symmetrically involved, 
 and which fulfils the condition already specified. Such a function is 
 \(x+y], and when this is adopted, we have ^ xy =v^ (x+9 ^ x , y > 
 
 347. This form of the joint life Commutation Table was proposed 
 by Professor De Morgan, first in the " Philosophical Magazine," 
 and, subsequently in his paper already cited, in the " Companion to the 
 Almanac" for 1842; and in the only tables of the kind since published, 
 viz. those of Mr. Farr, in the Sixth Report of the Registrar- General, 
 this form has been adopted. In most of the previously published 
 tables, including these of Mr. Davies, Mr. Jones, &c., the exponent of 
 v in D xy is the greater of the two,, x y y. It is to be hoped that all 
 future tables will be of the form shown to be the more advantageous. 
 
128 
 
 In what follows, however, the construction of both forms will be ex- 
 plained. 
 
 PROBLEM XXIV. 
 
 348. To construct Column D for two joint lives. 
 
 349. (1.) When log-vp xy for all ages has been formed (124). 
 
 and D xy =v a l 
 
 therefore, in both cases, 
 
 D. 
 
 whence, 
 
 or, in logarithms, 
 
 or = 
 
 x+l.y+l 
 
 + !.,+ ! 
 
 350. By this formula, commencing at the youngest age, the column 
 may be constructed in logarithms, and verification will be obtained by 
 means of the proper one of the fundamental expressions, 
 
 log.D, y = \og.l x -f log./y -|- log.v*, 
 or, \og.T> xy =\og.l x +log.l y + log.v^ x+ y\ 
 
 351. The following are examples : 
 
 Formation of Column D. Two Lives. 
 
 First Method. 
 I. DAVIES'S FORM. D xy =:v x l f ... 
 
 log.Z 000000 
 logJ 000000 
 log.vo 000000 
 
 log.? 000000 
 log./, 927422 
 log.t? 987163 
 
 x y=2. 
 
 log./ 000000 
 log./ 2 890924 
 log.02 974326 
 
 o.o 000000 i oooooooo 
 842007 
 
 i.o 914585 82145700 
 
 878087 
 
 2.0 865250 73324600 
 885433 
 
 i.i 842007 69503500 
 914167 
 
 2.1 792672 62040000 
 921513 
 
 3.1 750683 56322600 
 933866 
 
 2.2 756174 57039200 
 928860 
 
 3.2 714185 51782800 
 941213 ' 
 
 4.2 684549 48367000 
 945355 
 
 3.3 685034 48421000 
 953565 
 
 4.3 655398 45227000 
 957707 
 
 5.3 629904 42648500 
 962563 
 
 4.4 638599 43511000 
 961849 
 
 5.4 613105 41030300 
 966705 
 
 6.4 592467 39126200 
 969148 
 
 5.5 600448 39 8 5i8oo 
 971561 
 
 6.5 579810 38002300 
 973994 
 
 7.5 561605 36442300 
 975525 
 
 6.6 579009 37325800 
 976428 
 
 7.6 553804 3 5793 5 oo 
 
 8.6 537130 344453 oo 
 
 7.7 548437 353539 
 979489 
 
 8.7 531763 34022200 
 980459 
 
 9.7 516059 32814000 
 981114 
 
 8.8 527926 337 2 3 
 981429 
 
 9 .8 512222 3^5400 
 
 10.8 497173 31417600 
 982342 
 
 o.o 509355 32311400 
 982739 
 
 10.9 494306 31210900 
 982997 
 
 11.9 479515 30165800 
 982851 
 
 10.10 492094 31052300 
 
 * * * 
 
 1 1. 10 477303 30012600 
 
 * * * 
 
 12. 10 462366 28997900 
 
 * * * 
 
129 
 II. DE MORGAN'S FORM. D 
 
 x y=i.* 
 
 log.7 000000 
 log.fj 927422 
 log.* 993581 
 
 logJ 
 log./ 
 log-* 
 
 2.0 
 4 .2 
 
 5-3 
 6.4 
 
 7-5 
 8.6 
 
 9-7 
 10.8 
 
 11.9 
 
 i.o 921003 83368700 
 
 878087 
 
 2.1 799090 62963700 
 921513 
 
 3.2 720603 52553600 
 941213 
 
 4.3 661816 45900300 
 957707 
 
 5.4 619523 41641200 
 966705 
 
 6.5 586228 38568100 
 973994 
 
 7.6 560222 36326300 
 977959 
 
 8.7 538181 34528800 
 980459 
 
 0.8 518640 33009600 
 982084 
 
 10.9 500724 31675500 
 982997 
 
 1 1. 10 483721 30459400 12. K 
 * * * # 
 
 And examples of the meth 
 
 DAVIES'S FORM. 
 log./ 12 =806180 
 log.J 10 =810233 
 log.t? 12 =845953 
 
 log.Di2.io=4 62366 
 
 890924 
 
 ,o m 755,4300 
 
 763520 58012300 
 933866 
 
 4.2 697386 49818000 
 945355 
 
 5.3 642741 43928000 
 
 6.4 605304 40299900 
 969138 
 
 7.5 574442 37535500 
 975525 
 
 8.6 549967 35478700 
 978929 
 
 9.7 528896 33798400 
 
 10.8 510010 32360100 
 982342 
 
 1 1. o 492352 31070800 
 982851 
 
 12.10 475203 29867800 
 
 log. TO 000000 
 log.Z 3 861773 
 f log.0 980744 
 
 3.0 842517 
 897786 
 
 4.1 740303 
 938008 
 
 5.2 678311 
 950211 
 
 6.3 628522 
 964996 
 
 7.4 593518 
 970669 
 
 8.5 564187 
 976495 
 
 9.6 540682 
 979584 
 
 10.7 520266 
 981371 
 
 1 1. 8 501637 
 982197 
 
 12.9 483834 
 982774 
 
 13.10 466608 
 
 69585200 
 54992400 
 47677200 
 42513000 
 39220900 
 36659500 
 34728200 
 33 T 33400 
 31742200 
 30467300 
 
 29282500 
 * 
 
 DE MORGAN'S FORM. 
 log.fo =806180 
 log.7 w =810233 
 log.0 11 =858791 
 
 log.D 12 .io=4 75204 
 
 352. (2.) If \og.vp (xy) have not been formed, we may avail ourselves 
 of \og.vp x , (101), or co\og.vp x , (106), and construct the table by either 
 of the following formulae : 
 
 lo S- D *+ 1 . y + 1 = kg-I 
 
 commencing in the first case with the younger ages, and so proceeding 
 to the older, and in the second case beginning with the older, and pro- 
 ceeding thence to the younger. 
 
 353. Some examples of this method of construction follow : 
 
 * When x y=0 the two forms of the table coincide. It is therefore un- 
 necessary to repeat the construction for difference 0. 
 
130 
 
 Formation of Column D. Two Lives. 
 Second Method. 
 
 I. Commencing w 
 DAVIES'S FORM. 
 
 logJ 000000 
 log.fc, 890924 
 log.i? 2 974326 
 
 ith the Younger Ages. 
 DB MORGAN'S FORM. 
 
 log.Zo 000000 
 log./ 2 890924 
 log.v 987163 
 
 II. Commencing \ 
 DAVIES'S FORM. 
 
 log.* 000000 
 log./ 102 698970 
 log.w 104 664929 
 
 nth the Older Ages. 
 DE MORGAN'S FORM. 
 
 Iog./io4 000000 
 Iog./io2 698970 
 log.0 103 677766 
 
 2.0 865250 
 914584 
 970849 
 
 3.1 750683 
 950665 
 983201 
 
 2.0 878087 
 914584 
 970849 
 
 104.102 363899 
 489958 
 146128 
 
 104.102 376736 
 489958 
 146128 
 
 3.1 763520 
 950665 
 983201 
 
 103.101 999985 
 234687 
 109145 
 
 103.101 012822 
 234687 
 109145 
 
 4.2 684549 
 958012 
 987343 
 
 4.2 697386 
 958012 
 987343 
 
 102.100 343817 
 158965 
 087150 
 
 102.100 356654 
 158965 
 087150 
 
 5.3 629904 
 970364 
 992199 
 
 5.3 642741 
 970364 
 992199 
 
 6.4 605304 
 974505 
 994633 
 
 101.99 589932 
 121982 
 104735 
 
 101.99 602769 
 121982 
 104735 
 
 6.4 592467 
 974505 
 994633 
 
 7.5 561605 
 979362 
 996163 
 
 109.98 816649 
 099987 
 109145 
 
 99.97 025781 
 117572 
 106455 
 
 100.98 829486 
 099986 
 109145 
 
 7.5 574442 
 979362 
 996163 
 
 99.97 038617 
 117573 
 106455 
 
 8.6 537130 
 * * 
 
 8.6 549967 
 * * 
 
 98.96 249808 
 * * 
 
 98.96 262645 
 
 * * 
 
 354. (3.) If neither log.vp^^ nor log.^ have been formed, the 
 table may still be constructed by either of the following formula : 
 
 -,+ 1 .9 + 1 - 
 tog-D,., = D *+ 1 . y + 1 + colog.jp, + oology + log. (l + r). 
 
 355. It will be proper, in using this method, to employ the con- 
 stant logarithm, v, or 1 + r, as the case may be, to two places more 
 than are to be finally retained (43). The first of these additional 
 places also being, in the initial value, increased by 5, the mere cutting 
 off of two figures at the close will leave the values formed true to the 
 nearest unit in the last place retained (45). 
 
 356. The following are examples of this method of construction : 
 
 Formation of Column D. Two Lives. 
 Third Method. 
 
 I. Commencing wii 
 DAVIES'S FORM. 
 
 log./o 0000005 
 log.Z 2 890924 
 log.v 2 974325 56 
 
 h the Younger Ages. 
 DE MORGAN'S FORM. 
 
 log.Zo 0000005 
 log.7 2 890924 
 log.0 98/16278 
 
 II. Commencing ^ 
 DAVIES'S FORM. 
 
 log./iM 0000005 
 Iog./i 02 698970 
 log.?; 104 664929 12 
 
 ith the Older Ages. 
 DE MORGAN'S FORM. 
 
 Iog./,o4 0000005 
 logJ, M 698970 
 log.v 103 677766 34 
 
 2-0 86525006 
 927422 
 970849 
 987162 78 
 
 2.0 87808728 
 927422 
 970849 
 987162 78 
 
 104.102 363899 62 
 4/7121 
 146128 
 012837 22 
 
 104.102 376736 84 
 477121 
 146128 
 012837 22 
 
 3.1 750683 84 
 
 3.1 763521 06 
 
 103.101 99998584 
 
 103.101 012823 06 
 
4-2 
 
 5-3 
 
 6. 4 
 
 750683 84 
 963502 
 983201 
 987162 78 
 
 3-i 763521 06 
 963502 
 983201 
 987162 78 
 
 131 
 
 103.101 
 
 684549 62 
 970849 
 987343 
 987162 78 
 
 4 .2 697386 84 
 970849 
 987343 
 987162 78 
 
 IO2.IOO 
 
 629904 40 
 983201 
 992199 
 987162 78 
 
 5-3 64274162 
 983201 
 992199 
 987162 78 
 
 101.99 
 
 592467 18 
 * 
 
 6.4 605304 40 
 * * 
 
 100.99 
 * 
 
 999985 84 
 221849 
 109145 
 012837 22 
 34381/06 
 146128 
 087150 
 012837 22 
 589932 28 
 109145 
 104735 
 012837 22 
 816649 50 
 * 
 
 I03-IOI 012823 06 
 221849 
 109145 
 012837 22 
 
 102.100 356654 28 
 146128 
 087150 
 012837 22 
 
 iox.99 602/69 50 
 109145 
 104735 
 012837 22 
 
 100.98 829486 72 
 
 357. If tables have to be constructed for several values of x y, it 
 will obviously be well worth while, as a preliminary step, to form log. 
 vp x . The saving of labour effected by the use of this series is visible 
 on comparison of the second and third methods of construction just 
 explained. The trouble of writing down the constant logarithm for 
 each value may, however, be spared by setting it down once for all at 
 the bottom of a card, so that, by being held over the result of each 
 addition, it may easily be taken account of in the next. 
 
 358. (4.) A fourth method of constructing Column D, which, 
 when tables for all values of x y are to be formed, is preferable to 
 every other, will now be explained. It will be perceived that it is in 
 entire analogy with the method employed in (108) and (124) for the 
 formation of the series log.p( ) and log.vjt?^ ). 
 
 359. For Davies's form of the table we have, (x >y), 
 
 D=** and 
 
 n i -^ 
 Or, in logarithms, 
 
 ,* + 1 
 
 360. It thus appears that by the continuous addition to certain 
 initial values of the successive terms of the tabulated series colog.p y , 
 log. D xy for every combination of ages will be formed. The working 
 may be arranged as in the following specimen, except that in practice 
 it will be proper to leave a blank space between every two columns to 
 receive the natural numbers corresponding to the several logarithmic 
 values. It will be noticed also, that here, as in (110), &c., the terms 
 corresponding to each value of xy, and consequently belonging to 
 the same Column D, are found ranged in horizontal lines. Moreover, 
 the terms of colog.j y being only 103 in number, they may be put on 
 cards similar to that shown in fig. 1, (102), and so the whole labour 
 of the operation will be reduced to that of a series of additions of two 
 
132 
 
 lines. The horizontal initial series, and other horizontal series, for 
 verification at suitable intervals, may be formed by some one or other 
 of the methods already described, and ought to be inserted in their 
 places before commencing the principal operation. 
 
 361. Formation of Column D. Two Lives. 
 
 Fourth Method. 
 I. DAVIES'S FORM. D =#*/ . 
 
 x-y 
 o 
 
 i 
 
 2 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 9 
 
 10 
 
 y 
 104 
 
 103 
 1 02 
 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 
 664929 
 477121 
 
 y 
 
 103 
 
 IO2 
 IOI 
 IOO 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 * 
 
 632008 
 221849 
 
 y 
 
 IO2 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 92 
 
 * 
 
 088544 
 146128 
 
 y 
 
 IOI 
 IOO 
 
 99 
 98 
 
 97 
 
 96 
 
 95 
 94 
 
 93 
 92 
 
 #=IOI 
 
 393637 
 109145 
 
 y 
 
 IOO 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 92 
 
 90 
 
 a?=ioo 
 624764 
 087150 
 
 ; 
 
 98 
 
 97 
 96 
 
 95 
 94 
 93 
 92 
 
 90 
 
 89 
 * 
 
 811901 
 104735 
 
 * 
 * 
 * 
 * 
 * 
 
 * 
 
 * 
 
 142050 
 221849 
 
 853857 
 146128 
 
 234672 
 109145 
 
 502782 
 087150 
 
 711914 
 104735 
 
 916636 
 109145 
 
 363899 
 146128 
 
 999985 
 109145 
 
 343817 
 087150 
 
 589932 
 104735 
 
 816649 
 109145 
 
 025781 
 106455 
 
 510027 
 109145 
 
 109130 
 087150 
 
 430967 
 104735 
 
 694667 
 109145 
 
 925794 
 106455 
 
 132236 
 115393 
 
 619172 
 087150 
 
 196280 
 104735 
 
 535702 
 109145 
 
 803812 
 106455 
 
 032249 
 115393 
 
 247629 
 124939 
 
 706322 
 104735 
 
 301015 
 109145 
 
 644847 
 106455 
 
 910267 
 115393 
 
 147642 
 124939 
 
 372568 
 130334 
 
 811057 
 109145 
 
 410160 
 106455 
 
 751302 
 115393 
 
 025660 
 124939 
 
 150599 
 130334 
 
 272581 
 130334 
 
 502902 
 142667 
 
 920202 
 106455 
 
 516615 
 115393 
 
 866695 
 124939 
 
 402915 
 142667 
 
 645569 
 146128 
 
 026657 
 115393 
 
 632008 
 124939 
 
 991634 
 130334 
 
 280933 
 142667 
 
 545582 
 146128 
 
 791697 
 131099 
 
 142050 
 124939 
 
 756947! 
 130334 
 
 121968 
 142667 
 
 423600 
 146128 
 
 691710 
 131099 
 
 922796 
 105390 
 
 266989 
 
 887281 
 
 * 
 
 264635 
 * 
 
 569728 
 
 822809 
 
 028186 
 * 
 
 362. Again, for De Morgan's form we have, 
 
 Hence, in logarithms, 
 
 362. Colog.\/t;.jt? y is the same as ^log.(H-r) + colog./? y , and the 
 mode of its formation for each age was described in (107). Hence, 
 colog./> y in the formula of (359) being replaced by colog. ^/v.p y , a like 
 operation to that last exhibited will produce the logarithms of ~D xy 
 according to De Morgan's form. The remarks in (360) also are equally 
 applicable here, and the initial values in both cases are the same, 
 since, when x=y, v x =v* (x+y \ The following is a specimen of the type 
 of operation: 
 
363. 
 
 133 
 II. DE MORGAN'S FORM. D,. =1 
 
 xy 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 6 
 
 7 
 8 
 
 9 
 
 10 
 
 * 
 
 9 
 104 
 
 2 
 
 1 02 
 
 TOI 
 100 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 
 #=104 
 664929 
 483540 
 
 rl 
 103 
 
 1 02 
 
 101 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 . 
 
 94 
 93 
 
 ^=103! 
 632008] 
 
 228267 
 
 L 
 
 IOI 
 IOO 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 
 92 
 
 # = 102 
 
 088544 
 152547 
 
 y 
 
 IOI 
 IOO 
 
 99 
 
 98 
 
 97 
 96 
 
 95 
 94 
 
 93 
 92 
 
 I 1 
 
 #=ioi 
 393637 
 115563 
 
 y 
 
 IOO 
 
 99 
 98 
 
 97 
 96 
 
 95 
 94 
 93 
 92 
 
 9 1 
 90 
 
 #=100 
 
 624764 
 093569 
 
 y 
 
 99 
 
 98 
 
 97 
 
 96 
 
 95 
 94 
 93 
 92 
 
 9 1 
 90 
 
 S S 
 
 #=99 
 811901 
 111154 
 
 * 
 * 
 * 
 * 
 
 * 
 * 
 
 * 
 
 * 
 
 * 
 * 
 
 148469 
 228267 
 376736 
 152547 
 
 860275 
 152547 
 
 241091 
 115563 
 
 509200 
 093569 
 
 718333 
 111154 
 
 923055 
 115563 
 
 012822 
 115563 
 
 356654 
 093569 
 
 602769 
 111154 
 
 829487 
 115563 
 
 038618 
 112874 
 
 529283 
 115563 
 
 128385 
 093569 
 
 450223 
 111154 
 
 713923 
 115563 
 
 945050 
 112874 
 
 151492 
 121811 
 
 644846 
 093569 
 
 221954 
 111154 
 
 561377 
 115563 
 
 829486 
 112874 
 
 057924 
 121811 
 
 273303 
 131358 
 
 738415 
 111154 
 
 333108 
 115563 
 
 676940 
 112874 
 
 942360 
 121811 
 
 179735 
 131358 
 
 404661 
 136753 
 
 849569 
 115563 
 
 448671 
 112874 
 
 789814 
 121811 
 
 064171 
 131358 
 
 311093 
 136753 
 
 541414 
 149085 
 
 965132 
 1128/4 
 
 561545 
 121811 
 
 911625 
 131358 
 
 195529 
 136753 
 
 447846 
 149085 
 
 690499 
 152547 
 
 078006 
 121811 
 
 683356 
 131358 
 
 042983 
 136753 
 
 332282 
 149085 
 
 596931 
 152547 
 
 843046 
 137518 
 
 199817 
 131358 
 
 814715 
 136/53 
 
 179736 
 149085 
 
 481367 
 152547 
 
 749478 
 137518 
 
 980564 
 111808 
 
 331175 
 
 951467 
 
 328821 
 
 633914 
 
 886996 
 * 
 
 092372 
 
 364. If in either of the two modes of construction last explained it 
 be preferred to commence the computation at the other end of the 
 several series, that is, with the terms belonging to the younger lives, 
 and proceeding thence to those belonging to the older, it is obvious 
 that the series to be added will be the arithmetical complements of the 
 series employed when the computation is conducted as above directed. 
 This order of procedure will be found in various respects less conve- 
 nient than the other. 
 
 364. COROLLARY. The construction of the Columns N and S in 
 numbers, readily follows, as in (320), when that of D has been effected. 
 This is shown, as regards Column N, in the following examples : 
 
 Formation of Column N. Two Lives. 
 
 In Numbers. 
 I. DAVIES'S FORM. 
 
 # y=o. 
 103.103 '04623 
 .42856 
 
 # y=i. 
 103.102 '13869 
 71426 
 
 # y=2. 
 I03.IOI -23115 
 '99997 
 
 102.102 '47479 
 1-22615 
 
 102. IOI '85295 
 
 1*71661 
 
 IO2'IOO 1-23112 
 2*20707 
 
 101. 101 1-70094 
 2'47535 
 
 lonoo 2-56956 
 3-18260 
 
 101*99 3'438i9 
 
 3-88985 
 
 100.100 4-17629 
 
 4-21467 
 
 100.99 5-75216 
 
 100.98 7-32804 
 6-55615 
 
 99-99 8-39096 
 6-48486 
 
 99.98 10-90343 
 8-25346 
 
 99.97 13-88419 
 io'6n6 
 
 98.98 14-87582 
 * * 
 
 98.97 19-15689 
 
 * * 
 
 98-96 24*4958 
 * * 
 
134 
 
 II. DE MORGAN'S FORM. 
 
 x y=i. 
 
 103.102 
 
 IO2.IOI 
 
 IOI.IOO 
 
 IOO.99 
 
 99.98 
 
 98.97 
 
 14076 
 
 72490 
 
 '86566 
 
 I'742I7 
 
 2-60783 
 
 3*22998 
 
 5-8378I 
 
 5-22797 
 
 11-06578 
 
 8-37636 
 
 19-44214 
 
 103.101 
 
 IO2.IOO 
 
 100.98 
 
 99-97 
 
 98.96 
 
 / = 2. 
 
 23809 
 1*02996 
 
 1*26805 
 
 2-27328 
 
 IO3.IOO 
 102.99 
 101.98 
 100.97 
 99.96 
 
 98.95 
 ^, 
 
 33829 
 I*34396 
 
 I-68225 
 2-81983 
 
 4-50208 
 5-I75I5 
 
 3-54133 
 4-00654 
 
 7*54787 
 6 -75285 
 
 9-67723 
 8-8II50 
 
 14-30072 
 IO-9300 
 
 25-2307 
 
 * 
 
 18-48873 
 I4'I740 
 
 32-6687 
 
 * 
 
 PROBLEM XXV. 
 
 365. To construct Column N in logarithms. 
 
 366. This is effected in precisely the same way as the construction 
 of the corresponding column in the table for single lives. And the 
 type of operation in Problem XXII. will serve equally for the present 
 problem if for x there we write x.y, and also, of course, x+ l.y+ 1 
 for #+ 1. The remarks following the type in question are also appli- 
 cable here. A few examples follow : 
 
 367. Formation of Column N. Two Lives. 
 
 In Logarithms. 
 I. DAVIES'S FORM. 
 
 x y=o. 
 103.103 664929 -046231 
 
 xy=i. 
 103.102 142050 1-38692 
 
 xy=2. 
 103.101 363899 -231153 
 
 632008 0-967079 
 032921 
 011570 
 
 8538570711807 
 288193 
 
 788875 
 
 999985 0-636086 
 363914 
 726401 
 
 102.102 676499 '474787 
 
 102-101 930925 -852952 
 
 I02.ioo090300 1-23112 
 
 088544 412045 
 587955 
 554189 
 
 234672 303747 
 696253 
 478934 
 
 343817 253517 
 746483 
 446029 
 
 loi.ioi 230688 1-70093 
 
 101. 100 409859 2-56956 
 
 101.99 536329 3*43818 
 
 393637 162949 
 837051 
 390102 
 
 502782 092923 
 907077 
 349972 
 
 589932 053603 
 946397 
 328658 
 
 100.100 620790 4-17628 
 
 100.99 759831 5-75216 
 
 100.98 864987 7-32804 
 
 624764 003974 
 996026 
 303022 
 
 711914 047917 
 325640 
 9 
 
 816649 048338 
 325851 
 20 
 
 99.99 923812 8-39096 
 
 99.98 037563 10-9034 
 
 99.97 142520 13-8842 
 
 811901 111911 
 360574 
 6 
 
 916636 120927 
 365674 
 15 
 
 025781 116739 
 363288 
 22 
 
 98.98 172481 14-8758 
 * # * 
 
 98.97 282325 19-1569 
 * * * 
 
 98.96 389091 24*4958 
 * * * 
 
135 
 
 II. DE MORGAN'S FORM. 
 
 103. IO2 148469 '140756 
 
 x y=2. 
 103.101 376736 -238087 
 
 e 9=3. 
 
 I03.ioo529283 -338285 
 
 860275 0-71180.6 
 
 288194 
 
 788874 
 
 012822 0-636086 
 363914 
 726401 
 
 128385 0-599102 
 400898 
 696606 
 
 I02.IOI 937343 -865652 
 
 102.100 103137 1-26805 
 
 102.99 225889 1-68224 
 
 241091 303748 
 696252 
 478935 
 
 356654 253517 
 746483 
 446029 
 
 450223 224334 
 775666 
 427523 
 
 loi.ioo 416278 2-60782 
 
 101.99 549166 3-54133 
 
 101.98 653412 4-50207 
 
 509200 092922 
 907078 
 349971 
 
 602769 053603 
 946397 
 328658 
 
 713923 060511 
 939489 
 332339 
 
 100.99 766249 5-83780 
 
 718333 047917 
 325640 
 9 
 
 100.98 877824 7-54787 
 
 829487 048337 
 325851 
 20 
 
 100.97 985751 9*67723 
 945050 040701 
 321857 
 1 
 
 99.98 043982 11.0658 
 
 99.97 155358 14.3007 
 
 99.96 266908 18-4888 
 
 923055 120927 
 365674 
 15 
 
 038618 116740 
 363288 
 23 
 
 151492 115416 
 362552 
 9 
 
 98.97 288744 19.4422 
 * * * 
 
 98.96 401929 25.2307 
 * * * 
 
 98.95 514053 32-6628 
 * * * 
 
 368. The logarithmic annuities here formed, (335), may be com- 
 pared with those for the same ages in (207). 
 
 369. The same relations subsisting amongst the various columns in 
 the joint life table as in the single life table, it is obvious that by 
 means of (305), all the questions in regard to one or two lives that 
 either table enables us to solve, may be solved by the use of Columns 
 D and N alone, with the exception of those relating to benefits that 
 vary in their amount from year to year, and these can be treated by 
 means of Column S. If, therefore, the last mentioned class of benefits 
 be excluded, as when pairs of lives are concerned they do not often 
 require to be considered, we may say that all the purposes of a joint 
 life table can be served by the Columns D and N alone. This pro- 
 perty is taken advantage of to reduce the extent of the joint life tables, 
 a complete set of which, especially if constructed for different rates of 
 interest, would otherwise be of stupendous magnitude, on account of 
 the great number of combinations that two lives of different ages, 
 from one year to about a hundred years, admit of. It is not usual 
 therefore, in the case of such tables, to form more than Columns D 
 and N. The construction of the assurance columns indeed would be 
 a matter of great labour, as Column C for two lives does not admit of 
 a mode of formation corresponding to that adopted in the case of the 
 
136 
 
 same column for the single life table.* The formulae for the use of 
 the joint life tables are accordingly adapted to these circumstances, 
 and are always given in terms of D and N only. 
 
 370. Before leaving the subject of commutation tables for two lives 
 we offer one or two remarks. 
 
 (1.) Since Column D, and, in the case of single lives, Column C 
 also, are formed without the intervention of our Tables I. and II., it is 
 unnecessary, and perhaps also undesirable, to restrict ourselves in 
 their formation to logarithms of six places. The use of seven-figure 
 logarithms, if the elementary values have been formed to that extent, 
 will be just as easy, and will give one figure more in the natural 
 numbers. 
 
 371. (2.) If it be preferred to form commutation tables without the 
 aid of logarithms, which seems to have been the course pursued by 
 Mr. Jones in the construction of his valuable and extensive set of 
 tables of this kind, methods analogous to those that have been ex- 
 plained for the logarithmic formation, may be devised, by which the 
 operation will be simplified, and rendered very much more easy and 
 certain than if recourse were had to- the direct multiplication of the 
 factors composing the several values to be determined. On this 
 branch of the subject, however, it is not necessary here to enter. 
 
 372. (3.) If from these tables we omit the element v, which has 
 reference to the interest of money, they become adapted to questions 
 having reference to the mean duration of life and probabilities of life 
 and survivorship. Thus we have for the 
 
 curtate mean duration 1 N, 
 
 of one life.. .. JD/ ( 
 curtate mean duration "I N 
 
 of two lives.. .. ' ( 
 
 complete mean dura-1 K N ,-i.g-i + KN,_^ + N, y _ J + N,J 
 
 tion of two lives. . J V xy 
 
 probability that (x) UP^+N^-N^.J 
 
 will die before (y) J D^ 
 
 By means of these formula, taking the values from Mr. Jones's Table 
 XL., the results of the operations in (146) and (181) may be verified. 
 
 373. Mr. Farr, in the Appendix to the Sixth Report of the Regis- 
 trar-General, has recently devised and exemplified various ingenious 
 
 * If nevertheless it be chosen to encounter the labour of forming the assurance 
 as well as the annuity columns, it will be best to commence with Column M, 
 using the formula (305), 
 
 and applying it by means of either Table II. or the common tables. 
 
137 
 
 modifications of these tables, by means of which a number of most 
 important questions relating to the progress of population and cognate 
 subjects, are determined with facility. But we cannot farther refer to 
 them in this place. 
 
 374. (4.) The present value of a survivorship assurance (249,) is 
 assigned with facility by means of the commutation table for two 
 joint lives. Thus we have : 
 
 By these formulae the values deduced in (255) may be verified. And 
 it will here be observed that, in accordance with (346), when the 
 tables employed are of Davies's form, two formulae are requisite to 
 include every case,* while one suffices when the tables have the form 
 suggested by Mr. De Morgan. 
 
 375. On the subject of commutation tables for three joint lives it 
 is not necessary to enter. If ever the construction of such should be 
 undertaken, a guide to the formation of Column D will be found 
 in the explanation and examples that have been given (120 . . 123) 
 (126 . . 128) of the formation of the series log./?^^), between which 
 and the formation in question the same analogy subsists as we have seen 
 to subsist between the operations in (124) and those in (361 . . 363). 
 One thing is of the utmost importance to be attended to. ^ xyz being 
 equal to v k l xyj ,, the value most advantageous to be chosen for k is 
 J (x + y + z) . By the employment of any other value, as remarked by 
 Professor De Morgan, (whose suggestion this isf,) the use of the tables 
 " would become intolerable, from the large number of cases into which 
 different orders of survivorship would require formulae to be divided." 
 
 * Perhaps we ought to say three, since neither of the two includes the case 
 x=y. 
 
 t In the " Philosophical Magazine." 
 
CHAPTER VIII. 
 
 OF THE HISTORY OF THE COMMUTATION METHOD. 
 
 376. WE close with a few general remarks on the history and com- 
 parative merits of the Commutation Method, also called " Barrett's 
 Method/' of computation. 
 
 377. The first account we have of this method is in a paper by Mr. 
 Baily, read before the Royal Society in 1812, but not inserted in the 
 " Philosophical Transactions/' although, as Professor De Morgan re- 
 marks, it would have done honour to any one of the many eminent 
 inquirers whose writings on the subject of life contingencies are scat- 
 tered over the volumes of that work. It was given to the world, some- 
 what amplified, by Mr. Baily, in 1813, as an Appendix to his " Doc- 
 trine of Life Annuities and Assurances." Mr. Baily states that the 
 method was the invention of Mr. George Barrett, of Petworth, in 
 Sussex, who had devoted no less than twenty-five years of his life to 
 the construction of a most voluminous set of tables in accordance with 
 it. For want of encouragement those tables were never published, and 
 the manuscripts are understood to be now in the possession of Mr. 
 Babbage. Mr. Baily gives a full account of the nature and properties 
 of the tables, and applies a small selection from them, which accom- 
 panies his paper, to the solution of various problems. 
 
 378. Mr. Barrett's tables differ in several particulars as to form, 
 from those with which we are now familiar. (1.) They contain only 
 three columns, A, B, and C, corresponding, with the variations to be 
 immediately noted, to our D, N, and S, respectively. (2.) A. x is not 
 v x l x , but v*- w l x , where w is the oldest tabular age. (3.) B^ is not 
 
 A x+1 H-A, c+2 -f , but A^-f- A.,. + l + (4.) They commence with 
 
 the oldest age instead of the youngest. (5.) Mr. Barrett does not use 
 the decimal point, but denotes its position by prefixing to each tabular 
 value the index of its logarithm. 
 
 379. It is to Mr. Griffith Davies that we owe the alterations by 
 which the tables have assumed their present and now well known form. 
 None of these changes, it will be noticed, are essential the principle 
 of the method remains intact. With the exception of problems relat- 
 ing to assurances on single lives, the solution of which is much facili- 
 tated by the addition of the assurance columns, every problem that 
 
139 
 
 admits of solution by a table of Mr. Davies's form, admits of solution 
 with the same facility by a table of Mr. Barrett's form. The propriety 
 of, or at least the necessity for, some of the modifications introduced 
 by Mr. Davies may be questioned;* but it is not to be denied that the 
 method owes much to him. Besides the alterations in the structure 
 of the tables to which reference has been made, and the addition of 
 the assurance columns, he effected considerable improvement in the 
 notation, and gave many new formula?. 
 
 380. But while cheerfully according to Mr. Davies the merit to 
 which he is unquestionably entitled as an improver and extender of 
 this method, his claims must not be suffered to over-ride or supersede 
 those of his predecessor, Mr. Barrett, so as to warrant calling him the 
 inventor of the method. Something like this is, however, in effect, 
 not unfrequently done. Thus, Mr. David Jones, while throughout his 
 work he calls the method under consideration " Davies' s Method," 
 makes mention of Mr. Barrett's name only twice, and that in a casual 
 manner, (p. 117,) and has, at p. 116, these words : " Mr. Griffith 
 Davies was the FIRST who computed tables of annuities on the above 
 plan," the plan referred to being a portion of the formation of Co- 
 lumns D and N ; and Mr. Jenkin Jones, in his work founded on the 
 Actuaries' Data, says, p. 51 : "The D and N system was FIRST em- 
 ployed by Mr. Griffith Davies, the actuary of the Guardian Assurance 
 Company, and the formula used by him are somewhat analogous to 
 those originally pointed out by the late Mr. Barrett." 
 
 381. It seems barely possible that justification of such statements 
 may be sought in the circumstance that, to the eye, Mr. Davies' s for- 
 mulae are different from those of Mr. Barrett. Since, however, this 
 difference vanishes when the formula? of either author are translated 
 into the notation of the other, it is obvious that, having given Mr. 
 Davies credit for his improvements in this respect, there remain no 
 grounds whatsoever whereon to found a claim for him to the origina- 
 tion of the method, or to be considered the first who used it. 
 
 382. The work of Mr. David Jones, just cited, which was published 
 under the superintendence of the Useful Knowledge Society, contains 
 a most extensive and valuable set of tables of Mr. Davies' s form, for 
 both single and joint lives. Mr. Jones gives also several new formulae, 
 and effects some farther improvements in the notation. 
 
 383. But it is from two papers by Mr. De Morgan, in the " Com- 
 panion to the Almanac" for 1840 and 1842, that the most systematic 
 and complete view, by far, of this method is to be obtained. Along 
 
 * We notice that Mr. Farr, in his recently published tables, reverts to Mr. 
 Barrett's relation between Columns N and D. 
 
140 
 
 with a collection of formulae, unexampled in extent and variety, these 
 papers contains such a development of the principles of their applica- 
 tion, that no one, with a little previous knowledge of the subject,* 
 needs be at a loss in regard to the treatment of any case that may be 
 presented to him. In Mr. De Morgan's hands also, the notation of 
 the method has received several most important extensions. On the 
 whole, the possession of the papers in question is indispensable on the 
 part of every one who would obtain a complete mastery of the subject. 
 
 384. It is proper, before going farther, to notice an attempt that 
 has recently been made to deprive Mr. Barrett of the merit of having 
 originated the method of computation which bears his name. The 
 author of this attempt is Mr. E. J. Farren, who, in an ingenious work, 
 entitled "Historical Essay on the Rise and Early Progress of the 
 Doctrine of Life Contingencies in England,"t has, at pp. 64, 65, 
 the following passage : 
 
 " It may be here allowable to incidentally notice that the original 
 framing of the Column system of calculation referred to, has generally 
 been attributed to the late Mr. Barrett, whose method was published 
 in the form of an Appendix to Mr. Baily's Treatise on Annuities, as 
 re-issued in 1813. That we are principally indebted to this source 
 for the prevalent adoption of the system is undoubted, but a prior 
 embodiment of it will be found in the Treatise of the late Mr. W. 
 Morgan, published as early as the year 1779. The reader generally 
 conversant with such matters, will at once admit the correctness of 
 this assertion, by consulting p. 64 of Mr. Morgan's Treatise, in which 
 will be found a table of the following form, which, though primarily 
 offered merely for the purposes of rectification, has nearly all the 
 inherent qualities of modern arrangements. 
 
 First Table of the Values of Single Lives. 
 
 
 Age. 
 
 Values of Lives. 
 
 Values of 1. 
 payable, &c. 
 
 Sums. 
 
 
 
 9 2 
 
 "OOOOOO 
 
 0000332 
 
 ooooooo 
 
 
 
 91 
 
 480769 
 711908 
 
 "0000690 
 '0001436 
 
 0000332 
 '0001022 
 
 
 
 
 1-097377 
 
 '0002244 
 
 0002458 
 
 
 
 
 i'5i 2 53i 
 
 0003106 
 
 0004698 
 
 
 
 8? 
 
 1-93271 
 
 0004038 
 
 "0007804 
 
 
 
 86 
 
 2-16983 
 
 '0005460 
 
 '0011842 
 
 
 
 &c. 
 
 &c. 
 
 &C. 
 
 &C. 
 
 
 * Perhaps the writer may be allowed here to refer to a series of papers of his 
 own, on the subject of this method, printed in the " Mechanics' Magazine " for 
 November and December, 1842. They were intended as an introduction to Mr. 
 De Morgan's papers, which contain, avowedly, almost nothing of demonstration. 
 
 t Smith, Elder, and Co., 1844. 12mo. 
 
141 
 
 " Thus, the nature of the third column being represented, (modern 
 notation [Milne's] ) by n av n , that of the fourth may be considered as 
 S( n+l av n+1 ) -, consequently the value of complete and temporary life 
 annuities, &c., would be nearly as readily determinable by these means 
 as if the columns had represented the values of the numerators 
 n av n and S ("?*), as at present in use. (Columns D and N.) 
 
 "In p. 70 the tabular values are adapted to the common denomi- 
 nator 1000, &c. ; and in p. 74, the system is extended to Joint Lives ; 
 Mr. Morgan concluding the chapter by observing, ' as by these methods 
 the calculations are rendered pleasant as well as expeditious, I hope 
 that ere long some person will undertake them, choosing for his guide 
 the Northampton Table of Observations, which, perhaps, is better 
 fitted for common use than any other/ Mr. Barrett appears to have 
 pursued this recommendation, and to have incidentally detected that 
 the system was capable of further extension and uses, though his 
 unnecessary adoption of the reversal of ages, clearly indicates Mr. 
 Morgan's Treatise to have been the suggestive source/' 
 
 385. The purport of the foregoing extract seems to be this : That 
 the " original framing of the column system of calculation/' which 
 has been hitherto attributed to Mr. Barrett, ought to be attributed to 
 Mr. Morgan, in whose treatise " a prior embodiment of it " is to be 
 found ; and that, although Mr. Barrett " appears to have incidentally 
 detected that the system was capable of further extension and uses/' 
 yet the arrangement of the ages adopted by him " clearly indicates 
 Mr. Morgan's treatise to have been the suggestive source." This 
 must be examined in detail. 
 
 386. First then, we admit that Mr. Morgan's values, when arranged 
 in columns as in the extract, possess, not nearly all, but the whole of 
 the " inherent qualities " of columns D and N in the modern tables. 
 Mr. Farren's account of the constitution of the column answering to D 
 is not quite correct. The value opposite the age of A, according to 
 Mr. Milne's notation, is not n av n , but n a n .v n , where n is the difference 
 between the age of A and one year. In our notation the value oppo- 
 site age x would be denoted by Pi x ..\v x ~ l ) which may be written 
 
 this again is qual to l x v x , the value acccording to the 
 
 modern form, multiplied by the constant factor \--l l v. Whence the 
 possession of the "qualities " in question is manifest. (309. 346). 
 
 387. But, secondly, although Mr. Morgan's columns possess the 
 properties of columns I) and N, this was unknown to Mr. Morgan. 
 He constructs them simply to prove the correctness of his annuity 
 values, deduced by another process. This he does by multiplying 
 
142 
 
 together the annuity value and the corresponding value in the adjoin- 
 ing column, and finding the result equal to the corresponding value 
 in the last column. He then, with what now we cannot but denomi- 
 nate singular improvidence, throws away his columns, and makes no 
 farther use of them. Now surely this affords but slender grounds for 
 claiming for Mr. Morgan the origination of the " column system of 
 calculation." Not only does he show himself unacquainted with the 
 principal and most valuable properties of his columns, but we have no 
 evidence that he arranged his materials in columns at all, except 
 merely for the purpose of exemplification. Indeed it is pretty certain 
 that he did not. The form they would naturally assume may be seen 
 in Price, vol. i., p. 215, and is very different from that in which 
 they appear in the type quoted by Mr. Farren. We may admit 
 therefore that Mr. Morgan's work contains an " embodiment " of the 
 materials of the column system ; but that it contains an "embodiment" 
 of the system itself we must take leave to deny. 
 
 388. Thirdly. Mr. Morgan must be supposed to have known better 
 than any one else what his rights in the matter were. Was his con- 
 duct then, when Mr. Baily's paper descriptive of the new method 
 became known to him, that of a man who felt that his ideas had been 
 appropriated without acknowledgment ? It was not. The idea of 
 claiming the method or its origination seems never to have presented 
 itself to his mind. He appears indeed to have failed to appreciate its 
 value, and to have done what in him lay to prevent its being given to 
 the public. This we have abundant evidence to prove. It has been 
 already mentioned that Mr. Baily^s paper, giving an account of Mr. 
 Barrett's method, although read before the Royal Society, was refused 
 a place in the "Philosophical Transactions." Now it is the known 
 practice of that learned body to refer to the Council, to be adjudicated 
 upon by them, all papers read before the Society ; and Mr. Baily's 
 paper was therefore, of course, subjected to this ordeal. We are 
 furthermore warranted in assuming, that in the exercise of their 
 function of adjudicators, the Council would be guided in a great 
 measure, if not solely, in reference to a particular paper, by the opinion 
 of those of their number who are most conversant with the subject 
 treated of. A list of the council for 1812, to whom it fell to adjudicate 
 on Mr. Baily's paper, is given by that gentleman in his Appendix ; 
 and the only names which can be recognised in it, as those of indi- 
 viduals possessing a competent acquaintance with the subject of life 
 contingencies, are, " William Morgan, Esq.," and " Dr. Thomas 
 Young, For. Sec." With Mr. Morgan and Dr. Young therefore must 
 rest, not only the share belonging to them as members of the Council, 
 
of the responsibility of rejecting Mr. Baily's paper, but also the entire 
 responsibility of having counselled its rejection. The supposition is 
 not to be entertained that in this proceeding these gentlemen were 
 actuated by unworthy or improper motives, and we therefore conclude 
 that they did not see, what has been since universally admitted, the 
 value of the method proposed for their approval. This view, as regards 
 Mr. Morgan, receives confirmation from the circumstance that, in the 
 second edition of his treatise, published in 1821, eight years after the 
 appearance of Mr. Baily's Appendix, he took no notice of Mr. Barrett's 
 method, but presented his columns in the same form as in the first 
 edition, and with no other change in the details than the substitution 
 of the new Northampton Table for the old. 
 
 389. Seeing then that, when this method was presented to him as 
 the production of another, Mr. Morgan made no claim to it ; that he 
 pronounced it unworthy of publication in the " Philosophical Transac- 
 tions ;" and that, when it had been eight years before the public, and 
 its merits to some extent recognised, he took no notice of it, although 
 he had an opportunity of doing so ; a claim to the origination of it, 
 advanced on his behalf now, cannot possibly be admitted. 
 
 390. But, fourthly, Mr. Farren says, that the inversion of the ages 
 in Mr. Barrett's Table, corresponding to their order in Mr. Morgan's 
 columns, " clearly indicates Mr. Morgan's treatise to have been the 
 suggestive source." The inverse order of the ages arises so naturally 
 from the manner of constructing the tables, that we think no one who 
 has attempted this operation for himself, will consider Mr. Barrett's 
 adoption of that order as at all decisive of the question of his having 
 borrowed the idea of his columns from Mr. Morgan. To have de- 
 tected the possession by Mr. Morgan's columns of the properties now 
 known to belong to Columns D and N, would indicate, on the part of 
 Mr. Barrett, the exercise of a degree of perspicacity of a very high 
 order, seeing that, with Mr. Barrett's exemplification of the principle 
 before them since 1813, no one is publicly* known to have made the 
 discovery before Mr. Farren. 
 
 391. The present case is strikingly illustrative pf what frequently 
 happens in the progress of discovery. Few great discoveries have 
 been made regarding which it has not afterwards been found that they 
 ought to have been made a great deal earlier. Previous investigators 
 are then seen to have approached the discovery so nearly, that our 
 
 * The writer claims to have noticed the identity of Mr. Morgan's columns 
 with Columns D and N before the publication of Mr. Farren's Essay. He would 
 hardly have claimed the credit of this now, however, had he not mentioned the 
 circumstance at the time to one gentleman at least, whom he could name if 
 necessary. 
 
144 
 
 wonder is often excited at their having missed it. It happened in this 
 case, however, as has not often happened in other cases, that the dis- 
 covery was perfected and promulgated during the life of the investi- 
 gator, whose writings are now said to have contained not merely a 
 germ of that discovery, but the discovery itself. And the silence of 
 that investigator, under these circumstances, must be held for ever to 
 operate as a bar to the setting up in his behalf of a claim to the dis- 
 covery, or at least to the according of such claim if it should be set up. 
 
 392. We have now a few remarks to offer on the comparative merits 
 of the two methods of computation, the old and the new, the former 
 as developed in Mr. Milne's treatise, and the latter as exemplified in 
 Mr. De Morgan's papers in the " Companion to the Almanac/' The 
 materiel of the old method consists of three columns, namely, a x , l x , 
 and v*, for every age, or the logarithms of the last two. By means of 
 these the value of any uniform benefit, whether temporary, deferred, 
 or for the whole of life, may be determined. In every case, however, 
 except that of an annuity for the whole of life, an arithmetical opera- 
 tion is necessary, and this operation is more or less complex according 
 to circumstances. The same results are obtainable, according to the 
 new method, by the use of two columns only, namely, J) x and N^., but 
 here in every case an arithmetical operation is necessary. This appa- 
 rently shows an advantage on the part of the old method ; but it is 
 more appearance than reality. It is not often that the value of a 
 whole life annuity is wanted as a final result, and therefore the com- 
 parative merits of the two methods must rest on the comparative faci- 
 lities of the operations requisite in the other cases. Exhibition of the 
 formulae for a few of the cases of most common occurrence will, we 
 think, go far to show with which method the advantage lies. 
 
 393. Present value of annuity on (a;), deferred n years. 
 
 Annual premium for the same (n-\-l payments. 
 
 N 
 
 x+n 
 
 N,T; 
 
 Present value of annuity to last n years. 
 
 Present value of assurance for life. 
 v (1 v)a x ; or 1 (1 v)(l +aj 
 
 IX, 
 
145 
 
 Annual premium for the same. 
 1 
 
 N 
 
 , 
 
 Present value of assurance deferred n years. 
 
 Annual premium for the same payable for life. 
 
 ^^,^-N 
 
 + 0* 
 
 Present value of assurance to last n years. 
 
 Annual premium for the same m payments. 
 
 1 -fftj - (1 -f)(,-trp, A+ J t>(N,_,-N, + ._,)-(N,-N t+ J 
 
 - L \r-l ^x+m-l 
 
 The same when the number of payments is n. 
 
 N.-N.+- 
 
 t / ~~ -XT TVT 
 
 394. In regard to these formulae we remark, first, that if applied 
 to more than one life two additional tabular entries, at least, will be 
 requisite in the first set for each additional life, while in the second 
 set no increase will be necessary. Secondly, in the commutation for- 
 mulae we have restricted ourselves to the use of Columns D and N 
 only. Had Column M been employed the effect would have been a 
 most material simplification in all the assurance formulae. Thirdly, 
 by the addition to the commutation table of Columns S and R, one or 
 both, we acquire a power of treating cases in which either the benefit 
 or the payment, or both, vary from year to year. The like power is 
 gained for the old method by the addition to the data of a table of 
 increasing annuities. But although the manner of constructing such 
 a table exemplified in (269) is simpler than any that could have been 
 heretofore adopted, it is much less simple than the method of forming 
 Column S or Column R, as shown in (320.329). The formulae for 
 the variable benefits being, with the additional data, analogous in both 
 methods to those for the corresponding uniform benefits, the examples 
 just given afford the means of forming a judgment as to the compara- 
 tive facilities of the two methods in their application to benefits which 
 increase or decrease from year to year. 
 
 u 
 
146 
 
 395. The considerations just stated, and comparison of the two sets 
 of formulae suffice, we think, to determine the question as to the relative 
 facilities of the two methods. In every instance the superiority belongs 
 to the new method Barrett's Method. Not only is the number of 
 quantities to be dealt with in each case fewer according to this method 
 than according to the other, but the amount of operation to be per- 
 formed upon those quantities is also less, and usually simpler in kind. 
 All this will be more clearly seen from the following example of the 
 arithmetical translation of the formulae in a single case. 
 
 396. Required the present value of an assurance of 1. for the 
 next ten years upon a life now aged 30. Carlisle, 3 per cent. 
 
 397. The formula by the old method is 
 
 and the operation is as follows : 
 
 ko (Jones, p. 290) log. 37054360 
 
 Jao ( ' ) colog. 4-2485669 
 
 v" ( p. 103) log- 1-8716278 
 
 t>>3cuo = '6693153 1-8256307 
 
 l-0>3<uo = -3306847 1-5194141 
 
 v (Jones, p. 103) 1*9871628 
 
 1-5065769 
 
 f>3o.io (as above) log. f-8256307 
 
 a 40 (Jones, p. 311) 1-2340720 
 
 >3o.io4o=ir47368 1-0597027 
 
 8-08326 log. 0-9075865 
 l-r=w(76) 2-4642841 
 (l-tf)(3o-fl>3o.io04o)='2354348 1-3718706 
 0(1 10 j03<uo)=-32io53i 
 
 0856 1 83= value required. 
 
 398. The formula by Barrett's method is, 
 
 M 30 -M 4 o 
 
 30 
 
 and the arithmetical operation is as follows : 
 
 M.TO = 932-6869 Jones, p. 291 
 ^40 = 733' 6 73 
 
 199-0139 log. 2-2988834 
 Dao = 2324-429 3-3663164 
 
 Value required=-o856i838 2^325670 
 
147 
 
 So that if the sum assured were 1000., the present value would be 
 85-618, or 85. 12s. 4d. 
 
 399. To appreciate fully the advantage which the new method has 
 obviously over the old, the operations must be followed out step by 
 step, the mere number of figures set down in the two cases furnishing 
 a very insufficient index to their comparative facilities. 
 
POSTSCRIPT. 
 
 400. WE remarked, (371,) that if it were preferred to construct com- 
 mutation tables without the aid of logarithms, methods might be 
 devised for this purpose much more easy and certain than that by 
 actual multiplication of the factors composing the values to be deter- 
 mined. On farther consideration of the methods in view when the 
 foregoing remark was made, certain of them appear to present facilities 
 quite sufficient to entitle them to a place here. We therefore add two 
 problems illustrative of two of the methods in question. 
 
 PROBLEM XXVI. 
 
 401. To construct Column D, for single lives. 
 Since, (305,) C x =vV x - 
 
 whence, (76,) D,,.: 
 
 Hence if Column C have been constructed, and a single value in Co- 
 lumn D be known, all the D's corresponding to younger ages will be 
 formed in succession, the operation for each value being the addition of 
 a C to the last formed D, and the multiplication of the sum by 1 + r, 
 (37.) 
 
 402. We give an example, in which the values formed are true to 
 the nearest figure in the last place. 
 
 Formation of Column D. One Life. 
 
 CUM '0448840 
 
 x 03 13465 
 
 Di04 '0462305 
 
 Clo3 924610 
 
 '1386915 
 
 DIM '1428522 
 
 2380871 
 
 x-03 7 T 426 
 
 D,02 -2452297 
 
 doi 980919 
 
 In Numbers. 
 
 3433216 
 X'03 102996 
 
 1010347 
 
 4546559 
 
 136397 
 
 4682956 
 
 1040657 
 
 5723613 
 171708 
 
 D 
 
 7503136 
 
 D98 
 
 D 97 
 
 7503136 
 225094 
 
 ;77282 
 
 728230 
 208066 
 
 9936296 
 
 298089 
 
 1-0234385 
 
 2842884 
 
 1-3077269 
 392318 
 
 1*3469587 
 
149 
 
 403. The initial value is found thus. When x is the oldest tabular 
 age, D x+l ~Q -, so that, in the case of the Carlisle Table, we have, 
 
 404). The operation is an exceedingly easy one, and it possesses 
 besides the following advantages. First, it gives us throughout nearly 
 the whole of Column D one figure more, and in some cases two figures 
 more, than there are in the C's employed in the construction. So 
 that if these last have been formed to seven places, (as they may be by 
 the use of seven-figure logarithms,) the values in Column D will con- 
 sist generally of eight figures at least. This is a degree of accuracy 
 unattainable by the logarithmic method of construction. Secondly, 
 when the method now under consideration is employed we obtain 
 more readily and at an earlier stage, verification of the values in the 
 course of being formed ; since, the operation being continuous, and 
 the values admitting also of independent formation, such formation of 
 a few D's, at suitable intervals, furnishes us with verifications of both 
 the D's and the C's, as the work proceeds. When the logarithmic 
 method is employed, we must wait for verification of the natural 
 numbers till both N and M are formed, or at least are in course of 
 formation, (330.) 
 
 PROBLEM XXVII. 
 
 405. To construct Column D for two joint lives. 
 
 406. When x is not less than y, we always have, in Mr. Davies's 
 form of the tables, (349,) 
 
 that is, all the D's of the joint life table, in which x is constant, are 
 multiples of the ~D X of the single life table ; and the formation of those 
 D's is consequently nothing else than the formation of a series of mul- 
 tiples of the quantity ~D X by all the values of l y in which y does not 
 exceed x. The operation moreover will be simplified, and also ren- 
 dered continuous, if, instead of forming those multiples independently 
 of each other, we form each by adding to that last formed, a multiple 
 of the fundamental quantity D , by the difference between the values 
 of l y that enter into these multiples, respectively. Thus, since 
 
 That is, we pass from D^ y to B a . y _ 1 by adding to the former the pro- 
 duct of D r by l y _ l l yj which is the number in the column of decre- 
 ments of the mortality table opposite age y 1. 
 
150 
 407. These principles may be applied as follows : 
 
 Formation of Columns D and N. Two Lives. 
 DAVIES'S FORM. 
 
 T) f and its multiples. 
 
 
 D 104 
 
 D 103 
 
 D 102 
 
 Dioi 
 
 i 
 
 046230504 
 
 14285225 
 
 24522971 
 
 35362124 
 
 2 
 
 092461008 
 
 28570450 
 
 49045942 
 
 70724248 
 
 3 
 
 138691512 
 
 42855675 
 
 73568913 
 
 1-06086372 
 
 4 
 
 184922016 
 
 57140900 
 
 98091884 
 
 1-41448496 
 
 5 
 
 231152520 
 
 71426125 
 
 1-22614855 
 
 1-76810620 
 
 6 
 
 277383024 
 
 857H350 
 
 1-47137826 
 
 2-12172744 
 
 7 
 
 323613528 
 
 99996575 
 
 1-71660797 
 
 2.47534868 
 
 8 
 
 369844032 
 
 1*14281800 
 
 1-96183768 
 
 2*82896992 
 
 9 
 
 416074536 
 
 1-28567025 
 
 2*20706739 
 
 3-18259116 
 
 10 '462305040 
 
 1-42852250 
 
 2-45229710 
 
 3-53621240 
 
 
 D *y' 
 
 
 
 y 
 
 #=104 
 
 #=103 
 
 #=102 
 
 a?=ioi 
 
 104 i 
 
 0462305 
 
 
 
 
 2 
 
 924610 
 
 
 
 
 103 3 
 
 1386915 
 
 4285568 
 
 
 
 2 
 
 924610 
 
 2857045 
 
 
 
 102 5 
 
 2 
 
 2311525 
 924610 
 
 7142613 
 2857045 
 
 1*2261486 
 4904594 
 
 
 101 7 
 
 '3236135 
 
 9999658 
 
 1*7166080 
 
 2-4753487 
 
 2 
 
 924610 
 
 2857045 
 
 4904594 
 
 7072425 
 
 loo 9 
 
 4160745 
 
 1-2856703 
 
 2*2070674 
 
 3-1825912 
 
 2 
 
 924610 
 
 2857045 
 
 4904594 
 
 7072425 
 
 99 ii 
 3 
 
 -5085355 
 1386915 
 
 I-57I3748 
 4285568 
 
 2-6975268 
 7356891 
 
 3-8898337 
 1-0608637 
 
 98 14 
 
 6472270 
 
 1-9999316 
 
 3-4332159 
 
 4-9506974 
 
 _4 
 
 1849220 
 
 5714090 
 
 9809188 
 
 1-4144850 
 
 97 18 
 
 8321490 
 
 2-5713406 
 
 4-4141347 
 
 6-3651824 
 
 5 
 
 2311525 
 
 7142613 
 
 1-2261486 
 
 1*7681062 
 
 96 23 
 
 1-0633015 
 
 3-2856019 
 
 5-6402833 
 
 8-1332886 
 
 * * * 
 
 * 
 
 * 
 
 * 
 
 
 
 
 
 * * * 
 
 * 
 
 * 
 
 * 
 
 80 953 
 
 44-057670 
 
 136-138195 
 
 233-703915 
 
 337*001042 
 
 TOO 
 
 4-623050 
 
 14-285225 
 
 24-522971 
 
 35*362124 
 
 20 
 
 924610 
 
 2*857045 
 
 4-904594 
 
 7-072425 
 
 8 
 
 369844 
 
 1-142818 
 
 1-961838 
 
 2*828970 
 
 79 1081 
 
 49-975I74 
 
 154-423283 
 
 265-093318 
 
 382-264561 
 
 IOO 
 
 30 
 
 4-623050 
 1-386915 
 
 14-285225 
 
 4-285568 
 
 24-522971 
 7-356891 
 
 35-362124 
 10-608637 
 
 2 
 
 092461 
 
 285705 
 
 490459 
 
 707242 
 
 78 1213 
 
 56-077600 
 
 173-279781 
 
 297-463639 
 
 428-942564 
 
 IOO 
 
 4-623050 
 
 14-285225 
 
 24-522971 
 
 35-362124 
 
 40 
 
 1-849220 
 
 5-714090 
 
 9-809188 
 
 14-144850 
 
 6 
 
 277383 
 
 857114 
 
 I-47I378 
 
 2-121727 
 
 77 1359 
 
 62-827253 
 
 194-136210 
 
 333-267I76 
 
 480-571265 
 
 * * * 
 
 * 
 
 * 
 
 * 
 
 * * * 
 
 * 
 
 * 
 
 * 
 
151 
 
 7274 
 500 
 
 5 
 
 336-28069 
 23*1 1525 
 23115 
 359-62709 
 27-73830 
 3-69844 
 09246 
 
 1039-1073 
 71-4261 
 
 "7 J 43 
 
 1783-8009 
 122-6149 
 1-2261 
 
 2572-2409 
 170*8106 
 1-7681 
 
 7779 
 8o 
 
 2 
 
 1111-2477 
 85-7114 
 11-4282 
 
 2857 
 
 1907-6419 
 147-1378 
 19*6184 
 49 5 
 2074-8886 
 245-2297 
 122*6149 
 
 7'35 6 9 
 2-2071 
 
 2750-8196 
 212-1727 
 28-2897 
 '7072 
 2991-9892 
 353-6212 
 176-8106 
 10-6086 
 3-1826 
 
 8 4 6l 
 1000 
 
 500 
 30 
 
 9 
 
 391-15629 
 46-23050 
 
 23*11525 
 
 1-38692 
 41607 
 
 1208-6730 
 142-8523 
 71-4261 
 4-2856 
 1-2856 
 
 10,000 
 
 462-30503 
 
 1428-5226 
 
 2452-2972 
 
 3536-2122 
 
 N, 
 
 x y=o. 
 103.103 -0462305 
 
 4285568 
 
 x 
 
 xy=i. 
 103.102 -1386915 
 7142613 
 
 f 
 37^=24. 
 
 103.79 44-05767 
 154-42328 
 
 x- 
 
 103.2 
 
 102.102 -4747873 
 1*2261486 
 
 102. 101 '8529528 
 1*7166080 
 
 102.78 198-48095 
 297-46364 
 
 IO2.I 
 
 loi.ioi 1*7009359 
 
 2-4753487 
 
 101.100 2*5695608 
 
 3-1825912 
 
 101.77 495'94459 
 480*57127 
 
 IOI.O 
 
 100.100 4*1762846 
 * * 
 
 100.99 5-752l5 2 
 
 * * 
 
 100.76 976*51586 
 
 * * 
 
 100.- 
 
 y=ioi 
 
 i 
 
 1111-2477 
 
 1447-5284 
 
 2074-8886 
 
 3522-4170 
 3536-2122 
 7058-6292 
 
 408. The example consists of three portions. The first contains 
 the first ten multiples of D^ for the successive values of x. These 
 are formed by addition, as in (42), and they are verified by the visible 
 correspondence of the tenth multiple with the first. 
 
 409. The second portion of the example shows the formation of 
 T) xy . Four columns are given, (to complete the table 105 will be 
 requisite,) in each of which x is constant, having the value placed at the 
 top. The values of y are in the first side column, and each is constant 
 in the horizontal line on which it is placed. The second side column 
 contains the values of l y and of l y _ l l y , by the addition of the last of 
 which to the former we pass to l y _ r Opposite each value in this 
 column are formed in the several principal columns the products, by 
 that value, of the single life D's belonging to those columns respectively; 
 the products corresponding to the values of / _ j I being taken from 
 the auxiliary table at the top, and those of the values of l y formed by 
 addition. 
 
 410. The third portion shows the formation of N^; in which it 
 will be observed that the D's, by the addition of which the N's are 
 formed, are found in the successive columns above, ranging diagonally 
 downwards. 
 
 411. Verification of the D's is obtained by forming independently, 
 by means of the small auxiliary tables, a few values at intervals in each 
 
152 
 
 column; and when, as in the Carlisle and most other tables, 1 Q is a 
 power of 10, a final and visible verification is had at the close, by the 
 correspondence of T) x with D x . 
 
 412. In practically applying this method it will be advisable to 
 carry on four or five columns at the same time, and to have the aux- 
 iliary tables belonging to them, side by side, on a separate slip of 
 paper, which may be moved down as the work proceeds. Also, the 
 multiples in the auxiliary tables, corresponding to the same multiplier, 
 being on the same horizontal line, and the multiples as abstracted 
 holding like relative positions, they may be advantageously taken out 
 in that order ; so that a single reference to the index columns will 
 suffice for as many multiples as there are columns in course of forma- 
 tion at one time.* 
 
 413. It is obvious that the facility of this method depends in a great 
 measure on the number of figures in the several values of l y _ l l y . In 
 the case of the Carlisle Table this number in only a single instance 
 exceeds three. If it were as many as five or six the labour of construc- 
 tion would in consequence be considerably increased. In such a case it 
 might be advisable to form the series of addends by logarithms. The 
 logarithmic function to be found for each column then would be, log. 
 D x (l y _ l l y ) =log.D_ r + log.(/ y _ 1 l y ) for the value of x in that co- 
 lumn, and each value of y that does not exceed it. And these might 
 be formed continuously by adding to the first value the differences of 
 lg-(^_i~ l y ) f r the successive values of y. So many examples have 
 been given of this mode of construction that we do not occupy space 
 by giving another here. 
 
 414. Mr. De Morgan's form of the joint life table does not admit of 
 the advantageous application of the methods just laid down. 
 
 * This method has been successfully applied to the completion of Mr. Jones's 
 Table XXVIL, which, as given by him, extends only to x y=60. 
 
 [TABLES. 
 
TABLES. 
 
TABLE I. 
 
 f x* -oof- 
 
 '(J + k *?(> Ol 
 
 Log 
 
 Log(l+tf) 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3-00 
 
 00 0434 
 
 ")43o 
 
 0436 
 
 0437 
 
 0438 
 
 0439 
 
 0440 
 
 0441 
 
 0442 
 
 0443 
 
 
 01 
 
 0444 
 
 0445 
 
 0446 
 
 0447 
 
 0448 
 
 0449 
 
 0450 
 
 0451 
 
 0452 
 
 0453 
 
 
 02 
 
 0455 
 
 0456 
 
 0457 
 
 0458 
 
 0459 
 
 0460 
 
 0461 
 
 0462 
 
 0463 
 
 0464 
 
 i 
 
 03 
 
 0465 
 
 0466 
 
 047 
 
 0468 
 
 0469 
 
 0470 
 
 0472 
 
 )473 
 
 0474 
 
 0475 
 
 
 04 
 
 0476 
 
 0477 
 
 0478 
 
 0479 
 
 0480 
 
 0481 
 
 0483 
 
 0484 
 
 0485 
 
 0486 
 
 
 05 
 
 0487 
 
 0488 
 
 0489 
 
 0490 
 
 0492 
 
 0493 
 
 0494 
 
 0495 
 
 0496 
 
 0497 
 
 
 06 
 
 0498 
 
 0499 
 
 0501 
 
 0502 
 
 0503 
 
 0504 
 
 0505 
 
 0506 
 
 0508 
 
 0509 
 
 
 07 
 
 0510 
 
 0511 
 
 0512 
 
 0513 
 
 0515 
 
 0516 
 
 0517 
 
 0518 
 
 0519 
 
 0521 
 
 i 
 
 08 
 
 0522 
 
 0523 
 
 0524 
 
 0525 
 
 0527 
 
 0528 
 
 0529 
 
 0530 
 
 0532 
 
 0533 
 
 
 09 
 
 0534 
 
 0535 
 
 0536 
 
 0538 
 
 0539 
 
 0540 
 
 0541 
 
 0543 
 
 0544 
 
 0545 
 
 
 10 
 
 00 0546 
 
 0548 
 
 0549 
 
 0550 
 
 0551 
 
 0553 
 
 0554 
 
 0555 
 
 0557 
 
 0558 
 
 
 11 
 
 0559 
 
 0560 
 
 0562 
 
 0563 
 
 0564 
 
 0566 
 
 0567 
 
 0568 
 
 0570 
 
 0571 
 
 
 12 
 
 0572 
 
 0573 
 
 0575 
 
 0576 
 
 0577 
 
 0579 
 
 0580 
 
 0581 
 
 0583 
 
 0584 
 
 i 
 
 13 
 
 0585 
 
 0587 
 
 0588 
 
 0590 
 
 0591 
 
 0592 
 
 0594 
 
 0595 
 
 0596 
 
 0598 
 
 
 14 
 
 0599 
 
 0600 
 
 0602 
 
 0603 
 
 0605 
 
 0606 
 
 0607 
 
 0609 
 
 0610 
 
 0612 
 
 
 .15 
 
 0613 
 
 0614 
 
 0616 
 
 0617 
 
 0619 
 
 0620 
 
 0622 
 
 0623 
 
 0624 
 
 0626 
 
 
 16 
 
 0627 
 
 0629 
 
 0630 
 
 0632 
 
 0633 
 
 0635 
 
 0636 
 
 0637 
 
 0639 
 
 0640 
 
 
 17 
 
 0642 
 
 0643 
 
 0645 
 
 0646 
 
 0648 
 
 0649 
 
 0651 
 
 0652 
 
 0654 
 
 0655 
 
 2 
 
 18 
 
 0657 
 
 0658 
 
 0660 
 
 0661 
 
 0663 
 
 0664 
 
 0666 
 
 0667 
 
 0669 
 
 0671 
 
 
 19 
 
 0672 
 
 0674 
 
 0675 
 
 0677 
 
 0678 
 
 0680 
 
 0681 
 
 0683 
 
 0685 
 
 0686 
 
 
 20 
 
 00 0688 
 
 0689 
 
 0691 
 
 0693 
 
 0694 
 
 0696 
 
 0697 
 
 0699 
 
 0701 
 
 0702 
 
 
 21 
 
 0704 
 
 0705 
 
 0707 
 
 0709 
 
 0710 
 
 0712 
 
 0714 
 
 0715 
 
 0717 
 
 0718 
 
 
 22 
 
 0720 
 
 0722 
 
 0723 
 
 0725 
 
 0727 
 
 0728 
 
 0730 
 
 0732 
 
 0734 
 
 0735 
 
 2 
 
 23 
 
 0737 
 
 0739 
 
 0740 
 
 0742 
 
 0744 
 
 0745 
 
 0747 
 
 0749 
 
 0751 
 
 0752 
 
 
 24 
 
 0754 
 
 0756 
 
 0758 
 
 0759 
 
 0761 
 
 0763 
 
 0765 
 
 0766 
 
 0768 
 
 0770 
 
 
 25 
 
 0772 
 
 0773 
 
 0775 
 
 0777 
 
 0779 
 
 0781 
 
 0782 
 
 0784 
 
 0786 
 
 0788 
 
 
 26 
 
 0790 
 
 0791 
 
 0793 
 
 0795 
 
 0797 
 
 0799 
 
 0801 
 
 0802 
 
 0804 
 
 0806 
 
 
 27 
 
 0808 
 
 0810 
 
 0812 
 
 0814 
 
 0815 
 
 0817 
 
 0819 
 
 0821 
 
 0823 
 
 0825 
 
 2 
 
 28 
 
 0827 
 
 0829 
 
 0831 
 
 0832 
 
 0834 
 
 0836 
 
 0838 
 
 0840 
 
 0842 
 
 0844 
 
 
 29 
 
 0846 
 
 0848 
 
 0850 
 
 0852 
 
 0854 
 
 0856 
 
 0858 
 
 0860 
 
 0862 
 
 0864 
 
 
 30 
 
 00 0866 
 
 0868 
 
 0870 
 
 0872 
 
 0874 
 
 0876 
 
 0878 
 
 0880 
 
 0882 
 
 0884 
 
 
 31 
 
 0886 
 
 0888 
 
 0890 
 
 0892 
 
 0894 
 
 0896 
 
 0898 
 
 0900 
 
 0902 
 
 0904 
 
 
 32 
 
 0906 
 
 0909 
 
 0911 
 
 0913 
 
 0915 
 
 0917 
 
 0919 
 
 0921 
 
 0923 
 
 0925 
 
 2 
 
 33 
 
 0928 
 
 0930 
 
 0932 
 
 0934 
 
 0936 
 
 0938 
 
 0940 
 
 0943 
 
 0945 
 
 094" 
 
 
 34 
 
 0949 
 
 0951 
 
 0953 
 
 0956 
 
 0958 
 
 0960 
 
 0962 
 
 0965 
 
 0967 
 
 0969 
 
 
 35 
 
 0971 
 
 0973 
 
 0976 
 
 0978 
 
 0980 
 
 0982 
 
 0985 
 
 0987 
 
 0989 
 
 0991 
 
 
 36 
 
 0994 
 
 0996 
 
 0998 
 
 1001 
 
 1003 
 
 1005 
 
 1008 
 
 1010 
 
 1012 
 
 1015 
 
 
 37 
 
 1017 
 
 1019 
 
 1022 
 
 1024 
 
 1026 
 
 1029 
 
 1031 
 
 1033 
 
 1036 
 
 1038 
 
 2 
 
 38 
 
 1041 
 
 1043 
 
 1045 
 
 1048 
 
 1050 
 
 1053 
 
 1055 
 
 1057 
 
 1060 
 
 1062 
 
 
 39 
 
 1065 
 
 1067 
 
 1070 
 
 1072 
 
 1075 
 
 1077 
 
 1080 
 
 ld82 
 
 1085 
 
 1087 
 
 
 40 
 
 00 1090 
 
 1092 
 
 1095 
 
 109" 
 
 1100 
 
 1102 
 
 1105 
 
 1107 
 
 1110 
 
 1112 
 
 
 41 
 
 1115 
 
 111" 
 
 1120 
 
 1123 
 
 1125 
 
 1128 
 
 1130 
 
 1133 
 
 1136 
 
 1138 
 
 
 42 
 
 1141 
 
 1143 
 
 1146 
 
 1149 
 
 1151 
 
 1154 
 
 1157 
 
 1159 
 
 1162 
 
 1165 
 
 3 
 
 43 
 
 1167 
 
 1170 
 
 1173 
 
 1175 
 
 1178 
 
 1181 
 
 1184 
 
 1186 
 
 1189 
 
 1192 
 
 
 44 
 
 1195 
 
 1197 
 
 1200 
 
 1203 
 
 1206 
 
 1208 
 
 1211 
 
 1214 
 
 1217 
 
 1219 
 
 
 45 
 
 1222 
 
 122f 
 
 1228 
 
 1231 
 
 1234 
 
 1236 
 
 1239 
 
 1242 
 
 1245 
 
 1248 
 
 
 46 
 
 1251 
 
 1254 
 
 1256 
 
 1259 
 
 1262 
 
 1265 
 
 1268 
 
 1271 
 
 1274 
 
 1277 
 
 
 47 
 
 1280 
 
 1283 
 
 1286 
 
 1289 
 
 1292 
 
 1295 
 
 1298 
 
 1301 
 
 1304 
 
 130- 
 
 3 
 
 48 
 
 1310 
 
 1313 
 
 1316 
 
 1319 
 
 1322 
 
 1325 
 
 1328 
 
 1331 
 
 1334 
 
 133" 
 
 
 49 
 
 1340 
 
 1343 
 
 1346 
 
 1349 
 
 1352 
 
 1356 
 
 1359 
 
 1362 
 
 1365 
 
 1368 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLE I. 
 
 Log 
 
 Log (l + x) 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3-50 
 
 O'OO 1371 
 
 1374 1378 1381 
 
 1384 
 
 1387 
 
 1390 
 
 1393 
 
 1397 
 
 1400 
 
 
 '51 
 
 1403 
 
 1406 1410! 1413 
 
 1416 
 
 1419 
 
 1423 1426 
 
 1429 
 
 1432 
 
 
 52 
 
 1436 
 
 1439 
 
 1442 1446 
 
 1449 
 
 1452 
 
 1456 
 
 1459 
 
 1462 
 
 1466 
 
 3 
 
 53 
 
 1469 
 
 1472 
 
 1476 1479 
 
 1483 
 
 1486 
 
 1489 
 
 1493 
 
 1496 
 
 1500 
 
 
 54 
 
 1503 
 
 1507 
 
 1510 
 
 1514 
 
 1517 
 
 1521 
 
 1524 
 
 1528 
 
 1531 
 
 1535 
 
 
 55 
 
 1538 1542 
 
 1545 
 
 1549 
 
 1552 
 
 1556 
 
 1560 
 
 1563 
 
 1567 
 
 1570 
 
 
 56 
 
 1574 1 1578 
 
 1581 
 
 1585 
 
 1589 
 
 1592 
 
 1596 
 
 1600 
 
 1603 
 
 1607 
 
 
 57 
 
 1611 1614 
 
 1618 1622 
 
 1625 
 
 1629 
 
 1633 
 
 1637 
 
 1640 
 
 1644 
 
 4 
 
 58 
 
 1648 1652 
 
 1656 1659 1663 
 
 1667 
 
 1671 
 
 1675 
 
 1679 
 
 1682 
 
 
 59 
 
 1686 
 
 1690 
 
 1694 1698 1702 
 
 1706 
 
 1710 
 
 1714 
 
 1718 
 
 1722 
 
 
 60 
 
 00 1726 
 
 1729 
 
 1733 1737 
 
 1741 
 
 1745 
 
 1749 
 
 1754 
 
 1758 
 
 1762 
 
 
 61 
 
 1766 
 
 1770 
 
 1774 1778 
 
 1782 
 
 1786 
 
 1790 
 
 1794 
 
 1798 
 
 1803 
 
 
 62 
 
 1807 
 
 1811 
 
 1815 
 
 1819 
 
 1823 
 
 1828 
 
 1832 1836 
 
 1840 
 
 1844 
 
 4 
 
 63 
 
 1849 
 
 1853 
 
 1857 
 
 1861 
 
 1866 
 
 1870 
 
 1874 1879 
 
 1883 
 
 1887 
 
 
 64 
 
 1892 
 
 1896 
 
 1900 
 
 1905 
 
 1909 
 
 1913 
 
 1918 1922 
 
 1927 
 
 1931 
 
 
 65 
 
 1936 
 
 1940 
 
 1945 
 
 1949 
 
 1953 
 
 1958 
 
 1962 1967 
 
 1972 
 
 1976 
 
 
 66 
 
 1981 
 
 1985 
 
 1990 
 
 1994 
 
 1999 
 
 2003 
 
 20081 2013 
 
 2017 
 
 2022 
 
 
 67 
 
 2027 
 
 2031 
 
 2036 
 
 2041 
 
 2045 
 
 2050 
 
 2055 2059 
 
 2064 
 
 2069 
 
 5 
 
 68 
 
 2074 
 
 2078 
 
 2083 
 
 2088 
 
 2093 
 
 2098 
 
 2102! 2107 
 
 2112 
 
 2117 
 
 
 69 
 
 2122 2127 
 
 2132 
 
 2137 
 
 2141 
 
 2146 
 
 2151 
 
 2156 
 
 2161 
 
 2166 
 
 
 70 
 
 002171 
 
 2176 
 
 2181 
 
 2186 
 
 2191 
 
 2196 
 
 2201 
 
 2206 
 
 2211 
 
 2217 
 
 
 71 
 
 2222 2227 
 
 2232 
 
 2237 
 
 2242 
 
 2247 
 
 2252 
 
 2258 
 
 2263 
 
 2268 
 
 
 72 
 
 2273, 2278 
 
 2284 
 
 2289 
 
 2294 
 
 2300 
 
 2305 
 
 2310 
 
 2315 
 
 2321 
 
 5 
 
 73 
 
 2326 2331 
 
 2337 
 
 2342 2348 
 
 2353 
 
 2358 
 
 2364 
 
 2369 
 
 2375 
 
 
 74 
 
 2380 
 
 2386 
 
 2391 
 
 2397 
 
 2402 
 
 2408 
 
 2413 
 
 2419 
 
 2424 
 
 2430 
 
 
 75 
 
 2435 
 
 2441 
 
 2447 
 
 2452 
 
 2458 
 
 2464 
 
 2469 
 
 2475 
 
 2481 
 
 2486 
 
 
 76 
 
 2492 2498 
 
 2503 
 
 2509 
 
 2515 
 
 2521 
 
 2527J 2532 
 
 2538 
 
 2544 
 
 
 77 
 
 2550; 2556 
 
 2562 
 
 2567 
 
 2573 
 
 2579 
 
 2585 2591 
 
 2597 
 
 2603 
 
 6 
 
 78 
 
 2609! 2615 
 
 2621 
 
 2627 
 
 2633 
 
 2639 
 
 2645 2651 
 
 2657 
 
 2663 
 
 
 79 
 
 2670 2676 
 
 2682 
 
 2688 
 
 2694 
 
 2700 
 
 2707 
 
 2713 
 
 2719 
 
 2725 
 
 
 80 
 
 00 2732 2738 
 
 2744 
 
 2750 
 
 2757 
 
 2763 2769 
 
 2776 
 
 2782 
 
 2789 
 
 
 81 
 
 2795 2801 
 
 2808 
 
 2814 
 
 2821 
 
 2827 2834: 2840 
 
 2847 
 
 2853 
 
 
 82 
 
 2860 2866 
 
 2873 
 
 2880 
 
 2886 
 
 2893 2900 2906 
 
 2913 
 
 2920 
 
 7 
 
 83 
 
 2926 2933i 2940 
 
 2947| 2953 
 
 2960 2967 2974 
 
 2981 
 
 2987 
 
 
 84 
 
 2994 3001 
 
 3008 
 
 3015J 3022 
 
 3029 3036 
 
 3043 
 
 3050 
 
 3057 
 
 
 85 
 
 3064 3071 
 
 3078 
 
 3085 3092 
 
 3099 
 
 3106 
 
 3113 
 
 3120 
 
 3128 
 
 
 86 
 
 3135! 3142 3149 
 
 3156, 3164 
 
 3171 
 
 3178 3186 
 
 3193 
 
 3200 
 
 
 87 
 
 3208 3215 3222 
 
 3230! 3237 
 
 3245 3252 
 
 3260| 3267 
 
 3275 
 
 8 
 
 88 
 
 3282 3290 3297 
 
 3305 
 
 3312 
 
 3320 3326 
 
 3335 
 
 3343 
 
 3350 
 
 
 89 
 
 3358 
 
 3366 
 
 3374 
 
 3381 
 
 3389 
 
 3397 
 
 3405 
 
 3413 
 
 3420 
 
 3428 
 
 ' 
 
 90 
 
 00 3436 
 
 3444 
 
 3452 
 
 3460 
 
 3468 
 
 3476 
 
 3484 
 
 3492 
 
 3500 
 
 3508 
 
 
 91 
 
 3516 
 
 3524 
 
 3532 
 
 3540 
 
 3548 
 
 3556 3565 
 
 3573 
 
 3581 
 
 3589 
 
 
 92 
 
 3597 
 
 3606 
 
 3614 
 
 3622 
 
 363! 
 
 3639 3647 
 
 3656 
 
 3664 
 
 3672 
 
 9 
 
 93 
 
 3681 
 
 3689 
 
 3698 
 
 3706 
 
 3715 
 
 3723 
 
 3732 
 
 3740 
 
 3749 
 
 3758 
 
 
 94 
 
 3766 
 
 3775 
 
 3783 
 
 3792 
 
 3801 
 
 3810 
 
 3818 
 
 3827 
 
 3836 
 
 3845 
 
 
 95 
 
 3854 
 
 3862 
 
 3871 
 
 3880 
 
 3889 
 
 3898 
 
 3907 
 
 3916 
 
 3925 
 
 3934 
 
 
 96 
 
 3943 
 
 3952 
 
 3961 
 
 3970 
 
 3979 
 
 3988 
 
 3997 
 
 4007 
 
 4016 
 
 4025 
 
 
 97 
 
 4034 
 
 4044 
 
 4053 
 
 4062 
 
 4071 
 
 4081 
 
 4090 
 
 4100 
 
 4109 
 
 4118 
 
 10 
 
 98 
 
 4128 
 
 4137 4147 
 
 4156 
 
 4166 
 
 4175 4185 
 
 4195 
 
 4204 
 
 4214 
 
 
 99 
 
 4223 
 
 4233 
 
 4243 
 
 4253 
 
 4262 
 
 4272 4282 
 
 4292 
 
 4302 
 
 4311 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 [3] 
 
TABLE I. 
 
 Log 
 
 Log(l+a?) 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 2-00 
 
 O'OO 4321 
 
 4331 
 
 4341 
 
 4351 
 
 4361 
 
 4371 
 
 4381 
 
 4391 
 
 4401 
 
 4411 
 
 10 
 
 01 
 
 4422 
 
 4432 
 
 4442 
 
 4452 
 
 4462 
 
 4472 
 
 4483 
 
 4493 
 
 4503 
 
 4514 
 
 10 
 
 02 
 
 4524 
 
 4534 
 
 4545 
 
 4555 
 
 4566 
 
 4576 
 
 4587 
 
 4597 
 
 4608 
 
 4618 
 
 10 
 
 '03 
 
 4629 
 
 4639 
 
 4650 
 
 4661 
 
 4671 
 
 4682 
 
 4693 
 
 4704 
 
 4714 
 
 4725 
 
 11 
 
 04 
 
 4736 
 
 4747 
 
 4758 
 
 4769 
 
 4780 
 
 4791 
 
 4802 
 
 4813 
 
 4824 
 
 4835 
 
 11 
 
 05 
 
 4846 
 
 4857 
 
 4868 
 
 4879 
 
 4890 
 
 4902 
 
 4913 
 
 4924 
 
 4935 
 
 4947 
 
 11 
 
 06 
 
 4956 
 
 4969 
 
 4981 
 
 4992 
 
 5004 
 
 5015 
 
 5027 
 
 5038 
 
 5050 
 
 5061 
 
 12 
 
 07 
 
 5073 
 
 5084 
 
 5096 
 
 5108 
 
 5119 
 
 5131 
 
 5143 
 
 5155 
 
 5167 
 
 5178 
 
 12 
 
 08 
 
 5190 
 
 5202 
 
 5214 
 
 5226 
 
 5238 
 
 5250 
 
 5262 
 
 5274 
 
 5286 
 
 5298 
 
 12 
 
 09 
 
 5310 
 
 5323 
 
 5335 
 
 5347 
 
 5359 
 
 5372 
 
 5384 
 
 5396 
 
 5409 
 
 5421 
 
 12 
 
 10 
 
 00 5433 
 
 5446 
 
 5458 
 
 5471 
 
 5483 
 
 5496 
 
 5508 
 
 5521 
 
 5534 
 
 5546 
 
 13 
 
 11 
 
 5559 
 
 5572 
 
 5585 
 
 5597 
 
 5610 
 
 5623 
 
 5636 
 
 5649 
 
 5662 
 
 5675 
 
 13 
 
 12 
 
 5688 
 
 5701 
 
 5714 
 
 5727 
 
 5740 
 
 5753 
 
 5766 
 
 5780 
 
 5793 
 
 5806 
 
 13 
 
 13 
 
 5819 
 
 5833 
 
 5846 
 
 5859 
 
 5873 
 
 5886 
 
 5900 
 
 5913 
 
 5927 
 
 5940 
 
 13 
 
 14 
 
 5954 
 
 5968 
 
 5981 
 
 5995 
 
 6009 
 
 6022 
 
 6036 
 
 6050 
 
 6064 
 
 6078 
 
 14 
 
 15 
 
 6092 
 
 6106 
 
 6120 
 
 6134 
 
 6148 
 
 6162 
 
 6176 
 
 6190 
 
 6204 
 
 6218 
 
 14 
 
 16 
 
 6233 
 
 6247 
 
 6261 
 
 6275 
 
 6290 
 
 6304 
 
 6319J 6333 
 
 6348 
 
 636^ 
 
 14 
 
 17 
 
 6377 
 
 6391 
 
 6406 
 
 6421 
 
 6435 
 
 6450 
 
 6465 1 6479 
 
 6494 
 
 6509 
 
 15 
 
 18 
 
 6524 
 
 6539 
 
 6554 
 
 6569 
 
 6584 
 
 6599 
 
 6614 6629 
 
 6644 
 
 6660 
 
 lo 
 
 19 
 
 6675 
 
 6690 
 
 6705 
 
 6721 
 
 6736 
 
 6752 
 
 6767: 6782 
 
 6798 
 
 6814 
 
 15 
 
 20 
 
 00 6829 6845 
 
 6860 
 
 6876 
 
 6892 
 
 6908 
 
 6923 6939 
 
 6955 
 
 6971 
 
 16 
 
 21 
 
 6987 7003 
 
 7019 
 
 7035 
 
 7051 
 
 7067 
 
 7083 7100 
 
 7116 
 
 7132 
 
 16 
 
 22 
 
 7148' 7165 
 
 7181 
 
 7197 
 
 7214 
 
 7230 
 
 7247i 7264 
 
 7280 
 
 7297 
 
 16 
 
 23 
 
 7313 7330 
 
 7347 
 
 7364 
 
 7381 
 
 7397 
 
 7414 7431 
 
 7448 
 
 7465 
 
 17 
 
 24 
 
 7482 7499 
 
 7517 
 
 7534 
 
 7551 
 
 7568 
 
 7586, 7603 
 
 7620 
 
 7638 
 
 17 
 
 
 
 
 
 
 
 
 
 
 
 
 25 
 
 7655 
 
 7673 
 
 7690 
 
 7708 
 
 7725 
 
 7743 
 
 7761: 7778 
 
 7796 
 
 7814 
 
 18 
 
 26 
 
 7832 
 
 7850 
 
 7868 
 
 7886 7904 
 
 7922 
 
 7940 7958 
 
 7976 
 
 7994 
 
 18 
 
 27 
 
 8013 
 
 8031 
 
 8049 
 
 8068 8086 
 
 8104 
 
 8123 8142 
 
 8160 
 
 8179 
 
 18 
 
 28 
 
 8197 
 
 8216 
 
 8235 
 
 8254 8273 
 
 8291 
 
 83 1 Ol 8329 
 
 8348 
 
 8367 
 
 19 
 
 29 
 
 8387 
 
 8406 
 
 8425 
 
 8444 
 
 8463 
 
 8483 
 
 8502 
 
 8522 
 
 8541 
 
 8560 
 
 19 
 
 30 
 
 00 8580 
 
 8600 
 
 8619 
 
 8639 
 
 8659 
 
 8678 
 
 8698 
 
 8718 
 
 8738 
 
 8758 
 
 20 
 
 31 
 
 8778 
 
 8798 
 
 8818 
 
 8838 
 
 8858 
 
 8878 
 
 8899 
 
 8919 
 
 8939 
 
 8960 
 
 20 
 
 32 
 
 8980 
 
 9001 
 
 9021 
 
 9042 
 
 9062 
 
 9083 9104 
 
 9125 
 
 9145 
 
 9166 
 
 21 
 
 33 
 
 9187 
 
 9208 
 
 9229 
 
 9250 
 
 9271 
 
 9292 
 
 9314 
 
 9335 
 
 9356 
 
 9378 
 
 21 
 
 34 
 
 9399 
 
 9420 
 
 9442 
 
 9463 
 
 9485 
 
 9507 
 
 9528 
 
 9550 
 
 9572 
 
 9594 
 
 22 
 
 35 
 
 9615 
 
 9637 
 
 9659 
 
 9681 
 
 9703 
 
 9726 
 
 9748 
 
 9770 
 
 9792 
 
 9814 
 
 22 
 
 36 
 
 9837 
 
 9859 
 
 9882 
 
 9904 
 
 9927 
 
 9949 
 
 9972 
 
 9995 
 
 0018 
 
 0040 
 
 23 
 
 37 
 
 01 0063 
 
 0086 
 
 0109 
 
 0132 
 
 0155 
 
 0179 
 
 0202 
 
 0225 
 
 0248 
 
 0272 
 
 23 
 
 38 
 
 0295 
 
 0318 
 
 0342 
 
 0366 
 
 0389 
 
 0413 
 
 0437 
 
 0460 
 
 0484 
 
 0508 
 
 24 
 
 39 
 
 0532 
 
 0556 
 
 0580 
 
 0604 
 
 0628 
 
 0652 
 
 0677 
 
 0701 
 
 0725 
 
 0750 
 
 24 
 
 40 
 
 01 0774 
 
 0799 
 
 0823 
 
 0848 
 
 0873 
 
 0897 
 
 0922 
 
 0947 
 
 0972 
 
 0997 
 
 25 
 
 41 
 
 1022 
 
 1047 
 
 1072 
 
 1097 
 
 1123 
 
 1148 
 
 1173 
 
 1199 
 
 1224 
 
 1250 
 
 25 
 
 42 
 
 1275 
 
 1301 
 
 1327 
 
 1353 
 
 1378 
 
 1404 
 
 1430 
 
 1456 
 
 1482 
 
 1508 
 
 26 
 
 43 
 
 1535 
 
 1561 
 
 1587 
 
 1614 
 
 1640 
 
 1666 
 
 1693 
 
 1720 
 
 1746 
 
 1773 
 
 26 
 
 44 
 
 1800 
 
 1827 
 
 1853 
 
 1880 
 
 1907 
 
 1934 
 
 1962 
 
 1989 
 
 2016 
 
 2043 
 
 27 
 
 45 
 
 2071 
 
 2098 
 
 2126 
 
 2153 
 
 2181 
 
 2209 
 
 2236 
 
 2264 
 
 2292 
 
 2320 
 
 28 
 
 46 
 
 2348 
 
 2376 
 
 2404 
 
 2432 
 
 2461 
 
 2489 
 
 2517 
 
 2546 
 
 2574 
 
 2603 
 
 28 
 
 47 
 
 2631 
 
 2660 
 
 2689 
 
 2718 
 
 2747 
 
 2776 
 
 2805 
 
 2834 
 
 2863 
 
 2892 
 
 29 
 
 48 
 
 2921 
 
 2951 
 
 2980 
 
 3010 
 
 3039 
 
 3069 
 
 3098 
 
 3128 3158 
 
 3188 
 
 30 
 
 49 
 
 3218 
 
 3248 
 
 3278 
 
 3308 
 
 3338 
 
 3369 
 
 3399 
 
 3429 
 
 3460 
 
 3490 
 
 30 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLE I. 
 
 Log 
 
 
 
 Lo 
 
 g(l 
 
 +*) 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 2-50 
 
 O'Ol 3521 
 
 3552 
 
 3582 
 
 3613 
 
 3644 
 
 3675 
 
 3706 
 
 3737 
 
 3768 
 
 3800 
 
 31 
 
 51 
 
 3831 
 
 3862 
 
 3894 
 
 3925 
 
 3957 
 
 3989 
 
 4020 
 
 4052 
 
 4084 
 
 4116 
 
 32 
 
 52 
 
 4148 
 
 4180 
 
 4212 
 
 4244 
 
 4277 
 
 4309 
 
 4341 
 
 4374 
 
 4407 
 
 4439 
 
 32 
 
 53 
 
 4472 
 
 4505 
 
 4538 
 
 4571 
 
 4604 
 
 4637 
 
 4670 
 
 4703 
 
 4737 
 
 4770 
 
 33 
 
 54 
 
 4803 
 
 4837 
 
 4871 
 
 4904 
 
 4938 
 
 4972 
 
 5006 
 
 5040 
 
 5074 
 
 5108 
 
 34 
 
 55 
 
 5142 
 
 5177 
 
 5211 
 
 5245 
 
 5280 
 
 5315 
 
 5349 
 
 53,84 
 
 5419 
 
 5454 
 
 35 
 
 56 
 
 5489 
 
 5524 
 
 5559 
 
 5594 
 
 5630 
 
 5665 
 
 5700 
 
 5736 
 
 5772 
 
 5807 
 
 35 
 
 57 
 
 5843 
 
 5879 
 
 5915 
 
 5951 
 
 5987 
 
 6023 
 
 6059 
 
 6096 
 
 6132 
 
 6169 
 
 36 
 
 58 
 
 6205 
 
 6242 
 
 6279 
 
 6316 
 
 6352 
 
 6389 
 
 6427 
 
 6464 
 
 6501 
 
 6538 
 
 37 
 
 59 
 
 6576 
 
 6613 
 
 6651 
 
 6688 
 
 6726 
 
 6764 
 
 6802 
 
 6840 
 
 6878 
 
 6916 
 
 38 
 
 60 
 
 01 6954 
 
 6993 
 
 7031 
 
 7070 
 
 7108 
 
 7147 
 
 7186 
 
 7224 
 
 7263 
 
 7302 
 
 39 
 
 61 
 
 7341 
 
 7381 
 
 7420 
 
 7459 
 
 7499 
 
 7538 
 
 7578 
 
 7618 
 
 7657 
 
 7697 
 
 40 
 
 62 
 
 7737 
 
 7777 
 
 7817 
 
 7858 
 
 7898 
 
 7938 
 
 7979 
 
 8020 
 
 8060 
 
 8101 
 
 41 
 
 63 
 
 8142 
 
 8183 
 
 8224 
 
 8265 
 
 8306 
 
 8348 
 
 8389 
 
 8430 
 
 8472 
 
 8514 
 
 42 
 
 64 
 
 8556 
 
 8597 
 
 8639 
 
 8681 
 
 8724 
 
 8766 
 
 8808 
 
 8851 
 
 8893 
 
 8936 
 
 43 
 
 65 
 
 8978 
 
 9021 
 
 9064 
 
 9107 
 
 9150 
 
 9193 
 
 9237 
 
 9280 
 
 9324 
 
 9367 
 
 43 
 
 66 
 
 9411 
 
 9455 
 
 9498 
 
 9542 
 
 9586 
 
 9631 
 
 9675 
 
 9719 
 
 9764 
 
 9808 
 
 44 
 
 67 
 
 9853 
 
 9897 
 
 9942 
 
 9987 
 
 0032 
 
 o077 
 
 o!23 
 
 0168 
 
 o213 
 
 0259 
 
 45 
 
 68 
 
 02 0305 
 
 0350 
 
 0396 
 
 0442 
 
 0488 
 
 0534 
 
 0580 
 
 0627 
 
 0673 
 
 0720 
 
 46 
 
 69 
 
 0766 
 
 0813 
 
 0860 
 
 0907 
 
 0954 
 
 1001 
 
 1048 
 
 1096 
 
 1143 
 
 1191 
 
 47 
 
 70 
 
 02 1238 
 
 1286 
 
 1334 
 
 1382 
 
 1430 
 
 1478 
 
 1527 
 
 1575 
 
 1624 
 
 1672 
 
 48 
 
 71 
 
 1721 
 
 1770 
 
 1819 
 
 1868 
 
 1917 
 
 1966 
 
 2016 
 
 2065 
 
 2115 
 
 2164 
 
 49 
 
 72 
 
 2214 
 
 2264 
 
 2314 
 
 2364 
 
 2414 
 
 2465 
 
 2515 
 
 2566 
 
 2617 
 
 2667 
 
 50 
 
 73 
 
 2718 
 
 2769 
 
 2820 
 
 2872 
 
 2923 
 
 2975 
 
 3026 
 
 3078 
 
 3130 
 
 3182 
 
 52 
 
 74 
 
 3234 
 
 3286 
 
 3338 
 
 3390 
 
 3443 
 
 3495 
 
 3548 
 
 3601 
 
 3654 
 
 3707 
 
 53 
 
 75 
 
 3760 
 
 3813 
 
 3867 
 
 3920 
 
 3974 
 
 4028 
 
 4082 
 
 4136 
 
 4190 
 
 4244 
 
 54 
 
 76 
 
 4298 
 
 4353 
 
 4408 
 
 4462 
 
 4517 
 
 4572 
 
 4627 
 
 4682 
 
 4738 
 
 4793 
 
 55 
 
 77 
 
 4849 
 
 4904 
 
 4960 
 
 5016 
 
 5072 
 
 5128 
 
 5184 
 
 5241 
 
 5297 
 
 5354 
 
 56 
 
 78 
 
 5411 
 
 5468 
 
 5525 
 
 5582 
 
 5639 
 
 5696 
 
 5754 
 
 5812 
 
 5869 
 
 5927 
 
 57 
 
 79 
 
 5985 
 
 6043 
 
 6102 
 
 6160 
 
 6219 
 
 6277 
 
 6336 
 
 6395 
 
 6454 
 
 6513 
 
 59 
 
 80 
 
 '02 6572 
 
 6632 
 
 6691 
 
 6751 
 
 6811 
 
 6871 
 
 6931 
 
 6991 
 
 7051 
 
 7112 
 
 60 
 
 81 
 
 7172 
 
 7233 
 
 7294 
 
 7355 
 
 7416 
 
 7477 
 
 7539 
 
 7600 
 
 7662 
 
 7724 
 
 61 
 
 82 
 
 7785 
 
 7847 
 
 7910 
 
 7972 
 
 8034 
 
 8097 
 
 8160 
 
 8223 
 
 8286 
 
 8349 
 
 63 
 
 83 
 
 8412 
 
 8475 
 
 8539 
 
 8603 
 
 8666 
 
 8730 
 
 8794 
 
 8859 
 
 8923 
 
 8987 
 
 64 
 
 84 
 
 9052 
 
 9117 
 
 9182 
 
 9247 
 
 9312 
 
 9377 
 
 9443 
 
 9508 
 
 9574 
 
 9640 
 
 65 
 
 85 
 
 9706 
 
 9772 
 
 9839 
 
 9905 
 
 9972 
 
 o039 
 
 o!05 
 
 0172 
 
 0240 
 
 o307 
 
 67 
 
 86 
 
 03 0374 
 
 0442 
 
 0510 
 
 0578 
 
 0646 
 
 0714 
 
 0782 
 
 0851 
 
 0920 
 
 0988 
 
 68 
 
 87 
 
 1057 
 
 1126 
 
 1196 
 
 1265 
 
 1335 
 
 1404 
 
 1474 
 
 1544 
 
 1614 
 
 1684 
 
 70 
 
 88 
 
 1755 
 
 1825 
 
 1896 
 
 1967 
 
 2038 
 
 2109 
 
 2181 
 
 2252 
 
 2324 
 
 2396 
 
 71 
 
 89 
 
 2468 
 
 2540 
 
 2612 
 
 2684 
 
 2757 
 
 2830 
 
 2903 
 
 2976 
 
 3049 
 
 3122 
 
 73 
 
 90 
 
 03 3196 
 
 3269 
 
 3343 
 
 3417 
 
 3491 
 
 3566 
 
 3640 
 
 3715 
 
 3789 
 
 3864 
 
 74 
 
 91 
 
 3939 
 
 4015 
 
 4090 
 
 4166 
 
 4241 
 
 4317 
 
 4393 
 
 4470 
 
 4546 
 
 4622 
 
 76 
 
 92 
 
 4699 
 
 4776 
 
 4853 
 
 4930 
 
 5008 
 
 5085 
 
 5163 
 
 5241 
 
 5319 
 
 5397 
 
 78 
 
 93 
 
 5475 
 
 5554 
 
 5632 
 
 5711 
 
 5790 
 
 5870 
 
 5949 
 
 6028 
 
 6108 
 
 6188 
 
 79 
 
 94 
 
 6268 
 
 6348 
 
 6429 
 
 6509 
 
 6590 
 
 6671 
 
 6752 
 
 6833 
 
 6914 
 
 6996 
 
 81 
 
 95 
 
 7078 
 
 7160 
 
 7242 
 
 7324 
 
 7406 
 
 7489 
 
 7572 
 
 7655 
 
 7738 
 
 7821 
 
 83 
 
 96 
 
 7905 
 
 7988 
 
 8072 
 
 8156 
 
 8241 
 
 8325 
 
 8409 
 
 8494 
 
 8579 
 
 8664 
 
 85 
 
 97 
 
 8749 
 
 8835 
 
 8921 
 
 9006 
 
 9092 
 
 9179 
 
 9265 
 
 9351 
 
 9438 
 
 9525 
 
 87 
 
 98 
 
 9612 
 
 9699 
 
 9787 
 
 9874 
 
 9962 
 
 0050 
 
 0138 
 
 0227 
 
 0315 
 
 0404 
 
 88 
 
 99 
 
 04 0493 
 
 0582 
 
 0671 
 
 0761 
 
 0851 
 
 0941 
 
 1031 
 
 1121 
 
 1211 
 
 1302 
 
 90 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 [5] 
 
TABLE I. 
 
 Log 
 
 
 Log(l + z) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9 
 
 10 
 
 11 
 
 12 
 
 fooo 
 
 0'04 1393 
 
 1402 
 
 1411 
 
 1420 
 
 1429 
 
 1438 
 
 1447 
 
 1456 
 
 1465 
 
 1475 
 
 00 
 
 
 
 
 
 
 
 
 
 001 
 
 1484 
 
 1493 
 
 1502 
 
 1511 
 
 1520 
 
 1529 
 
 1538 
 
 1548 
 
 1557 
 
 1566 
 
 01 
 
 
 
 
 
 
 
 
 
 002 
 
 1575 
 
 1584 
 
 1593 
 
 1602 
 
 1611 
 
 1621 
 
 1630 
 
 1639 
 
 1648 
 
 1657 
 
 02 
 
 
 
 
 
 
 
 
 
 003 
 
 1666 
 
 1075 
 
 1685 
 
 1694 
 
 1703 
 
 1712 
 
 1721 
 
 1730 
 
 1740 
 
 1749 
 
 03 
 
 
 
 
 
 
 
 
 
 004 
 
 1758 
 
 1767 
 
 1776 
 
 1785 
 
 1795 
 
 1804 
 
 1813 
 
 1822 
 
 1831 
 
 1840 
 
 04 
 
 
 
 
 
 
 
 
 
 005 
 
 1850 
 
 1859 
 
 1868 
 
 1877 
 
 1886 
 
 1896 
 
 1905 
 
 1914 
 
 1923 
 
 1932 
 
 05 
 
 o 
 
 
 
 
 1 
 
 006 
 
 1942 
 
 1951 
 
 I960 1969 
 
 1978 
 
 1988 
 
 1997 
 
 2006 
 
 2015 
 
 2025 
 
 on i 
 
 1 
 
 
 1 
 
 007 
 
 2034 
 
 2043 
 
 2052 2061 
 
 2071 
 
 K is. , 
 
 2089 
 
 2098 2108 
 
 2117 
 
 07 1 
 
 1 
 
 
 1 
 
 008 
 
 2126 2135 214.", \: } \ 
 
 2163 
 
 2172 
 
 2182 
 
 2191 
 
 2200 
 
 2209 
 
 08 1 
 
 1 
 
 
 1 
 
 009 
 
 2219 
 
 2228 22371 2246 
 
 2256 
 
 2205 
 
 2274 
 
 2284 
 
 2293 
 
 2302 
 
 09J 1 
 
 1 
 
 
 i 
 
 010 
 
 042311 
 
 2321 
 
 2330 2339 
 
 2348 
 
 2358 
 
 2367 
 
 2376 
 
 2386 
 
 2395 
 
 10 
 
 1 
 
 1 
 
 
 1 
 
 on 
 
 2404 
 
 2414 
 
 2423 2432 
 
 2442 
 
 2451 
 
 2460 
 
 2469 
 
 2479 
 
 2488 
 
 11 
 
 1 
 
 1 
 
 
 1 
 
 012 
 
 2497 
 
 2507 
 
 2516 2525 
 
 2535 
 
 2544 
 
 2553 
 
 2563 
 
 2572 
 
 258J 
 
 12 
 
 1 
 
 1 
 
 
 1 
 
 013 
 
 2591 
 
 2600 
 
 2609 
 
 2619 
 
 2628 
 
 2637 
 
 2647 
 
 2656 
 
 2666 
 
 2675 
 
 L3 
 
 1 
 
 1 
 
 
 2 
 
 014 
 
 2684 
 
 2694 
 
 2703 
 
 2712 
 
 2722 
 
 2731 
 
 2740 
 
 2750 
 
 2759 
 
 2769 
 
 14 
 
 1 
 
 1 
 
 2 
 
 2 
 
 015 
 
 2778 
 
 2787 
 
 2797 
 
 2806 
 
 2815 
 
 2825 
 
 2834 
 
 2844 
 
 2853 
 
 2862 
 
 15 
 
 1 
 
 2 
 
 2 
 
 2 
 
 016 
 
 2872 
 
 2881 
 
 2891 
 
 2900 
 
 2909 
 
 2919 
 
 2928 
 
 2938 
 
 2947 
 
 2957 
 
 if; 
 
 1 
 
 2 
 
 2 
 
 2 
 
 017 
 
 2966 
 
 2975 
 
 2985 
 
 2994 
 
 3004 
 
 3013 
 
 3022 
 
 3032 
 
 3041 
 
 3051 
 
 17 
 
 2 
 
 2 
 
 2 
 
 2 
 
 018 
 
 3060 
 
 3070 
 
 3079 
 
 3089 
 
 3098 
 
 3107 
 
 3117 
 
 3126 
 
 3136 
 
 3145 
 
 18 
 
 2 
 
 2 
 
 2 
 
 2 
 
 019 
 
 3155 
 
 3164 
 
 3174 
 
 3183 
 
 3193 
 
 3202 
 
 3212 
 
 3221 
 
 3230 
 
 3240 
 
 19 
 
 2 
 
 2 
 
 2 
 
 2 
 
 020 
 
 04 3249 
 
 3259 
 
 3268 
 
 3278 
 
 3287 
 
 3297 
 
 3306 
 
 3316 
 
 3325 
 
 3335 
 
 20 
 
 2 
 
 2 
 
 2 
 
 2 
 
 021 
 
 3344 
 
 3354 
 
 3363 
 
 3373 
 
 3382 
 
 3392 
 
 3401 
 
 3411 
 
 3420 
 
 3430 
 
 21 
 
 2 
 
 2 
 
 2 
 
 j 
 
 022 
 
 3439 
 
 3449 
 
 3458 
 
 3468 
 
 3477 
 
 3487 
 
 3497 
 
 3506 
 
 3516 
 
 3525 
 
 22 
 
 2 
 
 2 
 
 2 
 
 
 023 
 
 3535 
 
 3544 
 
 3564 
 
 3563 
 
 3573 
 
 3582 
 
 3592 
 
 3601 
 
 3611 
 
 3621 
 
 23 
 
 2 
 
 2 
 
 
 
 024 
 
 3630 
 
 3640 
 
 3649 
 
 3659 
 
 3668 
 
 3678 
 
 3688 
 
 3697 
 
 3707 
 
 3716 
 
 24 
 
 2 
 
 2 
 
 
 
 
 025 
 
 3726 
 
 3735 
 
 3745 
 
 3755 
 
 3764 
 
 3774 
 
 3783 
 
 3793 
 
 3803 
 
 3812 
 
 25 
 
 2 
 
 2 
 
 
 
 
 026 
 
 3822 
 
 3831 
 
 3841 
 
 3851 
 
 3860 
 
 3870 
 
 3879 
 
 3869 
 
 3899 
 
 3908 
 
 20 
 
 2 
 
 3 
 
 
 
 027 
 
 3918 
 
 3927 
 
 3937 
 
 3947 
 
 3956 
 
 3966 
 
 3976 
 
 3985 
 
 3995 
 
 4004 
 
 27 
 
 2 
 
 
 
 
 
 028 
 
 4014 
 
 4024 
 
 4033 
 
 4043 
 
 4053 
 
 4062 
 
 4072 
 
 4082 
 
 4091 
 
 4101 
 
 28 
 
 3 
 
 3 
 
 
 
 029 
 
 4111 
 
 4120 
 
 4130 
 
 4140 
 
 4149 
 
 4159 
 
 4169 
 
 4178 
 
 4188 
 
 4198 
 
 29 
 
 3 
 
 3 
 
 
 
 030 
 
 04 4207 
 
 4217 
 
 4227 
 
 4236 
 
 4246 
 
 4256 
 
 4265 
 
 4275 
 
 4285 
 
 4294 
 
 30 
 
 a 
 
 3 
 
 
 4 
 
 031 
 
 4304 
 
 4314 
 
 4324 
 
 4333 
 
 4343 
 
 4353 
 
 4362 
 
 4372 
 
 4382 
 
 4391 
 
 31 
 
 3 
 
 3 
 
 
 4 
 
 032 
 
 4401 
 
 4411 
 
 4421 
 
 4430 
 
 4440 
 
 4450 
 
 4460 
 
 4469 
 
 4479 
 
 4489 
 
 32 
 
 3 
 
 3 
 
 4 
 
 4 
 
 033 
 
 4498 
 
 4508 
 
 4518 
 
 4528 
 
 4537 
 
 4547 
 
 4557 
 
 4567 
 
 4576 
 
 4586 
 
 33 
 
 3 
 
 3 
 
 4 
 
 4 
 
 034 
 
 4596 
 
 4606 
 
 4615 
 
 4625 
 
 4635 
 
 4645 
 
 4655 
 
 4664 
 
 4674 
 
 4684 
 
 34 
 
 a 
 
 3 
 
 4 
 
 4 
 
 035 
 
 4694 
 
 4703 
 
 4713 
 
 4723 
 
 4733 
 
 4743 
 
 4752 
 
 4762 
 
 4772 
 
 4782 
 
 35 
 
 3 
 
 4 
 
 4 
 
 4 
 
 036 
 
 4792 
 
 4801 
 
 4811 
 
 4821 
 
 4831 
 
 4841 
 
 4850 
 
 4860 
 
 4870 
 
 4880 
 
 36 
 
 3 
 
 4 
 
 4 
 
 4 
 
 037 
 
 4890 
 
 4899 
 
 4909 
 
 4919 
 
 4929 
 
 4939 
 
 4949 
 
 4958 
 
 4968 
 
 4978 
 
 37 
 
 3 
 
 4 
 
 4 
 
 4 
 
 038 
 
 4988 
 
 4998 
 
 5008 
 
 5017 
 
 5027 
 
 5037 
 
 5047 
 
 5057 
 
 5067 
 
 5077 
 
 38 
 
 3 
 
 4 
 
 4 
 
 
 039 
 
 5086 
 
 5096 
 
 5106 
 
 5116 
 
 5126 
 
 5136 
 
 5146 
 
 5156 
 
 5165 
 
 5175 
 
 39 
 
 4 
 
 4 
 
 4 
 
 
 040 
 
 04 5185 
 
 5195 
 
 5205 
 
 5215 
 
 5225 
 
 5235 
 
 5244 
 
 5254 
 
 5264 
 
 5274 
 
 40 
 
 4 
 
 4 
 
 4 
 
 
 041 
 
 5284 
 
 5294 
 
 5304 
 
 5314 
 
 5324 
 
 5334 
 
 5344 
 
 5353 
 
 5:5(;:5 
 
 5373 
 
 41 
 
 4 
 
 4 
 
 5 
 
 
 042 
 
 5383 
 
 5393 
 
 5403 
 
 5413 
 
 5423 
 
 5433 
 
 5443 
 
 5453 
 
 5463 
 
 5473 
 
 42 
 
 4 
 
 4 
 
 5 
 
 
 043 
 
 5483 
 
 5492 
 
 5502 
 
 5512 
 
 5522 
 
 5532 
 
 5542 
 
 5552 
 
 5502 
 
 5572 
 
 43 
 
 4 
 
 4 
 
 5 
 
 
 044 
 
 5582 
 
 5592 
 
 5602 
 
 5612 
 
 5622 
 
 5632 
 
 5642 
 
 5652 
 
 5662 
 
 5672 
 
 44 
 
 4 
 
 4 
 
 5 
 
 
 045 
 
 5682 
 
 5092 
 
 5702 
 
 5712 
 
 5722 
 
 5732 
 
 5742 
 
 5752 
 
 5762 
 
 5772 
 
 45 
 
 4 
 
 4 
 
 5 
 
 
 046 
 
 5782 
 
 5792 
 
 5802 
 
 5812 
 
 5822 
 
 5832 
 
 5842 
 
 5852 
 
 5862 
 
 5872 
 
 46 
 
 4 
 
 5 
 
 5 
 
 
 
 047 
 
 5882 
 
 5892| 5902 
 
 5912 
 
 5922 
 
 5932 
 
 5912 
 
 5952 
 
 5962 
 
 5972 
 
 47 
 
 4 
 
 r 
 
 5 
 
 o 
 
 048 
 
 5982 
 
 5992 
 
 6002 
 
 6012 
 
 6022 
 
 0033 
 
 0043 
 
 6053 
 
 6063 
 
 6073 
 
 48 
 
 4 
 
 5 
 
 5 
 
 c 
 
 049 
 
 6083 
 
 6093 
 
 6103 
 
 6113 
 
 6123 
 
 6133 
 
 6143 
 
 6153 
 
 6163 
 
 6174 
 
 49 
 
 4 
 
 r 
 
 5 
 
 G 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9 
 
 10 
 
 11 
 
 12 
 
 [6] 
 
TABLE L 
 
 Log 
 
 Log (1 + x) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 6 7 8 
 
 9 9101112 
 
 r-o5o 
 
 046184 6194 
 
 6204 6214 6224 
 
 t;234 6244 625-, 
 
 
 4556 
 
 051 
 
 6285 6295 6305 6315 6325 
 
 >345 6355 6366 
 
 6376 51 5 5 6 6 
 
 
 6386 6396 6406 6416 6426 
 
 '^447 6457 6467 
 
 6477 52 5 5 6 6 
 
 053 
 
 (3497 650- 
 
 6538 6548; 6558 
 
 6579 53 : c 6 6 
 
 054 
 
 6559 6599 60 
 
 6619 
 
 6630 
 
 (j(3-^0 6*35'. 6660 '3670 
 
 6680 54 5 5 6 6 
 
 055 
 
 6691 6701 
 
 6711 
 
 6721 
 
 6731 
 
 6742 6752 6762 6772 
 
 07-2 V; - C 6 - 
 
 056 
 
 6793 6803 681; 
 
 6834 
 
 6844 6-54 6864 6874 
 
 6555 ; 56 5067 
 
 057 
 
 196 6905 6915 
 
 692(3 
 
 6936 
 
 6946 6956 6907 0977 
 
 '57 3007 
 
 
 6997 7008 7015 
 
 1 _- 
 
 
 7049 7059 70691 7080 
 
 7090 58 5 6 7 
 
 059 
 
 7100 7110 7121 
 
 7131 
 
 7141 
 
 7151 7162 7172 7182 
 
 7193 59| a 6j 6| 7 
 
 060 
 
 04 7203 7213 
 
 
 7244 
 
 -_- :_ -_" ~--~ 
 
 7296 60 1 5J el 7J 7 
 
 061 
 
 7306 7316 7327 
 
 7337 
 
 7347 
 
 7358 7368 7378 7389 
 
 7399161 5 61 7i 7 
 
 062 
 
 
 
 7430 
 
 7440 
 
 7451 
 
 7461 7471 746' " 
 
 - - . 
 
 063 
 
 7513 
 
 7523 
 
 
 7544 
 
 _.. 
 
 - - - 
 
 
 064 
 
 7617 
 
 ' 27 
 
 7637 
 
 7648 
 
 
 76691 7679 7689 7700 
 
 
 065 
 
 na 
 
 7731 7741 
 
 7752 
 
 
 7773 7783 7793 7804 
 
 
 066 
 
 067 
 
 7929 
 
 7846 
 7940 7950 
 
 7960 
 
 7971 
 
 -Q^| ygQO ^',0 '^01% 
 
 
 068 
 
 8034 
 
 8044 
 
 8065 
 
 
 8086 8097 8107 8118 
 
 
 069 
 
 5139 
 
 8149 
 
 8160 
 
 a - 
 
 
 8 1 .:*: 82 i -212 -223 
 
 8233 09, 6788 
 
 070 
 
 
 
 
 
 8286 8296! 8307 8317; 8328 
 
 8338 70 6 7 8 8 
 
 071 
 
 834, 
 
 -391 
 
 
 
 072 
 
 
 
 
 073 
 
 - ... , 1 .,;;- .. _; , ; .; ... V 
 
 505." 1 73 7789 
 
 074 
 
 8666 8< 7 *m 87081 8719 872^874^ 8751 
 
 - 7 r ~ ~ 7 - *"' 
 
 075 
 
 877: -:u 8-1 J -2' 8830 -,-' 8-"7 
 
 v<;^ 75 7859 
 
 076 
 
 ^v~v ^ v. , , v , i , , , ^ 
 
 8'.C- 7'' 7 8 8 I 1 
 
 (C7 
 
 
 - 77 7 - - T- 
 
 078 
 
 909:. 0145] 915Q 9ied 9177 
 
 
 079 
 
 9199 9209 9220 9231 9241 9252J 9263J 9274] 9284 
 
 
 080 
 
 04 9306 
 
 3 9349 
 
 936(J 937(J 9381 9392 
 
 . r - ~ ^ o ~ . 
 
 081 
 
 941: 
 
 9435 
 
 9446 
 
 
 9510i 8ll 7l 81 9 1( 
 
 52 
 
 9521 1 9532 
 
 
 9553 9564 
 
 
 9615 82 7 8 910 
 
 083 
 
 9029 
 
 9640 
 
 9650 
 
 9661 
 
 9683 9094 9704 9715 
 
 9720 5o 7 5 910 
 
 054 
 
 9737 
 
 
 9759 
 
 9769 9780 
 
 
 9834 84 8 8 910 
 
 08: 
 
 9845 
 
 
 9867 
 
 
 9899 991(J 9921 9932 
 
 ./.- 8:- 5 5 oi' 
 
 08C 
 067 
 
 9954 
 05 0063 
 
 0073 - 
 
 - 
 0095 0106 
 
 0117 0128 0139 015? 
 
 o052|8d 9 9 910 
 N 01 61187 8 9|1010 
 
 
 0172 
 
 
 0193 
 
 0204 
 
 0215 
 
 
 4o270|6d 8J 911011 
 
 - 
 
 
 0292 
 
 
 0335 0346 0357 0368 
 
 0379l8 ^ 9J1011 
 
 
 05 0390 
 
 0401 
 
 
 :~" 
 
 J0489J9o| 8J 910J11 
 
 09 
 
 0500 
 
 0511 0522 
 
 0533 
 
 
 0555 0500 0577 058^ 
 
 059991^8 91011 
 
 09 
 
 0610 
 
 0621 
 
 
 0643 
 
 
 070992 5 91011 
 
 
 OQ 
 
 75 
 BO 
 
 ' . 
 
 
 0-1993 8 91011 
 > 093094, 8 91011 
 
 09 
 09 
 
 ->2|o963|oi~ - 1018J102S 
 1 052 106a 1074 1085 1096 1 1141 
 
 hodd 9101011 
 
 Jll5ll9d 910|1112 
 
 09 
 
 IK- : 'I 1218| 1229J 1240| 1251 
 
 
 09 
 
 1274 
 
 
 12901307 13181 13291 13411 135211361 
 
 
 09 
 
 
 1390 
 
 1408 141C 
 
 148 
 
 1441J 1452J 1463 
 
 
 
 
 
 1 
 
 2 1 3 
 
 4 
 
 5 6 7 S 
 
 i 9 1 1*10,1112 
 
 [7] 
 
TABLE I. 
 
 Log 
 
 Log (l + o;) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 11 
 
 12 
 
 13 
 
 14 
 
 Tioo 
 
 0-05 1497 
 
 1508 
 
 1519 
 
 1530 
 
 1542 
 
 1553 
 
 1564 
 
 1575 
 
 1586 
 
 1598 
 
 00 
 
 
 
 
 
 
 
 
 
 101 
 
 1609 
 
 1620 
 
 1631 
 
 1642 
 
 1654 
 
 1665 
 
 1676 
 
 1687 
 
 1699 
 
 1710 
 
 01 
 
 
 
 
 
 
 
 
 
 102 
 
 1721 
 
 1732 
 
 1743 
 
 1755 
 
 1766 
 
 1777 
 
 1788 
 
 1800 
 
 1811 
 
 1822 
 
 02 
 
 
 
 
 
 
 
 
 
 103 
 
 1833 
 
 1845 
 
 1856 
 
 1867 
 
 1878 
 
 1890 
 
 1901 
 
 1912 
 
 1923 
 
 1935 
 
 03 
 
 
 
 
 
 
 
 
 
 104 
 
 1946 
 
 1957 
 
 1969 
 
 1980 
 
 1991 
 
 2002 
 
 2014 
 
 2025 
 
 2036 
 
 2048 
 
 04 
 
 
 
 
 
 1 
 
 1 
 
 105 
 
 2059 
 
 2070 
 
 2081 
 
 2093 
 
 2104 
 
 2115 
 
 2127 
 
 2138 
 
 2149 
 
 2161 
 
 05 
 
 1 
 
 1 
 
 1 
 
 1 
 
 106 
 
 2172 
 
 2183 
 
 2195 
 
 2206 
 
 2217 
 
 2229 
 
 2240 
 
 2251 
 
 2263 
 
 2274 
 
 06 
 
 1 
 
 1 
 
 1 
 
 1 
 
 107 
 
 2285 
 
 2297 
 
 2308 
 
 2319 
 
 2331 
 
 2342 
 
 2353 
 
 2365 
 
 2376 
 
 2387 
 
 07 
 
 1 
 
 1 
 
 1 
 
 1 
 
 108 
 
 2399 
 
 2410 
 
 2422 
 
 2433 
 
 2444 
 
 2456 
 
 2467 
 
 2478 
 
 2490 
 
 2501 
 
 08 
 
 1 
 
 1 
 
 1 
 
 1 
 
 109 
 
 2513 
 
 2524 
 
 2535 
 
 2547 
 
 2558 
 
 2570 
 
 2581 
 
 2592 
 
 2604 
 
 2615 
 
 09 
 
 1 
 
 1 
 
 1 
 
 1 
 
 110 
 
 05 2627 
 
 2638 
 
 2649 
 
 2661 
 
 2672 
 
 2684 
 
 2695 
 
 2707 
 
 2718 
 
 2729 
 
 10 
 
 1 
 
 1 
 
 1 
 
 1 
 
 111 
 
 2741 
 
 2752 
 
 2764 
 
 2775 
 
 2787 
 
 2798 
 
 2809 
 
 2821 
 
 2832 
 
 2844 
 
 11 
 
 1 
 
 1 
 
 1 
 
 '2 
 
 112 
 
 2855 
 
 2867 
 
 2878 
 
 2890 
 
 2901 
 
 2913 
 
 2924 
 
 2936 
 
 2947 
 
 2959 
 
 12 
 
 1 
 
 1 
 
 2 
 
 2 
 
 113 
 
 2970 
 
 2982 
 
 2993 
 
 3004 
 
 3016 
 
 3027 
 
 3039 
 
 3050 
 
 3062 
 
 3073 
 
 13 
 
 1 
 
 2 
 
 2 
 
 2 
 
 114 
 
 3085 
 
 3096 
 
 3108 
 
 3119 
 
 3131 
 
 3143 
 
 3154 
 
 3166 
 
 3177 
 
 3189 
 
 14 
 
 2 
 
 2 
 
 2 
 
 2 
 
 115 
 
 3200 
 
 3212 
 
 3223 
 
 3235 
 
 3246 
 
 3258 
 
 3269 
 
 3281 
 
 3292 
 
 3304 
 
 15 
 
 2 
 
 2 
 
 2 
 
 2 
 
 116 
 
 3316 
 
 3327 
 
 3339 
 
 3350 
 
 3362 
 
 3373 
 
 3385 
 
 3396 
 
 3408 
 
 3420 
 
 ie 
 
 2 
 
 2 
 
 
 2 
 
 117 
 
 3431 
 
 3443 
 
 3454 
 
 3466 
 
 3478 
 
 3489 
 
 3501 
 
 3512 
 
 3524 
 
 3535 
 
 17 
 
 2 
 
 2 
 
 
 
 118 
 
 3547 
 
 3559 
 
 3570 
 
 3582 
 
 3593 
 
 3605 
 
 3617 
 
 3628 
 
 3640 
 
 3652 
 
 18 
 
 2 
 
 2 
 
 
 
 119 
 
 3663 
 
 3675 
 
 3686 
 
 3698 
 
 3710 
 
 3721 
 
 3733 
 
 3745 
 
 3756 
 
 3768 
 
 19 
 
 2 
 
 2 
 
 
 
 120 
 
 05 3780 
 
 3791 
 
 3803 
 
 3814 
 
 3826 
 
 3838 
 
 3849 
 
 3861 
 
 3873 
 
 3884 
 
 20 
 
 ^ 
 
 2 
 
 
 
 121 
 
 3896 
 
 3908 
 
 3919 
 
 3931 
 
 3943 
 
 3955 
 
 3966 
 
 3978 
 
 3990 
 
 4001 
 
 21 
 
 2 
 
 3 
 
 
 
 122 
 
 4013 
 
 4025 
 
 4036 
 
 4048 
 
 4060 
 
 4071 
 
 4083 
 
 4095 
 
 4107 
 
 4118 
 
 22 
 
 2 
 
 3 
 
 
 
 123 
 
 4130 
 
 4142 
 
 4153 
 
 4165 
 
 4177 
 
 4189 
 
 4200 
 
 4212 
 
 4224 
 
 4236 
 
 23 
 
 
 3 
 
 
 
 124 
 
 4247 
 
 4259 
 
 4271 
 
 4283 
 
 4294 
 
 4306 
 
 4318 
 
 4330 
 
 4341 
 
 4353 
 
 24 
 
 
 3 
 
 
 
 125 
 
 4365 
 
 4377 
 
 4388 
 
 4400 
 
 4412 
 
 4424 
 
 4436 
 
 4447 
 
 4459 
 
 4471 
 
 25 
 
 
 3 
 
 
 
 126 
 
 4483 
 
 4494 
 
 4506 
 
 4518 
 
 4530 
 
 4542 
 
 4553 
 
 4565 
 
 4577 
 
 4589 
 
 26 
 
 
 3 
 
 
 /j 
 
 127 
 
 4601 
 
 4612 
 
 4624 
 
 4636 
 
 4648 
 
 4660 
 
 4672 
 
 4683 
 
 4695 
 
 4707 
 
 27 
 
 
 3 
 
 4 
 
 4 
 
 128 
 
 4719 
 
 4731 
 
 4743 
 
 4754 
 
 4766 
 
 4778 
 
 4790 
 
 4802 
 
 4814 
 
 4826 
 
 28 
 
 
 3 
 
 4 
 
 4 
 
 129 
 
 4837 
 
 4849 
 
 4861 
 
 4873 
 
 4885 
 
 4897 
 
 4909 
 
 4921 
 
 4932 
 
 4944 
 
 29 
 
 < 
 
 3 
 
 4 
 
 4 
 
 130 
 
 05 4956 
 
 4968 
 
 4980 
 
 4992 
 
 5004 
 
 5016 
 
 5028 
 
 5039 
 
 5051 
 
 5063 
 
 30 
 
 3 
 
 4 
 
 4 
 
 4 
 
 131 
 
 5075 
 
 5087 
 
 5099 
 
 5111 
 
 5123 
 
 5135 
 
 5147 
 
 5159 
 
 5171 
 
 5182 
 
 31 
 
 3 
 
 4 
 
 4 
 
 4 
 
 132 
 
 5194 
 
 5206 
 
 5218 
 
 5230 
 
 5242 
 
 5254 
 
 5266 
 
 5278 
 
 5290 
 
 5302 
 
 32 
 
 4 
 
 4 
 
 4 
 
 4 
 
 133 
 
 5314 
 
 5326 
 
 5338 
 
 5350 
 
 5362 
 
 5374 
 
 5386 
 
 5398 
 
 5410 
 
 5422 
 
 33 
 
 4 
 
 4 
 
 4 
 
 
 134 
 
 5434 
 
 5446 
 
 5458 
 
 5470 
 
 5482 
 
 5494 
 
 5506 
 
 5517 
 
 5530 
 
 5542 
 
 34 
 
 4 
 
 4 
 
 4 
 
 
 135 
 
 5554 
 
 5566 
 
 5578 
 
 5590 
 
 5602 
 
 5614 
 
 5626 
 
 5638 
 
 5650 
 
 5662 
 
 35 
 
 4 
 
 4 
 
 Ir 
 o 
 
 
 136 
 
 5674 
 
 5686 
 
 5698, 
 
 5710 
 
 5722 
 
 5734 
 
 5746 
 
 5758 
 
 5770 
 
 5782 
 
 36 
 
 4 
 
 4 
 
 5 
 
 
 137 
 
 5794 
 
 5806 
 
 5818 
 
 5830 
 
 5842 
 
 5854 
 
 5867 
 
 5879 
 
 5891 
 
 5903 
 
 37 
 
 4 
 
 4 
 
 r. 
 
 
 138 
 
 5915 
 
 5927 
 
 5939 
 
 5951 
 
 5963 
 
 5975 
 
 5987 
 
 5999 
 
 6012 
 
 6024 
 
 38 
 
 4 
 
 5 
 
 5 
 
 
 139 
 
 6036 
 
 6048 
 
 6060 
 
 6072 
 
 6084 
 
 6096 
 
 6108 
 
 6121 
 
 6133 
 
 6145 
 
 39 
 
 4 
 
 5 
 
 5 
 
 
 140 
 
 05 6157 
 
 6169 
 
 6181 
 
 6193 
 
 6205 
 
 6218 
 
 6230 
 
 6242 
 
 6254 
 
 6266 
 
 40 
 
 4 
 
 5 
 
 5 
 
 6 
 
 141 
 
 6278 
 
 6290 
 
 6303 
 
 6315 
 
 6327 
 
 6339 
 
 6351 
 
 6363 
 
 6376 
 
 6388 
 
 41 
 
 5 
 
 5 
 
 5 
 
 6 
 
 142 
 
 6400 
 
 6412 
 
 6424 
 
 6437 
 
 6449 
 
 6461 
 
 6473 
 
 6485 
 
 6498 
 
 6510 
 
 42 
 
 5 
 
 5 
 
 5 
 
 6 
 
 143 
 
 6522 
 
 6534 
 
 6546 
 
 6559 
 
 6571 
 
 6583 
 
 6595 
 
 6607 
 
 6620 
 
 6632 
 
 43 
 
 5 
 
 5 
 
 6 
 
 6 
 
 144 
 
 6644 
 
 6656 
 
 6669 
 
 6681 
 
 6693 
 
 6705 
 
 6717 
 
 6730 
 
 6742 
 
 6754 
 
 44 
 
 5 
 
 5 
 
 6 
 
 6 
 
 145 
 
 6766 
 
 6779 
 
 6791 
 
 6803 
 
 6816 
 
 6828 
 
 6840 
 
 6852 
 
 6865 
 
 6877 
 
 45 
 
 6 
 
 5 
 
 6 
 
 6 
 
 146 
 
 6889 
 
 6901 
 
 6914 
 
 6926 
 
 6938 
 
 6951 
 
 6963 
 
 6975 
 
 6987 
 
 7000 
 
 4(5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 147 
 
 7012 
 
 7024 
 
 7037 
 
 7049 
 
 7061 
 
 7074 
 
 7086 
 
 7098 
 
 7111 
 
 7123 
 
 47 
 
 5 
 
 6 
 
 6 
 
 7 
 
 148 
 
 7135 
 
 7148 
 
 7160 
 
 7172 
 
 7185 
 
 7197 
 
 7209 
 
 7222 
 
 7234 
 
 7246 
 
 48 
 
 5 
 
 6 
 
 6 
 
 7 
 
 149 
 
 7259 
 
 7271 
 
 7283 
 
 7296 
 
 7308 
 
 7320 
 
 7333 
 
 7345 
 
 7357 
 
 7370 
 
 49 
 
 5 
 
 6 
 
 6 
 
 7 
 
 
 
 
 1 
 
 9 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 11 
 
 L2 
 
 13 
 
 14 
 
 [8] 
 
TABLE I. 
 
 Log 
 
 Log(l + a) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 11 
 
 12 
 
 13 
 
 14 
 
 1-150 
 
 0'05 7382 
 
 7395 
 
 7407 
 
 7419 
 
 7432 
 
 7444 
 
 7457 
 
 7469 
 
 7481 
 
 7494 
 
 50 
 
 5 
 
 6 
 
 6 
 
 7 
 
 151 
 
 7506 
 
 7519 
 
 7531 
 
 7543 
 
 7556 
 
 7568 
 
 7581 
 
 7593 
 
 7605 
 
 7618 
 
 31 
 
 6 
 
 6 
 
 7 
 
 7 
 
 152 
 
 7630 
 
 7643 
 
 7655 
 
 7668 
 
 7680 
 
 7692 
 
 7705 
 
 7717 
 
 7730 
 
 7742 
 
 52 
 
 6 
 
 6 
 
 *- 
 t 
 
 7 
 
 153 
 
 7755 7767 
 
 7780 
 
 7792 
 
 7804 
 
 7817 
 
 7829 
 
 7842 
 
 7854 
 
 7867 
 
 53 
 
 6 
 
 6 
 
 7 
 
 7 
 
 154 
 
 7879 
 
 7892 
 
 7904 
 
 7917 
 
 7929 
 
 7942 
 
 7954 
 
 7967 
 
 7979 
 
 7992 
 
 54 
 
 6 
 
 6 
 
 7 
 
 8 
 
 155 
 
 8004 
 
 8017 
 
 8029 
 
 8042 
 
 8054 
 
 8067 
 
 8079 
 
 8092 
 
 8104 
 
 8117 
 
 55 
 
 6 
 
 7 
 
 7 
 
 8 
 
 156 
 
 8129 8142 
 
 8154 
 
 8167 
 
 8179 
 
 8192 
 
 8205 
 
 8217 
 
 8230 
 
 8242 
 
 56 
 
 6 
 
 H- 
 / 
 
 7 
 
 g 
 
 157 
 
 8255 8267 
 
 8280 
 
 8292 
 
 8305 
 
 8318 
 
 8330 
 
 8343 
 
 8355 
 
 8368 
 
 57 
 
 6 
 
 / 
 
 7 
 
 8 
 
 158 
 
 8380 8393 
 
 8406 
 
 8418 
 
 8431 
 
 8443 
 
 8456 
 
 8469 
 
 8481 
 
 8494 
 
 58 
 
 6 
 
 7 
 
 8 
 
 8 
 
 159 
 
 8506 8519 
 
 8532 
 
 8544 
 
 8557 
 
 8569 
 
 8582 
 
 8595 
 
 8607 
 
 8620 
 
 59 
 
 6 
 
 7 
 
 8 
 
 8 
 
 160 
 
 05 8632 
 
 8645 
 
 8658 
 
 8670 
 
 8683 
 
 8696 
 
 8708 
 
 8721 
 
 8734 
 
 8746 
 
 60 
 
 7 
 
 7 
 
 8 
 
 8 
 
 161 
 
 8759 8772 
 
 8784 
 
 8797 
 
 8810 
 
 8822 
 
 8835 
 
 8848 
 
 8860 
 
 8873 
 
 61 
 
 7 
 
 7 
 
 8 
 
 9 
 
 162 
 
 8886 8898 
 
 8911 
 
 8924 
 
 8936 
 
 8949 
 
 8962 
 
 8974 
 
 8987 
 
 9000 
 
 62 
 
 7 
 
 7 
 
 8 
 
 9 
 
 163 
 
 9012 9025 
 
 9038 
 
 9051 
 
 9063 
 
 9076 
 
 9089 
 
 9101 
 
 9114 
 
 9127 
 
 63 
 
 7 
 
 8 
 
 8 
 
 9 
 
 164 
 
 9140 
 
 9152 
 
 9165 
 
 9178 
 
 9190 
 
 9203 
 
 9216 
 
 9229 
 
 9242 
 
 9254 
 
 64 
 
 7 
 
 8 
 
 8 
 
 9 
 
 165 
 
 9267 
 
 9280 
 
 9293 
 
 9305 
 
 9318 
 
 9331 
 
 9344 
 
 9356 
 
 9369 
 
 9382 
 
 65 
 
 7 
 
 8 
 
 8 
 
 9 
 
 166 
 
 9395 
 
 9408 
 
 9420 
 
 9433 
 
 9446 
 
 9459 
 
 9472 
 
 9484 
 
 9497 
 
 9510 
 
 66 
 
 7 
 
 8 
 
 9 
 
 9 
 
 167 
 
 9523 
 
 9536 
 
 9548 
 
 9561 
 
 9574 
 
 9587 
 
 9600 
 
 9612 
 
 9625 
 
 9638 
 
 67 
 
 7 
 
 8 
 
 9 
 
 9 
 
 168 
 
 9651 
 
 9664 
 
 9677 
 
 9689 
 
 9702 
 
 9715 
 
 9728 
 
 9741 
 
 9754 
 
 9767 
 
 68 
 
 7 
 
 8 
 
 9 
 
 10 
 
 169 
 
 9779 
 
 9792 
 
 9805 
 
 9818 
 
 9831 
 
 9844 
 
 9857 
 
 9870 
 
 9882 
 
 9895 
 
 69 
 
 8 
 
 8 
 
 9 
 
 1C 
 
 170 
 
 05 9908 9921 
 
 9934 
 
 9947 
 
 9960 
 
 9973 
 
 9986 
 
 9998 
 
 oOll 
 
 0024 
 
 70 
 
 8 
 
 8 
 
 9 
 
 10 
 
 171 
 
 06 0037 0050 
 
 0063 
 
 0076 
 
 0089 
 
 0102 
 
 0115 
 
 0128 
 
 0141 
 
 0153 
 
 71 
 
 8 
 
 9 
 
 9 
 
 10 
 
 172 
 
 01661 0179 
 
 0192 
 
 0205 
 
 0218 
 
 0231 
 
 0244 
 
 0257 
 
 0270 
 
 0283 
 
 72 
 
 8 
 
 9 
 
 9 
 
 10 
 
 173 
 
 0296 
 
 0309 
 
 0322 
 
 0335 
 
 0348 
 
 0361 
 
 0374 
 
 0387 
 
 0400 
 
 0413 
 
 73 
 
 8 
 
 9 
 
 9 
 
 1C 
 
 174 
 
 0426 
 
 0439 
 
 0452 
 
 0465 
 
 0478 
 
 0491 
 
 0504 
 
 0517 
 
 0530 
 
 0543 
 
 74 
 
 8 
 
 c 
 
 10 
 
 1C 
 
 175 
 
 0556 
 
 0569 
 
 0582 
 
 0595 
 
 0608 
 
 0621 
 
 0634 
 
 0647 
 
 0660 
 
 0673 
 
 75 
 
 8 
 
 c 
 
 10 
 
 11 
 
 176 
 
 0686 
 
 0699 
 
 0712 
 
 0725 
 
 0738 
 
 0751 
 
 0764 
 
 0777 
 
 0790 
 
 0803 
 
 76 
 
 8 
 
 9 
 
 10 
 
 11 
 
 177 
 
 0816 0830 
 
 0843 
 
 0856 
 
 0869 
 
 0882 
 
 0895 
 
 0908 
 
 0921 
 
 0934 
 
 77 
 
 8 
 
 c 
 
 10 
 
 11 
 
 178 
 
 0947 
 
 0960 
 
 0973 
 
 0987 
 
 1000 
 
 1013 
 
 1026 
 
 1039 
 
 1052 
 
 1065 
 
 78 
 
 9 
 
 9 
 
 1C) 
 
 11 
 
 179 
 
 1078 
 
 1091 
 
 1105 
 
 1118 
 
 1131 
 
 1144 
 
 1157 
 
 1170 
 
 1183 
 
 1197 
 
 79 
 
 9 
 
 c 
 
 10 
 
 11 
 
 180 
 
 06 1210 
 
 1223 
 
 1236 
 
 1249 
 
 1262 
 
 1275 
 
 1289 
 
 1302 
 
 1315 
 
 1328 
 
 8C 
 
 9 
 
 10 
 
 10 
 
 11 
 
 181 
 
 1341 
 
 1354 
 
 1368 
 
 1381 
 
 1394 
 
 1407 
 
 1420 
 
 1434 
 
 1447 
 
 1460 
 
 81 
 
 9 
 
 1C 
 
 11 
 
 11 
 
 182 
 
 1473 
 
 1486 
 
 1500 
 
 1513 
 
 1526 
 
 1539 
 
 1552 
 
 1566 
 
 1579 
 
 1592 
 
 82 
 
 9 
 
 10 
 
 11 
 
 11 
 
 183 
 
 1605 
 
 1618 
 
 1632 
 
 1645 
 
 1658 
 
 1671 
 
 1685 
 
 1698 
 
 1711 
 
 1724 
 
 83 
 
 9 
 
 1C 
 
 11 
 
 12 
 
 184 
 
 1738 
 
 1751 
 
 1764 
 
 1777 
 
 1791 
 
 1804 
 
 1817 
 
 1830 
 
 1844 
 
 1857 
 
 84 
 
 9 
 
 1C 
 
 11 
 
 1$ 
 
 185 
 
 1870 
 
 1884 
 
 1897 
 
 1910 
 
 1923 
 
 1937 
 
 1950 
 
 1963 
 
 1977 
 
 1990 
 
 85 
 
 9 
 
 1C 
 
 11 
 
 12 
 
 186 
 
 2003 
 
 2016 
 
 2030 
 
 2043 
 
 2056 
 
 2070 
 
 2083 
 
 2096 
 
 2110 
 
 2123 
 
 86 
 
 9 
 
 1C 
 
 11 
 
 12 
 
 187 
 
 2136 
 
 2150 
 
 2163 
 
 2176 
 
 2190 
 
 2203 
 
 2216 
 
 2230 
 
 2243 
 
 2256 
 
 87 
 
 10|lO 
 
 11 
 
 12 
 
 188 
 
 2270 
 
 2283 
 
 2297 
 
 2310 
 
 2323 
 
 2337 
 
 2350 
 
 2363 
 
 2377 
 
 2390 
 
 88 
 
 1011 
 
 11 
 
 12 
 
 189 
 
 2404 
 
 2417 
 
 2430 
 
 2444 
 
 2457 
 
 2470 
 
 2484 
 
 2497 
 
 2511 
 
 2524 
 
 89 
 
 10-11 
 
 12 
 
 12 
 
 190 
 
 06 2537 
 
 2551 
 
 2564 
 
 2578 
 
 2591 
 
 2605 
 
 2618 
 
 2631 
 
 2645 
 
 2658 
 
 90 
 
 10 
 
 11 
 
 12J1 
 
 191 
 
 2672 
 
 2685 
 
 2699 
 
 2712 
 
 2726 
 
 2739 
 
 2752 
 
 2766 
 
 2779 
 
 2793 
 
 91 
 
 10 
 
 11 
 
 12 1 
 
 192 
 
 2806 
 
 2820 
 
 2833 
 
 2847 
 
 2860 
 
 2874 
 
 2887 
 
 2901 
 
 2914 
 
 2928 
 
 92 
 
 10 
 
 11 
 
 1212 
 
 193 
 
 2941 2955 
 
 2968 
 
 2982 
 
 2995 
 
 3009 
 
 3022 
 
 3036 
 
 3049 
 
 3063 
 
 93ilO 
 
 11 
 
 12 lc 
 
 194 
 
 3076 
 
 3090 
 
 3103 
 
 3117 
 
 3130 
 
 3144 
 
 3157 
 
 3171 
 
 3184 
 
 3198 
 
 94 
 
 10 
 
 11 
 
 i2ji 
 
 195 
 
 3211 
 
 3225 
 
 3238 
 
 3252 
 
 3266 
 
 3279 
 
 3293 
 
 3306 
 
 3320 
 
 3333 
 
 95 
 
 10 
 
 11 
 
 12 1 
 
 196 
 
 3347 
 
 3361 
 
 3374 
 
 3388 
 
 3401 
 
 3415 
 
 3428 
 
 3442 
 
 3456 
 
 3469 
 
 9( 
 
 11 
 
 1212U 
 
 197 
 
 3483 
 
 3496 
 
 3510 
 
 3524 
 
 3537 
 
 3551 
 
 3564 
 
 3578 
 
 3592 
 
 3605 
 
 97 
 
 11 
 
 121314 
 
 198 
 
 3619 
 
 3633 
 
 3646 
 
 366( 
 
 3673 
 
 3687 
 
 3701 
 
 3714 
 
 3728 
 
 3742 
 
 98 
 
 11 
 
 121314 
 
 199 
 
 3755 
 
 3769 
 
 3783 
 
 3796 
 
 3810 
 
 3824 
 
 3837 
 
 3851 
 
 3865 
 
 3878 
 
 99 
 
 11 
 
 121314 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 11 
 
 12 
 
 13 
 
 14 
 
 F91 
 
TABLE I. 
 
 Log 
 
 Log (l + x) 
 
 Pro. Parts. 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 1-20 
 
 0-06 3892 
 
 3906 
 
 3919 
 
 3933 
 
 3947 
 
 396 
 
 3974 
 
 3988 
 
 400 
 
 4015 
 
 00 
 
 
 
 ( 
 
 
 
 
 
 
 
 20 
 
 4029 
 
 4043 
 
 4056 
 
 4070 
 
 4084 
 
 409 
 
 4111 
 
 4125 
 
 4139 
 
 4152 
 
 01 
 
 
 
 
 
 
 
 
 
 
 
 202 
 
 4166 
 
 4180 
 
 4194 
 
 4207 
 
 4221 
 
 423 
 
 4249 
 
 4262 
 
 427 
 
 4290 
 
 02 
 
 
 
 
 
 
 
 
 
 o 
 
 203 
 
 4304 
 
 4317 
 
 4331 
 
 4345 
 
 4359 
 
 4373 
 
 4386 
 
 4400 
 
 4414 
 
 4428 
 
 03 
 
 
 
 ( 
 
 
 
 
 
 ] 
 
 204 
 
 4441 
 
 4455 
 
 4469 
 
 4483 
 
 4497 
 
 451 
 
 4524 
 
 4538 
 
 4552 
 
 4566 
 
 04 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ''. 
 
 205 
 
 4579 
 
 4593 
 
 4607 
 
 4621 
 
 4635 
 
 464 
 
 4662 
 
 4676 
 
 4690 
 
 4704 
 
 05 
 
 1 
 
 1 
 
 1 
 
 1 
 
 - 
 
 206 
 
 4718 
 
 4732 
 
 4745 
 
 4759 
 
 4773 
 
 478 
 
 4801 
 
 4815 
 
 4829 
 
 4842 
 
 06 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ] 
 
 207 
 
 4856 
 
 4870 
 
 4884 
 
 4898 
 
 4912 
 
 492 
 
 4940 
 
 4954 
 
 496 
 
 4981 
 
 07 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 208 
 
 4995 
 
 5009 
 
 5023 
 
 5037 
 
 5051 
 
 506 
 
 5079 
 
 5093 
 
 510 
 
 5120 
 
 08 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ] 
 
 209 
 
 5134 
 
 5148 
 
 5162 
 
 5176 
 
 5190 
 
 5204 
 
 5218 
 
 5232 
 
 524 
 
 5260 
 
 09 
 
 1 
 
 1 
 
 1 
 
 1 
 
 I 
 
 210 
 
 06 5274 
 
 5288 
 
 5302 
 
 5316 
 
 5330 
 
 5344 
 
 5358 
 
 5372 
 
 538 
 
 5399 
 
 10 
 
 1 
 
 1 
 
 1 
 
 2 
 
 o 
 
 211 
 
 5413 
 
 5427 
 
 5441 
 
 5455 
 
 5469 
 
 5483 
 
 5497 
 
 5511 
 
 552 
 
 5539 
 
 11 
 
 1 
 
 2 
 
 2 
 
 2 
 
 . 
 
 212 
 
 5553 
 
 5567 
 
 5581 
 
 5595 
 
 5609 
 
 5624 
 
 5638 
 
 5652 
 
 566 
 
 5680 
 
 12 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 213 
 
 5694 
 
 5708 
 
 5722 
 
 5736 
 
 5750 
 
 5764 
 
 5778 
 
 5792 
 
 580 
 
 5820 
 
 13 
 
 2 
 
 2 
 
 2 
 
 2 
 
 o 
 
 214 
 
 5834 
 
 5848 
 
 5862 
 
 5876 
 
 5890 
 
 5905 
 
 5919 
 
 5933 
 
 594" 
 
 5961 
 
 14 
 
 2 
 
 2 
 
 2 
 
 2 
 
 . 
 
 215 
 
 5975 
 
 5989 
 
 6003 
 
 6017 
 
 6031 
 
 6045 
 
 6060 
 
 6074 
 
 6088 
 
 6102 
 
 15 
 
 2 
 
 2 
 
 2 
 
 2 
 
 r 
 t 
 
 216 
 
 6116 
 
 6130 
 
 6144 
 
 6158 
 
 6173 
 
 6187 
 
 6201 
 
 6215 
 
 6229 
 
 6243 
 
 16 
 
 2 
 
 2 
 
 2 
 
 3 
 
 t 
 
 217 
 
 6257 
 
 6272 
 
 6286 
 
 6300 
 
 6314 
 
 6328 
 
 6342 
 
 6357 
 
 6371 
 
 6385 
 
 17 
 
 2 
 
 2 
 
 3 
 
 3 
 
 I 
 
 218 
 
 6399 
 
 6413 
 
 6427 
 
 6442 
 
 6456 
 
 6470 
 
 6484 
 
 6498 
 
 6513 
 
 6527 
 
 18 
 
 2 
 
 
 3 
 
 3 
 
 c 
 
 219 
 
 6541 
 
 6555 
 
 6569 
 
 6584 
 
 6598 
 
 6612 
 
 6626 
 
 6640 
 
 6655 
 
 6669 
 
 19 
 
 2 
 
 
 3 
 
 a 
 
 3 
 
 220 
 
 06 6683 
 
 6697 
 
 6712 
 
 6726 
 
 6740 
 
 6754 
 
 6769 
 
 6783 
 
 679" 
 
 6811 
 
 20 
 
 j 
 
 
 
 3 
 
 c 
 
 221 
 
 6826 
 
 6840 
 
 6854 
 
 6868 
 
 6883 
 
 6897 
 
 6911 
 
 6926 
 
 6940 
 
 6954 
 
 21 
 
 ', 
 
 
 
 3 
 
 / 
 
 222 
 
 6968 
 
 6983 
 
 6997 
 
 7011 
 
 7026 
 
 7040 
 
 7054 
 
 7068 
 
 7083 
 
 7097 
 
 22 
 
 J 
 
 
 
 4 
 
 4 
 
 223 
 
 7111 
 
 7126 
 
 7140 
 
 7154 
 
 7169 
 
 7183 
 
 7197 
 
 7212 
 
 7226 
 
 7240 
 
 23 
 
 ; 
 
 
 
 4 
 
 ^ 
 
 224 
 
 7255 
 
 7269 
 
 7283 
 
 7298 
 
 7312 
 
 7327 
 
 7341 
 
 7355 
 
 7370 
 
 7384 
 
 24 
 
 3 
 
 
 i 
 
 4 
 
 4 
 
 225 
 
 7398 
 
 7413 
 
 7427 
 
 7442 
 
 7456 
 
 7470 
 
 7485 
 
 7499 
 
 7513 
 
 7528 
 
 25 
 
 *. 
 
 
 / 
 
 4 
 
 4 
 
 226 
 
 7542 
 
 7557 
 
 7571 
 
 7585 
 
 7600 
 
 7614 
 
 7629 
 
 7643 
 
 7658 
 
 7672 
 
 26 
 
 i 
 
 4 
 
 4 
 
 4 
 
 4 
 
 227 
 
 7686 
 
 7701 
 
 7715 
 
 7730 
 
 7744 
 
 7759 
 
 7773 
 
 7788 
 
 7802 
 
 7816 
 
 27 
 
 4 
 
 4 
 
 L 
 
 4 
 
 5 
 
 228 
 
 7831 
 
 7845 
 
 7860 
 
 7874 
 
 7889 
 
 7903 
 
 7918 
 
 7932 
 
 7947 
 
 7961 
 
 28 
 
 i. 
 
 4 
 
 4 
 
 4 
 
 C 
 c 
 
 229 
 
 7976 
 
 7990 
 
 8005 
 
 8019 
 
 8034 
 
 8048 
 
 8063 
 
 8077 
 
 8092 
 
 8106 
 
 29 
 
 L 
 
 4 
 
 ^ 
 
 5 
 
 5 
 
 230 
 
 06 8121 
 
 8135 
 
 8150 
 
 8164 
 
 8179 
 
 8193 
 
 8208 
 
 8222 
 
 8237 
 
 8251 
 
 30 
 
 / 
 
 4 
 
 5 
 
 5 
 
 p 
 
 231 
 
 8266 
 
 8281 
 
 8295 
 
 8310 
 
 8324 
 
 8339 
 
 8353 
 
 8368 
 
 8382 
 
 8397 
 
 31 
 
 4 
 
 4 
 
 5 
 
 5 
 
 5 
 
 232 
 
 8412 
 
 8426 
 
 8441 
 
 8455 
 
 8470 
 
 8484 
 
 8499 
 
 8514 
 
 8528 
 
 8543 
 
 32 
 
 4 
 
 4 
 
 r 
 
 6 
 
 5 
 
 233 
 
 8557 
 
 8572 
 
 8587 
 
 8601 
 
 8616 
 
 8631 
 
 8645 
 
 8660 
 
 8674 
 
 8689 
 
 33 
 
 / 
 
 f: 
 t. 
 
 5 
 
 5 
 
 6 
 
 234 
 
 8704 
 
 8718 
 
 8733 
 
 8748 
 
 8762 
 
 8777 
 
 8791 
 
 8806 
 
 8821 
 
 8835 
 
 34 
 
 4 
 
 5 
 
 5 
 
 5 
 
 6 
 
 235 
 
 8850 
 
 8865 
 
 8879 
 
 8894 
 
 8909 
 
 8923 
 
 8938 
 
 8953 
 
 8967 
 
 8982 
 
 35 
 
 5 
 
 F 
 
 5 
 
 6 
 
 6 
 
 236 
 
 8997 
 
 9012 
 
 9026 
 
 9041 
 
 9056 
 
 9070 
 
 9085 
 
 9100 
 
 9114 
 
 9129 
 
 36 
 
 5 
 
 g 
 
 p. 
 c 
 
 6 
 
 6 
 
 237 
 
 9144 
 
 9159 
 
 9173 
 
 9188 
 
 9203 
 
 9218 
 
 9232 
 
 9247 
 
 9262 
 
 9276 
 
 37 
 
 K 
 
 5 
 
 6 
 
 6 
 
 6 
 
 238 
 
 9291 
 
 9306 
 
 9321 
 
 9335 
 
 9350 
 
 9365 
 
 9380 
 
 9395 
 
 9409 
 
 9424 
 
 38 
 
 5 
 
 e 
 tj 
 
 6 
 
 6 
 
 6 
 
 239 
 
 9439 
 
 9454 
 
 9468 
 
 9483 
 
 9498 
 
 9513 
 
 9528 
 
 9542 
 
 9557 
 
 9572 
 
 39 
 
 c 
 
 O 
 
 c 
 
 <j 
 
 6 
 
 6 
 
 7 
 
 240 
 
 06 9587 
 
 9602 
 
 9616 
 
 9631 
 
 9646 
 
 9661 
 
 9676 
 
 9690 
 
 9705 
 
 9720 
 
 40 
 
 r 
 
 6 
 
 6 
 
 6 
 
 7 
 
 241 
 
 9735 
 
 9750 
 
 9765 
 
 9779 
 
 9794 
 
 9809 
 
 9824 
 
 9839 
 
 9854 
 
 9869 
 
 11 
 
 r; 
 c 
 
 6 
 
 6 
 
 7 
 
 7 
 
 242 
 
 9883 
 
 9898 
 
 9913 
 
 9928 
 
 9943 
 
 9958 
 
 9973 
 
 9988 
 
 0002 
 
 0017 
 
 42 
 
 5 
 
 6 
 
 6 
 
 7 
 
 7 
 
 243 
 
 07 0032 
 
 0047 
 
 0062 
 
 0077 
 
 0092 
 
 0107 
 
 0122 
 
 0137 
 
 0151 
 
 0166 
 
 43 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 244 
 
 0181 
 
 0196 
 
 0211 
 
 0226 
 
 0241 
 
 0256 
 
 0271 
 
 0286 
 
 0301 
 
 0316 
 
 44 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 245 
 
 0331 
 
 0346 
 
 0361 
 
 0376 
 
 0390 
 
 0405 
 
 0420 
 
 0435 
 
 0450 
 
 0465 
 
 45 
 
 6 
 
 6 
 
 7 
 
 7 
 
 8 
 
 246 
 
 0480 
 
 0495 
 
 0510 
 
 0525 
 
 0540 
 
 0555 
 
 0570 
 
 0585 
 
 0600 
 
 0615 
 
 46 
 
 6 
 
 6 
 
 7 
 
 7 
 
 8 
 
 247 
 
 0630 
 
 0645 
 
 0660 
 
 0675 
 
 0690 
 
 0705 
 
 0720 
 
 0735 
 
 0750 
 
 0765 
 
 47 
 
 6 
 
 7 
 
 7 
 
 8 
 
 8 
 
 248 
 
 0780 
 
 0796 
 
 0811 
 
 0826 
 
 0841 
 
 0856 
 
 0871 
 
 0886 
 
 0901 
 
 0916 
 
 48 
 
 6 
 
 7 
 
 7 
 
 8 
 
 8 
 
 249 
 
 0931 
 
 0946 
 
 0961 
 
 0976 
 
 0991 
 
 1006 
 
 1021 
 
 1037 
 
 1052 
 
 1067 
 
 49 
 
 6 
 
 7 
 
 7 
 
 8 
 
 8 
 
 
 
 
 l|l 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 13 
 
 14 
 
 15 
 
 L6 
 
 17 
 
 [10] 
 
TABLE I. 
 
 Log 
 
 Log (1+aj) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 T250 
 
 0'07 1082 
 
 1097 
 
 1112 
 
 1127 
 
 1142 
 
 1157 
 
 1172 
 
 1188 
 
 1203 
 
 1218 
 
 50 
 
 6 
 
 7 
 
 7 
 
 8 
 
 8 
 
 251 
 
 1233 
 
 1248 
 
 1263 
 
 1278 
 
 1294 
 
 1309 
 
 1324 
 
 1339 
 
 1354 
 
 1369 
 
 51 
 
 7 
 
 7 
 
 8 
 
 8 
 
 9 
 
 252 
 
 1384 
 
 1400 
 
 1415 
 
 1430 
 
 1445 
 
 1460 
 
 1475 
 
 1491 
 
 1506 
 
 1521 
 
 52 
 
 7 
 
 K 
 / 
 
 8 
 
 8 
 
 9 
 
 253 
 
 1536 
 
 1551 
 
 1567 
 
 1582 
 
 1597 
 
 1612 
 
 1627 
 
 1643 
 
 1658 
 
 1673 
 
 53 
 
 7 
 
 7 
 
 8 
 
 8 
 
 9 
 
 254 
 
 1688 
 
 1703 
 
 1719 
 
 1734 
 
 1749 
 
 1764 
 
 1779 
 
 1795 
 
 1810 
 
 1825 
 
 54 
 
 7 
 
 8 
 
 8 
 
 9 
 
 9 
 
 255 
 
 1840 
 
 1856 
 
 1871 
 
 1886 
 
 1901 
 
 1917 
 
 1932 
 
 1947 
 
 1963 
 
 1978 
 
 55 
 
 7 
 
 8 
 
 8 
 
 9 
 
 9 
 
 256 
 
 1993 
 
 2008 
 
 2024 
 
 2039 
 
 2054 
 
 2069 
 
 2085 
 
 2100 
 
 2115 
 
 2131 
 
 56 
 
 7 
 
 8 
 
 8 
 
 9 
 
 10 
 
 257 
 
 2146 
 
 2161 
 
 2177 
 
 2192 
 
 2207 
 
 2223 
 
 2238 
 
 2253 
 
 2269 
 
 2284 
 
 57 
 
 7 
 
 8 
 
 9 
 
 9 
 
 10 
 
 258 
 
 2299 
 
 2315 
 
 2330 
 
 2345 
 
 2361 
 
 2376 
 
 2391 
 
 2407 
 
 2422 
 
 2437 
 
 58 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 259 
 
 2453 
 
 2468 
 
 2483 
 
 2499 
 
 2514 
 
 2530 
 
 2545 
 
 2560 
 
 2576 
 
 2591 
 
 59 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 260 
 
 07 2606 
 
 2622 
 
 2637 
 
 2653 
 
 2668 
 
 2684 
 
 2699 
 
 2714 
 
 2730 
 
 2745 
 
 60 
 
 8 
 
 8 
 
 9 
 
 10 
 
 10 
 
 261 
 
 2761 
 
 2776 
 
 2791 
 
 2807 
 
 2822 
 
 2838 
 
 2853 
 
 2869 
 
 2884 
 
 2900 
 
 61 
 
 8 
 
 9 
 
 9 
 
 10 
 
 10 
 
 262 
 
 2915 
 
 2930 
 
 2946 
 
 2961 
 
 2977 
 
 2992 
 
 3008 
 
 3023 
 
 3039 
 
 3054 
 
 62 
 
 8 
 
 9 
 
 9 
 
 10 
 
 11 
 
 263 
 
 3070 
 
 3085 
 
 3101 
 
 3116 
 
 3132 
 
 3147 
 
 3163 
 
 3178 
 
 3194 
 
 3209 
 
 63 
 
 8 
 
 9 
 
 9 
 
 10 
 
 11 
 
 264 
 
 3225 
 
 3240 
 
 3256 
 
 3271 
 
 3287 
 
 3302 
 
 3318 
 
 3333 
 
 3349 
 
 3364 
 
 64 
 
 8 
 
 9 
 
 10 
 
 10 
 
 11 
 
 265 
 
 3380 
 
 3396 
 
 3411 
 
 3427 
 
 3442 
 
 3458 
 
 3473 
 
 3489 
 
 3504 
 
 3520 
 
 65 
 
 8 
 
 9 
 
 10 
 
 10 
 
 11 
 
 266 
 
 3536 
 
 3551 
 
 3567 
 
 3582 
 
 3598 
 
 3614 
 
 3629 
 
 3645 
 
 3660 
 
 3676 
 
 66 
 
 9 
 
 9 
 
 10 
 
 11 
 
 11 
 
 267 
 
 3692 
 
 3707 
 
 3723 
 
 3738 
 
 3754 
 
 3770 
 
 3785 
 
 3801 
 
 3816 
 
 3832 
 
 67 
 
 9 
 
 9 
 
 10 
 
 11 
 
 11 
 
 268 
 
 3848 
 
 3863 
 
 3879 
 
 3895 
 
 3910 
 
 3926 
 
 3942 
 
 3957 
 
 3973 
 
 3989 
 
 68 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 269 
 
 4004 
 
 4020 
 
 4036 
 
 4051 
 
 4067 
 
 4083 
 
 4098 
 
 4114 
 
 4130 
 
 4145 
 
 69 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 270 
 
 074161 
 
 4177 
 
 4193 
 
 4208 
 
 4224 
 
 4240 
 
 4255 
 
 4271 
 
 4287 
 
 4303 
 
 70 
 
 9 
 
 10 
 
 11 
 
 11 
 
 12 
 
 271 
 
 4318 
 
 4334 
 
 4350 
 
 4365 
 
 4381 
 
 4397 
 
 4413 
 
 4428 
 
 4444 
 
 4460 
 
 71 
 
 9 
 
 10 
 
 11 
 
 11 
 
 12 
 
 272 
 
 4476 
 
 4491 
 
 4507 
 
 4523 
 
 4539 
 
 4555 
 
 4570 
 
 4586 
 
 4602 
 
 4618 
 
 72 
 
 9 
 
 10 
 
 11 
 
 12 
 
 12 
 
 273 
 
 4633 
 
 4649 
 
 4665 
 
 4681 
 
 4697 
 
 4712 
 
 4728 
 
 4744 
 
 4760 
 
 4776 
 
 73 
 
 9 
 
 10 
 
 11 
 
 12 
 
 12 
 
 274 
 
 4791 
 
 4807 
 
 4823 
 
 4839 
 
 4855 
 
 4871 
 
 4886 
 
 4902 
 
 4918 
 
 4934 
 
 74 
 
 10 
 
 10 
 
 11 
 
 12 
 
 13 
 
 275 
 
 4950 
 
 4966 
 
 4982 
 
 4997 
 
 5013 
 
 5029 
 
 5045 
 
 5061 
 
 5077 
 
 5093 
 
 75 
 
 10 
 
 11 
 
 11 
 
 12 
 
 13 
 
 276 
 
 5108 
 
 5124 
 
 5140 
 
 5156 
 
 5172 
 
 5188 
 
 5204 
 
 5220 
 
 5236 
 
 5252 
 
 76 
 
 10 
 
 11 
 
 11 
 
 12 
 
 13 
 
 277 
 
 5267 
 
 5283 
 
 5299 
 
 5315 
 
 5331 
 
 5347 
 
 5363 
 
 5379 
 
 5395 
 
 5411 
 
 77 
 
 10 
 
 11 
 
 12 
 
 12 
 
 13 
 
 278 
 
 5427 
 
 5443 
 
 5459 
 
 5475 
 
 5491 
 
 5506 
 
 5522 
 
 5538 
 
 5554 
 
 5570 
 
 78 
 
 10 
 
 11 
 
 12 
 
 12 
 
 13 
 
 279 
 
 5586 
 
 5602 
 
 5618 
 
 5634 
 
 5650 
 
 5666 
 
 5682 
 
 5698 
 
 5714 
 
 5730 
 
 79 
 
 10 
 
 11 
 
 12 
 
 13 
 
 13 
 
 280 
 
 07 5746 
 
 5762 
 
 5778 
 
 5794 
 
 5810 
 
 5826 
 
 5842 
 
 5858 
 
 5874 
 
 5890 
 
 80 
 
 10 
 
 11 
 
 1213 
 
 14 
 
 281 
 
 5906 
 
 5922 
 
 5938 
 
 5955 
 
 5971 
 
 5987 
 
 6003 
 
 6019 
 
 6035 
 
 6051 
 
 81 
 
 11 
 
 11 
 
 12 
 
 13 
 
 14 
 
 282 
 
 6067 
 
 6083 
 
 6099 
 
 6115 
 
 6131 
 
 6147 
 
 6163 
 
 6179 
 
 6196 
 
 6212 
 
 82 
 
 11 
 
 11 
 
 12 
 
 13 
 
 14 
 
 283 
 
 6228 
 
 6244 
 
 6260 
 
 6276 
 
 6292 
 
 6308 
 
 6324 
 
 6341 
 
 6357 
 
 6373 
 
 83 
 
 11 
 
 12 
 
 12 
 
 13 
 
 14 
 
 284 
 
 6389 
 
 6405 
 
 6421 
 
 6437 
 
 6453 
 
 6470 
 
 6486 
 
 6502 
 
 6518 
 
 6534 
 
 84 
 
 11 
 
 12 
 
 13 
 
 13 
 
 14 
 
 285 
 
 6550 
 
 6566 
 
 6583 
 
 6599 
 
 6615 
 
 6631 
 
 6647 
 
 6664 
 
 6680 
 
 6696 
 
 85 
 
 11 
 
 12 
 
 13 
 
 14 
 
 14 
 
 286 
 
 6712 
 
 6728 
 
 6744 
 
 6761 
 
 6777 
 
 6793 
 
 6809 
 
 6826 
 
 6842 
 
 6858 
 
 86 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 287 
 
 6874 
 
 6890 
 
 6907 
 
 6923 
 
 6939 
 
 6955 
 
 6972 
 
 6988 
 
 7004 
 
 7020 
 
 87 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 288 
 
 7037 
 
 7053 
 
 7069 
 
 7085 
 
 7102 
 
 7118 
 
 7134 
 
 7150 
 
 7167 
 
 7183 
 
 88 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 289 
 
 7199 
 
 7216 
 
 7232 
 
 7248 
 
 7264 
 
 7281 
 
 7297 
 
 7313 
 
 7330 
 
 7346 
 
 89 
 
 12 
 
 12 
 
 13 
 
 14 
 
 15 
 
 290 
 
 07 7362 
 
 7379 
 
 7395 
 
 7411 
 
 7428 
 
 7444 
 
 7460 
 
 7477 
 
 7493 
 
 7509 
 
 90 
 
 12 
 
 13 
 
 13 
 
 14 
 
 15 
 
 291 
 
 7526 
 
 7542 
 
 7558 
 
 7575 
 
 7591 
 
 7607 
 
 7624 
 
 7640 
 
 7656 
 
 7673 
 
 91 
 
 12 
 
 13 
 
 14 
 
 15 
 
 15 
 
 292 
 
 7689 
 
 7706 
 
 7722 
 
 7738 
 
 7755 
 
 7771 
 
 7788 
 
 7804 
 
 7820 
 
 7837 
 
 92 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 293 
 
 7853 
 
 7870 
 
 7886 
 
 7902 
 
 7919 
 
 7935 
 
 7952 
 
 7968 
 
 7985 
 
 8001 
 
 93 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 294 
 
 8017 
 
 8034 
 
 8050 
 
 8067 
 
 8083 
 
 8100 
 
 8116 
 
 8133 
 
 8149 
 
 8166 
 
 94 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 295 
 
 8182 
 
 8199 
 
 8215 
 
 8231 
 
 8248 
 
 8264 
 
 8281 
 
 8297 
 
 8314 
 
 8330 
 
 95 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 296 
 
 8347 
 
 8363 
 
 8380 
 
 8396 
 
 8413 
 
 8430 
 
 8446 
 
 8463 
 
 8479 
 
 8496 
 
 96 
 
 12 
 
 1314 
 
 15 
 
 16 
 
 297 
 
 8512 
 
 8529 
 
 8545 
 
 8562 
 
 8578 
 
 8595 
 
 8611 
 
 8628 
 
 8645 
 
 8661 
 
 97 
 
 13 
 
 1415 
 
 16 
 
 16 
 
 298 
 
 8678 
 
 8694 
 
 8711 
 
 8727 
 
 8744 
 
 8761 
 
 8777 
 
 8794 
 
 8810 
 
 8827 
 
 98 
 
 13 
 
 1415 
 
 16 
 
 17 
 
 299 
 
 8844 
 
 8860 
 
 8877 
 
 8893 
 
 8910 
 
 8927 
 
 8943 
 
 8960 
 
 8976 
 
 8993 
 
 99 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 [11] 
 
TABLE I. 
 
 Log 
 
 Log(l+a?) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 K 
 
 17 
 
 18 
 
 1920 
 
 I-;3()( 
 
 0-07 9010 
 
 9026 
 
 9043 
 
 9060 
 
 9076 
 
 9093 
 
 9110 9126 
 
 9143 9160 
 
 00 
 
 ( 
 
 ( 
 
 
 
 
 
 301 
 
 9176 
 
 9193 
 
 9210 
 
 9226 
 
 9243 
 
 9260 
 
 9276 9293 
 
 9310 9326 
 
 01 
 
 C 
 
 ( 
 
 o 
 
 
 
 
 
 302 
 
 9343 
 
 9360 
 
 9376 
 
 9393 
 
 9410 
 
 9427 
 
 9443 9460 9477 9493 
 
 02 
 
 ( 
 
 ( 
 
 
 
 ( 
 
 
 
 303 
 
 9510 
 
 9527 
 
 9544 
 
 9560 
 
 9577 
 
 9594 
 
 9611 
 
 9G27 9644 9661 
 
 03 
 
 
 
 1 
 
 1 
 
 
 1 
 
 304 
 
 9678 
 
 9694 
 
 9711 
 
 9728 
 
 9745 
 
 9762 
 
 9778 
 
 9795 
 
 9812 
 
 9829 
 
 04 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 305 
 
 9845 
 
 9862 
 
 9879 
 
 9896 
 
 9913 
 
 9929 
 
 9946 
 
 9963 
 
 9980 
 
 9997 
 
 05 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 306 
 
 08 0014 
 
 0030 
 
 0047 
 
 0064 
 
 0081 
 
 0098 
 
 0115 
 
 0131 
 
 0148 
 
 0165 
 
 06 
 
 .1 
 
 1 
 
 1 
 
 
 1 
 
 307 
 
 0182 
 
 0199 
 
 0216 
 
 0233 
 
 0249 
 
 0266 0283 
 
 0300 
 
 0317 
 
 0334 
 
 07 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 308 
 
 0351 
 
 0368 
 
 0384 
 
 0401 
 
 0418 
 
 04351 0452 
 
 0469 
 
 0486 
 
 0503 
 
 08 
 
 1 
 
 I 
 
 1 
 
 
 2 
 
 309 
 
 0520 
 
 0537 
 
 0554 
 
 0571 
 
 0587 
 
 0604 
 
 0621 
 
 0638 
 
 0655 
 
 0672 
 
 09 
 
 1 
 
 .> 
 
 2 
 
 
 2 
 
 310 
 
 08 0689 
 
 0706 
 
 0723 
 
 0740 
 
 0757 
 
 0774 
 
 0791 
 
 0808 
 
 0825 
 
 0842 
 
 K 
 
 9 
 
 j 
 
 2 
 
 
 2 
 
 311 
 
 0859 
 
 0876 
 
 0893 
 
 0910 
 
 0927 
 
 0944 
 
 0961 
 
 0978 
 
 0995 
 
 1012 
 
 11 
 
 , 
 
 c 
 
 2 
 
 
 2 
 
 312 
 
 1029 
 
 1046 
 
 1063 
 
 1080 
 
 1097 
 
 1114 
 
 1131 
 
 1148 
 
 1165 
 
 1182 
 
 12 
 
 i 
 
 c 
 
 2 
 
 
 2 
 
 31-8 
 
 1199 
 
 1216 1233 
 
 1250 
 
 1268 
 
 1285 
 
 1302 
 
 1319 
 
 1336 
 
 1353 
 
 13 
 
 <. 
 
 t 
 
 2 
 
 
 3 
 
 314 
 
 1370 
 
 1387 1404 
 
 1421 
 
 1438 
 
 1455 
 
 1473 
 
 1490 
 
 1507 
 
 1524 
 
 14 
 
 t 
 
 <. 
 
 3 
 
 
 3 
 
 315 
 
 1541 
 
 1558 1575 
 
 1592 
 
 1610 
 
 1627 
 
 1644 
 
 1661 
 
 1678 
 
 1695 
 
 16 
 
 i) 
 
 
 
 3 
 
 
 3 
 
 316 
 
 1712 
 
 1730 1747 
 
 1764 
 
 1781 
 
 1798 
 
 1815 
 
 1832 
 
 1850 
 
 1867 
 
 K 
 
 ; 
 
 ; 
 
 3 
 
 
 3 
 
 317 
 
 1884 
 
 1901 1918 
 
 1936 
 
 1953 
 
 1970 
 
 1987 
 
 2004 
 
 2022 
 
 2039 
 
 17 
 
 
 t, 
 
 ; 
 
 3 
 
 
 3 
 
 318 
 
 2056 20731 2090 
 
 2108 
 
 2125 
 
 2142 
 
 2159 2177 
 
 2194! 22 11 
 
 IS 
 
 * 
 ' 
 
 
 
 3 
 
 
 4 
 
 319 
 
 22281 2246 2263 1 2280 
 
 2297 
 
 2315 
 
 2332 2349 
 
 2366 23 s 1 
 
 19 
 
 * 
 
 
 
 3 
 
 
 4 
 
 
 I 1 i 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 320 
 
 08 2401 
 
 2418 2436 2453 
 
 2470 
 
 2487 
 
 2505 2522 
 
 2539 
 
 2557 
 
 20 
 
 * 
 
 
 
 4 
 
 
 4 
 
 321 
 
 2574 
 
 2591 2609 2<;2< 
 
 2643 
 
 2661 
 
 2678 2695 
 
 2713 
 
 2730 
 
 2\ 
 
 * 
 > 
 
 i 
 
 4 
 
 
 4 
 
 322 
 
 2747 
 
 2765 2782 2799 
 
 2817 
 
 2834 
 
 2851 
 
 2886 
 
 2904 
 
 22 
 
 
 4 
 
 4 
 
 
 4 
 
 323 
 
 2921 
 
 2938 29561 2973 
 
 2991 
 
 3008 
 
 3025 3043 
 
 3060 3078 
 
 2-.} 
 
 t 
 
 t 
 
 4 
 
 
 5 
 
 324 
 
 3095 
 
 3112 
 
 3130 
 
 3147 
 
 3165 
 
 3182 
 
 3199 3217 
 
 3234 
 
 3252 
 
 24 
 
 ^ 
 
 ~ 
 
 4 
 
 
 5 
 
 325 
 
 3269 
 
 3287 
 
 3304 
 
 3322 
 
 3339 
 
 3357 
 
 3374 ! 3391 
 
 3409 
 
 3426 
 
 26 
 
 4 
 
 i 
 
 4 
 
 
 5 
 
 326 
 
 3444 
 
 3461 
 
 3479 
 
 3496 
 
 3514 
 
 3531 
 
 3549 3566 
 
 3584 
 
 3601 
 
 26 
 
 ^ 
 
 4 
 
 5 
 
 
 5 
 
 327 
 
 3619 
 
 3636 
 
 3654! 3671 
 
 3689 
 
 3706 
 
 3724 3742 
 
 3759 
 
 3777 
 
 27 
 
 ^ 
 
 i 
 
 5 
 
 
 5 
 
 328 
 
 3794 
 
 3812 
 
 3829] 3847 
 
 3864 
 
 3882 
 
 3900 31)17 
 
 3935 
 
 3952 
 
 28 
 
 L. 
 
 r 
 
 5 
 
 5 
 
 6 
 
 329 
 
 3970 
 
 3987 
 
 4005 
 
 4023 
 
 4040 
 
 4058 
 
 4075 4093 
 
 4111 
 
 4128 
 
 29 
 
 I 
 
 ,- 
 t 
 
 
 
 G 
 
 6 
 
 330 
 
 08 4146 
 
 4163 
 
 4181 
 
 4199 
 
 4216 
 
 4234 
 
 4252 
 
 4269 
 
 4287 
 
 4304 
 
 30 
 
 t 
 
 r 
 
 5 
 
 e 
 
 6 
 
 331 
 
 4322 
 
 4340 4357 
 
 4375 
 
 4393 
 
 4410 
 
 4428 
 
 4446 
 
 4463 
 
 4481 
 
 31 
 
 1 
 
 I 
 
 r 
 i 
 
 6 
 
 6 
 
 6 
 
 332 
 
 4499 
 
 4516 4534 
 
 4552 
 
 4569 
 
 4587 
 
 4605 
 
 4623 
 
 4640 
 
 4658 
 
 32 
 
 1 
 
 r 
 
 (J 
 
 
 
 G 
 
 333 
 
 4676 
 
 4693 4711 
 
 4729 
 
 4747 
 
 4764 
 
 4782 
 
 4800 
 
 4818 
 
 4835 
 
 33 
 
 K 
 
 t 
 
 ( 
 
 6 
 
 6 
 
 7 
 
 334 
 
 4853 
 
 4871 4889 
 
 4906 
 
 4924 
 
 4942 
 
 4960 
 
 4977 
 
 4995 
 
 5013 
 
 34 
 
 - 
 
 ( 
 
 6 
 
 6 
 
 7 
 
 335 
 
 5031 
 
 5048 5066 
 
 5084 
 
 5102 
 
 5120 
 
 5137 
 
 5155 
 
 5173 
 
 5191 
 
 36 
 
 6 
 
 ( 
 
 
 
 / 
 
 7 
 
 336 
 
 5209 
 
 5226 5244 
 
 5262 
 
 5280 
 
 5298 
 
 5316 
 
 5333 
 
 5351 
 
 5369 
 
 36 
 
 ( 
 
 ( 
 
 6 
 
 7 
 
 / 
 
 337 
 
 5387 
 
 5405 5423 
 
 5441 
 
 5458 
 
 5476 
 
 5494 
 
 5512 
 
 5530 
 
 5548 
 
 37 
 
 ( 
 
 6 
 
 N 
 
 / 
 
 / 
 
 7 
 
 338 
 
 5566 
 
 5584 5601 
 
 5619 
 
 5637 
 
 5655 
 
 5673 
 
 5691 
 
 5709 
 
 5727 
 
 38 
 
 ( 
 
 6 
 
 7 
 
 7 
 
 8 
 
 339 
 
 5745 
 
 5763 
 
 5780 
 
 5798 
 
 5816 
 
 5834 
 
 5852 
 
 5870 
 
 5888 
 
 5906 
 
 39 
 
 ( 
 
 i 
 
 *w 
 
 / 
 
 7 
 
 8 
 
 340 
 
 08 5924 
 
 5942 
 
 5960 
 
 5978 
 
 5996 
 
 6014 
 
 6032 
 
 6050 
 
 6068 
 
 6086 
 
 40 
 
 ( 
 
 
 7 
 
 8 
 
 8 
 
 341 
 
 6104 
 
 6122 
 
 6140 
 
 6158 
 
 6176 
 
 6194 
 
 6212 
 
 6230 
 
 6248 
 
 6266 
 
 41 
 
 t 
 
 
 7 
 
 8 
 
 8 
 
 342 
 
 6284 
 
 6302 
 
 6320 
 
 6338 
 
 6356 
 
 6374 
 
 6392 
 
 6410 
 
 6428 
 
 6446 
 
 42 
 
 7 
 
 
 8 
 
 8 
 
 8 
 
 343 
 
 6464 
 
 6482 6500 
 
 6518 
 
 6536 
 
 6554 
 
 6572 
 
 6590 
 
 6609 
 
 6627 
 
 43 
 
 7 
 
 
 8 
 
 8 
 
 9 
 
 344 
 
 6645 
 
 6663 6681 
 
 6699 
 
 6717 
 
 6735 
 
 6753 
 
 6771 
 
 6789 
 
 6808 
 
 44 
 
 7 
 
 
 8 
 
 8 
 
 9 
 
 345 
 
 6826 
 
 6844 6862 
 
 6880 
 
 6898 
 
 6916 
 
 6935 
 
 6953 
 
 6971 
 
 6989 
 
 45 
 
 7 
 
 8 
 
 8 
 
 9 
 
 9 
 
 346 
 
 7007 
 
 7025 7043 
 
 7062 
 
 7080 
 
 7098 
 
 7116 
 
 7134 
 
 7152 
 
 7171 
 
 40 
 
 7 
 
 8 
 
 8 
 
 !) 
 
 9 
 
 347 
 
 7189 
 
 7207 7225 
 
 7243 
 
 7262 
 
 7280 
 
 7298 
 
 7316 
 
 7334 
 
 7353 
 
 47 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 348 
 
 7371 
 
 7389 7407 
 
 7426 
 
 7444 
 
 7462 
 
 7480 
 
 7499 
 
 7517 
 
 7535 
 
 48 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 349 
 
 7553 
 
 7572 7590 
 
 7608 
 
 7G2G 
 
 7645 
 
 7663 
 
 7681 
 
 7699 
 
 7718 
 
 49 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 
 
 
 1 1 2 
 
 3 
 
 4 
 
 K 
 
 o 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 [12] 
 
TABLE I. 
 
 T ^ 
 
 Log (1+a) 
 
 Pro. Parts. 
 
 Log 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 16 
 
 17 
 
 18 
 
 1920 
 
 1-350 
 
 0-08 7736 7754 
 
 7773 
 
 7791 
 
 7809 
 
 7828 
 
 7846 
 
 7864 
 
 7882 
 
 7901 
 
 50 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 351 
 
 7919 7937 
 
 7956 
 
 7974 
 
 7992 
 
 8011 
 
 8029 
 
 8048 
 
 8066 
 
 8084 
 
 51 
 
 8 
 
 9 
 
 9 
 
 10 
 
 10 
 
 352 
 
 8103 8121 
 
 8139 
 
 8158 
 
 8176 
 
 8194 
 
 8213 
 
 8231 
 
 8250 
 
 8268 
 
 52 
 
 8 
 
 9 
 
 9 
 
 10 
 
 10 
 
 353 
 
 8286 8305 
 
 8323 
 
 8342 
 
 8360 
 
 8378 
 
 8397 
 
 8415 
 
 8434 
 
 8452 
 
 53 
 
 8 
 
 9 
 
 10 
 
 10 
 
 11 
 
 354 
 
 8470 8489 
 
 8507 
 
 8526 
 
 8544 
 
 8563 
 
 8581 
 
 8600 8618 
 
 8636 
 
 54 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 355 
 
 8655 
 
 8673 
 
 8692 
 
 8710 
 
 8729 
 
 8747 
 
 8766 
 
 8784 
 
 8803 
 
 8821 
 
 55 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 356 
 
 8840 
 
 8858 8877 
 
 8895 
 
 8914 
 
 8932 
 
 8951 
 
 8969 8988 
 
 9006 
 
 56 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 357 
 
 9025 
 
 9043 9062 9081 
 
 9099 
 
 9118 
 
 9136 
 
 9155 
 
 9173 
 
 9192 
 
 57 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 358 
 
 9210 
 
 9229 9248 9266 
 
 9285 
 
 9303 
 
 9322 
 
 9341 
 
 9359 
 
 9378 
 
 58 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 359 
 
 9396 
 
 9415 9434 9452 
 
 9471 
 
 9489 
 
 9508 
 
 9527 9545 
 
 9564 
 
 59 
 
 910 
 
 11 
 
 11 
 
 12 
 
 360 
 
 08 9583 
 
 9601 9620 9638 
 
 9657 
 
 9676 
 
 9694 
 
 9713 9732 
 
 9750 
 
 60 10 10 
 
 11 
 
 11 
 
 12 
 
 361 
 
 9769 
 
 9788 1 9806 9825 
 
 9844 
 
 9863 
 
 9881 
 
 9900 9919 
 
 9937 
 
 6110 
 
 10 
 
 11 
 
 12 
 
 12 
 
 362 
 
 9956 
 
 9975 ! 9993 0012 
 
 0031 
 
 0050 
 
 o068 
 
 0087 0106 
 
 0125 
 
 6210 
 
 11 
 
 11 
 
 12 
 
 12 
 
 363 
 
 09 0143 
 
 0162 0181 1 0200 
 
 0218 
 
 0237 
 
 0256 
 
 0275 
 
 0293 
 
 0312 
 
 63 
 
 10 
 
 11 
 
 11 
 
 12 
 
 13 
 
 364 
 
 0331 
 
 0350 
 
 0368 0387 
 
 0406 
 
 0425 
 
 0444 
 
 0462 0481 
 
 0500 
 
 64 
 
 10 
 
 11 
 
 12 
 
 12 
 
 13 
 
 365 
 366 
 
 0519 
 0707 
 
 0538 
 0726 
 
 0556 0575 
 0745 0764 
 
 0594 
 0783 
 
 0613 
 0801 
 
 0632 
 0820 
 
 0651 
 0839 
 
 0669 
 0858 
 
 0688 
 
 0877 
 
 65 
 66 
 
 10 
 
 11 
 
 11 
 11 
 
 12 
 12 
 
 1213 
 1313 
 
 367 
 
 0896 
 
 0915 
 
 0934 0953 
 
 0971 
 
 0990 
 
 1009 
 
 1028 1047 
 
 1066 
 
 67 
 
 11 
 
 11 
 
 121313 
 
 368 
 
 1085 
 
 1104 1123 1142 
 
 1161 
 
 1180 
 
 1198 
 
 1217, 1236 
 
 1255 
 
 68 
 
 11 
 
 12 
 
 121314 
 
 369 
 
 1274 
 
 1293; 1312 1331 
 
 1350 
 
 1369 
 
 1388 
 
 1407 1426 
 
 1445 
 
 69 
 
 11 
 
 12 
 
 1213)14 
 
 370 
 
 09 1464 
 
 1483 
 
 1502 1521 
 
 1540 
 
 1559 
 
 1578 
 
 1597| 1616 
 
 1635 
 
 7011 
 
 12 
 
 13 
 
 13 
 
 14 
 
 371 
 
 1654 1673 1692 1711 
 
 1730 
 
 1749 
 
 1768 
 
 1787; 1806 
 
 1825 
 
 71 
 
 11 
 
 12 
 
 1313 
 
 14 
 
 372 
 
 1844 1864J 1883 1902 
 
 1921 
 
 1940 
 
 1959 
 
 1978 1997 
 
 2016 
 
 72 
 
 12 
 
 12 
 
 1314 
 
 14 
 
 373 
 
 20351 2054! 2073 2093 
 
 2112 
 
 2131 
 
 2150 
 
 2169) 2188 
 
 2207 
 
 73 
 
 12 
 
 12 
 
 13 
 
 14 
 
 15 
 
 374 
 
 2226 
 
 2246 2265; 2284 
 
 2303 
 
 2322 
 
 2341 
 
 2360 2380 
 
 2399 
 
 74 
 
 12 
 
 13 
 
 13 
 
 14 
 
 15 
 
 375 
 
 2418 
 
 2437 
 
 2456 2475 
 
 2495 
 
 2514 
 
 2533 
 
 2552 2571 
 
 2591 
 
 75 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 376 
 
 2610! 2629 
 
 2648| 2667 
 
 2687 
 
 2706 
 
 2725 
 
 2744 2764 
 
 2783 
 
 76 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 377 
 
 2802 2821 
 
 2840, 2860 
 
 2879 
 
 2898 
 
 2918 
 
 2937 2956 
 
 2975 
 
 77 
 
 12 
 
 13 
 
 14 
 
 15 
 
 15 
 
 378 
 
 2995 3014 
 
 3033 3052 
 
 3072 
 
 3091 
 
 3110 
 
 3130 3149 
 
 3168 
 
 78 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 379 
 
 3188 3207 
 
 3226 3245 
 
 3265 
 
 3284 
 
 3303 
 
 3323 3342 
 
 3361 
 
 79 
 
 13 
 
 13 
 
 14 
 
 15 
 
 16 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 380 
 
 093381 3400 
 
 3420 3439 
 
 3458 
 
 3478 
 
 3497 
 
 3516 3536 
 
 3555 
 
 8013 
 
 14 
 
 14 
 
 15 
 
 16 
 
 381 
 
 3574 3594 
 
 3613 3633 
 
 3652 
 
 3671 
 
 3691 
 
 3710 3730 
 
 3749 
 
 8113 
 
 14 
 
 15 
 
 15 
 
 16 
 
 382 
 
 3768 378,^ 
 
 3807 
 
 3827 
 
 3846 
 
 3866 
 
 3885 
 
 3904 3924 
 
 3943 
 
 8213. 
 
 14 
 
 15 
 
 16 
 
 16 
 
 383 
 
 3963 3982 
 
 4002 
 
 4021 
 
 4041 
 
 4060 
 
 4080 
 
 4099 4119 
 
 4138 
 
 8313 
 
 14 
 
 15 
 
 16 
 
 17 
 
 384 
 
 4158 
 
 4177 
 
 4197 
 
 4216 
 
 4236 
 
 4255 
 
 4275 
 
 4294 4314 
 
 4333 
 
 84 ! 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 385 
 386 
 
 4353i 4372 
 4548 4568 
 
 4392 
 4587 
 
 4411 4431 
 4607 4626 
 
 4450 
 4646 
 
 4470 
 4666 
 
 4489 4509 
 4685 4705 
 
 4529 
 4724 
 
 85,14 
 8614 
 
 14 
 15 
 
 15 
 15 
 
 16 
 16 
 
 17 
 17 
 
 387 
 
 4744 4764 
 
 4783 
 
 4803 
 
 4822 
 
 4842 
 
 4862 
 
 4881 
 
 4901 
 
 4920 
 
 87J14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 388 
 
 4940 4960 
 
 4979 4999 5019 
 
 5038 
 
 5058 
 
 5078 5097 
 
 5117 
 
 8814 
 
 15 
 
 16 
 
 17 
 
 18 
 
 389 
 
 5137 
 
 5156 
 
 5176 5196 
 
 5215 
 
 5235 
 
 5255 
 
 5274 5294 
 
 5314 
 
 8914 
 
 15 
 
 16 
 
 17 
 
 18 
 
 390 
 
 09 5334 
 
 5353 
 
 5373 5393 
 
 5412 
 
 5432 
 
 5452 
 
 5472 5491 
 
 5511 
 
 9o|l4 
 
 15 
 
 16 
 
 17J18 
 
 391 
 
 5531 
 
 5551 
 
 5570 5590 
 
 5610 
 
 5630 
 
 5649 
 
 5669 
 
 5689 
 
 5709 
 
 9115 
 
 15 
 
 16 
 
 17)18 
 
 392 
 
 5728 
 
 5748 
 
 5768 5788 
 
 5808 
 
 5827 
 
 5847 
 
 5867 
 
 5887 
 
 5907 
 
 92' 15 
 
 16 
 
 17 
 
 17J18 
 
 393 
 
 5926 
 
 5946 
 
 5966 5986 
 
 6006 
 
 6026 
 
 6045 
 
 6065 
 
 6085 
 
 6105 
 
 93| 15 
 
 16 
 
 17 
 
 1819 
 
 394 
 
 6125 
 
 6145 
 
 6165 
 
 6184 
 
 6204 
 
 6224 
 
 6244 
 
 6264 
 
 6284 
 
 6304 
 
 9415 
 
 16 
 
 17 
 
 1819 
 
 395 
 
 6324 
 
 6343 
 
 6363 
 
 6383 
 
 6403 
 
 6423 
 
 6443 
 
 6463 
 
 6483 
 
 6503 
 
 9515 
 
 16 
 
 17 
 
 1819 
 
 396 
 
 6523 
 
 6543 
 
 6563 
 
 6583 
 
 6602 
 
 6622 
 
 6642 
 
 6662 
 
 6682 
 
 6702 
 
 9615 
 
 16 
 
 17 
 
 1819 
 
 397 
 
 6722 
 
 6742 
 
 6762 
 
 6782 
 
 6802 
 
 6822 
 
 6842 
 
 6862 
 
 6882 
 
 6902 
 
 9716 
 
 16 
 
 17 
 
 Its 10 
 
 398 
 
 6922 
 
 6942 
 
 6962 
 
 6982 
 
 7002 
 
 7022 
 
 7042 
 
 7062 
 
 7082 
 
 7102 
 
 98116 
 
 17 
 
 18 
 
 1920 
 
 399 
 
 7122 
 
 7142 
 
 7162 
 
 7182 
 
 7202 
 
 7222 7243 
 
 7263 
 
 7283 
 
 7303 
 
 99l 16 
 
 17 
 
 18 
 
 1920 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 16 
 
 17 
 
 18 
 
 1920 
 
 [13] 
 
TABLE I. 
 
 Log 
 
 
 Log(l- 
 
 M 
 
 
 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 T400 
 
 0-09 7323 
 
 7343 
 
 7363 
 
 7383 
 
 7403 
 
 7423 
 
 7443 
 
 7463 
 
 7484 
 
 7504 
 
 )() 
 
 
 
 
 
 
 
 
 
 
 
 401 
 
 7524 
 
 7544 
 
 7564 
 
 7584 
 
 7604 
 
 7624 
 
 7644 
 
 7665 
 
 7685 
 
 7705 
 
 )1 
 
 
 
 
 
 
 
 
 
 
 
 402 
 
 7725 
 
 7745 
 
 7765 
 
 7786 
 
 7806 
 
 7826 
 
 7846 
 
 7866 
 
 7886 
 
 7907 
 
 )2 
 
 
 
 
 
 
 
 
 
 
 
 403 
 
 7927 
 
 7947 
 
 7967 
 
 7987 
 
 8008 
 
 8028 
 
 8048 
 
 8068 
 
 8088 
 
 8109 
 
 )3 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 404 
 
 8129 
 
 8149 
 
 8169 
 
 8189 
 
 8210 
 
 8230 
 
 8250 
 
 8270 
 
 8291 
 
 8311 
 
 04 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 405 
 
 8331 
 
 8351 
 
 8372 
 
 8392 
 
 8412 
 
 8433 
 
 8453 
 
 8473 
 
 8493 
 
 8514 
 
 05 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 406 
 
 8534 
 
 8554 
 
 8575 
 
 8595 
 
 8615 
 
 8636 
 
 8656 
 
 8676 
 
 8697 
 
 8717 
 
 )6 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 407 
 
 8737 
 
 8758 
 
 8778 
 
 8798 
 
 8819 
 
 8839 
 
 8859 
 
 8880 
 
 8900 
 
 8920 
 
 )7 
 
 1 
 
 1 
 
 2 
 
 2 
 
 2 
 
 408 
 
 8941 
 
 8961 
 
 8981 
 
 9002 
 
 9022 
 
 9043 
 
 9063 
 
 9083 
 
 9104 
 
 9124 
 
 )8 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 409 
 
 9145 
 
 9165 
 
 9185 
 
 9206 
 
 9226 
 
 9247 
 
 9267 
 
 9288 
 
 9308 
 
 9329 
 
 )9 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 410 
 
 09 9349 
 
 9369 
 
 9390 
 
 9410 
 
 9431 
 
 9451 
 
 9472 
 
 9492 
 
 9513 
 
 9533 
 
 10 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 411 
 
 9554 
 
 9574 
 
 9595 
 
 9615 
 
 9636 
 
 9656 
 
 9677 
 
 9697 
 
 9718 
 
 9738 
 
 11 
 
 2 
 
 2 
 
 2 
 
 3 
 
 2 
 
 412 
 
 9759 
 
 9779 
 
 9800 
 
 9820 
 
 9841 
 
 9861 
 
 9882 
 
 9902 
 
 9923 
 
 9944 
 
 12 
 
 2 
 
 3 
 
 3 
 
 3 
 
 2 
 
 413 
 
 9964 
 
 9985 
 
 0005 
 
 0026 
 
 0046 
 
 0067 
 
 0088 
 
 0108 
 
 0129 
 
 0150 
 
 13 
 
 3 
 
 3 
 
 :3 
 
 3 
 
 ^ 
 
 414 
 
 100170 
 
 0190 
 
 0211 
 
 0232 
 
 0252 
 
 0273 
 
 0294 
 
 0314 
 
 0335 
 
 0355 
 
 14 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 415 
 
 0376 
 
 0397 
 
 0417 
 
 0438 
 
 0459 
 
 0479 
 
 0500 
 
 0521 
 
 0541 
 
 0562 
 
 15 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 416 
 
 0583 
 
 0603 
 
 0624 
 
 0645 
 
 0665 
 
 0686 
 
 0707 
 
 0727 
 
 0748 
 
 0769 
 
 10 
 
 3 
 
 3 
 
 4 
 
 4 
 
 4 
 
 417 
 
 0790 
 
 0810 
 
 0831 
 
 0852 
 
 0872 
 
 0893 
 
 0914 
 
 0935 
 
 0955 
 
 0976 
 
 17 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 418 
 
 0997 
 
 1018 
 
 1038 
 
 1059 
 
 1080 
 
 1101 
 
 1121 
 
 1142 
 
 1163 
 
 1184 
 
 18 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 419 
 
 1205 
 
 1225 
 
 1246 
 
 1267 
 
 1288 
 
 1308 
 
 1329 
 
 1350 
 
 1371 
 
 1392 
 
 18 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 420 
 
 10 1413 
 
 1433 
 
 1454 
 
 1475 
 
 1496 
 
 1517 
 
 1538 
 
 1558 
 
 1579 
 
 1600 
 
 20 
 
 4 
 
 4 
 
 4 
 
 5 
 
 5 
 
 421 
 
 1621 
 
 1642 
 
 1663 
 
 1684 
 
 1704 
 
 1725 
 
 1746 
 
 1767 
 
 1788 
 
 1809 
 
 21 
 
 4 
 
 4 
 
 r 
 
 5 
 
 5 
 
 422 
 
 1830 
 
 1851 
 
 1872 
 
 1893 
 
 1913 
 
 1934 
 
 1955 
 
 1976 
 
 1997 
 
 2018 
 
 22 
 
 4 
 
 5 
 
 6 
 
 5 
 
 5 
 
 423 
 
 2039 
 
 2060 
 
 2081 
 
 2102 
 
 2123 
 
 2144 
 
 2165 
 
 2186 
 
 2207 
 
 2228 
 
 22 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 424 
 
 2249 
 
 2270 
 
 2291 
 
 2312 
 
 2333 
 
 2354 
 
 2375 
 
 2396 
 
 2417 
 
 2438 
 
 24 
 
 5 
 
 5 
 
 r 
 
 6 
 
 6 
 
 425 
 
 2459 
 
 2480 
 
 2501 
 
 2522 
 
 2543 
 
 2564 
 
 2585 
 
 2606 
 
 2627 
 
 2648 
 
 25 
 
 5 
 
 5 
 
 r. 
 
 6 
 
 6 
 
 426 
 
 2669 
 
 2690 
 
 2711 
 
 2732 
 
 2753 
 
 2774 
 
 2795 
 
 2816 
 
 2837 
 
 2859 
 
 26 
 
 o 
 
 5 
 
 6 
 
 6 
 
 6 
 
 427 
 
 2880 
 
 2901 
 
 2922 
 
 2943 
 
 2964 
 
 2985 
 
 3006 
 
 3027 
 
 3049 
 
 3070 
 
 27 
 
 5 
 
 6 
 
 6 
 
 G 
 
 6 
 
 428 
 
 3091 
 
 3112 
 
 3133 
 
 3154 
 
 3175 
 
 3196 
 
 3218 
 
 3239 
 
 3260 
 
 3281 
 
 28 
 
 6 
 
 6 
 
 6 
 
 G 
 
 7 
 
 429 
 
 3302 
 
 3323 
 
 3345 
 
 3366 
 
 3387 
 
 3408 
 
 3429 
 
 3451 
 
 3472 
 
 3493 
 
 29 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 430 
 
 103514 
 
 3535 
 
 3557 
 
 3578 
 
 3599 
 
 3620 
 
 3641 
 
 3663 
 
 3684 
 
 3705 
 
 30 
 
 6 
 
 6 
 
 
 7 
 
 7 
 
 431 
 
 3726 
 
 3748 
 
 3769 
 
 3790 
 
 3811 
 
 3833 
 
 3854 
 
 3875 
 
 3897 
 
 3918 
 
 31 
 
 6 
 
 7 
 
 
 7 
 
 7 
 
 432 
 
 3939 
 
 3960 
 
 3982 
 
 4003 
 
 4024 
 
 4046 
 
 4067 
 
 4088 
 
 4109 
 
 4131 
 
 32 
 
 6 
 
 7 
 
 
 7 
 
 8 
 
 433 
 
 4152 
 
 4173 
 
 4195 
 
 4216 
 
 4237 
 
 4259 
 
 4280 
 
 4301 
 
 4323 
 
 4344 
 
 33 
 
 7 
 
 7 
 
 
 8 
 
 8 
 
 434 
 
 4366 
 
 4387 
 
 4408 
 
 4430 
 
 4451 
 
 4472 
 
 4494 
 
 4515 
 
 4537 
 
 4558 
 
 34 
 
 7 
 
 7 
 
 
 8 
 
 8 
 
 435 
 
 4579 
 
 4601 
 
 4622 
 
 4644 
 
 4665 
 
 4686 
 
 4708 
 
 4729 
 
 4751 
 
 4772 
 
 35 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 436 
 
 4794 
 
 4815 
 
 4836 
 
 4858 
 
 4879 
 
 4901 
 
 4922 
 
 4944 
 
 4965 
 
 4987 
 
 36 
 
 7 
 
 8 
 
 8 
 
 8 
 
 c 
 
 437 
 
 5008 
 
 5030 
 
 5051 
 
 5073 
 
 5094 
 
 5116 
 
 5137 
 
 5159 
 
 5180 
 
 5202 
 
 37 
 
 i 
 
 8 
 
 8 
 
 9 
 
 c 
 
 438 
 
 5223 
 
 5245 
 
 5266 
 
 5288 
 
 5309 
 
 5331 
 
 5352 
 
 5374 
 
 5395 
 
 5417 
 
 38 
 
 8 
 
 8 
 
 8 
 
 9 
 
 c 
 
 439 
 
 5438 
 
 5460 
 
 5482 
 
 5503 
 
 5525 
 
 5546 
 
 5568 
 
 5589 
 
 5611 
 
 5633 
 
 30 
 
 8 
 
 8 
 
 9 
 
 9 
 
 c 
 
 440 
 
 10 5654 
 
 5676 
 
 5697 
 
 5719 
 
 5741 
 
 5762 
 
 5784 
 
 5805 
 
 5827 
 
 5849 
 
 40 
 
 8 
 
 8 9 
 
 9 
 
 10 
 
 441 
 
 5870 
 
 5892 
 
 5914 
 
 5935 
 
 5957 
 
 5979 
 
 6000 
 
 6022 
 
 6044 
 
 6065 
 
 41 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 442 
 
 6087 
 
 6109 
 
 6130 
 
 6152 
 
 6174 
 
 6195 
 
 6217 
 
 6239 
 
 6260 
 
 6282 
 
 42 
 
 8 
 
 9 
 
 9 
 
 10 
 
 10 
 
 443 
 
 6304 
 
 6326 
 
 634" 
 
 6369 
 
 6391 
 
 6412 
 
 6434 
 
 6456 
 
 6478 
 
 6499 
 
 43 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 444 
 
 6521 
 
 6543 
 
 6565 
 
 6586 
 
 6608 
 
 663U 
 
 6652 
 
 6673 
 
 6695 
 
 6717 
 
 44 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 445 
 
 6739 
 
 6761 
 
 6782 
 
 6804 
 
 6826 
 
 6848 
 
 6870 
 
 6891 
 
 6913 
 
 6935 
 
 45 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 446 
 
 6957 
 
 6979 
 
 7001 
 
 7022 
 
 7044 
 
 706G 
 
 7088 
 
 7110 
 
 7132 
 
 7154 
 
 46 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 447 
 
 7175 
 
 719" 
 
 7219 
 
 7241 
 
 7263 
 
 7285 
 
 7307 
 
 7329 
 
 7350 
 
 7372 
 
 47 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 448 
 
 7394 
 
 7416 
 
 743P 
 
 746t 
 
 7482 
 
 7504 
 
 7526 
 
 7548 
 
 7570 
 
 7592 
 
 48 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 449 
 
 7614 
 
 7636 
 
 765" 
 
 767S 
 
 7701 
 
 7723 
 
 7745 
 
 7767 
 
 7789 
 
 7811 
 
 48 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 20 
 
 2122 
 
 23 
 
 24 
 
TABLE I. 
 
 Log- 
 
 Log (1 + 3) 
 
 Pro. Parts. 
 
 cc 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 1-450 
 
 010 7833 
 
 7855 
 
 7877 
 
 7899 
 
 7921 
 
 7943 
 
 7965 
 
 7987 
 
 8009 
 
 8031 
 
 50 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 451 
 
 8053 
 
 8075 
 
 8097 
 
 8119 
 
 8141 
 
 8163 
 
 8186 
 
 8208 
 
 8230 
 
 8252 
 
 51 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 452 
 
 8274 
 
 8296 
 
 8318 
 
 8340 
 
 8362 
 
 8384 
 
 8406 
 
 8428 
 
 8450 
 
 8473 
 
 52 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 453 
 
 8495 
 
 8517 
 
 8539 
 
 8561 
 
 8583 
 
 8605 
 
 8627 
 
 8649 
 
 8672 
 
 8694 
 
 53 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 454 
 
 8716 
 
 8738 
 
 8760 
 
 8782 
 
 8805 
 
 8827 
 
 8849 
 
 8871 
 
 8893 
 
 8915 
 
 54 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 455 
 
 8938 
 
 8960 
 
 8982 
 
 9004 
 
 9026 
 
 9049 
 
 9071 
 
 9093 
 
 9115 
 
 9137 
 
 55 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 456 
 
 9160 
 
 9182 
 
 9204 
 
 9226 9249 
 
 9271 
 
 9293 
 
 9315 
 
 9338 
 
 9360 
 
 56 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 457 
 
 9382 
 
 9404 
 
 9427 
 
 0449 9471 
 
 9493 
 
 9516 
 
 9538 
 
 9560 
 
 9583 
 
 57 
 
 1112 
 
 13 
 
 13 
 
 14 
 
 458 
 
 9605 
 
 9627 
 
 9650 
 
 9672 9694 
 
 9716 
 
 9739 
 
 9761 
 
 9783 
 
 9806 
 
 581212 
 
 13 
 
 13 
 
 14 
 
 459 
 
 9828 
 
 9850 
 
 9873 
 
 9895 9918 
 
 9940 
 
 9962 
 
 9985 
 
 0007 
 
 0029 
 
 591212 
 
 13 
 
 14 
 
 14 
 
 460 
 
 11 0052 
 
 0074 
 
 0097 
 
 0119 0141 
 
 0164 
 
 0186 
 
 0209 
 
 0231 
 
 0253 
 
 6012 
 
 13 
 
 13 
 
 14 
 
 14 
 
 461 
 
 0276 
 
 0298 
 
 0321 
 
 0343 0366 
 
 0388 
 
 0410 
 
 0433 
 
 0455 
 
 0478 
 
 6112 
 
 13 
 
 13 
 
 14 
 
 15 
 
 462 
 
 0500 
 
 0523 
 
 0545 
 
 0568 0590 
 
 0613 
 
 0635 
 
 0658 
 
 0680 
 
 0703 
 
 6212 
 
 13 
 
 14 
 
 14 
 
 15 
 
 463 
 
 0725 
 
 0748 
 
 0770 
 
 0793 0815 
 
 0838 
 
 0860 
 
 0883 
 
 0905 
 
 0928 
 
 63 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 464 
 
 0950 
 
 0973 
 
 0995 
 
 1018 
 
 1041 
 
 1063 
 
 1086 
 
 1108 
 
 1131 
 
 1153 
 
 64 
 
 13 
 
 13 
 
 14 
 
 15 
 
 15 
 
 465 
 
 1176 
 
 1199 
 
 1221 
 
 1244 
 
 1266 
 
 1289 
 
 1312 
 
 1334 
 
 1357 
 
 1379 
 
 65 
 
 13 
 
 14 
 
 14 
 
 15 
 
 16 
 
 466 
 
 1402 
 
 1425 
 
 1447 
 
 1470 1493 
 
 1515 
 
 1538 
 
 1561 
 
 1583 
 
 1606 
 
 66 
 
 13 
 
 14 
 
 15 
 
 15 
 
 16 
 
 467 
 
 1629 
 
 1651 
 
 1674 
 
 1697 
 
 1719 
 
 1742 
 
 1765 
 
 1787 
 
 1810 
 
 1833 
 
 67 
 
 13 
 
 14 
 
 15 
 
 15 
 
 16 
 
 468 
 
 1855 
 
 1878 
 
 1901 
 
 1924 
 
 1946 
 
 1969 
 
 1992 
 
 2014 
 
 2037 
 
 2060 
 
 68 
 
 14 
 
 14 
 
 15 
 
 16 
 
 16 
 
 469 
 
 2083 
 
 2105 
 
 2128 
 
 2151 
 
 2174 
 
 2196 
 
 2219 
 
 2242 
 
 2265 
 
 2288 
 
 69 
 
 14 
 
 14 
 
 15 
 
 16 
 
 17 
 
 470 
 
 11 2310 
 
 2333 
 
 2356 
 
 2379 
 
 2402 
 
 2424 
 
 2447 
 
 2470 
 
 2493 
 
 2516 
 
 70 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 471 
 
 2538 
 
 2561 
 
 2584 
 
 2607 
 
 2630 
 
 2653 
 
 2675 
 
 2698 
 
 2721 
 
 2744 
 
 71 
 
 14 
 
 15 
 
 16 
 
 16 
 
 17 
 
 472 
 
 2767 
 
 2790 
 
 2813 
 
 2835 1 2858 
 
 2881 
 
 2904 
 
 2927 
 
 2950 
 
 2973 
 
 72 
 
 14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 473 
 
 2996 
 
 3019 
 
 3042 3065 3087 
 
 3110 
 
 3133 
 
 3156 
 
 3179 
 
 3202 
 
 73 
 
 15 
 
 15 
 
 16 
 
 17 
 
 18 
 
 474 
 
 3225 
 
 3248 
 
 3271 
 
 3294 3317 
 
 3340 
 
 3363 
 
 3386 
 
 3409 
 
 3432 
 
 74 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 475 
 476 
 
 3455 
 
 3685 
 
 3478 
 3708 
 
 3501 
 3731 
 
 3524 3547 
 3754! 3777 
 
 3570 
 3800 
 
 3593 
 3823 
 
 3616 
 
 3846 
 
 3639 
 3869 
 
 3662 
 3892 
 
 75 
 76 
 
 15 
 15 
 
 16 
 16 
 
 17 
 
 17 
 
 17 
 17 
 
 18 
 
 18 
 
 477 
 
 3915 
 
 3938 
 
 3962 
 
 3985 4008 
 
 4031 
 
 4054 
 
 4077 
 
 4100 
 
 4123 
 
 77 
 
 15 
 
 16 
 
 17 
 
 18 
 
 18 
 
 478 
 
 4146 
 
 4169 
 
 4193 
 
 4216 4239 
 
 4262 
 
 4285 
 
 4308 
 
 4331 
 
 4354 
 
 78 
 
 16 
 
 16 
 
 17 
 
 18 
 
 19 
 
 479 
 
 4378 
 
 4401 
 
 4424 
 
 4447 
 
 4470 
 
 4493 
 
 4517 
 
 4540 
 
 4563 
 
 4586 
 
 79 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 480 
 
 114609 
 
 4633 
 
 4656 
 
 4679 
 
 4702 
 
 4725 
 
 4749 
 
 4772 
 
 4795 
 
 4818 
 
 80 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 481 
 
 4842 
 
 4865 
 
 4888 
 
 4911 
 
 4935 
 
 4958 
 
 4981 
 
 5004 
 
 5028 
 
 5051 
 
 81 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 482 
 
 5074 
 
 5097 
 
 5121 
 
 5144 
 
 5167 
 
 5191 
 
 5214 
 
 5237 
 
 5260 
 
 5284 
 
 82 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 483 
 
 5307 
 
 5330 
 
 5354 
 
 5377 
 
 5400 
 
 5424 
 
 5447 
 
 5470 
 
 5494 
 
 5517 
 
 83 
 
 17 
 
 17 
 
 18 
 
 19 
 
 20 
 
 484 
 
 5540 
 
 5564J 5587 
 
 5611 
 
 5634 
 
 5657 
 
 5681 
 
 5704 
 
 5727 
 
 5751 
 
 84 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 485 
 
 5774 
 
 5798 5821 
 
 5844! 5868 
 
 5891 
 
 5915 
 
 5938 
 
 5962 
 
 5985 
 
 85 
 
 17 
 
 18 
 
 19 
 
 20 
 
 20 
 
 486 
 
 6008 
 
 6032 6055 
 
 6079 6102 
 
 6126 
 
 6149 
 
 6173 
 
 6196 
 
 6220 
 
 86 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 487 
 
 6243 
 
 6267 6290 6314 6337 
 
 6361 
 
 6384 
 
 6408 
 
 6431 
 
 6455 
 
 87 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 488 
 489 
 
 6478 
 6714 
 
 6502 
 6737 
 
 6525 
 6761 
 
 6549 6572 
 6784 6808 
 
 6596 
 6831 
 
 6619 
 6855 
 
 6643 
 6879 
 
 6666 
 6902 
 
 6690 
 6926 
 
 88 
 89 
 
 18 
 18 
 
 18 
 19 
 
 19 
 
 20 
 
 20 
 2C 
 
 21 
 21 
 
 490 
 
 11 6949 
 
 6973: 6997 
 
 7020 
 
 7044 
 
 7068 
 
 7091 
 
 7115 
 
 7138 
 
 7162 
 
 9018ll9 
 
 2021 
 
 22 
 
 491 
 
 7186 
 
 7209 
 
 7233 
 
 7257 
 
 7280 
 
 7304 
 
 7328 
 
 7351 
 
 7375 
 
 7399 
 
 91 
 
 18H9 
 
 2021 
 
 22 
 
 492 
 
 7422 
 
 7446 
 
 7470 7494 
 
 7517 
 
 7541 
 
 7565 
 
 7588 
 
 7612 
 
 7636 
 
 92 
 
 1819 
 
 20'21 
 
 22 
 
 493 
 
 7660 
 
 7683 
 
 7707 
 
 7731 
 
 7755 
 
 7778 
 
 7802 
 
 7826 
 
 7850 
 
 7873 
 
 93 
 
 1920 
 
 2021 
 
 22 
 
 494 
 
 7897 
 
 7921 
 
 7945 
 
 7968 
 
 7992 
 
 8016 
 
 8040 
 
 8064 
 
 8087 
 
 8111 
 
 94 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 495 
 
 8135 
 
 8159 
 
 8183 
 
 8206 
 
 8230 
 
 8254 
 
 8278 
 
 8302 8326 
 
 8350 
 
 95 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 496 
 497 
 
 8373 
 8612 
 
 8397 
 8636 
 
 8421 
 8660 
 
 8445 
 8684 
 
 8469 
 8708 
 
 8493 
 
 8732 
 
 8517 
 8756 
 
 8540 8564 
 8780 8804 
 
 8588 
 8827 
 
 96 
 97 
 
 19 
 19 
 
 20 
 20 
 
 2122 
 21122 
 
 23 
 23 
 
 498 
 
 8851 
 
 8875 
 
 8899 
 
 8923 
 
 8947 
 
 8971 
 
 8995 
 
 9019 9043 
 
 9067 
 
 9820 
 
 21 
 
 2223 
 
 24 
 
 499 
 
 9091 
 
 9115 
 
 9139 
 
 9163 
 
 9187 
 
 9211 
 
 9235 
 
 9259 
 
 9283 
 
 9307 
 
 99 
 
 20 
 
 21 
 
 2223 
 
 24 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 20 
 
 21 
 
 2223 
 
 24 
 
 [15] 
 
TABLE I. 
 
 Log 
 
 
 
 Log (1 + aj) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 24 
 
 2 
 
 26 
 
 27 
 
 2 
 
 29 
 
 1-500 
 
 Oil 9331 
 
 9355 
 
 9379 
 
 9403 
 
 9427 
 
 9451 
 
 9475 
 
 9499 
 
 952C 
 
 9547 
 
 OC 
 
 C 
 
 C 
 
 C 
 
 C 
 
 C 
 
 
 501 
 
 9572 
 
 9596 
 
 9620 
 
 9644 
 
 9668 
 
 9692 
 
 9716 
 
 9740 
 
 9764 
 
 9788 
 
 01 
 
 C 
 
 C 
 
 C 
 
 C 
 
 C 
 
 
 502 
 
 9812 
 
 9837 
 
 9861 
 
 9885 
 
 9909 
 
 9933 
 
 9957 
 
 9981 
 
 oOO 
 
 0030 
 
 02 
 
 C 
 
 C 
 
 1 
 
 ] 
 
 1 
 
 
 503 
 
 120054 
 
 0078 
 
 0102 
 
 0126 
 
 0150 
 
 0175 
 
 0199 
 
 0223 
 
 0247 
 
 0271 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 504 
 
 0295 
 
 0320 
 
 0344 
 
 0368 
 
 0392 
 
 0416 
 
 0441 
 
 0465 
 
 048 
 
 0513 
 
 04 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 505 
 
 0538 
 
 0562 
 
 0586 
 
 0610 
 
 0635 
 
 0659 
 
 0682 
 
 0707 
 
 0735 
 
 0756 
 
 05 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 506 
 
 0780 
 
 0804 
 
 0829 
 
 0853 
 
 0877 
 
 0902 
 
 0926 
 
 0950 
 
 097 
 
 0999 
 
 06 
 
 1 
 
 2 
 
 a 
 
 e 
 
 
 
 507 
 
 1023 
 
 1047 
 
 1072 
 
 1096 
 
 1120 
 
 1145 
 
 1169 
 
 119; 
 
 121 
 
 1242 
 
 07 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 2 
 
 508 
 
 1267 
 
 1291 
 
 1315 
 
 1340 
 
 1364 
 
 1388 
 
 1413 
 
 1437 
 
 1465 
 
 1486 
 
 08 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 2 
 
 509 
 
 1510 
 
 1535 
 
 1559 
 
 1584 
 
 1608 
 
 1632 
 
 1657 
 
 1681 
 
 1706 
 
 1730 
 
 09 
 
 2 
 
 2 
 
 2 
 
 c 
 
 
 3 
 
 510 
 
 12 1755 
 
 1779 
 
 1804 
 
 1828 
 
 1853 
 
 1877 
 
 1901 
 
 1926 
 
 1950 
 
 1975 
 
 ]( 
 
 2 
 
 2 
 
 f 
 
 t. 
 
 
 3 
 
 511 
 
 1999 
 
 2024 
 
 2048 
 
 2073 
 
 2097 
 
 2122 
 
 2146 
 
 2171 
 
 91 o^ 
 
 2990 
 
 11 
 
 3 
 
 } 
 
 o 
 
 <> 
 
 
 
 512 
 
 2244 
 
 2269 
 
 2294 
 
 2318 
 
 2343 
 
 2367 
 
 2392 
 
 2416 
 
 AXtTt 
 
 2441 
 
 HuJHXAn 
 
 2465 
 
 12 
 
 3 
 
 2 
 
 g 
 
 tl 
 
 ^ 
 
 3 
 
 513 
 
 2490 
 
 2515 
 
 2539 
 
 2564 
 
 2588 
 
 2613 
 
 2638 
 
 2662 
 
 2687 
 
 2711 
 
 13 
 
 3 
 
 r 
 
 4] 
 
 4 
 
 4 
 
 i 
 
 514 
 
 2736 
 
 2761 
 
 2785 
 
 2810 
 
 2835 
 
 2859 
 
 2884 
 
 2908 
 
 2933 
 
 2958 
 
 14 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 515 
 
 2982 
 
 3007 
 
 3032 
 
 3056 
 
 3081 
 
 3106 
 
 3130 
 
 3155 
 
 3180 
 
 3205 
 
 15 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 < 
 
 516 
 
 3229 
 
 3254 
 
 3279 
 
 3303 
 
 3328 
 
 3353 
 
 33781 3402 
 
 3427 
 
 3452 
 
 16 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 517 
 
 3476 
 
 3501 
 
 3526 
 
 3551 
 
 3576 
 
 3600 
 
 3625! 3650 
 
 3675 
 
 3699 
 
 17 
 
 4 
 
 4 
 
 4 
 
 r. 
 *j 
 
 5 
 
 5 
 
 518 
 
 3724 
 
 3749 
 
 3774 
 
 3799 
 
 3823 
 
 3848 
 
 3873 
 
 3898 
 
 3923 
 
 3947 
 
 18 
 
 4 
 
 4 
 
 5 
 
 TJ 
 
 5 
 
 K 
 
 519 
 
 3972 
 
 3997 
 
 4022 
 
 4047 
 
 4072 
 
 4097 
 
 4121 
 
 4146 
 
 4171 
 
 4196 
 
 19 
 
 5 
 
 5 
 
 r; 
 
 G 
 
 K 
 
 . 
 
 5 
 
 6 
 
 520 
 
 124221 
 
 4246 
 
 4271 
 
 4295 
 
 4320 
 
 4345 
 
 4370 
 
 4395 
 
 4420 
 
 4445 
 
 20 
 
 i 
 
 5 
 
 5 
 
 f\ 
 
 6 
 
 6 
 
 521 
 
 4470 
 
 4495 
 
 4520 
 
 4545 
 
 4570 
 
 4594 
 
 4619 
 
 4644 
 
 4669 
 
 4694 
 
 21 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 522 
 
 4719 
 
 4744 
 
 4769 
 
 4794 
 
 4819 
 
 4844 
 
 4869 
 
 4894 
 
 4919 
 
 4944 
 
 22 
 
 K 
 . 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 523 
 
 4969 
 
 4994 
 
 5019 
 
 5044 
 
 5069 
 
 5094 
 
 5119 
 
 5144 
 
 5169 
 
 5194 
 
 23 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 524 
 
 5219 
 
 5244 
 
 5269 
 
 5294 
 
 5320 
 
 5345 
 
 5370 
 
 5395 
 
 5420 
 
 5445 
 
 24 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 *" 
 
 525 
 
 5470 
 
 5495 
 
 5520 
 
 5545 
 
 5570 
 
 5596 
 
 5621 
 
 5646 
 
 5671 
 
 5696 
 
 25 
 
 6 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 526 
 
 5721 
 
 5746 
 
 5771 
 
 5797 
 
 5822 
 
 5847 
 
 5872 
 
 5897 
 
 5922 
 
 5948 
 
 26 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 8 
 
 527 
 
 5973 
 
 5998 
 
 6023 
 
 6048 
 
 6073 
 
 6099 
 
 6124 
 
 6149 
 
 6174 
 
 6200 
 
 27 
 
 (i 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 528 
 
 6225 
 
 6250 
 
 6275 
 
 6300 
 
 6326 
 
 6351 
 
 6376 
 
 6401 
 
 6427 
 
 6452 
 
 28 
 
 / 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 529 
 
 6477 
 
 6502 
 
 6528 
 
 6553 
 
 6578 
 
 6604 
 
 6629 
 
 6654 
 
 6679 
 
 6705 
 
 29 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 530 
 
 12 6730 
 
 6755 
 
 6781 
 
 6806 
 
 6831 
 
 6857 
 
 6882 
 
 6907 
 
 6933 
 
 6958 
 
 30 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 531 
 
 6983 
 
 7009 
 
 7034 
 
 7059 
 
 7085 
 
 7110 
 
 7136 
 
 7161 
 
 7186 
 
 7212 
 
 31 
 
 7 
 
 8 
 
 8 
 
 8 
 
 9 
 
 c 
 
 532 
 
 7237 
 
 7262 
 
 7288 
 
 7313 
 
 7339 
 
 7364 
 
 7390 
 
 7415 
 
 7440 
 
 7466 
 
 32 
 
 8 
 
 6 
 
 8 
 
 9 
 
 9 
 
 9 
 
 533 
 
 7491 
 
 7517 
 
 7542 
 
 7568 
 
 7593 
 
 7619 
 
 7644 
 
 7669 
 
 7695 
 
 7720 
 
 33 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 10 
 
 534 
 
 7746 
 
 7771 
 
 7797 
 
 7822 
 
 7848 
 
 7873 
 
 7899 
 
 7924 
 
 7950 
 
 7975 
 
 34 
 
 8 
 
 8 
 
 Q 
 
 9 
 
 10 
 
 10 
 
 535 
 
 8001 
 
 8026 
 
 8052 
 
 8078 
 
 8103 
 
 8129 
 
 8154 
 
 8180 
 
 8205 
 
 8231 
 
 35 
 
 8 
 
 9 
 
 C 
 
 9 
 
 10 
 
 10 
 
 536 
 
 8256 
 
 8282 
 
 8308 
 
 8333 
 
 8359 
 
 8384 
 
 8410 
 
 8436 
 
 8461 
 
 8487 
 
 36 
 
 9 
 
 c 
 
 9 
 
 10 
 
 10 
 
 10 
 
 537 
 
 8512 
 
 8538 
 
 8564 
 
 8589 
 
 8615 
 
 8640 
 
 8666 
 
 8692 
 
 8717 
 
 8743 
 
 37 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 538 
 
 8769 
 
 8794 
 
 8820 
 
 8846 
 
 8871 
 
 8897 
 
 8923 
 
 8948 
 
 8974 
 
 9000 
 
 38 
 
 9 
 
 10 
 
 10 
 
 1C) 
 
 11 
 
 11 
 
 539 
 
 9025 
 
 9051 
 
 9077 
 
 9103 
 
 9128 
 
 9154 
 
 9180 
 
 9206 
 
 9231 
 
 9257 
 
 39 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 540 
 
 12 9283 
 
 9308 
 
 9334 
 
 9360 
 
 9386 
 
 9412 
 
 9437 
 
 9463 
 
 9489 
 
 9515 
 
 40 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 541 
 
 9540 
 
 9566 
 
 9592 
 
 9618 
 
 9644 
 
 9669 
 
 9695 
 
 9721 
 
 9747 
 
 9773 
 
 41 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 12 
 
 542 
 
 9799 
 
 9824 
 
 9850 
 
 9876 
 
 9902 
 
 9928 
 
 9954 
 
 9980 
 
 0005 
 
 0031 
 
 42 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 543 
 
 13 0057 
 
 0083 
 
 0109 
 
 0135 
 
 0161 
 
 0187 
 
 0212 
 
 0238 
 
 0264 
 
 0290 
 
 4310 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 544 
 
 0316 
 
 0342 
 
 0368 
 
 0394 
 
 0420 
 
 0446 
 
 0472 
 
 0498 
 
 0524 
 
 0550 
 
 44 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 545 
 
 0576 
 
 0602 
 
 0628 
 
 0654 
 
 0679 
 
 0705 
 
 0731 
 
 0757 
 
 0783 
 
 0809 
 
 45 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 546 
 
 0835 
 
 0861 
 
 0888 
 
 0914 
 
 0940 
 
 0966 
 
 0992 
 
 1018 
 
 1044 
 
 1070 
 
 46 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 547 
 
 1096 
 
 1122 
 
 1148 
 
 1174 
 
 1200 
 
 1226 
 
 1252 
 
 1278 
 
 1304 
 
 1330 
 
 :7 
 
 11 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 548 
 
 1357 
 
 1383 
 
 1409 
 
 1435 
 
 1461 
 
 1487 
 
 1513 
 
 1539 
 
 1566 
 
 1592 
 
 48 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 549 
 
 1618 
 
 1644 
 
 1670 
 
 1696 
 
 1722 
 
 1749 
 
 1775 
 
 1801 
 
 1827 
 
 1853 
 
 49 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 [16] 
 
TABLE I. 
 
 Log 
 
 Log(l + a) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 1-550 
 
 0-13 1879 
 
 1906 
 
 1932 
 
 1958 
 
 1984 
 
 2010 
 
 2037 
 
 2063 
 
 2089 
 
 2115 
 
 50 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 551 
 
 2142 
 
 2168 
 
 2194 
 
 2220 
 
 2247 
 
 2273 
 
 2299 
 
 2325 
 
 2352 
 
 2378 
 
 51 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 552 
 
 2404 
 
 2430 
 
 2457 
 
 2483 
 
 2509 
 
 2536 
 
 2562 
 
 2588 
 
 2615 
 
 2641 
 
 52 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 553 
 
 2667 
 
 2693 
 
 2720 
 
 2746 
 
 2772 
 
 2799 
 
 2825 
 
 2852 
 
 2878 
 
 2904 
 
 53 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 554 
 
 2931 
 
 2957 
 
 2983 
 
 3010 
 
 3036 
 
 3063 
 
 3089 
 
 3115 
 
 3142 
 
 3168 
 
 54 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 555 
 
 3195 
 
 3221 
 
 3247 
 
 3274 
 
 3300 
 
 3327 
 
 3353 
 
 3380 
 
 3406 
 
 3432 
 
 55 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 556 
 
 3459 
 
 3485 
 
 3512 
 
 3538 
 
 3565 
 
 3591 
 
 3618 
 
 3644 
 
 3671 
 
 3697 
 
 56 
 
 13 
 
 H 
 
 15 
 
 15 
 
 16 
 
 16 
 
 557 
 
 3724| 3750 
 
 3777 
 
 3803 
 
 3830 
 
 3856 
 
 3883 
 
 3909 
 
 3936 
 
 3962 
 
 57 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 558 
 
 3989| 4015 
 
 4042 
 
 4069 
 
 4095 
 
 4122 
 
 4148 
 
 4175 
 
 4201 
 
 4228 
 
 58 
 
 14 
 
 14 
 
 15 
 
 16 
 
 16 
 
 17 
 
 559 
 
 4255 
 
 4281 
 
 4308 
 
 4334 
 
 4361 
 
 4388 
 
 4414 
 
 4441 
 
 4467 
 
 4494 
 
 59 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 17 
 
 560 
 
 13 4521 
 
 4547 
 
 4574 
 
 4601 
 
 4627 
 
 4654 
 
 4681 
 
 4707 
 
 4734 
 
 4761 
 
 60 
 
 14 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 561 
 
 4787 
 
 4814 
 
 4841 
 
 4867 
 
 4894 
 
 4921 
 
 4947 
 
 4974 
 
 5001 
 
 5028 
 
 61 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 562 
 
 5054 
 
 5081 
 
 5108 
 
 5135 
 
 5161 
 
 5188 
 
 5215 
 
 5242 
 
 5268 
 
 5295 
 
 62 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 563 
 
 5322 
 
 5349 
 
 5375 
 
 5402 
 
 5429 
 
 5456 
 
 5483 
 
 5509 
 
 5536 
 
 5563 
 
 63 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 18 
 
 564 
 
 5590 
 
 5617 
 
 5643 
 
 5670 
 
 5697 
 
 5724 
 
 5751 
 
 5778 
 
 5804 
 
 5831 
 
 64 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 565 
 
 5858 
 
 5885 
 
 5912 
 
 5939 
 
 5966 
 
 5993 
 
 6019 
 
 6046 
 
 6073 
 
 6100 
 
 65 
 
 16 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 566 
 
 6127 
 
 6154 
 
 6181 
 
 6208 
 
 6235 
 
 6262 
 
 6289 
 
 6316 
 
 6342 
 
 6369 
 
 66 
 
 16 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 567 
 
 6396 
 
 6423 
 
 6450 
 
 6477 
 
 6504 
 
 6531 
 
 6558 
 
 6585 
 
 6612 
 
 6639 
 
 67 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 568 
 
 6666 
 
 6693 
 
 6720 
 
 6747 
 
 6774 
 
 6801 
 
 6828 
 
 6855 
 
 6882 
 
 6909 
 
 68 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 569 
 
 6936 
 
 6963 
 
 6990 
 
 7017 
 
 7045 
 
 7072 
 
 7099 
 
 7126 
 
 7153 
 
 7180 
 
 69 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 570 
 
 13 7207 
 
 7234 
 
 7261 
 
 7288 
 
 7315 
 
 7342 
 
 7370 
 
 7397 
 
 7424 
 
 7451 
 
 70 
 
 17 
 
 18 
 
 18 
 
 1920 
 
 20 
 
 571 
 
 7478 
 
 7505 
 
 7532 
 
 7560 
 
 7587 
 
 7614 
 
 7641 
 
 7668 
 
 7695 
 
 7722 
 
 71 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 21 
 
 572 
 
 7750 
 
 7777 
 
 7804 
 
 7831 
 
 7858 
 
 7886 
 
 7913 
 
 7940 
 
 7967 
 
 7994 
 
 72 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 573 
 
 8022 
 
 8049 
 
 8076 
 
 8103 
 
 8131 
 
 8158 
 
 8185 
 
 8212 
 
 8240 
 
 8267 
 
 73 
 
 18 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 574 
 
 8294 
 
 8321 
 
 8349 
 
 8376 
 
 8403 
 
 8431 
 
 8458 
 
 8485 
 
 8512 
 
 8540 
 
 74 
 
 18 
 
 18 
 
 19 
 
 20 
 
 21 
 
 21 
 
 575 
 
 8567 
 
 8594 
 
 8622 
 
 8649 
 
 8676 
 
 8704 
 
 8731 
 
 8758 
 
 8786 
 
 8813 
 
 75 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 576 
 
 8841 
 
 8868 
 
 8895 
 
 8923 
 
 8950 
 
 8977 
 
 9005 
 
 9032 
 
 9060 
 
 9087 
 
 76 
 
 18 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 577 
 
 9114 
 
 9142 
 
 9169 
 
 9197 
 
 9224 
 
 9251 
 
 9279 
 
 9306 
 
 9334 
 
 9361 
 
 77 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 22 
 
 578 
 
 9389 
 
 9416 
 
 9444 
 
 9471 
 
 9499 
 
 9526 
 
 9553 
 
 9581 
 
 9608 
 
 9636 
 
 78 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 23 
 
 579 
 
 9663 
 
 9691 
 
 9718 
 
 9746 
 
 9773 
 
 9801 
 
 9829 
 
 9856 
 
 9884 
 
 9911 
 
 79 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 580 
 
 13 9939 
 
 9966 
 
 9994 
 
 0021 
 
 0049 
 
 0076 
 
 0104 
 
 0132 
 
 0159 
 
 0187 
 
 80 
 
 1920 
 
 21 
 
 22 
 
 22 
 
 23 
 
 581 
 
 140214 
 
 0242 
 
 0270 
 
 0297 
 
 0325 
 
 0352 
 
 0380 
 
 0408 
 
 0435 
 
 0463 
 
 81 
 
 1920 
 
 21 
 
 22 
 
 23 
 
 23 
 
 582 
 
 0491 
 
 0518 
 
 0546 
 
 0573 
 
 0601 
 
 0629 
 
 0656 
 
 0684 
 
 0712 
 
 0739 
 
 82 
 
 2020 
 
 21 
 
 22 
 
 23 
 
 24 
 
 583 
 
 0767 
 
 0795 
 
 0823 
 
 0850 
 
 0878 
 
 0906 
 
 0933 
 
 0961 
 
 0989 
 
 1016 
 
 83 
 
 2021 
 
 22 
 
 22 
 
 23 
 
 24 
 
 584 
 
 1044 
 
 1072 
 
 1100 
 
 1127 
 
 1155 
 
 1183 
 
 1211 
 
 1238 
 
 1266 
 
 1294 
 
 84 
 
 2021 
 
 22 
 
 23 
 
 24 
 
 24 
 
 585 
 
 1322 
 
 1350 
 
 1377 
 
 1405 
 
 1433 
 
 1461 
 
 1488 
 
 1516 
 
 1544 
 
 1572 
 
 85 
 
 2021 
 
 22 
 
 23 
 
 24 
 
 25 
 
 586 
 
 1600 
 
 1628 
 
 1655 
 
 1683 
 
 1711 
 
 1739 
 
 1767 
 
 1795 
 
 1822 
 
 1850 
 
 86 
 
 21 22 22 
 
 23 
 
 24 
 
 25 
 
 587 
 
 1878 
 
 1906 
 
 1934 
 
 1962 
 
 1990 
 
 2018 
 
 2046 
 
 2073 
 
 2101 
 
 2129 
 
 87 
 
 212223123 
 
 24 
 
 25 
 
 588 
 
 2157 
 
 2185 
 
 2213 
 
 2241 
 
 2269 
 
 2297 
 
 2325 
 
 2353 
 
 2381 
 
 2409 
 
 88 
 
 21 122 23124 
 
 25 
 
 26 
 
 589 
 
 2437 
 
 2464 
 
 2492 
 
 2520 
 
 2548 
 
 2576 
 
 2604 
 
 2632 
 
 2660 
 
 2688 
 
 89 
 
 2122 
 
 2324 
 
 25 
 
 26 
 
 590 
 
 142716 
 
 2744 
 
 2772 
 
 2800 
 
 2828 
 
 2856 
 
 2884 
 
 2913 
 
 2941 
 
 2969 
 
 90 
 
 22J22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 591 
 
 2997 
 
 3025 
 
 3053 
 
 3081 
 
 3109 
 
 3137 
 
 3165 
 
 3193 
 
 3221 
 
 3249 
 
 91 
 
 2223 
 
 24 
 
 25 
 
 25 
 
 26 
 
 592 
 
 3277 
 
 3306 
 
 3334 
 
 3362 
 
 3390 
 
 3418 
 
 3446 
 
 3474 
 
 3502 
 
 3531 
 
 92 
 
 2223 
 
 24 
 
 2o 
 
 26 
 
 27 
 
 593 
 
 3559 
 
 3587 
 
 3615 
 
 3643 
 
 3671 
 
 3699 
 
 3728 
 
 3756 
 
 3784 
 
 3812 
 
 93 
 
 2223 
 
 24 
 
 25 
 
 26 
 
 27 
 
 594 
 
 3840 
 
 3869 
 
 3897 
 
 3925 
 
 3953 
 
 3981 
 
 4010 
 
 4038 
 
 4066 
 
 4094 
 
 94 
 
 23 
 
 24 
 
 24 
 
 25 
 
 26 
 
 27 
 
 595 
 
 4123 
 
 4151 
 
 4179 
 
 4207 
 
 4236 
 
 4264 
 
 4292 
 
 4320 
 
 4349 
 
 4377 
 
 95 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 596 
 
 4405 
 
 4434 
 
 4462 
 
 4490 
 
 4518 
 
 4547 
 
 4575 
 
 4603 
 
 4632 
 
 4660 
 
 96 
 
 23 
 
 2425 
 
 26 
 
 27 
 
 28 
 
 597 
 
 4688 
 
 4717 
 
 4745 
 
 4773 
 
 4802 
 
 4830 
 
 4858 
 
 4887 
 
 4915 
 
 4944 
 
 97 
 
 23 
 
 2425 
 
 26 
 
 27 
 
 28 
 
 598 
 
 4972 
 
 5000 
 
 5029 
 
 5057 
 
 5085 
 
 5114 
 
 5142 
 
 5171 
 
 5199 
 
 5228 
 
 98 
 
 24 
 
 2425 
 
 26 
 
 27 
 
 28 
 
 599 
 
 5256 
 
 5284 
 
 5313 
 
 5341 
 
 5370 
 
 5398 
 
 5427 
 
 5455 
 
 5484 
 
 5512 
 
 99 
 
 24 
 
 2526 
 
 27 
 
 28 
 
 29 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 26 
 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 24 
 
 25 
 
 27 
 
 28 
 
 29 
 
TABLE I. 
 
 Log 
 
 
 
 
 Log (1+aj) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 28 
 
 20 
 
 30 
 
 31 
 
 32 
 
 33 
 
 f-600 
 
 014 5540 
 
 5569 
 
 5597 
 
 5626 
 
 5654 
 
 5683 
 
 5711 
 
 5740 
 
 5768 
 
 5797 
 
 0( 
 
 
 
 
 
 
 
 
 
 
 
 
 
 601 
 
 5825 
 
 5854 
 
 5882 
 
 5911 
 
 5940 
 
 5968 
 
 5997 
 
 6025 
 
 6054 
 
 6082 
 
 01 
 
 
 
 
 
 o 
 
 o 
 
 
 
 
 
 602 
 
 6111 
 
 6139 
 
 6168 
 
 6197 
 
 6225 
 
 6254 
 
 6282 
 
 6311 
 
 6340 
 
 6368 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 603 
 
 6397 
 
 6425 
 
 6454 
 
 6483 
 
 6511 
 
 6540 
 
 6569 
 
 6597 
 
 6626 
 
 6655 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 604 
 
 6683 
 
 6712 
 
 6741 
 
 6769 
 
 6798 
 
 6827 
 
 6855 
 
 6884 
 
 6913 
 
 6941 
 
 04 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 605 
 
 6970 
 
 6999 
 
 7028 
 
 7056 
 
 7085 
 
 7114 
 
 7142 
 
 7171 
 
 7200 
 
 7229 
 
 06 
 
 1 
 
 1 
 
 1 
 
 2 
 
 2 
 
 2 
 
 606 
 
 7257 
 
 7286 
 
 7315 
 
 7344 
 
 7372 
 
 7401 
 
 7430 
 
 7459 
 
 7488 
 
 7516 
 
 0( 
 
 9 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 607 
 
 7545 
 
 7574 
 
 7603 
 
 7632 
 
 7660 
 
 7689 
 
 7718 
 
 7747 
 
 7776 
 
 7805 
 
 07 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 608 
 
 7833 
 
 7862 
 
 7891 
 
 7920 
 
 7949 
 
 7978 
 
 8007 
 
 8036 
 
 8064 
 
 8093 
 
 OS 
 
 2 
 
 2 
 
 2 
 
 2 
 
 3 
 
 3 
 
 609 
 
 8122 
 
 8151 
 
 8180 
 
 8209 
 
 8238 
 
 8267 
 
 s2'.)G 
 
 8325 
 
 8354 
 
 8383 
 
 09 
 
 a 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 610 
 
 148411 
 
 8440 
 
 8469 
 
 8498 
 
 8527 
 
 8556 
 
 8585 
 
 Sfil4 
 
 8643 
 
 8672 
 
 LI 
 
 3 
 
 3 
 
 3 
 
 <> 
 > 
 
 3 
 
 3 
 
 611 
 
 8701 
 
 8730 
 
 8759 
 
 8788 
 
 8817 
 
 8846 
 
 8875 
 
 8904 
 
 8933 
 
 8962 
 
 1 1 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 612 
 
 8991 
 
 9020 
 
 9049 
 
 9078 
 
 9108 
 
 9137 
 
 9166 
 
 9195 
 
 9224 
 
 9253 
 
 12 
 
 3 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 613 
 
 9282 
 
 9311 
 
 9340 
 
 9369 
 
 9398 
 
 9427 
 
 9457 
 
 9486 
 
 9515 
 
 9544 
 
 L3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 614 
 
 9573 
 
 9602 
 
 9631 
 
 9661 
 
 9690 
 
 9719 
 
 9748 
 
 9777 
 
 9806 
 
 9836 
 
 14 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 615 
 
 9865 
 
 9894 
 
 9923 
 
 9952 
 
 9981 
 
 oOll 
 
 0040 
 
 0069 
 
 0098 
 
 0128 
 
 15 
 
 4 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 616 
 
 150157 
 
 0186 
 
 0215 
 
 0244 
 
 0274 
 
 0303 
 
 0332 
 
 0361 
 
 0391 
 
 0420 
 
 1C 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 617 
 
 0449 
 
 0479 
 
 0508 
 
 0537 
 
 0566 
 
 0596 
 
 0625 
 
 0654 
 
 0684 
 
 0713 
 
 17 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 618 
 
 0742 
 
 0772 
 
 0801 
 
 0830 
 
 0860 
 
 0889 
 
 0918 
 
 0948 
 
 0977 
 
 loon 
 
 is 
 
 ff 
 
 6 
 
 6 
 
 c 
 
 (i 
 
 6 
 
 619 
 
 1036 
 
 1065 
 
 1095 
 
 1124 
 
 1153 
 
 1183 
 
 1212 
 
 1242 
 
 1271 
 
 1300 
 
 10 
 
 5 
 
 6 
 
 8 
 
 <; 
 
 6 
 
 6 
 
 620 
 
 15 1330 
 
 1359 
 
 1389 
 
 1418 
 
 1448 
 
 1477 
 
 1506 
 
 1536 
 
 1565 
 
 1595 
 
 20 
 
 6 
 
 6 
 
 6 
 
 (, 
 
 6 
 
 7 
 
 621 
 
 1624 
 
 1654 
 
 1683 
 
 1713 
 
 1742 
 
 1772 
 
 1801 
 
 1831 
 
 1860 
 
 1890 
 
 21 
 
 8 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 622 
 
 1919 
 
 1949 
 
 1978 
 
 2008 
 
 2037 
 
 2067 
 
 2096 
 
 2126 
 
 2156 
 
 2185 
 
 1-1 
 
 (5 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 623 
 
 2215 
 
 2244 
 
 2274 
 
 2303 
 
 2333 
 
 2363 
 
 2392 
 
 2422 
 
 2451 
 
 2481 
 
 23 
 
 8 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 624 
 
 2511 
 
 2540 
 
 2570 
 
 2599 
 
 2629 
 
 2659 
 
 2688 
 
 2718 
 
 2748 
 
 2777 
 
 24 
 
 7 
 
 7 
 
 7 
 
 7 
 
 s 
 
 8 
 
 625 
 
 2807 
 
 2837 
 
 2866 
 
 2896 
 
 2926 
 
 2955 
 
 2985 
 
 3015 
 
 3044 
 
 3074 
 
 25 
 
 7 
 
 7 
 
 7 
 
 8 
 
 s 
 
 8 
 
 626 
 
 3104 
 
 3133 
 
 3163 
 
 3193 
 
 3223 
 
 3252 
 
 3282 
 
 3312 
 
 3342 
 
 3371 
 
 26 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 627 
 
 3401 
 
 3431 
 
 3461 
 
 3490 
 
 3520 
 
 3550 
 
 3580 
 
 3610 
 
 3639 
 
 3669 
 
 27 
 
 8 
 
 8 
 
 8 
 
 8 
 
 
 
 9 
 
 628 
 
 3699 
 
 3729 
 
 3759 
 
 3788 
 
 3818 
 
 3848 
 
 3878 
 
 3908 
 
 3938 
 
 3967 
 
 28 
 
 6 
 
 6 
 
 8 
 
 
 
 
 
 9 
 
 629 
 
 3997 
 
 4027 
 
 4057 
 
 4087 
 
 4117 
 
 4147 
 
 4176 
 
 4206 
 
 4236 
 
 4266 
 
 29 
 
 8 
 
 8 
 
 9 
 
 
 
 9 
 
 10 
 
 630 
 
 15 4296 
 
 4326 
 
 4356 
 
 4386 
 
 4416 
 
 4446 
 
 4475 
 
 4505 
 
 4535 
 
 4565 
 
 30 
 
 8 
 
 9 
 
 o 
 
 
 
 10 
 
 10 
 
 631 
 
 4595 
 
 4625 
 
 4655 
 
 4685 
 
 4715 
 
 4745 
 
 4775 
 
 4805 
 
 4835 
 
 4865 
 
 31 
 
 9 
 
 9 
 
 9 
 
 1( 
 
 1010 
 
 632 
 
 4895 
 
 4925 
 
 4955 
 
 4985 
 
 5015 
 
 5045 
 
 5075 
 
 5105 
 
 5135 
 
 5165 
 
 32 
 
 9 
 
 9 
 
 10 
 
 1( 
 
 10 
 
 11 
 
 633 
 
 5195 
 
 5225 
 
 5255 
 
 5285 
 
 5315 
 
 5346 
 
 5376 
 
 5406 
 
 5436 
 
 5466 
 
 33 
 
 9 
 
 10 
 
 10 
 
 1( 
 
 11 
 
 11 
 
 634 
 
 5496 
 
 5526 
 
 5556 
 
 5586 
 
 5616 
 
 5646 
 
 5677 
 
 5707 
 
 5737 
 
 5767 
 
 34 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 635 
 
 5797 
 
 5827 
 
 5857 
 
 5888 
 
 5918 
 
 5948 
 
 5978 
 
 6008 
 
 6038 
 
 6069 
 
 35 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 12 
 
 636 
 
 6099 
 
 6129 
 
 6159 
 
 6189 
 
 6220 
 
 6250 
 
 6280 
 
 6310 
 
 6341 
 
 6371 
 
 36 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 12 
 
 637 
 
 6401 
 
 6431 
 
 6461 
 
 6492 
 
 6522 
 
 6552 
 
 6583 
 
 6613 
 
 6643 
 
 6673 
 
 37 
 
 10 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 638 
 
 6704 
 
 6734 
 
 6764 
 
 6795 
 
 6825 
 
 6855 
 
 6885 
 
 6916 
 
 6946 
 
 6976 
 
 38 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 13 
 
 639 
 
 7007 
 
 7037 
 
 7067 
 
 7098 
 
 7128 
 
 7159 
 
 7189 
 
 7219 
 
 7250 
 
 7280 
 
 39 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 640 
 
 15 7310 
 
 7341 
 
 7371 
 
 7402 
 
 7432 
 
 7462 
 
 7493 
 
 7523 
 
 7554 
 
 7584 
 
 4( 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 641 
 
 7615 
 
 7645 
 
 7675 
 
 7706 
 
 7736 
 
 7767 
 
 7797 
 
 7828 
 
 7858 
 
 7889 
 
 41 
 
 II 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 642 
 
 7919 
 
 7950 
 
 7980 
 
 8011 
 
 8041 
 
 8072 
 
 8102 
 
 8133 
 
 8163 
 
 8194 
 
 42 
 
 L2 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 643 
 
 8224 
 
 8255 
 
 8285 
 
 8316 
 
 8346 
 
 8377 
 
 8408 
 
 8438 
 
 8469 
 
 8499 
 
 43 
 
 12 
 
 12 
 
 13 
 
 13 
 
 14 
 
 14 
 
 644 
 
 8530 
 
 8560 
 
 8591 
 
 8622 
 
 8652 
 
 8683 
 
 8713 
 
 8744 
 
 8775 
 
 8805 
 
 44 
 
 [-2 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 645 
 
 8836 
 
 8867 
 
 8897 
 
 8928 
 
 8958 
 
 8989 
 
 9020 
 
 9050 
 
 9081 
 
 9112 
 
 45 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 15 
 
 646 
 
 9142 
 
 9173 
 
 9204 
 
 9234 
 
 9265 
 
 9296 
 
 9327 
 
 9357 
 
 9388 
 
 9419 
 
 46 
 
 L3 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 647 
 
 9449 
 
 9480 
 
 9511 
 
 9542 
 
 9572 
 
 9603 
 
 9634 
 
 9665 
 
 9695 
 
 9726 
 
 47 
 
 L3 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 648 
 
 9757 
 
 9788 
 
 9819 
 
 9849 
 
 9880 
 
 9911 
 
 9942 
 
 9973 
 
 o003 
 
 o034 
 
 48 
 
 13 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 649 
 
 16 0065 
 
 0096 
 
 0127 
 
 0158 
 
 0188 
 
 0219 
 
 0250 
 
 0281 
 
 0312 
 
 0343 
 
 49 
 
 14 
 
 14 
 
 15 
 
 15 
 
 10 
 
 16 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 2s 
 
 20 
 
 30 
 
 31 
 
 12 
 
 33 
 
 [18] 
 
TABLE I. 
 
 Log 
 
 Log (l+x) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 T650 
 
 0'16 0374 
 
 0404 
 
 0435 
 
 0466 
 
 0497 
 
 0528 
 
 0559 
 
 0590 
 
 0621 
 
 0652 
 
 50 
 
 14 
 
 14 
 
 15 
 
 15 
 
 10 
 
 16 
 
 651 
 
 0683 
 
 0713 
 
 0744 
 
 0775 
 
 0806 
 
 0837 
 
 0868 
 
 0899 
 
 0930 
 
 0961 
 
 51 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 652 
 
 0992 
 
 1023 
 
 1054 
 
 1085 
 
 1116 
 
 1147 
 
 1178 
 
 1209 
 
 1240 
 
 1271 
 
 5215 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 653 
 
 1302 
 
 1333 
 
 1364 
 
 1395 
 
 1426 
 
 1457 
 
 1488 
 
 1519 
 
 1550 
 
 1581 
 
 5315 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 654 
 
 1613 
 
 1644 
 
 1675 
 
 1706 
 
 1737 
 
 1768 
 
 1799 
 
 1830 
 
 1861 
 
 1892 
 
 54 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 655 
 
 1924 
 
 1955 
 
 1986 
 
 2017 
 
 2048 
 
 2079 
 
 2110 
 
 2142 
 
 2173 
 
 2204 
 
 55 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 656 
 
 2235 
 
 2266 
 
 2297 
 
 2329 
 
 2360 
 
 2391 
 
 2422 
 
 2453 
 
 2485 
 
 2516 
 
 56 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 657 
 
 2547 
 
 2578 
 
 2609 
 
 2641 
 
 2672 
 
 2703 
 
 2734 
 
 2766 
 
 2797 
 
 2828 
 
 57 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 658 
 
 2859 
 
 2891 
 
 2922 
 
 2953 
 
 2985 
 
 3016 
 
 3047 
 
 3078 
 
 3110 
 
 3141 
 
 58 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 659 
 
 3172 
 
 3204 
 
 3235 
 
 3266 
 
 3298 
 
 3329 
 
 3360 
 
 3392 
 
 3423 
 
 3454 
 
 59 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 660 
 
 16 3486 
 
 3517 
 
 3549 
 
 3580 
 
 3611 
 
 3643 
 
 3674 
 
 3706 
 
 3737 
 
 3768 
 
 GO 
 
 17 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 661 
 
 3800 
 
 3831 
 
 3863 
 
 3894 
 
 3926 
 
 3957 
 
 3988 
 
 4020 
 
 4051 
 
 4083 
 
 01 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 20 
 
 662 
 
 4114 
 
 4146 
 
 4177 
 
 4209 
 
 4240 
 
 4272 
 
 4303 
 
 4335 
 
 4366 
 
 4398 
 
 G2 
 
 17 
 
 18 
 
 19 
 
 1920 
 
 20 
 
 663 
 
 4429 
 
 4461 
 
 4492 
 
 4524 
 
 4555 
 
 4587 
 
 4618 
 
 4650 
 
 4681 
 
 4713 
 
 63 
 
 18 
 
 18 
 
 19 
 
 2020 
 
 21 
 
 664 
 
 4745 
 
 4776 
 
 4808 
 
 4839 
 
 4871 
 
 4903 
 
 4934 
 
 4966 
 
 4997 
 
 5029 
 
 04 
 
 18 
 
 19 
 
 19 
 
 2020 
 
 21 
 
 665 
 
 5061 
 
 5092 
 
 5124 
 
 5155 
 
 5187 
 
 5219 
 
 5250 
 
 5282 
 
 5314 
 
 5345 
 
 65 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 21 
 
 666 
 
 5377 
 
 5409 
 
 5440 
 
 5472 
 
 5504 
 
 5535 
 
 5567 
 
 5599 
 
 5630 
 
 5662 
 
 60 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 667 
 
 5694 
 
 5726 
 
 5757 
 
 5789 
 
 5821 
 
 5853 
 
 5884 
 
 5916 
 
 5948 
 
 5980 
 
 G7 
 
 19 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 668 
 
 6011 
 
 6043 
 
 6075 
 
 6107 
 
 6138 
 
 6170 
 
 6202 
 
 6234 
 
 6266 
 
 6297 
 
 68 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 22 
 
 669 
 
 6329 
 
 6361 
 
 6393 
 
 6425 
 
 6457 
 
 6488 
 
 6520 
 
 6552 
 
 6584 
 
 6616 
 
 69 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 670 
 
 16 6648 
 
 6680 
 
 6711 
 
 6743 
 
 6775 
 
 6807 
 
 6839 
 
 6871 
 
 6903 
 
 6935 
 
 70 
 
 20 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 671 
 
 6967 
 
 6999 
 
 7030 
 
 7062 
 
 7094 
 
 7126 
 
 7158 
 
 7190 
 
 7222 
 
 7254 
 
 71 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 23 
 
 672 
 
 7286 
 
 7318 
 
 7350 
 
 7382 
 
 7414 
 
 7446 
 
 7478 
 
 7510 
 
 7542 
 
 7574 
 
 72 
 
 20 
 
 21 
 
 22 
 
 22 
 
 23 
 
 24 
 
 673 
 
 7606 
 
 7638 
 
 7670 
 
 7702 
 
 7734 
 
 7766 
 
 7798 
 
 7830 
 
 7862 
 
 7894 
 
 73 
 
 20 
 
 21 
 
 22 
 
 23 
 
 23 
 
 24 
 
 674 
 
 7926 
 
 7958 
 
 7991 
 
 8023 
 
 8055 
 
 8087 
 
 8119 
 
 8151 
 
 8183 
 
 8215 
 
 74 
 
 21 
 
 21 
 
 22 
 
 23 
 
 24 
 
 24 
 
 675 
 
 8247 
 
 8279 
 
 8312 
 
 8344 
 
 8376 
 
 8408 
 
 8440 
 
 8472 
 
 8504 
 
 8537 
 
 75 
 
 21 
 
 22 
 
 23 
 
 23 
 
 24 
 
 25 
 
 676 
 
 8569 
 
 8601 
 
 8633 
 
 8665 
 
 8697 
 
 8730 
 
 8762 
 
 8794 
 
 8826 
 
 8858 
 
 76 
 
 21 
 
 22 
 
 23 
 
 24 
 
 24 
 
 25 
 
 677 
 
 8891 
 
 8923 
 
 8955 
 
 8987 
 
 9020 
 
 9052 
 
 9084 
 
 9116 
 
 9149 
 
 9181 
 
 77 
 
 22 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 678 
 
 9213 
 
 9245 
 
 9278 
 
 9310 
 
 9342 
 
 9375 
 
 9407 
 
 9439 
 
 9471 
 
 9504 
 
 78 
 
 22 
 
 23 
 
 23 
 
 24 
 
 25 
 
 26 
 
 679 
 
 9536 
 
 9568 
 
 9601 
 
 9633 
 
 9665 
 
 9698 
 
 9730 
 
 9762 
 
 9795 
 
 9827 
 
 79 
 
 22 
 
 23 
 
 24 
 
 24 
 
 25 
 
 26 
 
 680 
 
 16 9860 
 
 9892 
 
 9924 
 
 9957 
 
 9989 
 
 0021 
 
 0054 
 
 0086 
 
 0119 
 
 0151 
 
 80 
 
 2223 
 
 24 
 
 25 
 
 20 
 
 26 
 
 681 
 
 170183 
 
 0216 
 
 0248 
 
 0281 
 
 0313 
 
 0346 
 
 0378 
 
 0411 
 
 0443 
 
 0475 
 
 81 
 
 23 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 682 
 
 0508 
 
 0540 
 
 0573 
 
 0605 
 
 0638 
 
 0670 
 
 0703 
 
 0735 
 
 0768 
 
 0800 
 
 82 
 
 23 
 
 24 
 
 25 
 
 25 
 
 26 
 
 27 
 
 683 
 
 0833 
 
 0865 
 
 0898 
 
 0930 
 
 0963 
 
 0996 
 
 1028 
 
 1061 
 
 1093 
 
 1126 
 
 83 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 27 
 
 684 
 
 1158 
 
 1191 
 
 1224 
 
 1256 
 
 1289 
 
 1321 
 
 1354 
 
 1386 
 
 1419 
 
 1452 
 
 84 
 
 24 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 685 
 
 1484 
 
 1517 
 
 1550 
 
 1582 
 
 1615 
 
 1648 
 
 1680 
 
 1713 
 
 1745 
 
 1778 
 
 85 
 
 24 
 
 25 
 
 25 
 
 26 
 
 27 
 
 28 
 
 686 
 
 1811 
 
 1843 
 
 1876 
 
 1909 
 
 1942 
 
 1974 
 
 2007 
 
 2040 
 
 2072 
 
 2105 
 
 86 
 
 24 
 
 25 
 
 20 
 
 27 
 
 28 
 
 28 
 
 687 
 
 2138 
 
 2171 
 
 2203 
 
 2236 
 
 2269 
 
 2301 
 
 2334 
 
 2367 
 
 2400 
 
 2433 
 
 87 
 
 24 
 
 25 
 
 20 
 
 27 
 
 28 
 
 29 
 
 688 
 
 2465 
 
 2498 
 
 2531 
 
 2564 
 
 2596 
 
 2629 
 
 2662 
 
 2695 
 
 2728 
 
 2760 
 
 88 
 
 25 
 
 2G 
 
 26 
 
 27 
 
 28 
 
 29 
 
 689 
 
 2793 
 
 2826 
 
 2859 
 
 2892 
 
 2925 
 
 2957 
 
 2990 
 
 3023 
 
 3056 
 
 3089 
 
 89 
 
 25 
 
 26 
 
 27 
 
 28 
 
 2829 
 
 690 
 
 173122 
 
 3155 
 
 3188 
 
 3220 
 
 3253 
 
 3286 
 
 3319 
 
 3352 
 
 3385 
 
 3418 
 
 90 
 
 25 
 
 26 
 
 2728 
 
 29 
 
 30 
 
 691 
 
 3451 
 
 3484 
 
 3517 
 
 3550 
 
 3583 
 
 3616 
 
 3648 
 
 3681 
 
 3714 
 
 3747 
 
 91 
 
 25 
 
 20 
 
 2728 
 
 29 
 
 30 
 
 692 
 
 3780 
 
 3813 
 
 3846 
 
 3879 
 
 3912 
 
 3945 
 
 3978 
 
 4011 
 
 4044 
 
 4077 
 
 92 
 
 26 
 
 27 
 
 2829 
 
 29 
 
 30 
 
 693 
 
 4110 
 
 4143 
 
 4176 
 
 4209 
 
 4243 
 
 4276 
 
 4309 
 
 4342 
 
 4375 
 
 4408 
 
 93 
 
 26 
 
 27 
 
 2829 
 
 30 
 
 31 
 
 694 
 
 4441 
 
 4474 
 
 4507 
 
 4540 
 
 4573 
 
 4606 
 
 4639 
 
 4673 
 
 4706 
 
 4739 
 
 94 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 695 
 
 4772 
 
 4805 
 
 4838 
 
 4871 
 
 4905 
 
 4938 
 
 4971 
 
 5004 
 
 5037 
 
 5070 
 
 95 
 
 27 
 
 28 
 
 29 
 
 29 
 
 30 
 
 31 
 
 696 
 
 5104 
 
 5137 
 
 5170 
 
 5203 
 
 5236 
 
 5269 
 
 5303 
 
 5336 
 
 5369 
 
 5402 
 
 96 
 
 27 
 
 28 
 
 2930 
 
 31 
 
 32 
 
 697 
 
 5436 
 
 5469 
 
 5502 
 
 5535 
 
 5569 
 
 5602 
 
 5635 
 
 5668 
 
 5702 
 
 5735 
 
 97 
 
 27 
 
 28 
 
 2930 
 
 31 
 
 32 
 
 698 
 
 5768 
 
 5801 
 
 5835 
 
 5868 
 
 5901 
 
 5935 
 
 5968 
 
 6001 
 
 6035 
 
 6068 
 
 98 
 
 27 
 
 28 
 
 2930 
 
 31 
 
 32 
 
 699 
 
 6101 
 
 6135 
 
 6168 
 
 6201 
 
 6235 
 
 6268 
 
 6301 
 
 6335 
 
 6368 
 
 6401 
 
 99 
 
 28 
 
 29 
 
 3031 
 
 32 
 
 33 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 28 
 
 29 
 
 30|31 
 
 32 
 
 33 
 
TABLE I. 
 
 Log 
 
 Log(l + a) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 r-700 
 
 017 6435 
 
 6468 
 
 6502 
 
 6535 
 
 6568 
 
 6602 
 
 6635 
 
 6669 
 
 6702 
 
 6736 
 
 )() 
 
 
 
 
 
 
 
 
 
 
 
 
 
 701 
 
 6769 
 
 6802 
 
 6836 
 
 6869 
 
 6903 
 
 6936 
 
 6970 
 
 7003 
 
 7037 
 
 7070 
 
 )1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 702 
 
 7104 
 
 7137 
 
 7171 
 
 7204 
 
 7238 
 
 7271 
 
 7305 
 
 7338 
 
 7372 
 
 7405 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 703 
 
 7439 
 
 7472 
 
 7506 
 
 7539 
 
 7573 
 
 7607 
 
 7640 
 
 7674 
 
 7707 
 
 7741 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 704 
 
 7774 
 
 7808 
 
 7842 
 
 7875 
 
 7909 
 
 7942 
 
 7976 
 
 8010 
 
 8043 
 
 8077 
 
 04 
 
 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 2 
 
 705 
 
 8111 
 
 8144 
 
 8178 
 
 8212 
 
 8245 
 
 8279 
 
 8313 
 
 8346 
 
 8380 
 
 8414 
 
 05 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 706 
 
 8447 
 
 8481 
 
 8515 
 
 8548 
 
 8582 
 
 8616 
 
 8650 
 
 8683 
 
 8717 
 
 8751 
 
 06 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 707 
 
 8784 
 
 8818 
 
 8852 
 
 8886 
 
 8919 
 
 8953 
 
 8987 
 
 9021 
 
 9055 
 
 9088 
 
 07 
 
 2 
 
 2 
 
 3 
 
 3 
 
 3 
 
 3 
 
 708 
 
 9122 
 
 9156 
 
 9190 
 
 9224 
 
 9257 
 
 9291 
 
 9325 
 
 9359 
 
 9393 
 
 9427 
 
 08 
 
 o 
 o 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 709 
 
 9460 
 
 9494 
 
 9528 
 
 9562 
 
 9596 
 
 9630 
 
 9664 
 
 9697 
 
 9731 
 
 9765 
 
 09 
 
 g 
 
 3 
 
 3 
 
 3 
 
 8 
 
 4 
 
 710 
 
 17 9799 
 
 9833 
 
 9867 
 
 9901 
 
 9935 
 
 9969 
 
 0003 
 
 0037 
 
 0071 
 
 0104 
 
 10 
 
 3 
 
 3 
 
 4 
 
 4 
 
 4 
 
 4 
 
 711 
 
 18 0138 
 
 0172 
 
 0206 
 
 0240 
 
 0274 
 
 0308 
 
 0342 
 
 0376 
 
 0410 
 
 0444 
 
 11 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 712 
 
 0478 
 
 0512 
 
 0546 
 
 0580 
 
 0614 
 
 0648 
 
 0682 
 
 0716 
 
 0750 
 
 0784 
 
 12 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 5 
 
 713 
 
 0818 
 
 0853 
 
 0887 
 
 0921 
 
 0955 
 
 0989 
 
 1023 
 
 1057 
 
 1091 
 
 1125 
 
 13 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 714 
 
 1159 
 
 1193 
 
 1227 
 
 1262 
 
 1296 
 
 1330 
 
 1364 
 
 1398 
 
 1432 
 
 1466 
 
 14 
 
 r> 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 715 
 
 1501 
 
 1535 
 
 1569 
 
 1603 
 
 1637 
 
 1671 
 
 1706 
 
 1740 
 
 1774 
 
 1808 
 
 15 
 
 r. 
 o 
 
 5 
 
 5 
 
 6 
 
 e 
 
 6 
 
 716 
 
 1842 
 
 1877 
 
 1911 
 
 1945 
 
 1979 
 
 2014 
 
 2048 
 
 2082 
 
 2116 
 
 2151 
 
 If. 
 
 r-j 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 717 
 
 2185 
 
 2219 
 
 2253 
 
 2288 
 
 2322 
 
 2356 
 
 2390 
 
 2425 
 
 2459 
 
 2493 
 
 17 
 
 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 718 
 
 2528 
 
 2562 
 
 2596 
 
 2631 
 
 2665 
 
 2699 
 
 2734 
 
 2768 
 
 2802 
 
 2837 
 
 18 
 
 () 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 719 
 
 2871 
 
 2905 
 
 2940 
 
 2974 
 
 3009 
 
 3043 
 
 3077 
 
 3112 
 
 3146 
 
 3181 
 
 19 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 720 
 
 183215 
 
 3249 
 
 3284 
 
 3318 
 
 3353 
 
 3387 
 
 3422 
 
 3456 
 
 3491 
 
 3525 
 
 20 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 721 
 
 3559 
 
 3594 
 
 3628 
 
 3663 
 
 3697 
 
 3732 
 
 3766 
 
 3801 
 
 3835 
 
 3870 
 
 21 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 722 
 
 3904 
 
 3939 
 
 3973 
 
 4008 
 
 4043 
 
 4077 
 
 4112 
 
 4146 
 
 4181 
 
 4215 
 
 22 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 723 
 
 4250 
 
 4284 
 
 4319 
 
 4354 
 
 4388 
 
 4423 
 
 4457 
 
 4492 
 
 4527 
 
 4561 
 
 23 
 
 8 
 
 8 
 
 8 
 
 9 
 
 g 
 
 9 
 
 724 
 
 4596 
 
 4631 
 
 4665 
 
 4700 
 
 4734 
 
 4769 
 
 4804 
 
 4838 
 
 4873 
 
 4908 
 
 24 
 
 8 
 
 8 
 
 9 
 
 9 
 
 <: 
 
 9 
 
 725 
 
 4942 
 
 4977 
 
 5012 
 
 5046 
 
 5081 
 
 5116 
 
 5151 
 
 5185 
 
 5220 
 
 5255 
 
 25 
 
 8 
 
 9 
 
 9 
 
 9 
 
 () 
 
 10 
 
 726 
 
 5289 
 
 5324 
 
 5359 
 
 5394 
 
 5428 
 
 5463 
 
 5498 
 
 5533 
 
 5567 
 
 5602 
 
 26 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 727 
 
 5637 
 
 5672 
 
 5707 
 
 5741 
 
 5776 
 
 5811 
 
 5846 
 
 5881 
 
 5915 
 
 5950 
 
 27 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 728 
 
 5985 
 
 6020 
 
 6055 
 
 6090 
 
 6124 
 
 6159 
 
 6194 
 
 6229 
 
 6264 
 
 6299 
 
 28 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 729 
 
 6334 
 
 6369 
 
 6404 
 
 6438 
 
 6473 
 
 6508 
 
 6543 
 
 6578 
 
 6613 
 
 6648 
 
 29 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 730 
 
 18 6683 
 
 6718 
 
 6753 
 
 6788 
 
 6823 
 
 6858 
 
 6893 
 
 6928 
 
 6963 
 
 6998 
 
 30 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 731 
 
 7033 
 
 7068 
 
 7103 
 
 7138 
 
 7173 
 
 7208 
 
 7243 
 
 7278 
 
 7313 
 
 7348 
 
 31 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 732 
 
 7383 
 
 7418 
 
 7453 
 
 7488 
 
 7523 
 
 7558 
 
 7593 
 
 7628 
 
 7663 
 
 7698 
 
 32 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 12 
 
 733 
 
 7733 
 
 7768 
 
 7804 
 
 7839 
 
 7874 
 
 7909 
 
 7944 
 
 7979 
 
 8014 
 
 8049 
 
 33 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 734 
 
 8085 
 
 8120 
 
 8155 
 
 8190 
 
 8225 
 
 8260 
 
 8296 
 
 8331 
 
 8366 
 
 8401 
 
 34 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 735 
 
 8436 
 
 8472 
 
 8507 
 
 8542 
 
 8577 
 
 8612 
 
 8648 
 
 8683 
 
 8718 
 
 8753 
 
 35 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 736 
 
 8789 
 
 8824 
 
 8859 
 
 8894 
 
 8930 
 
 8965 
 
 9000 
 
 9036 
 
 9071 
 
 9106 
 
 36 
 
 1 Q 
 
 1? 
 
 1? 
 
 IS 
 
 "M 
 
 14 
 
 737 
 
 9141 
 
 9177 
 
 9212 
 
 9247 
 
 9283 
 
 9318 
 
 9353 
 
 9389 
 
 9424 
 
 9459 
 
 37 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 738 
 
 9495 
 
 9530 
 
 9566 
 
 9601 
 
 9636 
 
 9672 
 
 9707 
 
 9742 
 
 9778 
 
 9813 
 
 38 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 15 
 
 739 
 
 9849 
 
 9884 
 
 9919 
 
 9955 
 
 9990 
 
 0026 
 
 0061 
 
 0097 
 
 0132 
 
 0168 
 
 a 
 
 13 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 740 
 
 190203 
 
 0239 
 
 0274 
 
 0309 
 
 0345 
 
 0380 
 
 0416 
 
 0451 
 
 0487 
 
 0522 
 
 40 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 741 
 
 0558 
 
 0593 
 
 0629 
 
 0665 
 
 0700 
 
 0736 
 
 0771 
 
 0807 
 
 0842 
 
 0878 
 
 41 
 
 14 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 742 
 
 0913 
 
 0949 
 
 0985 
 
 1020 
 
 1056 
 
 1091 
 
 1127 
 
 1163 
 
 1198 
 
 1234 
 
 42 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 743 
 
 1269 
 
 1305 
 
 1341 
 
 1376 
 
 1412 
 
 1448 
 
 1483 
 
 1519 
 
 1555 
 
 1590 
 
 43 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 744 
 
 1626 
 
 1662 
 
 1697 
 
 1733 
 
 1769 
 
 1804 
 
 1840 
 
 1876 
 
 1911 
 
 1947 
 
 44 
 
 15 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 745 
 
 1983 
 
 2019 
 
 2054 
 
 2090 
 
 2126 
 
 2162 
 
 2197 
 
 2233 
 
 2269 
 
 2305 
 
 45 
 
 15 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 746 
 
 2340 
 
 2376 
 
 2412 
 
 2448 
 
 2484 
 
 2519 
 
 2555 
 
 2591 
 
 2627 
 
 2663 
 
 4f 
 
 16 
 
 16 
 
 17 
 
 17 
 
 17 
 
 18 
 
 747 
 
 2699 
 
 2734 
 
 2770 
 
 2806 
 
 2842 
 
 2878 
 
 2914 
 
 2949 
 
 2985 
 
 3021 
 
 47 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 748 
 
 3057 
 
 3093 
 
 3129 
 
 3165 
 
 3201 
 
 3237 
 
 3273 
 
 3308 
 
 3344 
 
 3380 
 
 48 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 749 
 
 3416 
 
 3452 
 
 3488 
 
 3524 
 
 3560 
 
 3596 
 
 3632 
 
 3668 
 
 3704 
 
 3740 
 
 49 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 [20] 
 
TABLE I. 
 
 Log 
 
 Log(l+x) 
 
 Pro. Parts. 
 
 X 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 1-750 
 
 0-19 3776 
 
 3812 
 
 3848 
 
 3884 
 
 3920 
 
 3956 
 
 3992 
 
 4028 
 
 4064 
 
 4100 
 
 50 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 751 
 
 4136 
 
 4172 
 
 4208 
 
 4244 
 
 4280 
 
 4316 
 
 4352 
 
 4389 
 
 4425 
 
 4461 
 
 51 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 20 
 
 752 
 
 4497 
 
 4533 
 
 4569 
 
 4605 
 
 4641 
 
 4677 
 
 4714 
 
 4750 
 
 4786 
 
 4822 
 
 52 
 
 18 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 753 
 
 4858 
 
 4894 
 
 4930 
 
 4967 
 
 5003 
 
 5039 
 
 5075 
 
 5111 
 
 5148 
 
 5184 
 
 53 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 754 
 
 5220 
 
 5256 
 
 5292 
 
 5329 
 
 5365 
 
 5401 
 
 5437 
 
 5473 
 
 5510 
 
 5546 
 
 54 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 21 
 
 '755 
 
 5582 
 
 5618 
 
 5655 
 
 5691 
 
 5727 
 
 5764 
 
 5800 
 
 5836 
 
 5872 
 
 5909 
 
 55 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 21 
 
 756 
 
 5945 
 
 5981 
 
 6018 
 
 6054 
 
 6090 
 
 6127 
 
 6163 
 
 6199 
 
 6236 
 
 6272 
 
 56 
 
 19 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 757 
 
 6308 
 
 6345 
 
 6381 
 
 6418 
 
 6454 
 
 6490 
 
 6527 
 
 6563 
 
 6600 
 
 6636 
 
 57 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 758 
 
 6672 
 
 6709 
 
 6745 
 
 6782 
 
 6818 
 
 6855 
 
 6891 
 
 6927 
 
 6964 
 
 7000 
 
 58 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 23 
 
 759 
 
 7037 
 
 7073 
 
 7110 
 
 7146 
 
 7183 
 
 7219 
 
 7256 
 
 7292 
 
 7329 
 
 7365 
 
 59 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 23 
 
 760 
 
 19 7402 
 
 7438 
 
 7475 
 
 7511 
 
 7548 
 
 7585 
 
 7621 
 
 7658 
 
 7694 
 
 7731 
 
 60 
 
 20 
 
 21 
 
 22 
 
 22 
 
 2323 
 
 761 
 
 7767 
 
 7804 
 
 7841 
 
 7877 
 
 7914 
 
 7950 
 
 7987 
 
 8024 
 
 8060 
 
 8097 
 
 61 
 
 21 
 
 21 
 
 22 
 
 23 
 
 2324 
 
 762 
 
 8133 
 
 8170 
 
 8207 
 
 8243 
 
 8280 
 
 8317 
 
 8353 
 
 8390 
 
 8427 
 
 8463 
 
 6221 
 
 22 
 
 22 
 
 23 
 
 24 
 
 24 
 
 763 
 
 8500 
 
 8537 
 
 8573 
 
 8610 
 
 8647 
 
 8684 
 
 8720 
 
 8757 
 
 8794 
 
 8830 
 
 63'21 
 
 2223 
 
 23 
 
 24 
 
 25 
 
 764 
 
 8867 
 
 8904 
 
 8941 
 
 8977 
 
 9014 
 
 9051 
 
 9088 
 
 9124 
 
 9161 
 
 9198 
 
 64 
 
 22 
 
 22 
 
 23 
 
 24 
 
 24 
 
 25 
 
 765 
 
 9235 
 
 9272 
 
 9308 
 
 9345 
 
 9382 
 
 9419 
 
 9456 
 
 9492 
 
 9529 
 
 9566 
 
 65 
 
 22 
 
 23 
 
 23 
 
 24 
 
 25 
 
 25 
 
 766 
 
 9603 
 
 9640 
 
 9677 
 
 9714 
 
 9750 
 
 9787 
 
 9824 
 
 9861 
 
 9898 
 
 9935 
 
 66,22 
 
 23 
 
 24 
 
 24 
 
 25 
 
 26 
 
 767 
 
 9972 
 
 0009 
 
 o046 
 
 0082 
 
 0119 
 
 0156 
 
 0193 
 
 0230 
 
 0267 
 
 0304 
 
 6723 
 
 23 
 
 24 
 
 25 
 
 25 
 
 26 
 
 768 
 
 20 0341 
 
 0378 
 
 0415 
 
 0452 
 
 0489 
 
 0526 
 
 0563 
 
 0600 
 
 0637 
 
 0674 
 
 6823 
 
 24 
 
 24 
 
 25 
 
 26 
 
 27 
 
 769 
 
 0711 
 
 0748 
 
 0785 
 
 0822 
 
 0859 
 
 0896 
 
 0933 
 
 0970 
 
 1007 
 
 1044 
 
 6923 
 
 24 
 
 25 
 
 26 
 
 26 
 
 27 
 
 770 
 
 20 1081 
 
 1118 
 
 1155 
 
 1192 
 
 1229 
 
 1267 
 
 1304 
 
 1341 
 
 1378 
 
 1415 
 
 70 
 
 24 
 
 25 
 
 25 
 
 26 
 
 27 
 
 27 
 
 771 
 
 1452 
 
 1489 
 
 1526 
 
 1563 
 
 1601 
 
 1638 
 
 1675 
 
 1712 
 
 1749 
 
 1786 
 
 71 
 
 24 
 
 25 
 
 26 
 
 26 
 
 27 
 
 28 
 
 772 
 
 1823 
 
 1861 
 
 1898 
 
 1935 
 
 1972 
 
 2009 
 
 2047 
 
 2084 
 
 2121 
 
 2158 
 
 72 
 
 24 
 
 25 
 
 26 
 
 27 
 
 27 
 
 28 
 
 773 
 
 2195 
 
 2233 
 
 2270 
 
 2307 
 
 2344 
 
 2382 
 
 2419 
 
 2456 
 
 2493 
 
 2531 
 
 73 
 
 25 
 
 26 
 
 26 
 
 27 
 
 28 
 
 28 
 
 774 
 
 2568 
 
 2605 
 
 2642 
 
 2680 
 
 2717 
 
 2754 
 
 2792 
 
 2829 
 
 2866 
 
 2904 
 
 74 
 
 25 
 
 26 
 
 27 
 
 27 
 
 28 
 
 29 
 
 775 
 
 2941 
 
 2978 
 
 3016 
 
 3053 
 
 3090 
 
 3128 
 
 3165 
 
 3202 
 
 3240 
 
 3277 
 
 75 
 
 26 
 
 26 
 
 27 
 
 28 
 
 29 
 
 29 
 
 776 
 
 3315 
 
 3352 
 
 3389 
 
 3427 
 
 3464 
 
 3501 
 
 3539 3576 
 
 3614 
 
 3651 
 
 76 
 
 26 
 
 27 
 
 27 
 
 28 
 
 29 
 
 30 
 
 777 
 
 3689 
 
 3726 
 
 3764 
 
 3801 
 
 3838 
 
 3876 
 
 3913 3951 
 
 3988 
 
 4026 
 
 77 
 
 26 
 
 27 
 
 28 
 
 28 
 
 29 
 
 30 
 
 778 
 
 4063 
 
 4101 
 
 4138 
 
 4176 
 
 4213 
 
 4251 
 
 4288 4326 
 
 4363 
 
 4401 
 
 78 
 
 27 
 
 27 
 
 28 
 
 29 
 
 30 
 
 30 
 
 779 
 
 4438 
 
 4476 
 
 4514 
 
 4551 
 
 4589 
 
 4626 
 
 4664 
 
 4701 
 
 4739 
 
 4777 
 
 79 
 
 27 
 
 28 
 
 28 
 
 29 
 
 30 
 
 31 
 
 780 
 
 20 4814 
 
 4852 
 
 4889 
 
 4927 
 
 4965 
 
 5002 
 
 5040 
 
 5078 
 
 5115 
 
 5153 
 
 80 
 
 27 
 
 2829 
 
 30 
 
 30 
 
 31 
 
 781 
 
 5190 
 
 5228 
 
 5266 
 
 5303 
 
 5341 
 
 5379 
 
 5416 5454 
 
 5492 
 
 5530 
 
 8128 
 
 2829 
 
 30 
 
 31 
 
 32 
 
 782 
 
 5567 
 
 5605 
 
 5643 
 
 5680 
 
 5718 
 
 5756 
 
 5794 
 
 5831 
 
 5869 
 
 5907 
 
 82 ; 28 
 
 29 
 
 30 
 
 30 
 
 31 
 
 32 
 
 783 
 
 5945 
 
 5982 
 
 6020 
 
 6058 
 
 6096 
 
 6133 
 
 6171 
 
 6209 
 
 6247 
 
 6285 
 
 8328 
 
 29 
 
 30 
 
 31 
 
 32 
 
 32 
 
 784 
 
 6323 
 
 6360 
 
 6398 
 
 6436 
 
 6474 
 
 6512 
 
 6549 
 
 6587 
 
 6625 
 
 6663 
 
 84 
 
 29 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 -785 
 
 6701 
 
 6739 
 
 6777 
 
 6815 
 
 6852 
 
 6890 
 
 6928 
 
 6966 
 
 7004 
 
 7042 
 
 85 
 
 29 
 
 30 
 
 31 
 
 31 
 
 32 
 
 33 
 
 786 
 
 7080 
 
 7118 
 
 7156 
 
 7194 
 
 7232 
 
 7270 
 
 7308 
 
 7346 
 
 7383 
 
 7421 
 
 86 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 787 
 
 7459 
 
 7497 
 
 7535 
 
 7573 
 
 7611 
 
 7649 
 
 7687 
 
 7725 
 
 7763 
 
 7801 
 
 87 
 
 30 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 788 
 
 7839 
 
 7878 
 
 7916 
 
 7954 
 
 7992 
 
 8030 
 
 8068 
 
 8106 
 
 8144 
 
 8182 
 
 88 
 
 30 
 
 31 
 
 32 
 
 33 
 
 33 
 
 34 
 
 789 
 
 8220 
 
 8258 
 
 8296 
 
 8334 
 
 8372 
 
 8411 
 
 8449 
 
 8487 
 
 8525 
 
 8563 
 
 89 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 790 
 
 20 8601 
 
 8639 
 
 8678 
 
 8716 
 
 8754 
 
 8792 
 
 8830 
 
 8868 
 
 8907 
 
 8945 
 
 90 
 
 31 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 791 
 
 8983 
 
 9021 
 
 9059 
 
 9098 9136 
 
 9174 
 
 9212 
 
 9250 
 
 9289 
 
 9327 
 
 91 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 35 
 
 792 
 
 9365 
 
 9403 
 
 9442 
 
 9480 
 
 9518 
 
 9556 
 
 9595! 9633 
 
 9671 
 
 9710 
 
 92 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 793 
 
 9748 
 
 9786 
 
 9825 
 
 9863 
 
 9901 
 
 9940 
 
 9978 0016 
 
 0055 
 
 0093 
 
 93 
 
 32 
 
 33 
 
 33 
 
 34 
 
 35 
 
 36 
 
 794 
 
 21 0131 
 
 0170 
 
 0208 
 
 0246 
 
 0285 
 
 0323 
 
 0361 
 
 0400 
 
 0438 
 
 0477 
 
 94 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 795 
 
 0515 
 
 0554 
 
 0592 
 
 0630 
 
 0669 
 
 0707 
 
 0746 
 
 0784 
 
 0823 
 
 0861 
 
 95 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 796 
 
 0900 
 
 0938 
 
 0976 
 
 1015 
 
 1053 
 
 1092 
 
 1130 
 
 1169 
 
 1207 
 
 1246 
 
 96 
 
 33 
 
 34 
 
 35 
 
 36 
 
 36 
 
 37 
 
 797 
 
 1284 
 
 1323 
 
 1362 
 
 1400 
 
 1439 
 
 1477 
 
 1516 
 
 1554 
 
 1593 
 
 1631 
 
 97 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 798 
 
 1670 
 
 1709 
 
 1747 
 
 1786 
 
 1824 
 
 1863 
 
 1902 
 
 1940 
 
 1979 
 
 2017 
 
 98 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 799 
 
 2056 
 
 2095 
 
 2133 
 
 2172 
 
 2211 
 
 2249 
 
 2288 
 
 2327 
 
 2365 
 
 2404 
 
 99 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 F211 
 
TABLE I. 
 
 Log 
 
 
 Log (l+x) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 T-800 
 
 0-21 2443 
 
 2481 
 
 2520 
 
 2559 
 
 2597 
 
 2636 
 
 2675 
 
 2714 
 
 2752 
 
 2791 
 
 00 
 
 
 
 Q 
 
 
 
 Q 
 
 
 
 
 
 801 
 
 2830 
 
 2868 
 
 2907 
 
 2946 
 
 2985 
 
 3024 
 
 3062 
 
 3101 
 
 3140 
 
 3179 
 
 01 
 
 
 
 
 
 
 
 
 
 
 
 
 
 802 
 
 3217 
 
 3256 
 
 3295 
 
 3334 
 
 3373 
 
 3411 
 
 3450 
 
 3489 
 
 3528 
 
 3567 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 803 
 
 3606 
 
 3645 
 
 3683 
 
 3722 
 
 3761 
 
 3800 
 
 3839 
 
 3878 
 
 3917 
 
 3956 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 804 
 
 3994 
 
 4033 
 
 4072 
 
 4111 
 
 4150 
 
 4189 
 
 4228 
 
 4267 
 
 4306 
 
 4345 
 
 04 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 805 
 
 4384 
 
 4423 
 
 4462 
 
 4501 
 
 4540 
 
 4579 
 
 4618 
 
 4657 
 
 4696 
 
 4735 
 
 06 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 806 
 
 4774 
 
 4813 
 
 4852 
 
 4891 
 
 4930 
 
 4969 
 
 5008 
 
 5047 
 
 5086 
 
 5125 
 
 06 
 
 2 
 
 2 
 
 2 
 
 3 
 
 3 
 
 3 
 
 807 
 
 5164 
 
 5203 
 
 5242 
 
 5281 
 
 5320 
 
 5359 
 
 5399 
 
 5438 
 
 5477 
 
 5516 
 
 07 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 808 
 
 5555 
 
 5594 
 
 5633 
 
 5672 
 
 5712 
 
 5751 
 
 5790 
 
 5829 
 
 5868 
 
 5907 
 
 08 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 
 
 809 
 
 5947 
 
 5986 
 
 6025 
 
 6064 
 
 6103 
 
 6142 
 
 6182 
 
 6221 
 
 6260 
 
 6299 
 
 09 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 810 
 
 21 6339 
 
 6378 
 
 6417 
 
 6456 
 
 6496 
 
 6535 
 
 6574 
 
 6613 
 
 6653 
 
 6692 
 
 10 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 811 
 
 6731 
 
 6770 
 
 6810 
 
 6849 
 
 6888 
 
 6928 
 
 6967 
 
 7006 
 
 7046 
 
 7085 
 
 11 
 
 4 
 
 4 
 
 5 
 
 5 
 
 5 
 
 5 
 
 812 
 
 7124 
 
 7164 
 
 7203 
 
 7242 
 
 7282 
 
 7321 
 
 7361 
 
 7400 
 
 7439 
 
 7479 
 
 12 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 813 
 
 7518 
 
 7557 
 
 7597 
 
 7636 
 
 7676 
 
 7715 
 
 7755 
 
 7794 
 
 7833 
 
 7873 
 
 13 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 814 
 
 7912 
 
 7952 
 
 7991 
 
 8031 
 
 8070 
 
 8110 
 
 8149 
 
 8189 
 
 8228 
 
 8268 
 
 14 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 815 
 
 8307 
 
 8347 
 
 8386 
 
 8426 
 
 8465 
 
 8505 
 
 8544 
 
 8584 
 
 8623 
 
 8663 
 
 18 
 
 6 
 
 a 
 
 6 
 
 6 
 
 6 
 
 7 
 
 816 
 
 8703 
 
 8742 
 
 8782 
 
 8821 
 
 8861 
 
 8900 
 
 8940 
 
 8980 
 
 9019 
 
 9059 
 
 16 
 
 6 
 
 Q 
 
 7 
 
 7 
 
 7 
 
 7 
 
 817 
 
 9098 
 
 9138 
 
 9178 
 
 9217 
 
 9257 
 
 9297 
 
 9336 
 
 9376 
 
 9416 
 
 9455 
 
 17 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 818 
 
 9495 
 
 9535 
 
 9574 
 
 9614 
 
 9654 
 
 9693 
 
 9733 
 
 9773 
 
 9812 
 
 9852 
 
 18 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 819 
 
 9892 
 
 9932 
 
 9971 
 
 oOll 
 
 0051 
 
 0091 
 
 0130 
 
 0170 
 
 0210 
 
 0250 
 
 19 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 820 
 
 22 0289 
 
 0329 
 
 0369 
 
 0409 
 
 0449 
 
 0488 
 
 0528 
 
 0568 
 
 0608 
 
 0648 
 
 20 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 821 
 
 0688 
 
 0727 
 
 0767 
 
 0807 
 
 0847 
 
 0887 
 
 0927 
 
 0967 
 
 1006 
 
 1046 
 
 21 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 822 
 
 1086 
 
 1126 
 
 1166 
 
 1206 
 
 1246 
 
 1286 
 
 1326 
 
 1366 
 
 1406 
 
 1446 
 
 2-2 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 10 
 
 823 
 
 1485 
 
 1525 
 
 1565 
 
 1605 
 
 1645 
 
 1685 
 
 1725 
 
 1765 
 
 1805 
 
 1845 
 
 2-3 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 824 
 
 1885 
 
 1925 
 
 1965 
 
 2005 
 
 2045 
 
 2085 
 
 2125 
 
 2165 
 
 2205 
 
 2246 
 
 24 
 
 9 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 825 
 
 2286 
 
 2326 
 
 2366 
 
 2406 
 
 2446 
 
 2486 
 
 2526 
 
 2566 
 
 2606 
 
 2646 
 
 25 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 826 
 
 2686 
 
 2727 
 
 2767 
 
 2807 
 
 2847 
 
 2887 
 
 2927 
 
 2967 
 
 3008 
 
 3048 
 
 2(5 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 827 
 
 3088 
 
 3128 
 
 3168 
 
 3208 
 
 3249 
 
 3289 
 
 3329 
 
 3369 
 
 3409 
 
 3450 
 
 27 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 828 
 
 3490 
 
 3530 
 
 3570 
 
 3611 
 
 3651 
 
 3691 
 
 3731 
 
 3772 
 
 3812 
 
 3852 
 
 28 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 829 
 
 3892 
 
 3933 
 
 3973 
 
 4013 
 
 4054 
 
 4094 
 
 4134 
 
 4175 
 
 4215 
 
 4255 
 
 2!) 
 
 11 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 830 
 
 22 4296 
 
 4336 
 
 4376 
 
 4417 
 
 4457 
 
 4497 
 
 4538 
 
 4578 
 
 4618 
 
 4659 
 
 30 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 831 
 
 4699 
 
 4740 
 
 4780 
 
 4820 
 
 4861 
 
 4901 
 
 4942 
 
 4982 
 
 5022 
 
 5063 
 
 31 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 832 
 
 5103 
 
 5144 
 
 5184 
 
 5225 
 
 5265 
 
 5306 
 
 5346 
 
 5387 
 
 5427 
 
 5468 
 
 32 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 833 
 
 5508 
 
 5549 
 
 5589 
 
 5630 
 
 5670 
 
 5711 
 
 5751 
 
 5792 
 
 5832 
 
 5873 
 
 33 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 15 
 
 834 
 
 5913 
 
 5954 
 
 5995 
 
 6035 
 
 6076 
 
 6116 
 
 6157 
 
 6197 
 
 6238 
 
 6279 
 
 34 
 
 13 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 835 
 
 6319 
 
 6360 
 
 6401 
 
 6441 
 
 6482 
 
 6522 
 
 6563 
 
 6604 
 
 6644 
 
 668*5 
 
 35 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 836 
 
 6726 
 
 6766 
 
 6807 
 
 6848 
 
 6888 
 
 6929 
 
 6970 
 
 7011 
 
 7051 
 
 7092 
 
 30 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 16 
 
 837 
 
 7133 
 
 7173 
 
 7214 
 
 7255 
 
 7296 
 
 7336 
 
 7377 
 
 7418 
 
 7459 
 
 7499 
 
 37 
 
 14 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 838 
 
 7540 
 
 7581 
 
 7622 
 
 7663 
 
 7703 
 
 7744 
 
 7785 
 
 7826 
 
 7867 
 
 7907 
 
 38 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 17 
 
 839 
 
 7948 
 
 7989 
 
 8030 
 
 8071 
 
 8112 
 
 8153 
 
 8193 
 
 8234 
 
 8275 
 
 8316 
 
 39 
 
 15 
 
 1C 
 
 16 
 
 16 
 
 17 
 
 17 
 
 840 
 
 22 8357 
 
 8398 
 
 8439 
 
 8480 
 
 8521 
 
 8562 
 
 8602 
 
 8643 
 
 8684 
 
 8725 
 
 40 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 841 
 
 8766 
 
 8807 
 
 8848 
 
 8889 
 
 8930 
 
 8971 
 
 9012 
 
 9053 
 
 9094 
 
 9135 
 
 41 
 
 16 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 842 
 
 9176 
 
 9217 
 
 9258 
 
 9299 
 
 9340 
 
 9381 
 
 9422 
 
 9463 
 
 9504 
 
 9545 
 
 42 
 
 16 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 843 
 
 9586 
 
 9627 
 
 9668 
 
 9709 
 
 9751 
 
 9792 
 
 9833 
 
 9874 
 
 9915 
 
 9956 
 
 43 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 19 
 
 844 
 
 9997 
 
 0038 
 
 0079 
 
 0120 
 
 0162 
 
 0203 
 
 0244 
 
 0285 
 
 0326 
 
 0367 
 
 44 
 
 17 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 845 
 
 23 0409 
 
 0450 
 
 0491 
 
 0532 
 
 0573 
 
 0614 
 
 0656 
 
 0697 
 
 0738 
 
 0779 
 
 45 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 20 
 
 846 
 
 0821 
 
 0862 
 
 0903 
 
 0944 
 
 0985 
 
 1027 
 
 1068 
 
 1109 
 
 1151 
 
 1192 
 
 46 
 
 18 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 847 
 
 1233 
 
 1274 
 
 1316 
 
 1357 
 
 1398 
 
 1440 
 
 1481 
 
 1522 
 
 1564 
 
 1605 
 
 47 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 848 
 
 1646 
 
 1688 
 
 1729 
 
 1770 
 
 1812 
 
 1853 
 
 1894 
 
 1936 
 
 1977 
 
 2018 
 
 48 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 21 
 
 849 
 
 2060 
 
 2101 
 
 2143 
 
 2184 
 
 2225 
 
 2267 
 
 2308 
 
 2350 
 
 2391 
 
 2433 
 
 49 
 
 19 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 [22] 
 
TABLE I. 
 
 
 Lo (l+x) 
 
 Pro. Parts. 
 
 Log 
 
 5 v / 
 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 39J40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 T850 
 
 23 2474 
 
 2516 
 
 2557 
 
 2598 
 
 2640 
 
 2681 
 
 2723 
 
 2764 2806 
 
 2847 
 
 
 
 1920 
 
 20 
 
 21 
 
 21 
 
 22 
 
 851 
 
 2889 
 
 2930 
 
 2972 
 
 3013 
 
 3055 
 
 3096 
 
 3138 
 
 3180 : 3221 3263 
 
 1 
 
 2020 
 
 21 
 
 21 
 
 22 
 
 22 
 
 852 
 
 3304 
 
 3346 
 
 3387 
 
 3429 
 
 3470 
 
 3512 
 
 3554 
 
 3595 3637 3678 
 
 2 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22123 
 
 853 
 
 3720 
 
 3762 
 
 3803 
 
 3845 
 
 3887 
 
 3928 
 
 3970 
 
 401 2 ! 4053 4095 
 
 3 
 
 21 
 
 21 
 
 22 
 
 22 
 
 2323 
 
 854 
 
 4137 
 
 4178 
 
 4220 
 
 4262 
 
 4303 
 
 4345 
 
 4387 
 
 4428 
 
 4470 4512 
 
 4 
 
 21 
 
 22 
 
 22 
 
 23 
 
 23 
 
 24 
 
 855 
 
 4554 
 
 4595 
 
 4637 
 
 4679 
 
 4721 
 
 4762 
 
 4804 
 
 4846 
 
 4888 4929 
 
 5 
 
 21 
 
 22 
 
 23 
 
 23 
 
 24 
 
 24 
 
 856 
 
 4971 
 
 5013 
 
 5055 
 
 6097 
 
 5138 
 
 5180 
 
 5222 
 
 5264 
 
 5306 
 
 5347 
 
 6 
 
 22 
 
 22 
 
 23 
 
 24 
 
 2425 
 
 857 
 
 5389 
 
 5431 
 
 5473 
 
 5515 
 
 5557 
 
 5599 5640 
 
 5682 
 
 5724 
 
 5766 
 
 o7 
 
 2223 
 
 23 
 
 24 
 
 25:25 
 
 858 
 
 5808 
 
 5850 
 
 5892 
 
 5934 
 
 5976 
 
 6018 
 
 6059 
 
 6101 
 
 6143 
 
 6185 
 
 58 
 
 2323 
 
 24 
 
 24 
 
 2526 
 
 859 
 
 6227 
 
 6269 
 
 6311 
 
 6353 
 
 6395 
 
 6437 
 
 6479 
 
 6521 
 
 6563 
 
 6605 
 
 59 
 
 2324 
 
 24 
 
 25 
 
 2526 
 
 860 
 
 23 6647 
 
 6689 
 
 6731 
 
 6773 
 
 6815 
 
 6857 
 
 6899 
 
 6941 
 
 6983 
 
 7025 
 
 ;o 
 
 23 
 
 24 
 
 2525 
 
 26 
 
 26 
 
 861 
 
 7067 
 
 7110 
 
 7152 
 
 7194 
 
 7236 
 
 7278 
 
 7320 
 
 7362 
 
 7404 
 
 7446 
 
 61 
 
 24 
 
 24 
 
 2526 
 
 26 
 
 27 
 
 862 
 
 7488 
 
 7531 
 
 7573 
 
 7615 
 
 7657 
 
 7699 7741 
 
 7783 
 
 7826 
 
 7868 
 
 r2 
 
 24 
 
 26 
 
 2526 
 
 27 
 
 27 
 
 863 
 
 7910 
 
 7952 
 
 7994 
 
 8036 
 
 8079 
 
 8121 
 
 8163 
 
 8205 
 
 8248 
 
 8290 
 
 33 
 
 25 
 
 25 
 
 2626 
 
 27 
 
 28 
 
 864 
 
 8332 
 
 8374 
 
 8416 
 
 8459 
 
 8501 
 
 8543 
 
 8585 
 
 8628 
 
 8670 
 
 8712 
 
 64 
 
 25 
 
 26 
 
 2627 
 
 28 
 
 28 
 
 865 
 
 8755 
 
 8797 
 
 8839 
 
 8881 
 
 8924 
 
 8966 
 
 9008 
 
 9051 
 
 9093 
 
 9135 
 
 56 
 
 25 
 
 26 
 
 2727 
 
 28 
 
 29 
 
 866 
 
 9178 
 
 9220 
 
 9262 
 
 9305 
 
 9347 
 
 9390 9432 
 
 9474 
 
 9517 
 
 9559 
 
 66 
 
 26 
 
 26 
 
 2728 
 
 28 
 
 29 
 
 867 
 
 9602 
 
 9644 
 
 9686 
 
 9729 
 
 9771 
 
 9814 9856 9898 
 
 9941 
 
 9983 
 
 j7 
 
 26 
 
 27 
 
 2728 
 
 29 
 
 29 
 
 868 
 
 24 0026 
 
 0068 
 
 0111 
 
 0153 
 
 0196 
 
 0238 
 
 0281 
 
 0323 
 
 0366 
 
 0408 
 
 68 
 
 27 
 
 27 
 
 2829 
 
 29 
 
 30 
 
 869 
 
 0451 
 
 0493 
 
 0536 
 
 0578 
 
 0621 
 
 0663 
 
 0706 
 
 0748 
 
 0791 
 
 0834 
 
 69 
 
 27 
 
 28 
 
 2829 
 
 30 
 
 30 
 
 870 
 
 24 0876 
 
 0919 
 
 0961 
 
 1004 
 
 1046 
 
 1089 
 
 1132 
 
 1174 
 
 1217 
 
 1260 
 
 70 
 
 27 
 
 28 
 
 2929 
 
 30 
 
 31 
 
 871 
 
 1302 
 
 1345 
 
 1387 
 
 1430 
 
 1473 
 
 1515 
 
 1558 
 
 1601 
 
 1643 
 
 1686 
 
 71 
 
 28 
 
 2829 
 
 30 
 
 31 
 
 31 
 
 872 
 
 1729 
 
 1771 
 
 1814 
 
 1857 
 
 1900 
 
 1942 
 
 1985 
 
 2028 
 
 2070 
 
 2113 
 
 72 
 
 28 
 
 29 
 
 30 
 
 30 
 
 31 
 
 32 
 
 873 
 
 2156 
 
 2199 
 
 2241 
 
 2284 
 
 2327 
 
 2370 
 
 2412 
 
 2455 
 
 2498 
 
 2541 
 
 73 
 
 28 
 
 29 
 
 30 
 
 31 
 
 31 
 
 32 
 
 874 
 
 2584 
 
 2626 
 
 2669 
 
 2712 
 
 2755 
 
 2798 
 
 2840 
 
 2883 
 
 2926 
 
 2969 
 
 74 
 
 29 
 
 30 
 
 30 
 
 31 
 
 32 
 
 33 
 
 875 
 
 3012 
 
 3055 
 
 3098 
 
 3140 
 
 3183 
 
 3226 
 
 3269 
 
 3312 
 
 3355 
 
 3398 
 
 75 
 
 29 
 
 30 
 
 31 
 
 32 
 
 32 
 
 33 
 
 876 
 
 3441 
 
 3484 
 
 3526| 3569 
 
 3612 
 
 3655 
 
 3698 
 
 3741 
 
 3784 
 
 3827 
 
 76 
 
 30 
 
 30 
 
 31 
 
 32 
 
 33 
 
 33 
 
 877 
 
 3870 
 
 3913 
 
 3956 
 
 3999 
 
 4042 
 
 4085 
 
 4128 
 
 4171 
 
 4214 
 
 4257 
 
 77 
 
 30 
 
 31 
 
 32 
 
 32 
 
 33 
 
 34 
 
 878 
 
 4300 
 
 4343 
 
 4386 
 
 4429 
 
 4472 
 
 4515 
 
 4558 
 
 4601 
 
 4644 
 
 4687 
 
 78 
 
 3( 
 
 31 
 
 32 
 
 33 
 
 34 
 
 34 
 
 879 
 
 4730 
 
 4774 
 
 4817 
 
 4860 
 
 4903 
 
 4946 
 
 4989 
 
 5032 
 
 5075 
 
 5118 
 
 79 
 
 31 
 
 32 
 
 32 
 
 33 
 
 34 
 
 35 
 
 880 
 
 24 5162 
 
 5205 
 
 5248 
 
 5291 
 
 5334 
 
 5377 
 
 5420 
 
 5464 
 
 5507 
 
 5550 
 
 80 
 
 31 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 881 
 
 5593 
 
 5636 
 
 5680 
 
 5723 
 
 5766 
 
 5809 
 
 5852 
 
 5896 
 
 5939 
 
 5982 
 
 81 
 
 3232 
 
 33 
 
 34 
 
 35 
 
 36 
 
 882 
 
 6025 
 
 6069 
 
 6112 
 
 6155 
 
 6198 
 
 6242 
 
 6285 
 
 6328 
 
 6372 
 
 6415 
 
 82 
 
 3233 
 
 34 
 
 34 
 
 35 
 
 36 
 
 883 
 
 6458 
 
 6501 
 
 6545 
 
 6588 
 
 6631 
 
 6675 
 
 6718 
 
 6761 
 
 6805 
 
 6848 
 
 83 
 
 32'33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 884 
 
 6891 
 
 6935 
 
 6978 
 
 7022 
 
 7065 
 
 7108 
 
 7152 
 
 7195 
 
 7239 
 
 7282 
 
 84 
 
 3334 
 
 34 
 
 35 
 
 36 
 
 37 
 
 885 
 
 7325 
 
 7369 
 
 7412 
 
 7456 
 
 7499 
 
 7543 
 
 7586 
 
 7629 
 
 7673 
 
 7716 
 
 85 
 
 333435 
 
 36 
 
 37 
 
 37 
 
 886 
 
 887 
 
 7760 
 8195 
 
 7803 
 
 8238 
 
 7847 
 8282 
 
 7890 
 8326 
 
 7934 
 
 8369 
 
 7977 
 
 8413 
 
 8021 
 8456 
 
 8064 
 8500 
 
 8108 
 8543 
 
 8151 
 
 8587 
 
 86 
 
 87 
 
 343435 
 34'3536 
 
 3( 
 37 
 
 37 
 37 
 
 38 
 38 
 
 888 
 
 8630 
 
 8674 
 
 8718 
 
 8761 
 
 8805 
 
 8849 
 
 8892 8936 
 
 8979 
 
 9023 
 
 88 
 
 3435'36 
 
 37 
 
 38 
 
 39 
 
 889 
 
 906" 
 
 9110 
 
 9154 
 
 9198 
 
 9241 
 
 9285 
 
 9329 
 
 9372 
 
 9416 
 
 9460 
 
 89 
 
 35|36j36 
 
 37 
 
 38 
 
 39 
 
 89C 
 
 249503 
 
 9547 
 
 9591 
 
 9635 
 
 9678 
 
 9722 
 
 9766 
 
 9809 
 
 9853 
 
 9897 
 
 90 
 
 35 
 
 36 
 
 37 
 
 38 
 
 3940 
 
 891 
 
 994 
 
 9984 
 
 0028 
 
 0072 
 
 0116 
 
 o!60 
 
 0203 
 
 o247 
 
 0291 
 
 0335 
 
 91 
 
 35 
 
 36 
 
 37 
 
 38 
 
 3940 
 
 892 
 
 25 037 
 
 0422 
 
 0466 
 
 0510 
 
 0554 
 
 0598 
 
 0642 
 
 0685 
 
 0729 
 
 0773 
 
 92 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 40 
 
 893 
 
 081 
 
 0861 
 
 0905 
 
 0949 
 
 0993 
 
 1036 
 
 1080 1124 
 
 1168 
 
 1212 
 
 93 
 
 36 
 
 37 
 
 3839 
 
 40 
 
 41 
 
 894 
 
 125 
 
 130C 
 
 1344 
 
 1388 
 
 1432 
 
 1476 
 
 1520 
 
 1564 
 
 1608 
 
 1652 
 
 94 
 
 37 
 
 38 
 
 39 
 
 39 
 
 40 
 
 41 
 
 89o 
 
 169 
 
 174C 
 
 1784 
 
 1828 
 
 1872 
 
 1916 
 
 1960 
 
 2004 
 
 2048 
 
 2092 
 
 95 
 
 37 
 
 38 
 
 39 
 
 4C 
 
 41 
 
 42 
 
 89 
 
 213 
 
 218C 
 
 2224 
 
 2268 
 
 2312 
 
 2356 
 
 2400 2444 
 
 2488 
 
 2532 
 
 96 
 
 37 
 
 3F 
 
 39 
 
 4C 
 
 41 
 
 42 
 
 89 
 
 257 
 
 262C 
 
 266 
 
 2709 
 
 2753 
 
 2797 
 
 2841 
 
 2885 
 
 2929 
 
 2973 
 
 97 
 
 38 
 
 39 
 
 4( 
 
 41 
 
 42 
 
 43 
 
 89 
 
 301 
 
 306$ 
 
 3106 
 
 3150 
 
 3194 
 
 3238 
 
 3283 3327 
 
 3371 
 
 3415 
 
 9b 
 
 38 
 
 39 
 
 4C 
 
 41 
 
 42 
 
 43 
 
 89 
 
 345 
 
 3504 
 
 354fc 
 
 3592 
 
 3636 
 
 3681 
 
 372 
 
 376 
 
 3813 
 
 3858 
 
 9S 
 
 39 
 
 4C 
 
 41 
 
 42 
 
 43 
 
 44 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 39 
 
 4C 
 
 41 
 
 45 
 
 43 
 
 44 
 
 [23] 
 
TABLE I. 
 
 Log 
 
 
 
 
 Log (l+x) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 Q 
 
 
 4546 
 
 47 
 
 48 
 
 49 
 
 50 
 
 1-900 
 
 0-25 3902 
 
 3946 
 
 3990 
 
 4035 
 
 4079 
 
 4123 
 
 4168 
 
 4212 
 
 4256 
 
 4301 
 
 OC 
 
 
 
 
 
 
 
 
 
 
 
 901 
 
 4345 
 
 4389 
 
 4434 
 
 4478 
 
 4522 
 
 4567 
 
 4611 
 
 4655 
 
 4700 
 
 4744 
 
 01 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 902 
 
 4788 
 
 4833 
 
 4877 
 
 4922 
 
 4966 
 
 5010 
 
 5055 
 
 5099 
 
 5144 
 
 5188 
 
 02 
 
 I 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 903 
 
 5233 
 
 5277 
 
 5321 
 
 5366 
 
 5410 
 
 5455 
 
 5499 
 
 5544 
 
 5588 
 
 5633 
 
 03 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 904 
 
 5677 
 
 5722 
 
 5766 
 
 5811 
 
 5855 
 
 5900 
 
 5944 
 
 5989 
 
 6033 
 
 6078 
 
 04 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 905 
 
 6122 
 
 6167 
 
 6212 
 
 6256 
 
 6301 
 
 6345 
 
 6390 
 
 6434 
 
 6479 
 
 6524 
 
 05 
 
 2 
 
 2 
 
 2 
 
 2 
 
 9 
 
 i 
 
 3 
 
 906 
 
 6568 
 
 6613 
 
 6657 
 
 6702 
 
 6747 
 
 6791 
 
 6836 
 
 6881 
 
 6925 
 
 6970 
 
 06 
 
 3 
 
 3 
 
 3 
 
 3 
 
 f 
 t 
 
 3 
 
 907 
 
 7015 
 
 7059 
 
 7104 
 
 7149 
 
 7193 
 
 7238 
 
 7283 
 
 7327 
 
 7372 
 
 7417 
 
 07 
 
 3 
 
 3 
 
 
 
 o 
 
 3 
 
 t. 
 
 3 
 
 908 
 
 7462 
 
 7506 
 
 7551 
 
 7596 
 
 7641 
 
 7685 
 
 7730 
 
 7775 
 
 7820 
 
 7864 
 
 08 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 909 
 
 . 7909 
 
 7954 
 
 7999 
 
 8043 
 
 8088 
 
 8133 
 
 8178 
 
 8223 
 
 8268 
 
 8312 
 
 09 
 
 4 
 
 4 
 
 4 
 
 4 
 
 / 
 
 K 
 
 J 
 
 910 
 
 25 8357 
 
 8402 
 
 8447 
 
 8492 
 
 8537 
 
 8581 
 
 8626 
 
 8671 
 
 8716 
 
 8761 
 
 10 
 
 4 
 
 5 
 
 5 
 
 5 
 
 F 
 
 5 
 
 911 
 
 8806 
 
 8851 
 
 8896 
 
 8941 
 
 8985 
 
 9030 
 
 9075 
 
 9120 
 
 9165 
 
 9210 
 
 LI 
 
 5 
 
 r; 
 
 ., 
 
 K 
 
 1.; 
 
 5 
 
 K 
 
 u 
 
 5 
 
 912 
 
 9255 
 
 9300 
 
 9345 
 
 9390 
 
 9435 
 
 9480 
 
 9525 
 
 9570 
 
 9615 
 
 9660 
 
 12 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 913 
 
 9705 
 
 9750 
 
 9795 
 
 9840 
 
 9885 
 
 9930 
 
 9975 
 
 0020 
 
 0065 
 
 ollO 
 
 13 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 7 
 
 914 
 
 26 0155 
 
 0200 
 
 0245 
 
 0290 
 
 0336 
 
 0381 
 
 0426 
 
 0471 
 
 0516 
 
 0561 
 
 14 
 
 6 
 
 6 
 
 7 
 
 7 
 
 "7 
 
 i 
 
 915 
 
 0606 
 
 0651 
 
 0696 
 
 0742 
 
 0787 
 
 0832 
 
 0877 
 
 0922 
 
 0967 
 
 1013 
 
 If 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 916 
 
 1058 
 
 1103 
 
 1148 
 
 1193 
 
 1238 
 
 1284 
 
 1329 
 
 1374 
 
 1419 
 
 1465 
 
 16 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 917 
 
 1510 
 
 1555 
 
 1600 
 
 1646 
 
 1691 
 
 1736 
 
 1781 
 
 1827 
 
 1872 
 
 1917 
 
 17 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 918 
 
 1962 
 
 2008 
 
 2053 
 
 2098 
 
 2144 
 
 2189 
 
 2234 
 
 2280 
 
 2325 
 
 2370 
 
 18 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 919 
 
 2416 
 
 2461 
 
 2506 
 
 2552 
 
 2597 
 
 2642 
 
 2688 
 
 2733 
 
 2779 
 
 2824 
 
 19 
 
 9 
 
 9 
 
 9 
 
 9 
 
 t 
 
 9 
 
 920 
 
 26 2869 
 
 2915 
 
 2960 
 
 3006 
 
 3051 
 
 3097 
 
 3142 
 
 3187 
 
 3233 
 
 3278 
 
 20 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 921 
 
 3324 
 
 3369 
 
 3415 
 
 3460 
 
 3506 
 
 3551 
 
 3597 
 
 3642 
 
 3688 
 
 3733 
 
 21 
 
 9 
 
 10 
 
 10 
 
 10 
 
 1C 
 
 11 
 
 922 
 
 3779 
 
 3824 
 
 3870 
 
 3915 
 
 3961 
 
 4006 
 
 4052 
 
 4098 
 
 4143 
 
 4189 
 
 22 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 923 
 
 4234 
 
 4280 
 
 4325 
 
 4371 
 
 4417 
 
 4462 
 
 4508 
 
 4553 
 
 4599 
 
 4645 
 
 23 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 924 
 
 4690 
 
 4736 
 
 4782 
 
 4827 
 
 4873 
 
 4919 
 
 4964 
 
 5010 
 
 5056 
 
 5101 
 
 24 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 925 
 
 5147 
 
 5193 
 
 5238 
 
 5284 
 
 5330 
 
 5376 
 
 5421 
 
 5467 
 
 5513 
 
 5558 
 
 25 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 13 
 
 926 
 
 5604 
 
 5650 
 
 5696 
 
 5741 
 
 5787 
 
 5833 
 
 5879 
 
 5925 
 
 5970 
 
 6016 
 
 20 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 927 
 
 6062 
 
 6108 
 
 6154 
 
 6199 
 
 6245 
 
 6291 
 
 6337 
 
 6383 
 
 6429 
 
 6475 
 
 27 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 13 
 
 928 
 
 6520 
 
 6566 
 
 6612 
 
 6658 
 
 6704 
 
 6750 
 
 6796 
 
 6842 
 
 6887 
 
 6933 
 
 28 
 
 13 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 929 
 
 6979 
 
 7025 
 
 7071 
 
 7117 
 
 7163 
 
 7209 
 
 7255 
 
 7301 
 
 7347 
 
 7393 
 
 29 
 
 1313 
 
 14 
 
 14 
 
 14 
 
 15 
 
 930 
 
 26 7439 
 
 7485 
 
 7531 
 
 7577 
 
 7623 
 
 7669 
 
 7715 
 
 7761 
 
 7807 
 
 7853 
 
 30 
 
 14 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 931 
 
 7899 
 
 7945 
 
 7991 
 
 8037 
 
 8083 
 
 8129 
 
 8175 
 
 8221 
 
 8267 
 
 8313 
 
 31 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 15 
 
 932 
 
 8360 
 
 8406 
 
 8452 
 
 8498 
 
 8544 
 
 8590 
 
 8636 
 
 8682 
 
 8728 
 
 8775 
 
 32 
 
 14 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 933 
 
 8821 
 
 8867 
 
 8913 
 
 8959 
 
 9005 
 
 9052 
 
 9098 
 
 9144 
 
 9190 
 
 9236 
 
 33 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 17 
 
 934 
 
 9283 
 
 9329 
 
 9375 
 
 9421 
 
 9467 
 
 9514 
 
 9560 
 
 9606 
 
 9652 
 
 9699 
 
 34 
 
 15 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 935 
 
 9745 
 
 9791 
 
 9837 
 
 9884 
 
 9930 
 
 9976 
 
 0023 
 
 0069 
 
 0115 
 
 0162 
 
 35 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 17 
 
 936 
 
 27 0208 
 
 0254 
 
 0301 
 
 0347 
 
 0393 
 
 0440 
 
 0486 
 
 0532 
 
 0579 
 
 0625 
 
 36 
 
 16 
 
 17 
 
 17 
 
 17 
 
 18 
 
 18 
 
 937 
 
 0671 
 
 0718 
 
 0764 
 
 0811 
 
 0857 
 
 0903 
 
 0950 
 
 0996 
 
 1043 
 
 1089 
 
 37 
 
 17 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 938 
 
 1135 
 
 1182 
 
 1228 
 
 1275 
 
 1321 
 
 1368 
 
 1414 
 
 1461 
 
 1507 
 
 1554 
 
 38 
 
 17 
 
 17 
 
 18 
 
 18 
 
 19 
 
 19 
 
 939 
 
 1600 
 
 1647 
 
 1693 
 
 1740 
 
 1786 
 
 1833 
 
 1879 
 
 1926 
 
 1972 
 
 2019 
 
 39 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 19 
 
 940 
 
 27 2065 
 
 2112 
 
 2158 
 
 2205 
 
 2252 
 
 2298 
 
 2345 
 
 2391 
 
 2438 
 
 2485 
 
 40 
 
 18 
 
 18 
 
 19 
 
 1920 
 
 20 
 
 941 
 
 2531 
 
 2578 
 
 2624 
 
 2671 
 
 2718 
 
 2764 
 
 2811 
 
 2858 
 
 2904 
 
 2951 
 
 41 
 
 18 
 
 19 
 
 19 
 
 2020 
 
 21 
 
 942 
 
 2998 
 
 3044 
 
 3091 
 
 3138 
 
 3184 
 
 3231 
 
 3278 
 
 3324 
 
 3371 
 
 3418 
 
 42 
 
 19 
 
 19 
 
 20 
 
 2021 
 
 21 
 
 943 
 
 3464 
 
 3511 
 
 3558 
 
 3605 
 
 3651 
 
 3698 
 
 3745 
 
 3792 
 
 3838 
 
 3885 
 
 43 
 
 19 
 
 20 
 
 20 
 
 2121 
 
 21 
 
 944 
 
 3932 
 
 3979 
 
 4026 
 
 4072 
 
 4119 
 
 4166 
 
 4213 
 
 4260 
 
 4306 
 
 4353 
 
 44 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 945 
 
 4400 
 
 4447 
 
 4494 
 
 4541 
 
 4587 
 
 4634 
 
 4681 
 
 4728 
 
 4775 
 
 4822 
 
 45 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 23 
 
 946 
 
 4869 
 
 4916 
 
 4963 
 
 5009 
 
 5056 
 
 5103 
 
 5150 
 
 5197 
 
 5244 
 
 5291 
 
 46 
 
 21 
 
 21 
 
 222223 
 
 23 
 
 947 
 
 5338 
 
 5385 
 
 5432 
 
 5479 
 
 5526 
 
 5573 
 
 5620 
 
 5667 
 
 5714 
 
 5761 
 
 47 
 
 21 
 
 22 
 
 22 
 
 2323 
 
 23 
 
 948 
 
 5808 
 
 5855 
 
 5902 
 
 5949 
 
 5996 
 
 6043 
 
 6090 
 
 6137 
 
 6184 
 
 6231 
 
 48 
 
 22 
 
 22 
 
 23 
 
 2324 
 
 24 
 
 949 
 
 6278 
 
 6325 
 
 6372 
 
 6419 
 
 6467 
 
 6514 
 
 6561 
 
 6608 
 
 6655 
 
 6702 
 
 49 
 
 22 
 
 23 
 
 23 
 
 24 
 
 24 
 
 25 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 K 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 45 
 
 46 
 
 47 
 
 48- 
 
 19 
 
 50 
 
 [24] 
 
ADDENDUM. 
 
 I PROPOSE here to give some account of the origin and the 
 object the raison d'etre of the preceding work, which was first 
 issued in 1849, upwards of twenty years ago. 
 
 Circumstances chiefly a desire to help a friend just com- 
 mencing the study having led me to take up the subject of Life 
 Contingencies, I found that there was one branch of it which had 
 not received the amount of attention in the books to which its 
 great importance entitles it. I refer to the construction of tables, 
 which are, in fact, nothing less than the tools of the actuary : in 
 their absence he is helpless, and unable to apply his theoretical 
 results to practical purposes. I found that the methods indicated 
 for their formation were generally of the most primitive character, 
 and all of them insufficiently illustrated ; and I thought I saw the 
 way to material improvement in these respects. Accordingly, by 
 availing myself of relations subsisting amongst the elementary 
 values of the Mortality Table, which had not been before noticed, 
 and by the application of a new logarithmic table, of a form first 
 suggested by the illustrious Gauss, I succeeded in organising 
 methods of construction which seemed to me, if I may venture to 
 say so, to leave little room for further improvement. 
 
 To utilize my results, it was, of course, necessary that they 
 should be placed within reach of those in a position to benefit by 
 them. Had there existed then such a medium of communication 
 amongst those interested in the science of Life Contingencies as 
 we now have in the Journal of the Institute of Actuaries, I should 
 have gladly offered for insertion in its pages a series of papers 
 explaining and illustrating the methods I had devised. But the 
 Institute had not been formed, and its Journal, consequently, 
 
154 
 
 had not been established; and there was no other publication to 
 which my communications could with propriety have been 
 addressed. There was therefore no resource open to me, but that 
 of separate and independent publication. 
 
 In desiring publicity, I was actuated mainly, I believe, as 
 already intimated, by a hope that I should thereby contribute 
 something to the advancement of a science in which I was much 
 interested; and also in a measure, I dare say why should I 
 hesitate to confess it ? by a wish to air my hobby in the sight of 
 those who might be expected to take an intelligent interest in its 
 gambols. 
 
 The reception accorded to my work on its appearance was such 
 as far to exceed any expectations I had ventured to form, and to 
 call for my grateful acknowledgments. In particular, I desire to 
 place on record my sense of the generous appreciation it met with at 
 the hands of the class to whom it was more specially addressed ; 
 an appreciation in no degree modified, so far as I am aware, 
 by the circumstance that the work was the production of an 
 amateur one who might have been considered, and I think in 
 other days would have been considered, as trespassing on a manor 
 with which he had no concern. Personally, also, I have been 
 frankly recognized as one "free of the craft," and have had an 
 honourable place conferred upon me in the roll of members of the 
 Institute. 
 
 I never anticipated a large sale for my work, and in this respect 
 my anticipation has been realised. The impression was by no 
 means a large one, and there remain a number of copies still 
 on hand. I solace myself by the belief that although, thanks 
 mainly to the labours and the influence of the Institute, the 
 number of students is now very much greater than it was in the 
 days of Milne, it is nevertheless not yet large enough to take off a 
 moderate edition of a purely scientific work (although practical 
 withal) in twenty years. I propose to use up the remaining copies 
 in the manner following : 
 
 I have long been aware that an extension of Table I. in the 
 negative direction was desirable.* The want of this extension, 
 although it does not necessarily limit the applicability of the table, 
 
 * See Preface, p. 15. 
 
155 
 
 yet renders the use of it, in certain circumstances, somewhat 
 inconvenient, or rather for the expression seems too strong less 
 convenient than it is in other circumstances. And it has thus, I 
 know, proved a hindrance to the employment of the table in cases 
 in which such employment would have been productive of saving 
 in both time and labour. 
 
 I have now made the requisite extension, by which the incon- 
 venience in question is removed ; and I utilize the remaining copies 
 of the work by issuing them with the table in a complete form. 
 
 To complete the copies already in circulation, the additional 
 matter will be supplied to their possessors, free of charge, by the 
 publishers, Messrs. Layton.* 
 
 The necessary explanations follow. 
 
 LONDON, 
 
 IQth August, 1870. 
 
 * Applications by post must enclose a penny stamp; and applications through the Trade 
 will require to be accompanied by the names of the parties on whose behalf they are made. 
 
ON THE EXTENSION OF TABLE I. 
 
 IN the preceding work it is shown (for the first time in regard to most 
 of them) that all the values of Life Contingencies which it is usual and 
 necessary to tabulate, may be formed in dependent series, in which each 
 value helps in finding the next. In some of the cases the formation is 
 effected by the continuous addition to an initial value, of the successive 
 terms of a series easily and independently formed; while in others a 
 tabular operation also is required for each value. It is with formations 
 of the latter class only that we are now concerned. The tabular 
 operation requisite in the class of formations referred to is in all cases 
 reducible to that of finding the logarithm of [unity plus a number 
 whose logarithm is known]. To effect this by the common tables presents 
 no difficulty; but it requires two entries, one inverse and the other 
 direct. Now, if we were provided with a table which should enable us to 
 pass by a single direct entry from, say, log.x to log.(l+#), we should 
 by its use effect a saving of at least one-half in the labour of this part of 
 the work. 
 
 We have such a table in the appended Table I. In it the argument 
 extends from 3 to +2 : it therefore affords the means of dealing 
 directly with any value of log.a; lying between those limits ; and this is a 
 range wide enough to include all that could well arise in the construction 
 of tables of the values of Life Contingencies. 
 
 In the work, as originally issued, Table I. extended only from 
 to + 2, and it therefore did not afford the means of so dealing with 
 negative values of log.x. It was consequently necessary, when such 
 values presented themselves, to have recourse to indirect processes, 
 which, besides involving an increase of labour, occasioned also a depar- 
 ture from the uniformity in the series of operations by which the several 
 constructions would have otherwise been characterised. 
 
 The defect which rendered this course necessary, has proved a greater 
 
 idrance to the employment of the table than was anticipated ; and 
 therefore it is that in the present re-issue it has been supplied. 
 
 A few words as to the manner in which the extension of the table has 
 been formed. It has not been a work of great labour. Very little 
 )mputation, properly so called, has been necessary. The first thousand 
 dues only, from 3 to 2, have required independent construction; 
 remainder, from 2 to 0, were obtained very simply from the 
 portion of the tables originally published, in virtue of a relation which 
 subsists between values which correspond to complementary arguments. 
 
158 
 
 We call complementary arguments a pair of terms in the argument series, 
 which, being arithmetical complements of each other having their sums 
 equal to nothing occupy places in the series on opposite sides of 0, at 
 equal distances from it. Thus TOOOO and I'OOOO are complementary 
 arguments; and so are 1-3628 and 0-6372. 
 
 Hence in a table of equal extent on both sides of 0, each argument on 
 one side will have its complementary argument on the other side ; and 
 generally, the portion of the series on one side of will be complemen- 
 tary, in reverse order, to the portion on the other side. 
 
 Complementary results are the tabular results which belong to com- 
 plementary arguments. The same things will obviously hold with regard 
 to them that we have just seen to hold with regard to complementary 
 arguments. Each result on either side of will have its complementary 
 result on the other side; and the portions of the series of results on 
 opposite sides of argument 0, will be complementary to each other in 
 reverse order. 
 
 The entire portions of the two series on opposite sides of being thus 
 complementary, it follows that any two corresponding parts of them are 
 also complementary, always in reverse order. Thus, referring to pages 
 [6] and 21, let each of them be supposed to be extended by the addition 
 of the next term in the argument column and in column 0. The two 
 pages are now complementary. The one commences with 1-0000 and 
 ends with 1-0500, while the other commences with 0-9500 and ends with 
 1-0000, which terms being complementary, in reverse order, the inter- 
 mediate terms are also complementary in the same order. 
 
 The columns in complementary pages are also complementary, each to 
 each. That is, each column in the one page has a column complemen- 
 tary to it in the other ; the relation consisting in this, that the results in 
 each of them are complementary to those in the other, in reverse order. 
 
 The complementary columns are those of which the headings are the 
 complements of each other to 10. Thus, the columns headed and 0, 
 1 and 9, 2 and 8, 3 and 7, &c., are complementary, each to each. 
 
 The relation subsisting between complementary results is hardly less 
 simple than that subsisting betweem complementary arguments. If we 
 used T as a functional symbol, to denote the tabular result correspond- 
 ing to the appended argument, the equation characteristic of the table 
 will be, 
 
 T[log.*]=log.(l+4 
 In this, write - for ar, and we get, 
 
 ., or -og.*= 
 
 =log.(l 4-*)log.#=T[log.a:] log,*. 
 
159 
 
 From this we learn that if from any result we subtract its argument, 
 the remainder is the complementary result. 
 For example : 
 
 T [0-6372]=0-727306 
 Subtract 0-637200 
 
 Kemainder, 0090106=T [1-3628]. 
 
 The complementary results are thus, 0727306 and 0-090106, cor- 
 responding to the complementary arguments, 0-6372 and 1*3628, re- 
 spectively. 
 
 It is apparent from what has just been shown, that if the portion of 
 the table on either side of is given, the portion on the opposite side 
 can be formed by subtraction. And this is, in effect, the method that 
 was employed for the extension of the table from I'OOOO to O'OOOO, the 
 labour of formal subtraction, however, being avoided in the way now to 
 be shown. 
 
 In the subtractions here requisite it is to be observed that while the 
 minuends and the remainders are complementary results, the subtra- 
 hends would be successive terms of the argument series. Now in this 
 series, considered as a series of six figure logarithms, the last two places 
 are always ciphers, in every tenth term the last three places are ciphers, 
 and in every hundredth term the last four places are ciphers. Comple- 
 mentary results therefore will agree in every term in their last two 
 places, in every tenth term in their last three places, and in every 
 hundredth term in their last four places. 
 
 To this extent then, that is to the extent of their last two, three, or 
 four places, as the case may be, the terms of the portion of the table to 
 be formed may be determined by inspection of their complementary 
 terms, and inserted in their places. And this operation will be much 
 facilitated by availing ourselves of the information we have acquired 
 as to the relative positions in the table occupied by complementary 
 results. 
 
 Examination and comparison of any two complementary pages will 
 afford sufficient illustration of what precedes. Refer again to pages [6] 
 and 21, and take any two complementary columns, say 1 and 9. 
 It will be perceived at once that the two terminal figures in either 
 column are the same as those in the other, in reverse order. And 
 the same will hold in the case of every pair of complementary 
 columns. In the two columns however we find a closer agree- 
 ment. In them the identity extends to three figures throughout, and 
 in every tenth term to four. It is easy now to see that when the 
 identical figures have been inserted in all the columns the completion 
 of the table is a very simple matter. 
 
160 
 
 The differences belonging to complementary results are complements 
 of each other to 100. This may be shown as follows : 
 
 Since T flog.-! =T[log.a?] log.* ; 
 
 therefore, AT [log.-] = AT[log.*]- Alog.s= AT[log.*]- 100, 
 since in our Table Alog.#=100. 
 
 Transposing we have, AT[log.s] AT flog.-J = 100 ; 
 
 which, since AT[log.#] and AT log.- are of opposite signs, establishes 
 
 the theorem. 
 
 For illustration refer to any two complementary openings, say pages 
 [6], [7], and 20, 21. In the first opening the differences are 9, 10, 11, 
 12, and in the second 88, 89, 90, 91 ; and the terms of these, in reverse 
 order, are respectively the complements of each other to 100. 
 
 A similar property holds with regard to the table of proportional 
 parts. Thus, opposite 80, for instance, we find in the former opening 
 7, 8, 9, 10, and in the latter 70, 71, 72, 73 ; and the terms of these, in 
 reverse order, are respectively the complements of each other to 80. 
 
 It will be obvious from this how the table of proportional parts has 
 been completed. 
 
 The portion of the table 2 to 1, is of only one-tenth the extent of 
 its complementary portion 1 to 2 ; and it consists therefore of the com- 
 plementary results of only every tenth term in the last-named portion, 
 the terms being those in column 0. These terms, it will be recollected, 
 agree in their last three places, and in every tenth term in their last 
 four places, with their complementary terms. Hence in the portion of 
 the table now under consideration, we have this agreement feo three 
 places throughout, and in column to four places. This will be apparent 
 on inspection. 
 
 The remaining portion from 3 to 2, having no complementary 
 portion on the opposite side of 0, was formed by interpolation. The 
 basis was ten terms constructed independently, true to ten places. 
 
 In the part of the table extending from 3 to 1 the differences 
 change so rapidly that there is no room for the insertion of proportional 
 parts. This is of no great importance as the differences are small, and 
 this part of the table will but seldom come into use. 
 
 The complete rule for the use of the differences is : Multiply the 
 proper difference by the number composed of the last three figures of the 
 argument, and strike off three figures from the right of the result. The 
 remainder is the required proportional part. But the differences are 
 so small that the proportional part will usually be determinable by 
 inspection. 
 
161 
 
 ON THE APPLICATION OF THE EXTENDED TABLE. 
 
 A Table in "which the result is related to its argument as in Table I., 
 can be most advantageously applied to the formation of the logarithm of 
 the sum of two numbers whose logarithms are known, and without the 
 necessity of first finding the numbers themselves. 
 
 For, in the characteristic equation of the table, 
 
 for x write a-^-b, and we get, 
 
 That is, Tpog.a log. 5]= log. (a + 5) -log. 3; 
 
 whence, Log. (a + b) =log.& + T[log. a log. b]. 
 
 We may now, if we please, interchange a and 5; but it is not necessary 
 as they are symmetrically involved, and there is no restriction as to their 
 relative magnitudes. 
 
 From the foregoing equation, if log. a and log. b are given, to form 
 log. (a + b) we have the following 
 
 Rule : Subtract either logarithm from the other, and to the subtrahend 
 add the tabular result corresponding to the remainder. The sum is the 
 logarithm required. 
 
 We have thus in all cases two methods of solution. Both are shown 
 in the following example. 
 
 Two logarithms are 2-370267 and 2-045757; required the logarithm 
 of the sum of the corresponding numbers 
 
 2-370267 2-045757 
 
 2-045757 0-324510 2-370267 1-675490 
 
 0-492915 0-168405 
 
 2-538672 2-538672 
 
 The given logarithms are those of 234-567 and 111 -111; the sum of 
 these numbers is 345-678, and its logarithm is 2-538672, as here found. 
 
 It is not easy to conceive a simpler operation than that by which this 
 result has been obtained, or one admitting of more commodious arrange- 
 ment. The given logarithms being set down one under the other, the 
 lower is subtracted from the upper, and the difference placed at the 
 side. The table is entered with this difference, and the result set down 
 under the subtrahend. Addition of the last two lines then gives the 
 required logarithm. 
 
 In the two methods of solution, the differences of the given logarithms, 
 which form the arguments for the table, are necessarily complementary 
 
162 
 
 to each other, one of them therefore being positive and the other 
 negative. The form of the operation is, nevertheless, the same in both 
 cases. This is the advantage we gain by the negative addition to the 
 table. It is no matter now in what order the two logarithms to be 
 operated upon present themselves, whether the] greater over the less, or 
 the less over the greater. The upper is always the minuend, the lower 
 is always the subtrahend, and the remainder always forms the tabular 
 argument. 
 
 On pp. 80, 81, the following Lemma is demonstrated: "If B denote 
 " the present value of a benefit of 1 upon a given life or combination 
 " of lives, and such that, in the case of a combination of lives the risk 
 " is determined by the failure of any one of them; and if Bj denote the 
 " present value of a similar benefit on a life or a combination of lives 
 " respectively one year older than those on which B depends; if, more- 
 " over, TT denote the probability of a payment of B being received in the 
 " first year, and II the probability of the single life, or of all the lives, 
 "on which that benefit depends, surviving a year; then will the 
 " following equation always subsist : 
 
 " B=cn+Bl '" 
 
 On passing to logarithms this equation becomes, 
 
 log.B=log.i;n + log.^ + B 1 ) ............... (A) 
 
 Now this is obviously a case suited for the application of Table I. 
 We pass from log.Bi to log.B by adding together log. til and logY^ + B! J. 
 
 Log. rll, log. and log.Bj are known, the two former being terms of series 
 
 independently formed, and log. B! being the result of a like operation 
 for the preceding age. We have therefore, from what precedes, 
 
 log.^+B 1 )=log.B I +T[log.j!j-log.:B I ] ) 
 or=log.+T[log.B,-log.g. 
 
 Adopting the second form as best suited for our purpose, we obtain by 
 substitution in (A), 
 
 log.B=log.t;n + log.g + T [log^ - log.jH . 
 
 And finally, changing the order of the first two terms of the right hand 
 side of this equation, we have for the type of the operation requisite to 
 pass from log.B! to log.B : 
 
163 
 
 A 
 
 log. B! 
 
 BI 
 
 A 
 
 log-Or-5- ) 
 
 Dt-A 
 
 B 
 
 log.rll 
 
 
 C 
 
 T[D,-X| 
 
 
 D 
 
 (A+B+C 
 t=:log.B 
 
 B 
 
 By giving to the different symbols in this general type suitable values 
 we produce the particular types adapted to the various benefits. 
 
 Thus, for an annuity on (x) we have B 1 =% +1 , ~B=a x ; ir=p x , and 
 H=p x , and consequently 7r-f-II=l, the logarithm of which is 0. The 
 type in this case therefore becomes : 
 
 A 
 B 
 
 T[C,] 
 
 (A + B 
 
 See p. 84. 
 
 And this again becomes changed into that for the curtate mean 
 duration of (x) by making v=l. See p. 63. 
 
 For an assurance on (x) we have B^A^!, B=A a: ; 7r=l #p and 
 IL=p x , .'. 7r-f-n=^ A .~ 1 1 ; and the type becomes : 
 
 A 
 
 A 
 B 
 C 
 
 D 
 
 log.t^ 
 
 (A + B+C 
 
 D.-A 
 
 See p. 92. 
 
 For further examples see pp. 75, 101, 115. 
 
164 
 
 We now repeat some of the formations already given, availing our- 
 selves of the facility afforded by the extension of the table. 
 
 Present Values of Annuities. Page 85. 
 One Life. Two Lives. 
 
 103 
 
 102 
 
 101 
 
 100 
 
 99 
 
 510042 
 
 765314 
 
 121755 
 
 10 
 
 887079 
 
 841035 
 
 248195 
 
 34 
 
 089264 
 
 878017 
 
 347916 
 
 35 
 
 225968 
 
 900013 
 
 428505 
 
 43 
 
 328561 
 
 882428 
 
 495626 
 
 41 
 
 378095 
 
 32363 
 
 77104 
 
 1-22818 
 
 1-68255 
 
 2-13089 
 
 2-38833 
 
 103 
 103 
 
 102 
 102 
 
 101 
 101 
 
 100 
 100 
 
 032921 -10788 
 
 543465 
 044489 
 2 
 
 587956 -38722 
 
 694907 
 
 142129 
 
 .16 
 
 837052 -68715 
 
 768872 
 
 227133 
 
 21 
 
 996026 -99089 
 
 812863 
 
 299035 
 
 13 
 
 111911 
 
 777693 
 
 360574 
 
 6 
 
 1-29392 
 
 98 138273 1-37491 
 
 103 
 
 102 
 
 288193 -19418 
 
 619186 
 072053 
 
 lOS) U 
 
 1Q1 I 696254 
 
 731890 
 175170 
 18 
 
 907078 
 
 101 
 100 
 
 790867 
 257015 
 
 99 j 047916 
 
 795278 
 325640 
 8 
 
 120926 
 
 49688 
 
 80738 
 
 11665 
 
 32107 
 
 98 
 
 773283 
 
 365674 
 
 15 
 
 138972 
 
 The cases in which Table I. comes into use divide themselves into two 
 classes. In the first class, which comprises Curtate Mean Durations, and 
 Annuities (certain, for terms of years, or on one or more lives) the 
 tabular argument in each operation is always the result of the previous 
 operation. The process is somewhat simpler than in the cases comprised 
 in the second class, in the operations belonging to which the argument is 
 the difference between two logarithms, one of which is the preceding result. 
 
 The annuity process has been chosen as the type of the first class of 
 cases, and comparison of the examples as given above with the same 
 examples in their original form, will show the improvement the operation 
 has undergone by the use of the extended table. 
 
 It will be observed that, as formerly, the proportional parts are still 
 made to occupy a separate line. Experience has amply confirmed the 
 propriety of this arrangement. It is found to contribute in a most 
 material degree to the facility and rapidity with which the table may be 
 used. The mental incorporation of the proportional parts, which is thus 
 avoided, is the portion of an operation which soonest induces fatigue 
 when a number of entries have to be made ; and it is also that in which 
 when it is used the greatest liability to error is found. 
 
165 
 
 The chief advantage of the annuity operation, when conducted by 
 means of the present table, over the same when conducted by means of 
 the common table, is, that here all the entries of the same kind direct 
 and inverse are brought together, the whole of the logarithms being 
 formed before a single natural number is taken out. We consequently 
 proceed right through the table, and as we proceed we find two, three, 
 four, and even as many as six or eight entries on the same opening. At 
 the close, moreover, the taking out of the numbers may, if necessary, 
 be turned over to an assistant. On the other hand, when the common 
 tables are used, direct and inverse entries alternate with each other, and 
 involve likewise a continual turning of the leaves backwards and 
 forwards, by which the process is rendered exceedingly irksome. 
 
 On the whole it may with safety be affirmed, that no one who will give 
 the new method of formation, with the additional facility now afforded, 
 a fail' trial, will ever again seek to have recourse to the old method. 
 
 It is sufficiently well understood that for the proper solution of a case 
 involving, say n lives, we require to know the value of an annuity, or 
 some other integral function, on those lives, and also, when orders of 
 survivorship are concerned, the values of the like on one, two, or more 
 similar adjoining combinations of lives. When n does not exceed 2, we 
 have the requisite values, according to various rates of interest, in the 
 tables of Jones and others : and therefore we can solve rigorously all 
 cases in which one or two lives are involved. But it is otherwise when 
 the number of lives is more than two. We have no tables suited to 
 such cases ; the reason being that the labour requisite for their construc- 
 tion is so great that no one hitherto has had the hardihood to undertake 
 it.* The consequence is, that when cases involving more than two 
 lives present themselves, as they frequently do, we are obliged to have 
 recourse to methods of approximation, the results of which have no 
 claim to be considered as other than vague guesses at the truth. 
 
 We are not likely then, proceeding on the present plan, ever to be 
 put in possession of the means of dealing satisfactorily with cases involv- 
 ing more than two lives. But we have recently had presented to us a 
 prospect of deliverance from this condition of comparative helpless- 
 ness. It has been shown to us how, by the aid of a small amount of 
 tables, we may acquire the power of treating rigorously cases involving 
 any number of lives whatsoever. 
 
 It is to Mr. W. M. Makeham that we owe this great advance. Mr. 
 M. propounds a theory of the law of mortality, which he shows to possess 
 strong claims to acceptance on physiological grounds, and also from the 
 accordance of its results with observed facts. It is a modification and 
 extension of the now famous theory of the late Mr. Gompertz, in a 
 
 * At a single rate of interest a complete annuity table for three Urea would comprise 
 161,700 values, and one for four lives, so many aa 3,921,225 values. 
 
166 
 
 direction in which Mr. Gompertz distinctly stated that his theory needed 
 modification and extension. 
 
 Mr. Makeham's theory may be applied to the adjustment of any 
 given table of mortality. An immediate result of this application, being 
 that with which we are here more immediately concerned, is, that if we 
 form from the table so adjusted a table of the values of annuities on all 
 the combinations of n lives (n being any integer), in which the ages are 
 equal, we shall be able to deduce thence, by an easy process, the value 
 of an annuity on any combination of the same number of lives that may 
 be proposed, and consequently differing among themselves anyhow as to 
 age. We thus, at the cost of the construction of an annuity table of not 
 more than a hundred values (the labour of construction being no greater 
 for a dozen lives than for one), acquire the power of legitimately dealing 
 with every case involving the same number of lives as that for which 
 the table has been constructed. 
 
 Surely this is a great improvement one, indeed, the importance of 
 which, in the present connexion, can hardly be over-estimated. It opens 
 a way for the effectual removal of that which has hitherto been the 
 opprobium of the science of Life Contingencies the inability to deal 
 rigorously with cases involving more than two lives ; and it cannot be 
 but that its value and utility will ere long be generally recognised. 
 
 When the anticipated period arrives, there will arise a necessity for 
 the construction of tables of annuities on many lives, at various rates of 
 interest, and according to different tables of mortality; and it is to 
 afford the requisite facility for the construction of these that Table I. 
 has been extended in the negative direction so far as 3-0000. For 
 two lives it will seldom, if ever, be necessary to go beyond 1 '0000 ; 
 but for so many as five or six, the portion of the table between 3 and 
 2 will come into use. 
 
 An example of the construction of a table of annuities on six lives 
 of equal ages is appended. The data are, as usual, the Carlisle Table of 
 Mortality, and 3 per cent, interest. The index of the initial term is 
 3, and of the next 2. It is hardly necessary to say that the 
 auxiliary series is log.vp XJUtM = log.u + 6 log./?,. 
 
 Annuities. Six Lives. Equal Ages. 
 
 103 
 
 102 
 
 101 
 
 124437 
 656069 
 000577 
 
 
 656646 
 
 110395 
 
 019237 
 
 28 
 
 129660 
 
 00133 
 
 04536 
 
 13479 
 
 101 
 
 100 
 
 129660 
 
 332292 
 054909 
 
 7 
 
 387208 
 
 24390 
 
 99 559048 -36228 
 
 99 559048 
 
 358753 
 
 134255 
 
 12 
 
 98 493020 '31119 
 
 3~32293 
 117660 
 6 
 
 97 449959 '28181 
 
167 
 
 Annuities. Six Lives. Equal Ages. (Continued) 
 
 97 449959 
 
 348432 
 
 107811 
 
 13 
 
 96 456256 '28593 
 
 294805 
 
 109204 
 
 12 
 
 95 404021 -25353 
 
 237529 
 098129 
 4 
 
 94 335662 '21660 
 
 205159 
 
 035137 
 
 11 
 
 93 290307 -19512 
 
 131160 
 077411 
 1 
 
 92 208572 -16165 
 
 110395 
 
 065065 
 
 10 
 
 91 175470 '14979 
 
 200569 
 060608 
 9 
 
 90 261186 -18247 
 
 354817 
 
 072776 
 
 13 
 
 89 427606 -26767 
 
 89 427606 
 
 340308 
 103006 
 1 
 
 88 443315 -27753 
 
 352339 
 106369 
 3 
 
 87 458711 '28755 
 
 426919 
 109761 
 2 
 
 86 536682 -34410 
 
 484999 
 
 128410 
 
 21 
 
 85 613430 -41061 
 
 536586 
 149398 
 
 84 685993 '48528 
 
 83 
 
 560971 
 
 171778 
 
 31 
 
 732780 
 
 592063 
 
 187628 
 
 28 
 
 54048 
 
 82 779719 -60217 
 
 612841 
 204701 
 
 7 
 
 81 817549 '65697 
 
 81 817549 
 
 648955 
 219297 
 19 
 
 80 868271 73837 
 
 658765 
 
 240111 
 
 30 
 
 79 898906 79233 
 
 686953 
 253415 
 3 
 
 78 940371 '87171 
 
 691015 
 
 272205 
 
 33 
 
 77 963253 -91887 
 
 703999 
 
 283020 
 
 25 
 
 76 987044 -97061 
 
 725550 
 
 294579 
 
 22 
 
 75 020151 1*04749 
 
 740928 
 
 311196 
 
 26 
 
 74 052150 1-12759 
 
 775218 
 327861 
 
 27 
 
 73 103106 1-26796 
 
 The effect of the faulty graduation of the Carlisle Table is here 
 strikingly manifest. The annuity value at 99 is greater than that at 
 86, and two-and-a-half times as great as that at 91. 
 
 The second class of cases in which Table I. comes into use, comprises 
 Complete Mean Durations (one life), Probabilities of Survivorship (two 
 lives), Assurances (one life), Survivorship Assurances (two lives), Sick 
 Allowances, &c. We have selected for illustration the examples of 
 Complete Mean Duration, p. 65, Assurance, whole life, p. 93, and Sur- 
 vivorship Assurance, two lives, equal ages, p. 103. 
 
 Assurance, on (a?) against 
 Complete Mean Duration. 
 One Life. 
 
 104 698970 -50000 
 
 301030 T'397940 
 522879 
 096902 
 8 
 
 103 920819 '83333 
 
 Assurance. 
 
 One Life. 
 
 104 987163 -970874 
 
 301030 Teseiss 
 
 510042 
 
 171843 
 
 11 
 
 103 982926 '961448 
 
 Equal Ages. 
 104 686133 -485437 
 
 602060 T084073 
 032921 
 049737 
 8 
 
 103 684726 '483867 
 
168 
 
 Complete Mean Duration. 
 One Life. 
 
 103 920819 
 
 Assurance. 
 One Life. 
 
 103 982926 
 
 Assurance, on (?) against 
 
 (y). 
 
 Equal Ages. 
 103 684726 
 
 124939 1795880 
 778151 
 210823 
 30 
 
 823909 0-159017 
 765314 
 387766 
 10 
 
 948848 1-735878 
 543465 
 188718 
 27 
 
 102 113943 1-30000 
 
 102 976999 -948415 
 
 102 681058 '479798 
 
 079181 0-034762 
 853872 
 318726 
 32 
 
 602060 374939 
 841035 
 527737 
 27 
 
 681241 999817 
 694907 
 300930 
 9 
 
 101 251811 176572 
 
 101 970859 -935101 
 
 101 677087 '475430 
 
 057992 193819 
 890855 
 408652 
 12 
 
 455932 514927 
 878017 
 630698 
 21 
 
 513924 163163 
 768872 
 390192 
 37 
 
 100 357511 2-17778 
 
 100 964668 -921866 
 
 100 673025 -471004 
 
 045757 311754 
 912850 
 484264 
 36 
 
 346787 617881 
 900013 
 711607 
 66 
 
 392544 280481 
 812863 
 463477 
 53 
 
 99 442907 2*77*73 
 
 99 958473 -908810 
 
 99 668937 -466592 
 
 055517 387390 
 895265 
 636495 
 64 
 
 435728 522745 
 882428 
 636685 
 35 
 
 491245 177692 
 777693 
 398846 
 55 
 
 98 487341 3-07143 
 
 98 954876 -901313 
 
 98 667839 -465413 
 
 By comparison of the above with the original operations, on the 
 pages cited, the advantage gained by the extension of the table will 
 become very apparent. 
 
 A very excellent six-figure table, suitable for use with the present 
 tables, is the following : " Logarithmisch-Trigonometrische Tafeln mit 
 Sechs Decimalstellen. Mit Rucksicht auf den Schulgebrauch bearbeitet 
 von Dr. C. Bremiker. Berlin, 1869." The short title in Latin is, 
 " Tabula Logarithmorum VI. Decimalium." 
 
 Like the seven-figure tables, this table has a five-figure argument j 
 and it is much more easily used than they when the seventh figure is 
 not wanted. 
 
 It contains a table of log.(l+#), to argument log.*, like Table I. 
 As this table extends no further in the positive direction than 0*2500, 
 its usefulness in annuity computations is extremely limited.* 
 
 * Bremiker's Work may be obtained through Messrs. Williams & Norgate, Henrietta 
 Street, Covent Garden, for the extremely moderate cost of three shillings and sixpence. 
 
TABLE I. 
 
 Log 
 
 Log (1+a?) 
 
 Pro. Parts. 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 1-950 
 
 0-27 6749 
 
 6796 
 
 6843 
 
 6891 
 
 6938 
 
 6985 
 
 7032 
 
 7079 
 
 7126 
 
 7174 
 
 50 
 
 22 
 
 23 
 
 23 
 
 24 
 
 24 
 
 25 
 
 951 
 
 7221 
 
 7268 
 
 7315 
 
 7362 
 
 7409 
 
 7457 
 
 7504 
 
 7551 
 
 7598 
 
 7646 
 
 51 
 
 23 
 
 23 
 
 24 
 
 2425 
 
 25 
 
 952 
 
 7693 
 
 7740 
 
 7787 
 
 7835 
 
 7882 
 
 7929 
 
 7976 
 
 8024 
 
 8071 
 
 8118 
 
 52 
 
 23 
 
 24 
 
 24 
 
 2525 
 
 26 
 
 953 
 
 8165 
 
 8213 
 
 8260 
 
 8307 
 
 8355 
 
 8402 
 
 8449 
 
 8497 
 
 8544 
 
 8591 
 
 53 
 
 24 
 
 24 
 
 25 
 
 25126 
 
 27 
 
 954 
 
 8639 
 
 8686 
 
 8733 
 
 8781 
 
 8828 
 
 8876 
 
 8923 
 
 8970 
 
 9018 
 
 9065 
 
 54 
 
 24 
 
 25 
 
 25 
 
 26 
 
 26 
 
 27 
 
 955 
 
 9113 
 
 9160 
 
 9207 
 
 9255 
 
 9302 
 
 9350 
 
 9397 
 
 9445 
 
 9492 
 
 9540 
 
 55 
 
 25 
 
 25 
 
 26 
 
 26 
 
 27 
 
 27 
 
 956 
 
 9587 
 
 9634 
 
 9682 
 
 9729 
 
 9777 
 
 9824 
 
 9872 
 
 9919 
 
 9967 
 
 0014 
 
 56 
 
 25 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 957 
 
 28 0062 
 
 0109 
 
 0157 
 
 0205 
 
 0252 
 
 0300 
 
 0347 
 
 0395 
 
 0442 
 
 0490 
 
 57 
 
 20 
 
 26 
 
 27 
 
 27 
 
 28 
 
 29 
 
 958 
 
 0538 
 
 0585 
 
 0633 
 
 0680 
 
 0728 
 
 0776 
 
 0823 
 
 0871 
 
 0918 
 
 0966 
 
 58 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 29 
 
 959 
 
 1014 
 
 1061 
 
 1109 
 
 1157 
 
 1204 
 
 1252 
 
 1300 
 
 1347 
 
 1395 
 
 1443 
 
 59 
 
 27 
 
 27 
 
 28 
 
 28 
 
 29 
 
 29 
 
 960 
 
 28 1490 
 
 1538 
 
 1586 
 
 1633 
 
 1681 
 
 1729 
 
 1777 
 
 1824 
 
 1872 
 
 1920 
 
 60 
 
 27 
 
 2828 
 
 29 
 
 20 
 
 30 
 
 961 
 
 1968 
 
 2015 
 
 2063 
 
 2111 
 
 2159 
 
 2206 
 
 2254 
 
 2302 
 
 2350 
 
 2398 
 
 61 
 
 27 
 
 2829 
 
 29 
 
 30 
 
 31 
 
 962 
 
 2445 
 
 2493 
 
 2541 
 
 2589 
 
 2637 
 
 2685 
 
 2732 
 
 2780 
 
 2828 
 
 2876 
 
 62 
 
 28 
 
 2929 
 
 30 
 
 3(. 
 
 31 
 
 963 
 
 2924 
 
 2972 
 
 3020 
 
 3068 
 
 3115 
 
 3163 
 
 3211 
 
 3259 
 
 3307 
 
 3355 
 
 63 
 
 28 
 
 2930 
 
 30 
 
 31 
 
 31 
 
 964 
 
 3403 
 
 3451 
 
 3499 
 
 3547 
 
 3595 
 
 3643 
 
 3691 
 
 3739 
 
 3787 
 
 3835 
 
 64 
 
 29 
 
 29 
 
 3C 
 
 31 
 
 31 
 
 32 
 
 965 
 
 3882 
 
 3930 
 
 3978 
 
 4026 
 
 4074 
 
 4122 
 
 4171 
 
 4219 
 
 4267 
 
 4315 
 
 65 
 
 29 
 
 30 
 
 31 
 
 31 
 
 32 
 
 33 
 
 966 
 
 4363 
 
 4411 
 
 4459 
 
 4507 
 
 4555 
 
 4603 
 
 4651 
 
 4699 
 
 4747 
 
 4795 
 
 66 
 
 30 
 
 30 
 
 31 
 
 3232 
 
 33 
 
 967 
 
 4843 
 
 4891 
 
 4940 
 
 4988 
 
 5036 
 
 5084 
 
 5132 
 
 5180 
 
 5228 
 
 5277 
 
 67 
 
 30 
 
 31 
 
 31 
 
 3233 
 
 33 
 
 968 
 
 5325 
 
 5373 
 
 5421 
 
 5469 
 
 5517 
 
 5566 
 
 5614 
 
 5662 
 
 5710 
 
 5758 
 
 68 
 
 31 
 
 31 
 
 32 
 
 3333 
 
 34 
 
 969 
 
 5807 
 
 5855 
 
 5903 
 
 5951 
 
 5999 
 
 6048 
 
 6096 
 
 6144 
 
 6192 
 
 6241 
 
 69 
 
 31 
 
 32 
 
 32 
 
 3334 
 
 35 
 
 970 
 
 28 6289 
 
 6337 
 
 6386 
 
 6434 
 
 6482 
 
 6530 
 
 6579 
 
 6627 
 
 6675 
 
 6724 
 
 70 
 
 32 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 971 
 972 
 
 6772 
 7256 
 
 6820 
 7304 
 
 6869 
 7352 
 
 6917 
 7401 
 
 6965 
 7449 
 
 7014 
 7498 
 
 7062 
 7546 
 
 7110 
 7594 
 
 7159 
 7643 
 
 7207 
 7691 
 
 71 
 
 72 
 
 3233 
 3233 
 
 33 
 
 34 
 
 34 
 35 
 
 35 
 35 
 
 35 
 36 
 
 973 
 
 7740 
 
 7788 
 
 7837 
 
 7885 
 
 7934 
 
 7982 
 
 8031 
 
 8079 
 
 8128 
 
 8176 
 
 73 
 
 33 
 
 34 
 
 34 
 
 35 
 
 36 
 
 37 
 
 974 
 
 8225 
 
 8273 
 
 8322 
 
 8370 
 
 8419 
 
 8467 
 
 8516 
 
 8564 
 
 8613 
 
 8661 
 
 74 
 
 33 
 
 34 
 
 35 
 
 36 
 
 30 
 
 37 
 
 975 
 
 8710 
 
 8758 
 
 8807 
 
 8856 
 
 8904 
 
 8953 
 
 9001 
 
 9050 
 
 9099 
 
 9147 
 
 75 
 
 34 
 
 34 
 
 35 
 
 36 
 
 37 
 
 37 
 
 976 
 
 9196 
 
 9244 
 
 9293 
 
 9342 
 
 9390 
 
 9439 
 
 9488 
 
 9536 
 
 9585 
 
 9634 
 
 76 
 
 34 
 
 35 
 
 36 
 
 36 
 
 37 
 
 38 
 
 977 
 
 9682 
 
 9731 
 
 9780 
 
 9828 
 
 9877 
 
 9926 
 
 9974 
 
 0023 
 
 0072 
 
 0121 
 
 77 
 
 35 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 978 
 
 29 0169 
 
 0218 
 
 0267 
 
 0316 
 
 0364 
 
 0413 
 
 0462 
 
 0511 
 
 0559 
 
 0608 
 
 78 
 
 35 
 
 36 
 
 37 
 
 37 
 
 38 
 
 39 
 
 979 
 
 0657 
 
 0706 
 
 0755 
 
 0803 
 
 0852 
 
 0901 
 
 0950 
 
 0999 
 
 1047 
 
 1096 
 
 79 
 
 36 
 
 36 
 
 37 
 
 38 
 
 30 
 
 39 
 
 980 
 
 29 1145 
 
 1194 
 
 1243 
 
 1292 
 
 1341 
 
 1389 
 
 1438 
 
 1487 
 
 1536 
 
 1585 
 
 80 
 
 3637 
 
 38 
 
 38 
 
 30 
 
 40 
 
 981 
 
 1634 
 
 1683 
 
 1732 
 
 1781 
 
 1830 
 
 1878 
 
 1927 
 
 1976 
 
 2025 
 
 2074 
 
 81 
 
 3637 
 
 38 
 
 39 
 
 40 
 
 41 
 
 982 
 
 2123 
 
 2172 
 
 2221 
 
 2270 
 
 2319 
 
 2368 
 
 2417 
 
 2466 
 
 2515 
 
 2564 
 
 82 
 
 37 
 
 38 
 
 30 
 
 30 
 
 40 
 
 41 
 
 983 
 
 2613 
 
 2662 
 
 2711 
 
 2760 
 
 2809 
 
 2858 
 
 2907 
 
 2956 
 
 3006 
 
 3055 
 
 83 
 
 37 
 
 38 
 
 30 
 
 40 
 
 41 
 
 41 
 
 984 
 
 3104 
 
 3153 
 
 3202 
 
 3251 
 
 3300 
 
 3349 
 
 3398 
 
 3447 
 
 3496 
 
 3546 
 
 84 
 
 38 
 
 39 
 
 30 
 
 40 
 
 41 
 
 42 
 
 985 
 
 3595 
 
 3644 
 
 3693 
 
 3742 
 
 3791 
 
 3841 
 
 3890 
 
 3939 
 
 3988 
 
 4037 
 
 85 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 986 
 
 4086 
 
 4136 
 
 4185 
 
 4234 
 
 4283 
 
 4332 
 
 4382 
 
 4431 
 
 4480 
 
 4529 
 
 86 
 
 30 
 
 40 
 
 40 
 
 41 
 
 42 
 
 43 
 
 987 
 
 4579 
 
 4628 
 
 4677 
 
 4726 
 
 4776 
 
 4825 
 
 4874 
 
 4924 
 
 4973 
 
 5022 
 
 87 
 
 30 
 
 40 
 
 41 
 
 42 
 
 43 
 
 43 
 
 988 
 
 5071 
 
 5121 
 
 5170 
 
 5219 
 
 5269 
 
 5318 
 
 5367 
 
 5417 
 
 5466 
 
 5515 
 
 88 
 
 40 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 989 
 
 5565 
 
 5614 
 
 5664 
 
 5713 
 
 5762 
 
 5812 
 
 5861 
 
 5911 
 
 5960 
 
 6009 
 
 69 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 990 
 
 29 6059 
 
 6108 
 
 6158 
 
 6207 
 
 6257 
 
 6306 
 
 6355 
 
 6405 
 
 6454 
 
 6504 
 
 90 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 991 
 
 6553 
 
 6603 
 
 6652 
 
 6702 
 
 6751 
 
 6801 
 
 6850 
 
 6900 
 
 6949 
 
 6999 
 
 91 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 45 
 
 992 
 
 7048 
 
 7098 
 
 7148 
 
 7197 
 
 7247 
 
 7296 
 
 7346 
 
 7395 
 
 7445 
 
 7495 
 
 92 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 993 
 
 7544 
 
 7594 
 
 7643 
 
 7693 
 
 7743 
 
 7792 
 
 7842 
 
 7891 
 
 7941 
 
 7991 
 
 93 
 
 42 
 
 43 
 
 44 
 
 45 
 
 40 
 
 47 
 
 994 
 
 8040 
 
 8090 
 
 8140 
 
 8189 
 
 8239 
 
 8289 
 
 8338 
 
 8388 
 
 8438 
 
 8487 
 
 94 
 
 42 
 
 43 
 
 44 
 
 45 
 
 40 
 
 47 
 
 995 
 
 8537 
 
 8587 
 
 8637 
 
 8686 
 
 8736 
 
 8786 
 
 8836 
 
 8885 
 
 8935 
 
 8985 
 
 95 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 47 
 
 996 
 
 9035 
 
 9084 
 
 9134 
 
 9184 
 
 9234 
 
 9284 
 
 9333 
 
 9383 
 
 9433 
 
 9483 
 
 96 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 997 
 
 9533 
 
 9582 
 
 9632 
 
 9682 
 
 9732 
 
 9782 
 
 9832 
 
 9882 
 
 9931 
 
 9981 
 
 07 
 
 44 
 
 45 
 
 46 
 
 47 
 
 4^ 
 
 49 
 
 998 
 
 30 0031 
 
 0081 
 
 0131 
 
 0181 
 
 0231 
 
 0281 
 
 0331 
 
 0380 
 
 0430 
 
 0480 
 
 98 
 
 44 
 
 45 
 
 46 
 
 ^7 
 
 48 
 
 49 
 
 999 
 
 0530 
 
 0580 
 
 0630 
 
 0680 
 
 0730 
 
 0780 
 
 0830 
 
 0880 
 
 0930 
 
 0980 
 
 99 
 
 45 
 
 40 
 
 47 
 
 48 
 
 49 
 
 49 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 45 
 
 4(1 
 
 47 
 
 48 
 
 49150 
 
TABLE I. 
 
 Log 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 O'QOO 
 
 0-30 1030 
 
 1080 
 
 1130 
 
 1180 
 
 1230 
 
 1280 
 
 1330 
 
 1380 
 
 1430 
 
 1480 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 001 
 
 1530 
 
 1580 
 
 1630 
 
 1680 
 
 1731 
 
 1781 
 
 1831 
 
 1881 
 
 1931 
 
 1981 
 
 01 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 002 
 
 2031 
 
 2081 
 
 2131 
 
 2182 
 
 2232 
 
 2282 
 
 2332 
 
 2382 
 
 2432 
 
 2482 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ] 
 
 003 
 
 2533 
 
 2583 
 
 2633 
 
 2683 
 
 2733 
 
 2784 
 
 2834 
 
 2884 
 
 2934 
 
 2984 
 
 03 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 9 
 
 004 
 
 3035 
 
 3085 
 
 3135 
 
 3185 
 
 3236 
 
 3286 
 
 3336 
 
 3386 
 
 3437 
 
 3487 
 
 04 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 005 
 
 3537 
 
 3587 
 
 3638 
 
 3688 
 
 3738 
 
 3789 
 
 3839 
 
 3889 
 
 3940 
 
 3990 
 
 06 
 
 2 
 
 3 
 
 a 
 
 3 
 
 3 
 
 3 
 
 006 
 
 4040 
 
 4091 
 
 4141 
 
 4191 
 
 4242 
 
 4292 4343 
 
 4393 
 
 4443 
 
 4494 
 
 06 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 j 
 
 007 
 
 4544 
 
 4595 
 
 4645 
 
 4695 
 
 4746 
 
 4796 
 
 4847 
 
 4897 
 
 4948 
 
 4998 
 
 07 
 
 4 
 
 4 
 
 4 
 
 '4 
 
 4 
 
 i 
 
 008 
 
 5048 
 
 5099 
 
 5149 
 
 5200 
 
 5250 
 
 5301 
 
 5351 
 
 5402 
 
 5452 
 
 5503 
 
 08 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 L 
 
 009 
 
 5553 
 
 5604 
 
 5654 
 
 5705 
 
 5755 
 
 5806 
 
 5857 
 
 5907 
 
 5958 
 
 6008 
 
 09 
 
 4 
 
 5 
 
 5 
 
 K 
 
 5 
 
 K 
 
 010 
 
 30 6059 
 
 6109 
 
 6160 
 
 6211 
 
 6261 
 
 6312 
 
 6362 
 
 6413 
 
 6464 
 
 6514 
 
 10 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 on 
 
 6565 
 
 6615 
 
 6666 
 
 6717 
 
 6767 
 
 6818 
 
 6869 
 
 6919 
 
 6970 
 
 7021 
 
 11 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 012 
 
 7071 
 
 7122 
 
 7173 
 
 7224 
 
 7274 
 
 7325 
 
 7376 
 
 7426 7477 
 
 7528 
 
 12 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 
 013 
 
 7579 
 
 7629 
 
 7680 
 
 7731 
 
 7782 
 
 7832 
 
 7883 
 
 79341 7985 
 
 8036 
 
 13 
 
 6 
 
 7 
 
 7 
 
 7 
 
 i 
 
 7 
 
 014 
 
 8086 
 
 8137 
 
 8188 
 
 8239 
 
 8290 
 
 8341 
 
 8391 
 
 8442 
 
 8493 
 
 8544 
 
 14 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 015 
 
 8595 
 
 8646 
 
 8696 
 
 8747 
 
 8798 
 
 8849 
 
 8900 
 
 8951 
 
 9002 
 
 9053 
 
 15 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 016 
 
 9104 
 
 9155 
 
 9206 
 
 9256 
 
 9307 
 
 9358 
 
 9409 
 
 9460 
 
 9511 
 
 9562 
 
 16 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 c 
 
 017 
 
 9613 
 
 9664 
 
 9715 
 
 9766 
 
 9817 
 
 9868 
 
 9919 
 
 9970 
 
 0021 
 
 0072 
 
 17 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 018 
 
 31 0123 
 
 0174 
 
 0225 
 
 0276 
 
 0327 
 
 0378 
 
 0430 
 
 0481 
 
 0532 
 
 0583 
 
 18 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 019 
 
 0634 
 
 0685 
 
 0736 
 
 0787 
 
 0838 
 
 0889 
 
 0941 
 
 0992 
 
 1043 
 
 1094 
 
 19 
 
 10 
 
 10|10|10 
 
 10 
 
 10 
 
 020 
 
 31 1145 
 
 1196 
 
 1247 
 
 1299 
 
 1350 
 
 1401 
 
 1452 
 
 1503 
 
 1555 
 
 1606 
 
 20 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 021 
 
 1657 
 
 1708 
 
 1759 
 
 1811 
 
 1862 
 
 1913 
 
 1964 
 
 2016 
 
 2067 
 
 2118 
 
 21 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 022 
 
 2169 
 
 2221 
 
 2272 
 
 2323 
 
 2374 
 
 2426 
 
 2477 
 
 2528| 2580 
 
 2631 
 
 22 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 023 
 
 2682 
 
 2734 
 
 2785 
 
 2836 
 
 2888 
 
 2939 
 
 2990 
 
 3042 
 
 3093 
 
 3144 
 
 23 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 024 
 
 3196 
 
 3247 
 
 3299 
 
 3350 
 
 3401 
 
 3453 
 
 3504 
 
 3556 
 
 3607 
 
 3658 
 
 24 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 025 
 
 3710 
 
 3761 
 
 3813 
 
 3864 
 
 3916 
 
 3967 
 
 4019 
 
 4070 
 
 4122 
 
 4173 
 
 25 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 026 
 
 4225 
 
 4276 
 
 4328 
 
 4379 
 
 4431 
 
 4482 
 
 4534 
 
 4585 
 
 4637 
 
 4688 
 
 26 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 14 
 
 027 
 
 4740 
 
 4791 
 
 4843 
 
 4894 
 
 4946 
 
 4998 
 
 5049 
 
 5101 
 
 5152 
 
 5204 
 
 27 
 
 14 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 028 
 
 5256 
 
 5307 
 
 5359 
 
 5410 
 
 5462 
 
 5514 
 
 5565 
 
 5617 
 
 5669 
 
 5720 
 
 28 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 15 
 
 029 
 
 5772 
 
 5824 
 
 5875 
 
 5927 
 
 5979 
 
 6030 
 
 6082 
 
 6134 
 
 6186 
 
 6237 
 
 29 
 
 14 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 030 
 
 31 6289 
 
 6341 
 
 6392 
 
 6444 
 
 6496 
 
 6548 
 
 6599 
 
 6651 
 
 6703 
 
 6755 
 
 30 
 
 1515 
 
 16J16 
 
 16 
 
 16 
 
 031 
 
 6807 
 
 6858 
 
 6910 
 
 6962 
 
 7014 
 
 7066 
 
 7117 
 
 7169 
 
 7221 
 
 7273 
 
 31 
 
 1616 
 
 1616 
 
 17 
 
 17 
 
 032 
 
 7325 
 
 7377 
 
 7428 
 
 7480 
 
 7532 
 
 7584 
 
 7636 
 
 7688 
 
 7740 
 
 7791 
 
 32 
 
 1616 
 
 17 
 
 17 
 
 17 
 
 18 
 
 033 
 
 7843 
 
 7895 
 
 7947 
 
 7999 
 
 8051 
 
 8103 
 
 8155 
 
 8207 
 
 8259 
 
 8311 
 
 33 
 
 1617 
 
 17 
 
 17 
 
 18 
 
 18 
 
 034 
 
 8363 
 
 8415 
 
 8467 
 
 8519 
 
 8571 
 
 8622 
 
 8674 
 
 8726 
 
 8778 
 
 8830 
 
 34 
 
 17^7 
 
 18 
 
 18 
 
 18 
 
 19 
 
 035 
 
 8882 
 
 8935 
 
 8987 
 
 9039 
 
 9091 
 
 9143 
 
 9195 
 
 9247 
 
 9299 
 
 9351 
 
 35 
 
 1818 
 
 18 
 
 19 
 
 19 
 
 19 
 
 036 
 
 9403 
 
 9455 
 
 9507 
 
 9559 
 
 9611 
 
 9663 
 
 9715 
 
 9768 
 
 9820 
 
 9872 
 
 36 
 
 1818 
 
 19 
 
 19 
 
 19 
 
 20 
 
 037 
 
 9924 
 
 9976 
 
 0028 
 
 0080 
 
 0132 
 
 0185 
 
 o237 
 
 o289 
 
 0341 
 
 0393 
 
 37 
 
 1819 
 
 19 
 
 20 
 
 20 
 
 20 
 
 038 
 
 32 0445 
 
 0498 
 
 0550 
 
 0602 
 
 0654 
 
 0706 
 
 0759 
 
 0811 
 
 0863 
 
 0915 
 
 38 
 
 1919 
 
 20 
 
 20 
 
 21 
 
 21 
 
 039 
 
 0968 
 
 1020 
 
 1072 
 
 1124 
 
 1177 
 
 1229 
 
 1281 
 
 1333 
 
 1386 
 
 1438 
 
 39 
 
 2020 
 
 20|21 
 
 21 
 
 21 
 
 040 
 
 32 1490 
 
 1543 
 
 1595 
 
 1647 
 
 1700 
 
 1752 
 
 1804 
 
 1857 
 
 1909 
 
 1961 
 
 40 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 
 041 
 
 2014 
 
 2066 
 
 2118 
 
 2171 
 
 2223 
 
 2276 
 
 2328 
 
 2380 
 
 2433 
 
 2485 
 
 41 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 23 
 
 042 
 
 2538 
 
 2590 
 
 2642 
 
 2695 
 
 2747 
 
 2800 
 
 2852 
 
 2905 
 
 2957 
 
 3009 
 
 42 
 
 21 
 
 21 
 
 2222 
 
 23 
 
 23 
 
 043 
 
 3062 
 
 3114 
 
 3167 
 
 3219 
 
 3272 
 
 3324 
 
 3377 
 
 3429 
 
 3482 
 
 3534 
 
 43 
 
 22 
 
 22 
 
 2223 
 
 23 
 
 24 
 
 044 
 
 3587 
 
 3640 
 
 3692 
 
 3745 
 
 3797 
 
 3850 
 
 3902 
 
 3955 
 
 4007 
 
 4060 
 
 44 
 
 22 
 
 2223 
 
 23 
 
 24 
 
 24 
 
 045 
 
 4113 
 
 4165 
 
 4218 
 
 4270 
 
 4323 
 
 4376 
 
 4428 
 
 4481 
 
 4533 
 
 4586 
 
 45 
 
 22 
 
 2323 
 
 24 
 
 24 
 
 25 
 
 046 
 
 4639 
 
 4691 
 
 4744 
 
 4797 
 
 4849 
 
 4902 
 
 4955 
 
 5007 
 
 5060 
 
 5113 
 
 46 
 
 23 
 
 23124 24 
 
 25 
 
 25 
 
 047 
 
 5165 
 
 5218 
 
 5271 
 
 5324 
 
 5376 
 
 5429 
 
 5482 
 
 5535 
 
 5587 
 
 5640 
 
 47 
 
 24 
 
 24 24 25 
 
 25 
 
 26 
 
 048 
 049 
 
 5693 
 6221 
 
 5746 
 6274 
 
 5798 
 6326 
 
 5851 
 6379 
 
 5904 
 6432 
 
 5957 
 6485 
 
 6009 
 6538 
 
 6062 
 6591 
 
 6115 
 6643 
 
 6168 
 6696 
 
 48 
 49 
 
 24 
 24 
 
 2425 
 2525 
 
 25 
 
 26 
 
 26 
 26 
 
 26 
 27 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 50 
 
 5152 
 
 5354 
 
 55 
 
TABLE I. 
 
 Log 
 
 Log(l+;r) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 0-050 
 
 0-32 674S 
 
 6802 
 
 6855 
 
 690S 
 
 6961 
 
 7014 
 
 7067 
 
 7119 
 
 7172 
 
 7225 
 
 50 
 
 25 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 051 
 
 7278 
 
 7331 
 
 7384 
 
 7437 
 
 7490 
 
 7543 
 
 7596 
 
 7649 
 
 7702 
 
 7755 
 
 51 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 052 
 
 7808 
 
 7861 
 
 7914 
 
 796? 
 
 8020 
 
 8073 
 
 8126 
 
 8179 
 
 8232 
 
 828o 
 
 52 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 29 
 
 053 
 
 8338 
 
 8391 
 
 8444 
 
 8497 
 
 8550 
 
 8603 
 
 8656 
 
 8709 
 
 8763 
 
 8816 
 
 53 
 
 26 
 
 27 
 
 28 
 
 2829 
 
 29 
 
 054 
 
 8869 
 
 8922 
 
 8975 
 
 902S 
 
 9081 
 
 9134 
 
 9187 
 
 9241 
 
 9294 
 
 9347 
 
 54 
 
 27 
 
 28 
 
 28 29 29 
 
 30 
 
 055 
 
 9400 
 
 9453 
 
 9506 
 
 9560 
 
 9613 
 
 9666 
 
 9719 
 
 9772 
 
 9826 
 
 9879 
 
 55 
 
 28 
 
 28 
 
 29(29 30 
 
 30 
 
 056 
 
 9932 
 
 9985 
 
 0038 
 
 0092 
 
 0145 
 
 0198 
 
 0251 
 
 0305 
 
 0358 
 
 0411 
 
 56 
 
 28 29 29 30 30 
 
 31 
 
 057 
 
 33 0464 
 
 0518 
 
 0571 
 
 0624 
 
 0678 
 
 0731 
 
 0784 
 
 0838 
 
 0891 
 
 0944 
 
 57 
 
 2812930 30:31 
 
 31 
 
 058 
 
 0998 1051 
 
 1104 
 
 1158 
 
 1211 
 
 1264 
 
 1318 
 
 1371 
 
 1424 
 
 1478 
 
 58 
 
 29303013131 
 
 32 
 
 059 
 
 1531 1585 
 
 1638 
 
 1691 
 
 1745 
 
 1798 
 
 1852 
 
 1905 
 
 1958 
 
 2012 
 
 59 
 
 30 
 
 30 
 
 31 
 
 31 
 
 32 
 
 32 
 
 060 
 
 332065 2119 
 
 2172 
 
 2226 
 
 2279 
 
 2333 
 
 2386 
 
 2440 
 
 2493 
 
 2547 
 
 60 
 
 30 
 
 31 
 
 3132 
 
 32 
 
 33 
 
 061 
 
 2600 2654 
 
 2707 
 
 2761 
 
 2814 
 
 2868 
 
 2921 
 
 2975 
 
 3028 
 
 3082 
 
 61 
 
 30 
 
 31 
 
 3232 
 
 33 
 
 34 
 
 062 
 
 313513189 
 
 3243 
 
 3296 3350 
 
 3403 
 
 3457 
 
 3511 
 
 3564 
 
 3618 
 
 62 
 
 31 
 
 32*323333 
 
 34 
 
 063 
 
 3671 
 
 3725 
 
 3779 
 
 3832 
 
 3886 
 
 3940 
 
 3993 
 
 4047 
 
 4101 
 
 4154 
 
 63 
 
 32 
 
 32 
 
 33 33 34 
 
 35 
 
 064- 
 
 4208 
 
 4262 
 
 4315 
 
 4369 
 
 4423 
 
 4476 
 
 4530 
 
 4584 
 
 4637 
 
 4691 
 
 64 
 
 32 
 
 33 
 
 33!34 35 
 
 35 
 
 065 
 
 4745 
 
 4799 
 
 4852 
 
 4906 
 
 4960 
 
 5014 
 
 5067 
 
 5121 
 
 5175 
 
 5229 
 
 65 
 
 32 
 
 33 
 
 3434 
 
 35 
 
 36 
 
 066 
 
 5283 
 
 5336 
 
 5390 
 
 5444 
 
 5498 
 
 5552 
 
 5605 
 
 5659 
 
 5713 
 
 5767 
 
 6633 
 
 34 
 
 3435 
 
 36 
 
 36 
 
 067 
 
 5821 
 
 5875 
 
 5928 
 
 5982 
 
 6036 
 
 6090 
 
 6144 
 
 6198 
 
 6252 
 
 6306 
 
 67 
 
 34 
 
 34 
 
 3536 
 
 36 
 
 37 
 
 068 
 
 6360 
 
 6413 
 
 6467 
 
 6521 
 
 6575 
 
 6629 
 
 6683 
 
 6737 
 
 6791 
 
 6845 
 
 68 
 
 34 
 
 35 
 
 3536 
 
 37 
 
 37 
 
 069 
 
 6899 
 
 6953 
 
 7007 
 
 7061 
 
 7115 
 
 7169 
 
 7223 
 
 7277 
 
 7331 
 
 7385 
 
 69 
 
 34 
 
 35 
 
 36 
 
 37 
 
 37 
 
 38 
 
 070 
 
 33 7439 
 
 7493 
 
 7547 
 
 7601 
 
 7655 
 
 7709 
 
 7763 
 
 7817 
 
 7871 
 
 7925 
 
 70 
 
 35 
 
 36 
 
 36 
 
 37 
 
 38 
 
 38 
 
 071 
 
 7979 
 
 8033 
 
 8087 
 
 8142 
 
 8196 
 
 8250 
 
 8304 
 
 8358 
 
 8412 
 
 8466 
 
 71 
 
 36 
 
 36 
 
 37 
 
 3838 
 
 39 
 
 072 
 
 8520 8575 
 
 8629 
 
 8683 
 
 8737 
 
 8791 
 
 8845 
 
 8899 
 
 8954 
 
 9008 
 
 72 
 
 36 
 
 37 
 
 37 
 
 38;39 40 
 
 073 
 
 9062 
 
 9116 
 
 9170 
 
 9225 
 
 9279 
 
 9333 
 
 9387 
 
 9441 
 
 9496 
 
 9550 
 
 73 
 
 36 
 
 37 
 
 38 
 
 39 
 
 39 
 
 40 
 
 074 
 
 9604 
 
 9658 
 
 9713 
 
 9767 
 
 9821 
 
 9876 
 
 9930 
 
 9984 
 
 0038 
 
 0093 
 
 74 
 
 37 
 
 38 
 
 38 
 
 39 
 
 40 
 
 41 
 
 075 
 
 34 0147 
 
 0201 
 
 0256 
 
 0310 
 
 0364 
 
 0419 
 
 0473 
 
 0527 
 
 0582 
 
 0636 
 
 75 
 
 38 
 
 38 
 
 39 
 
 40 
 
 41 
 
 41 
 
 076 
 
 0690 
 
 0745 
 
 0799 
 
 0853 
 
 0908 
 
 0962 
 
 1017 
 
 1071 
 
 1125 
 
 1180 
 
 76 
 
 38 
 
 39 
 
 4040 
 
 41 
 
 42 
 
 077 
 
 1234 
 
 1289 
 
 1343 
 
 1398 
 
 1452 
 
 1506 
 
 1561 
 
 1615 
 
 1670 
 
 1724 
 
 77 
 
 33 
 
 39 
 
 4041 
 
 42 
 
 42 
 
 078 
 
 1779 
 
 1833 
 
 1888 
 
 1942 
 
 1997 
 
 2051 
 
 2106 
 
 2160 
 
 2215 
 
 2269 
 
 78 
 
 3940 
 
 41 
 
 41 
 
 42 
 
 43 
 
 079 
 
 2324 
 
 2378 
 
 2433 
 
 2487 
 
 2542 
 
 2597 
 
 2651 
 
 2706 
 
 2760 
 
 2815 
 
 79 
 
 40 
 
 40 
 
 41 
 
 42 
 
 43 
 
 43 
 
 080 
 
 34 2869 
 
 2924 
 
 2979 
 
 3033 
 
 3088 
 
 3142 
 
 3197 
 
 3252 
 
 3306 
 
 3361 
 
 80 
 
 40 
 
 41 
 
 12 
 
 4243 
 
 44 
 
 081 
 
 3416 
 
 3470 
 
 3525 
 
 3580 
 
 3634 
 
 3689 
 
 3744 
 
 3798 
 
 3853 
 
 3908 
 
 81 
 
 40 
 
 41 
 
 12 
 
 1311 
 
 45 
 
 082 
 
 3962 
 
 4017 
 
 4072 
 
 4127 
 
 4181 
 
 4236 
 
 4291 
 
 4346 
 
 4400 
 
 4455 
 
 82 
 
 41 
 
 42!43 43 44 
 
 45 
 
 083 
 
 4510 
 
 4565 
 
 4619 
 
 4674 
 
 4729 
 
 4784 
 
 4838 
 
 4893 
 
 4948 
 
 5003 
 
 83 
 
 42 
 
 42 43:44 45 
 
 46 
 
 084 
 
 5058 
 
 5113 
 
 5167 
 
 5222 
 
 5277 
 
 5332 
 
 5387 
 
 5442 
 
 5496 
 
 5551 
 
 84 
 
 42 
 
 43 
 
 11 45 15 
 
 46 
 
 085 
 
 5606 
 
 5661 
 
 5716 
 
 5771 
 
 5826 
 
 5881 
 
 5936 
 
 5990 
 
 6045 
 
 6100 
 
 85 
 
 42 
 
 434445 
 
 46 
 
 47 
 
 086 
 
 6155 
 
 6210 
 
 6265 
 
 6320 
 
 6375 
 
 6430 
 
 6485 
 
 6540 
 
 6595 
 
 6650 
 
 86 
 
 43 
 
 4445464647 
 
 087 
 
 6705 
 
 6760 
 
 6815 6870 6925 
 
 6980 
 
 7035 
 
 7090 
 
 7145 
 
 7200 
 
 87 
 
 44 
 
 44 45 46 47 18 
 
 088 
 
 7255 
 
 7310 
 
 7365 
 
 7420 7475 
 
 7530 
 
 7585 
 
 7641 
 
 7696 7751 
 
 8844 
 
 45 46 47:48 
 
 48 
 
 089 
 
 7806 
 
 7861 
 
 7916 
 
 7971 
 
 8026 
 
 8081 
 
 8137 
 
 8192 
 
 8247 
 
 8302 
 
 89 
 
 44 
 
 45 
 
 16 17 18 
 
 49 
 
 090 
 
 34 8357 
 
 8412 
 
 8468 
 
 8523 
 
 8578 
 
 8633 
 
 8688 
 
 8743 
 
 8799 
 
 8854 
 
 90 
 
 45 
 
 46 
 
 47 
 
 48 
 
 19 
 
 50 
 
 091 
 092 
 
 8909 
 9462 
 
 8964 
 9517 
 
 9020 
 9572 
 
 9075 
 9627 
 
 9130 
 9683 
 
 9185 
 9738 
 
 9241 
 9793 
 
 9296 
 9849 
 
 9351 
 9904 
 
 9406 
 9959 
 
 91 
 92 
 
 46 
 46 
 
 46 
 47 
 
 47 48 49J50 
 48495051 
 
 093 
 
 35 0015 0070 
 
 0125 
 
 0181 
 
 0236 
 
 0291 
 
 0347 
 
 0402 
 
 0457 
 
 0513 
 
 93 
 
 46 
 
 47 
 
 18495051 
 
 094 
 
 0568 0624 
 
 0679 0734 
 
 0790 
 
 0845 
 
 0901 
 
 0956 
 
 1012 
 
 1067 
 
 94 
 
 47 
 
 48 
 
 495051 
 
 52 
 
 095 
 
 1122 
 
 1178 
 
 1233 1289 
 
 1344 
 
 1400 
 
 1455 
 
 1511 
 
 1566 
 
 1622 
 
 95 
 
 48484950,31 
 
 52 
 
 096 
 
 1677 
 
 1733 
 
 1788 
 
 1844 
 
 1899 
 
 1955 
 
 2010 
 
 2066 
 
 2121 
 
 2177 
 
 96 
 
 484950515253 
 
 097 
 
 2233 
 
 2288 
 
 2344 
 
 2399 
 
 2455 
 
 2510 
 
 2566 
 
 2622 
 
 2677 
 
 2733 
 
 97 
 
 48 
 
 195051 52 
 
 53 
 
 098 
 
 2788J 2844 
 
 2900 
 
 2955 
 
 3011 
 
 3067 
 
 3122 
 
 3178 
 
 3234 
 
 3289 
 
 98 
 
 19 
 
 50 
 
 jl 5253 
 
 54 
 
 099 
 
 3345 
 
 3401 
 
 3456 
 
 3512 
 
 3568 
 
 3623 
 
 3679 
 
 3735 
 
 3790 
 
 3846 
 
 99 
 
 50 
 
 50. 
 
 51-5253. 
 
 >4 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 50 
 
 51, 
 
 5253, 
 
 54, 
 
 55 
 
TABLE I. 
 
 Log 
 
 Lc 
 
 gO+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 61 
 
 o-ioo 
 
 0-35 3902 
 
 3958 
 
 4013 
 
 4069 
 
 4125 
 
 4181 
 
 4236 
 
 4292 
 
 4348 
 
 4404 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 101 
 
 4459 
 
 4515 
 
 4571 
 
 4627 
 
 4683 
 
 4738 
 
 4794 
 
 4850 
 
 4906 
 
 4962 
 
 01 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 j 
 
 102 
 
 5018 
 
 5073 
 
 5129 
 
 5185 
 
 5241 
 
 5297 
 
 5353 
 
 5409 
 
 5465 
 
 5520 
 
 02 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 103 
 
 5576 
 
 5632 
 
 5688 
 
 5744 
 
 5800 
 
 5856 
 
 5912 
 
 5968 
 
 6024 
 
 6080 
 
 03 
 
 2 
 
 
 
 2 
 
 2 
 
 
 2 
 
 104 
 
 6136 
 
 6192 
 
 6248 
 
 6304 
 
 6360 
 
 6416 
 
 6472 
 
 6528 
 
 6584 
 
 6640 
 
 04 
 
 2 
 
 9 
 
 2 
 
 2 
 
 
 9 
 
 105 
 
 6696 
 
 6752 
 
 6808 
 
 6864 
 
 6920 
 
 6976 
 
 7032 
 
 7088 
 
 7144 
 
 7200 
 
 05 
 
 r 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 106 
 
 7256 
 
 7312 
 
 7368 
 
 7424 
 
 7480 
 
 7536 
 
 7593 
 
 7649 
 
 7705 
 
 7761 
 
 06 
 
 | 
 
 f 
 
 3 
 
 4 
 
 / 
 
 L 
 
 107 
 
 7817 
 
 7873 
 
 7929 
 
 7985 
 
 8042 
 
 8098 
 
 8154 
 
 8210 
 
 8266 
 
 8322 
 
 07 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 L 
 
 108 
 
 8379 
 
 8435 
 
 8491 
 
 8547 
 
 8603 
 
 8660 
 
 8716 
 
 8772 
 
 8828 
 
 8884 
 
 08 
 
 4 
 
 t 
 
 5 
 
 5 
 
 c 
 
 5 
 
 109 
 
 8941 
 
 8997 
 
 9053 
 
 9109 
 
 9166 
 
 9222 
 
 9278 
 
 9335 
 
 9391 
 
 9447 
 
 09 
 
 5 
 
 5 
 
 5 
 
 5 
 
 
 c 
 
 110 
 
 35 9503 
 
 9560 
 
 9616 
 
 9672 
 
 9729 
 
 9785 
 
 9841 
 
 9898 
 
 9954 
 
 0010 
 
 10 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 111 
 
 36 0067 
 
 0123 
 
 0179 
 
 0236 
 
 0292 
 
 0349 
 
 0405 
 
 0461 
 
 0518 
 
 0574 
 
 11 
 
 6 
 
 6 
 
 6 
 
 6 
 
 r 
 
 r 
 
 112 
 
 0630 
 
 0687 
 
 0743 
 
 0800 
 
 0856 
 
 0913 
 
 0969 
 
 1026 
 
 1082 
 
 1138 
 
 12 
 
 r t 
 
 r 
 
 7 
 
 7 
 
 7 
 
 f 
 
 113 
 
 1195 
 
 1251 
 
 1308 
 
 1364 
 
 1421 
 
 1477 
 
 1534 
 
 1590 
 
 1647 
 
 1703 
 
 13 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 114 
 
 1760 
 
 1816 
 
 1873 
 
 1929 
 
 1986 
 
 2043 
 
 2099 
 
 2156 
 
 2212 
 
 2269 
 
 14 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 115 
 
 2325 
 
 2382 
 
 2439 
 
 2495 
 
 2552 
 
 2608 
 
 2665 
 
 2722 
 
 2778 
 
 2835 
 
 15 
 
 8 
 
 9 
 
 9 
 
 9 
 
 < 
 
 j 
 
 116 
 
 2891 
 
 2948 
 
 3005 
 
 3061 
 
 3118 
 
 3175 
 
 3231 
 
 3288 
 
 3345 
 
 3401 
 
 16 
 
 c 
 
 c 
 
 9 
 
 9 
 
 10 
 
 10 
 
 117 
 
 3458 
 
 3515 
 
 3572 
 
 3628 
 
 3685 
 
 3742 
 
 3798 
 
 3855 
 
 3912 
 
 3969 
 
 17 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 118 
 
 4025 
 
 4082 
 
 4139 
 
 4196 
 
 4252 
 
 4309 
 
 4366 
 
 4423 
 
 4480 
 
 4536 
 
 18 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 
 119 
 
 4593 
 
 4650 
 
 4707 
 
 4764 
 
 4820 
 
 4877 
 
 4934 
 
 4991 
 
 5048 
 
 5105 
 
 19 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 9 
 
 120 
 
 36 5162 
 
 5218 
 
 5275 
 
 5332 
 
 5389 
 
 5446 
 
 5503 
 
 5560 
 
 5617 
 
 5674 
 
 20 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 9 
 
 121 
 
 5730 
 
 5787 
 
 5844 
 
 5901 
 
 5958 
 
 6015 
 
 6072 
 
 6129 
 
 6186 
 
 6243 
 
 21 
 
 1 
 
 12 
 
 12 
 
 12 
 
 13 
 
 J 
 
 122 
 
 6300 
 
 6357 
 
 6414 
 
 6471 
 
 6528 
 
 6585 
 
 6642 
 
 6699 
 
 6756 
 
 6813 
 
 22 
 
 1 
 
 13 
 
 13 
 
 13 
 
 13 
 
 J 
 
 123 
 
 6870 
 
 6927 
 
 6984 
 
 7041 
 
 7098 
 
 7155 
 
 7212 
 
 7269 
 
 7326 
 
 7384 
 
 23 
 
 1 
 
 13 
 
 13 
 
 14 
 
 14 
 
 L 
 
 124 
 
 7441 
 
 7498 
 
 7555 
 
 7612 
 
 7669 
 
 7726 
 
 7783 
 
 7840 
 
 7898 
 
 7955 
 
 24 
 
 1 
 
 14 
 
 14 
 
 14 
 
 14 
 
 5 
 
 125 
 
 8012 
 
 8069 
 
 8126 
 
 8183 
 
 8240 
 
 8298 
 
 8355 
 
 8412 
 
 8469 
 
 8526 
 
 25 
 
 I 
 
 14 
 
 15 
 
 15 
 
 15 
 
 15 
 
 126 
 
 8584 
 
 8641 
 
 8698 
 
 8755 
 
 8812 
 
 8870 
 
 8927 
 
 8984 
 
 9041 
 
 9099 
 
 26 
 
 1 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 127 
 
 9156 
 
 9213 
 
 9270 
 
 9328 
 
 9385 
 
 9442 
 
 9500 
 
 9557 
 
 9614 
 
 9671 
 
 27 
 
 1 
 
 15 
 
 16 
 
 16 
 
 16 
 
 16 
 
 128 
 
 9729 
 
 9786 
 
 9843 
 
 9901 
 
 9958 
 
 0015 
 
 0073 
 
 0130 
 
 0187 
 
 0245 
 
 28 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 17 
 
 129 
 
 37 0302 
 
 0360 
 
 0417 
 
 0474 
 
 0532 
 
 0589 
 
 0646 
 
 0704 
 
 0761 
 
 0819 
 
 29 
 
 16 
 
 17 
 
 17 
 
 17 
 
 17 
 
 18 
 
 130 
 
 37 0876 
 
 0934 
 
 0991 
 
 1048 
 
 1106 
 
 1163 
 
 1221 
 
 1278 
 
 1336 
 
 1393 
 
 30 
 
 17 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 131 
 
 1451 
 
 1508 
 
 1566 
 
 1623 
 
 1681 
 
 1738 
 
 1796 
 
 1853 
 
 1911 
 
 1968 
 
 31 
 
 17 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 132 
 
 2026 
 
 2083 
 
 2141 
 
 2198 
 
 2256 
 
 2314 
 
 2371 
 
 2429 
 
 2486 
 
 2544 
 
 32 
 
 18 
 
 18 
 
 19 
 
 19 
 
 19 
 
 20 
 
 133 
 
 2602 
 
 2659 
 
 2717 
 
 2774 
 
 2832 
 
 2890 
 
 2947 
 
 3005 
 
 3062 
 
 3120 
 
 33 
 
 18 
 
 19 
 
 19 
 
 19 
 
 20 
 
 20 
 
 134 
 
 3178 
 
 3235 
 
 3293 
 
 3351 
 
 3408 
 
 3466 
 
 3524 
 
 3581 
 
 3639 
 
 3697 
 
 34 
 
 19 
 
 19 
 
 20 
 
 20 
 
 20 
 
 21 
 
 135 
 
 3755 
 
 3812 
 
 3870 
 
 3928 
 
 3985 
 
 4043 
 
 4101 
 
 4159 
 
 4216 
 
 4274 
 
 35 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 21 
 
 136 
 
 4332 
 
 4390 
 
 4448 
 
 4505 
 
 4563 
 
 4621 
 
 4679 
 
 4736 
 
 4794 
 
 4852 
 
 36 
 
 20 
 
 21 
 
 21 
 
 21 
 
 22 
 
 22 
 
 137 
 
 4910 
 
 4968 
 
 5026 
 
 5083 
 
 5141 
 
 5199 
 
 5257 
 
 5315 
 
 5373 
 
 5431 
 
 37|21 
 
 21 
 
 21 
 
 22 
 
 22 
 
 23 
 
 138 
 
 5488 
 
 5546 
 
 5604 
 
 5662 
 
 5720 
 
 5778 
 
 5836 
 
 5894 
 
 5952 
 
 6010 
 
 3821 
 
 22 
 
 2222 
 
 23 
 
 23 
 
 139 
 
 6067 
 
 6125 
 
 6183 
 
 6241 
 
 6299 
 
 6357 
 
 6415 
 
 6473 
 
 6531 
 
 6589 
 
 39 
 
 22 
 
 22 
 
 23 
 
 23 
 
 23 
 
 24 
 
 140 
 
 37 6647 
 
 6705 
 
 6763 
 
 6821 
 
 6879 
 
 6937 
 
 6995 
 
 7053 
 
 7111 
 
 7169 
 
 40 
 
 22 
 
 23 
 
 23 
 
 24 
 
 24 
 
 24 
 
 141 
 
 7227 
 
 7285 
 
 7343 
 
 7401 
 
 7459 
 
 7518 
 
 7576 
 
 7634 
 
 7692 
 
 7750 
 
 41 
 
 23 
 
 23 
 
 24 
 
 24 
 
 25 
 
 25 
 
 142 
 
 7808 
 
 7866 
 
 7924 
 
 7982 
 
 8040 
 
 8099 
 
 8157 
 
 8215 
 
 8273 
 
 8331 
 
 42 
 
 24 
 
 24 
 
 24 
 
 25 
 
 25 
 
 26 
 
 143 
 
 8389 
 
 8447 
 
 8506 
 
 8564 
 
 8622 
 
 8680 
 
 8738 
 
 8797 
 
 8855 
 
 8913 
 
 43 
 
 24 
 
 25 
 
 25 
 
 25 
 
 26 
 
 26 
 
 144 
 
 8971 
 
 9029 
 
 9088 
 
 9146 
 
 9204 
 
 9262 
 
 9321 
 
 9379 
 
 9437 
 
 9495 
 
 44 
 
 25 
 
 25 
 
 26 
 
 26 
 
 26 
 
 27 
 
 145 
 
 9554 
 
 9612 
 
 9670 
 
 9728 
 
 9787 
 
 9845 
 
 9903 
 
 9962 
 
 0020 
 
 0078 
 
 45 
 
 25 
 
 26 
 
 26 
 
 27 
 
 27 
 
 27 
 
 146 
 
 380137 
 
 0195 
 
 0253 
 
 0312 
 
 0370 
 
 0428 
 
 0487 
 
 0545 
 
 0603 
 
 0662 
 
 46 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 147 
 
 0720 
 
 0778 
 
 0837 
 
 0895 
 
 0954 
 
 1012 
 
 1070 
 
 1129 
 
 1187 
 
 1246 
 
 47 
 
 26 
 
 27 
 
 27 
 
 2828 
 
 29 
 
 148 
 
 1304 
 
 1363 
 
 1421 
 
 1480 
 
 1538 
 
 1596 
 
 1655 
 
 1713 
 
 1772 
 
 1830 
 
 48 
 
 27 
 
 27 
 
 28 
 
 2829 
 
 29 
 
 149 
 
 1889 
 
 1947 
 
 2006 
 
 2064 
 
 2123 
 
 2181 
 
 2240 
 
 2298 
 
 2357 
 
 2416 
 
 49 
 
 27 
 
 28 
 
 28 
 
 29 
 
 29 
 
 30 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 50 
 
 57 
 
 58 
 
 59 
 
 60 
 
 61 
 
TABLE I. 
 
 Log 
 
 Log (1+ar) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 50 
 
 56 57|58 
 28 29 29 
 
 59J6061 
 303031 
 
 0-150 
 
 0-38 2474 
 
 2533 
 
 2591 
 
 2650 
 
 2708 
 
 2767 
 
 2825 
 
 2884 
 
 2943 
 
 3001 
 
 151 
 
 3060 
 
 3118 
 
 3177 
 
 3236 
 
 3294 
 
 3353 
 
 3412 
 
 3470 
 
 3529 
 
 3588 
 
 51 
 
 292930!303li31 
 
 152 
 
 3646 
 
 3705 
 
 3764 
 
 3822 
 
 3881 
 
 3940 
 
 3998 
 
 4057 
 
 4116 
 
 4174 
 
 52293030!313l!32 
 
 153 
 
 4233 
 
 4292 
 
 4351 
 
 4409 
 
 4468 
 
 4527 
 
 4585 
 
 4644 
 
 4703 
 
 4762 
 
 5330;30313132 
 
 32 
 
 154 
 
 4821 
 
 4879 
 
 4938 
 
 4997 
 
 5056 
 
 5114 
 
 5173 
 
 5232 
 
 5291 
 
 5350 
 
 54 
 
 30 
 
 31 
 
 31 
 
 32,32 
 
 33 
 
 155 
 
 5409 
 
 5467 
 
 5526 
 
 5585 
 
 5644 
 
 5703 
 
 5762 
 
 5820 
 
 5879 
 
 5938 
 
 55 
 
 31 
 
 31 
 
 32 
 
 3233 
 
 34 
 
 156 
 
 5997 
 
 6056 
 
 6115 
 
 6174 
 
 6233 
 
 6292 
 
 6351 
 
 6409 
 
 6468 
 
 6527 
 
 5631 
 
 32132 
 
 3334 
 
 34 
 
 157 
 158 
 
 6586 
 7176 
 
 6645 
 7235 
 
 6704 
 7294 
 
 6763 
 7353 
 
 6822 
 7412 
 
 6881 
 7471 
 
 6940 
 7530 
 
 6999 
 
 7589 
 
 7058 
 7648 
 
 7117 
 7707 
 
 57 32 32: 33 34 34 
 583233343435 
 
 35 
 35 
 
 159 
 
 7766 
 
 7825 
 
 7884 
 
 7943 
 
 8002 
 
 8062 
 
 8121 
 
 8180 
 
 8239 
 
 8298 
 
 593334343535 
 
 36 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 160 
 161 
 
 38 8357 
 8948 
 
 8416 
 9007 
 
 8475 
 9067 
 
 8534 
 9126 
 
 8593 
 9185 
 
 8653 
 9244 
 
 8712 
 9303 
 
 8771 
 9363 
 
 8830 
 9422 
 
 8889 
 9481 
 
 603434353536 
 613435353637 
 
 37 
 37 
 
 162 
 
 9540 
 
 9599 
 
 9659 
 
 9718 
 
 9777 
 
 9836 
 
 9896 
 
 9955 
 
 0014 
 
 0073 
 
 6235'35 I 3637 
 
 37 
 
 38 
 
 163 
 
 39 0133 
 
 0192 
 
 0251 
 
 0311 
 
 0370 
 
 0429 
 
 0488 
 
 0548 
 
 0607 
 
 0666 
 
 6335363737 
 
 38 
 
 38 
 
 164 
 
 0726 
 
 0785 
 
 0844 
 
 0904 
 
 0963 
 
 1022 
 
 1082 
 
 1141 
 
 1201 
 
 1260 
 
 64363637 
 
 38 
 
 38 
 
 39 
 
 165 
 
 1319 
 
 1379 
 
 1438 
 
 1497 
 
 1557 
 
 1616 
 
 1676 
 
 1735 
 
 1795 
 
 1854 
 
 65 36 37 
 
 38 
 
 38 
 
 39 
 
 40 
 
 166 
 
 1913 
 
 1973 
 
 2032 
 
 2092 
 
 2151 
 
 2211 
 
 2270 
 
 2330 
 
 2389 
 
 2449 
 
 66^713838 
 
 39 
 
 40 
 
 40 
 
 167 
 
 2508 
 
 2568 
 
 2627 
 
 2687 
 
 2746 
 
 2806 
 
 2865 
 
 2925 
 
 2984 
 
 3044 
 
 67383839 
 
 40 
 
 40 
 
 41 
 
 168 
 
 3103 
 
 3163 
 
 3222 
 
 3282 
 
 3342 
 
 3401 
 
 3461 
 
 3520 
 
 3580 
 
 3640 
 
 68 38 39 39 
 
 40 
 
 41 
 
 41 
 
 169 
 
 3699 
 
 3759 
 
 3818 
 
 3878 
 
 3938 
 
 3997 
 
 4057 
 
 4117 
 
 4176 
 
 4236 
 
 69 
 
 39 
 
 3940 
 
 41 
 
 41 
 
 42 
 
 170 
 
 39 4296 
 
 4355 
 
 4415 
 
 4475 
 
 4534 
 
 4594 
 
 4654 
 
 4713 
 
 4773 
 
 4833 
 
 70 
 
 39 
 
 40 
 
 41 
 
 41 
 
 42 
 
 43 
 
 171 
 
 4892 
 
 4952 
 
 5012 
 
 5072 
 
 5131 
 
 5191 
 
 5251 
 
 5311 
 
 5370 
 
 5430 
 
 71 
 
 40 
 
 40 
 
 41 
 
 42 
 
 43 
 
 43 
 
 172 
 
 5490 
 
 5550 
 
 5609 
 
 5669 
 
 5729 
 
 5789 
 
 5849 
 
 5908 
 
 5968 
 
 6028 
 
 72 
 
 40 
 
 41 
 
 42 
 
 42 
 
 4344 
 
 173 
 
 6088 
 
 6148 
 
 6208 
 
 6267 
 
 6327 
 
 6387 
 
 6447 
 
 6507 
 
 6567 
 
 6627 
 
 7341 
 
 4242 
 
 43 
 
 44 
 
 45 
 
 174- 
 
 6686 
 
 6746 
 
 6806 
 
 6866 
 
 6926 
 
 6986 
 
 7046 
 
 7106 
 
 7166 
 
 7226 
 
 7441 
 
 4243 
 
 44 
 
 44 
 
 45 
 
 175 
 
 7286 
 
 7346 
 
 7405 
 
 7465 7525 
 
 7585 
 
 7645 
 
 7705 
 
 7765 
 
 7825 
 
 7542 
 
 4343 
 
 44 
 
 45 
 
 46 
 
 176 
 
 7885 
 
 7945 
 
 8005 
 
 8065 8125 
 
 8185 
 
 8245 
 
 8305 
 
 8365 
 
 8425 
 
 7643 
 
 4344 
 
 45 
 
 46 
 
 46 
 
 177 
 
 8485 
 
 8546 
 
 8606 
 
 8666 8726 
 
 8786 
 
 8846 
 
 8906 
 
 8966 
 
 9026 
 
 77 
 
 43 
 
 4445 
 
 45 
 
 46 
 
 47 
 
 178 
 
 9086 
 
 9146 
 
 9206 
 
 9267 9327 
 
 9387 
 
 9447 
 
 9507 
 
 9567 
 
 9627 
 
 78 
 
 44 
 
 4445 
 
 46 
 
 47 
 
 48 
 
 179 
 
 9688 
 
 9748 
 
 9808 
 
 9868 
 
 9928 
 
 9988 
 
 0049 
 
 0109 
 
 0169 
 
 0229 
 
 79 
 
 44 
 
 45 
 
 46 
 
 47 
 
 47 
 
 48 
 
 180 
 
 40 0289 
 
 0350 
 
 0410 
 
 0470 
 
 0530 
 
 0591 
 
 0651 
 
 0711 
 
 0771 
 
 0832 
 
 80 
 
 45 
 
 464647 
 
 48 
 
 49 
 
 181 
 
 0892 
 
 0952 
 
 1012 
 
 1073 
 
 1133 
 
 1193 
 
 1254 
 
 1314 
 
 1374 
 
 1435 
 
 81 
 
 45 
 
 464748 
 
 49 
 
 49 
 
 182 
 
 1495 
 
 1555 
 
 1616 
 
 1676 
 
 1736 
 
 1797 
 
 1857 
 
 1917 
 
 1978 
 
 2038 
 
 824647 
 
 4848 
 
 49 
 
 50 
 
 183 
 
 2098 
 
 2159 
 
 2219 
 
 2280 
 
 2340 
 
 2400 2461 
 
 2521 
 
 2582 
 
 2642 
 
 83 46 47 
 
 4849 
 
 50 
 
 51 
 
 184 
 
 2703 
 
 2763 
 
 2823 
 
 2884 
 
 2944 
 
 3005 
 
 3065 
 
 3126 
 
 3186 
 
 3247 
 
 844748 
 
 4950 
 
 50 
 
 51 
 
 185 
 
 3307 
 
 3368 
 
 3428 
 
 3489 
 
 3549 
 
 3610 
 
 3670 
 
 3731 
 
 3791 
 
 3852 
 
 8548 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 186 
 
 3912 
 
 3973 
 
 4033 
 
 4094 
 
 4155 
 
 4215 
 
 4276 
 
 4336 
 
 4397 
 
 4457 
 
 86 48 49 
 
 5051 
 
 52 
 
 52 
 
 187 
 
 4518 
 
 4579 
 
 4639 
 
 4700 
 
 4761 
 
 4821 
 
 4882 
 
 4942 
 
 5003 
 
 5064 
 
 87 49 50 
 
 50|51 
 
 5253 
 
 188 
 189 
 
 5124 
 5731 
 
 5185 
 5792 
 
 5246 
 5853 
 
 5306 
 5913 
 
 5367 
 5974 
 
 5428 
 6035 
 
 5488 
 6096 
 
 5549 
 6156 
 
 5610 
 6217 
 
 5670 
 6278 
 
 88 
 89 
 
 4950 
 5051 
 
 5152 
 5253 
 
 5354 
 5354 
 
 190 
 191 
 192 
 
 40 6339 
 6947 
 7555 
 
 6399 
 7007 
 7616 
 
 6460 
 7068 
 7677 
 
 6521 
 7129 
 
 7738 
 
 6582 
 7190 
 7799 
 
 6642 
 7251 
 7859 
 
 6703 
 7312 
 7920 
 
 6764 
 
 7372 
 7981 
 
 6825 
 7433 
 8042 
 
 6886 
 7494 
 8103 
 
 905051 
 
 915152 
 92 52 52 
 
 52 
 53 
 53 
 
 53 
 54 
 54 
 
 5455 
 5556 
 5556 
 
 193 
 
 8164 
 
 8225 
 
 8286 
 
 8347 
 
 840S 
 
 8469 
 
 8530 
 
 8591 
 
 8652 
 
 8713 
 
 935253 
 
 5455 
 
 56,57 
 
 194 
 
 8774 
 
 8835 
 
 8896 
 
 8957 
 
 9018 
 
 9079 
 
 9140 
 
 9201 
 
 9262 
 
 9323 
 
 945354 
 
 5555 
 
 56!57 
 
 195 
 
 9384 
 
 9445 
 
 9506 
 
 9567 
 
 9628 
 
 9689 
 
 9750 
 
 9811 
 
 9872 
 
 9933 
 
 955354 
 
 5556 
 
 57158 
 
 196 
 
 9994 
 
 0056 
 
 0117 
 
 0178 
 
 0239 
 
 0300 
 
 0361 
 
 o422 
 
 0483 
 
 0545 
 
 965455 
 
 5657 
 
 5859 
 
 197 
 
 41 0606 
 
 0667 
 
 0728 
 
 0789 
 
 0850 
 
 0911 
 
 0973 
 
 1034 
 
 1095 
 
 1156 
 
 97 
 
 5455 
 
 56'57 
 
 5859 
 
 198 
 
 1217 
 
 1279 
 
 1340 
 
 1401 
 
 1462 
 
 1524 
 
 1585 
 
 1646 
 
 1707 
 
 1768 
 
 98 55 56 57 58 ; 59 60 
 
 199 
 
 1830 
 
 1891 
 
 1952 
 
 2014 
 
 2075 
 
 2136 
 
 2197 
 
 2259 
 
 2320 
 
 2381 
 
 99 
 
 5556 
 
 57J5859:60 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 5657 
 
 58596061 
 
TABLE I. 
 
 Log 
 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 61 
 
 62|62 
 
 64 
 
 65 
 
 66 
 
 0200 
 
 0-41 2443 
 
 2504 
 
 2565 
 
 262 r 
 
 2688 
 
 274S 
 
 2811 
 
 2872 
 
 2933 
 
 299 
 
 01 
 
 C 
 
 C 
 
 C 
 
 C 
 
 
 
 i 
 
 201 
 
 3056 
 
 3117 
 
 3179 
 
 324C 
 
 3302 
 
 3362 
 
 3424 
 
 3486 
 
 3547 
 
 360 
 
 01 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 202 
 
 3670 
 
 3731 
 
 3793 
 
 3854 
 
 3916 
 
 397 r 
 
 403S 
 
 410C 
 
 4162 
 
 422 
 
 Ov 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 203 
 
 4284 
 
 4346 
 
 4407 
 
 446S 
 
 4530 
 
 4592 
 
 4652 
 
 4715 
 
 4776 
 
 483 
 
 03 
 
 
 
 2 
 
 f 
 
 q 
 
 2 
 
 
 204 
 
 4900 
 
 4961 
 
 5023 
 
 5084 
 
 5146 
 
 520" 
 
 526 
 
 533C 
 
 5392 
 
 545 
 
 04 
 
 1 
 
 2 
 
 \ 
 
 a 
 
 3 
 
 
 205 
 
 5515 
 
 5577 
 
 5638 
 
 570C 
 
 576 
 
 582: 
 
 5885 
 
 5946 
 
 6008 
 
 607 
 
 05 
 
 < 
 
 3 
 
 
 3 
 
 3 
 
 
 206 
 
 6131 
 
 6193 
 
 6255 
 
 6316 
 
 6378 
 
 644C 
 
 6501 
 
 6562 
 
 6625 
 
 668 
 
 00 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 - 
 
 207 
 
 6748 
 
 6810 
 
 6871 
 
 6933 
 
 6995 
 
 7056 
 
 711S 
 
 718C 
 
 7242 
 
 7303 
 
 0? 
 
 4 
 
 4 
 
 4 
 
 4 
 
 5 
 
 5 
 
 208 
 
 9ftQ 
 
 7365 
 
 7Qft*i 
 
 7427 
 004,^ 
 
 7489 
 
 S107 
 
 7550 
 
 s his 
 
 7612 
 
 fiQQf 
 
 7674 
 
 COQ5 
 
 7736 
 
 OOK/ 
 
 779S 
 
 S4.1 
 
 7859 
 
 ft4,7 
 
 792 
 
 QKQC 
 
 08 
 on 
 
 I 
 
 
 
 5 
 
 5 
 
 
 uy 
 
 / yot) 
 
 OUTi 
 
 O1U t 
 
 1 U" 
 
 OcOU 
 
 o<st/2 
 
 OOOT 
 
 OTlD 
 
 at; i 
 
 OOO 
 
 U.' 
 
 
 
 
 
 
 
 210 
 
 41 8601 
 
 8663 
 
 8725 
 
 8787 
 
 8849 
 
 8911 
 
 8972 
 
 9034 
 
 9096 
 
 9158 
 
 10 
 
 6 
 
 6 
 
 i 
 
 6 
 
 7 
 
 7 
 
 211 
 
 9220 
 
 9282 
 
 9344 
 
 9406 
 
 9468 
 
 9530 
 
 9592 
 
 9654 
 
 9716 
 
 9778 
 
 1] 
 
 
 
 
 1 
 
 
 
 212 
 
 9839 
 
 9901 
 
 9963 
 
 0025 
 
 0087 
 
 o!49 
 
 0211 
 
 0273 
 
 0335 
 
 0397 
 
 18 
 
 ? 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 213 
 
 42 0459 
 
 0521 
 
 0583 
 
 0646 
 
 0708 
 
 0770 
 
 0832 
 
 0894 
 
 0956 
 
 1018 
 
 13 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 214 
 
 1080 
 
 1142 
 
 1204 
 
 1266 
 
 1328 
 
 1390 
 
 1452 
 
 1515 
 
 1577 
 
 1639 
 
 14 
 
 9 
 
 9 
 
 ( 
 
 9 
 
 c 
 
 9 
 
 215 
 
 1701 
 
 1763 
 
 1825 
 
 1887 
 
 1949 
 
 2012 
 
 2074 
 
 2136 
 
 2198 
 
 2260 
 
 15 
 
 9 
 
 9 
 
 c 
 
 10 
 
 10 
 
 10 
 
 216 
 
 2323 
 
 2385 
 
 2447 
 
 2509 
 
 2571 
 
 2633 
 
 2696 
 
 2758 
 
 2820 
 
 2882 
 
 16 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 217 
 
 2945 
 
 3007 
 
 3069 
 
 3131 
 
 3194 
 
 3256 
 
 3318 
 
 3380 
 
 3443 
 
 3505 
 
 17 
 
 10 
 
 11 
 
 U 
 
 11 
 
 11 
 
 U 
 
 218 
 
 3567 
 
 3630 
 
 3692 
 
 3754 
 
 3816 
 
 3879 
 
 3941 
 
 4003 
 
 4066 
 
 4128 
 
 18 
 
 11 
 
 11 
 
 11 
 
 12 
 
 12 
 
 12 
 
 219 
 
 4190 
 
 4253 
 
 4315 
 
 4378 
 
 4440 
 
 4502 
 
 4565 
 
 4627 
 
 4689 
 
 4752 
 
 19 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 220 
 
 424814 
 
 4877 
 
 4939 
 
 5001 
 
 5064 
 
 5126 
 
 5189 
 
 5251 
 
 5314 
 
 5376 
 
 20 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 13 
 
 221 
 
 5438 
 
 5501 
 
 5563 
 
 5626 
 
 5688 
 
 5751 
 
 5813 
 
 5876 
 
 5938 
 
 6001 
 
 21 
 
 IS 
 
 
 
 13 
 
 13 
 
 14 
 
 14 
 
 222 
 
 6063 
 
 6126 
 
 6188 
 
 6251 
 
 6313 
 
 6376 
 
 6438 
 
 6501 
 
 6564 
 
 6626 
 
 22 
 
 13 
 
 14 
 
 14 
 
 14 
 
 14 
 
 15 
 
 223 
 
 6689 
 
 6751 
 
 6814 
 
 6876 
 
 6939 
 
 7001 
 
 7064 
 
 7127 
 
 7189 
 
 7252 
 
 O '< 
 
 14 
 
 14 
 
 14 
 
 
 
 15 
 
 1 K 
 
 224 
 
 7315 
 
 7377 
 
 7440 
 
 7502 
 
 7565 
 
 7628 
 
 7690 
 
 7753 
 
 7816 
 
 7878 
 
 > 
 
 15 
 
 15 
 
 15 
 
 : 
 
 16 
 
 16 
 
 225 
 
 7941 
 
 8004 
 
 8066 
 
 8129 
 
 8192 
 
 8254 
 
 8317 
 
 8380 
 
 8442 
 
 8505 
 
 ~ 
 u 
 
 15 
 
 16 
 
 16 
 
 16 
 
 16 
 
 1 7 
 
 226 
 
 8568 
 
 8631 
 
 8693 
 
 8756 
 
 8819 
 
 8882 
 
 8944 
 
 9007 
 
 9070 
 
 9133 
 
 8 
 
 16 
 
 16 
 
 16 
 
 7 
 
 17 
 
 :7 
 
 227 
 
 9195 
 
 9258 
 
 9321 
 
 9384 
 
 9447 
 
 9509 
 
 9572 
 
 9635 
 
 9698 
 
 9761 
 
 7 
 
 16 
 
 17 
 
 [7 
 
 7 
 
 18 
 
 18 
 
 228 
 
 9823 
 
 9886 
 
 9949 
 
 0012 
 
 0075 
 
 o!38 
 
 0201 
 
 0263 
 
 o326 
 
 0389 
 
 8 
 
 17 
 
 17 
 
 18 
 
 8 
 
 18 
 
 18 
 
 229 
 
 43 0452 
 
 0515 
 
 0578 
 
 0641 
 
 0704 
 
 0767 
 
 0829 
 
 0892 
 
 0955 
 
 1018 
 
 9 
 
 18 
 
 18 
 
 18 
 
 9 
 
 19 
 
 9 
 
 230 
 
 43 1081 
 
 1144 
 
 1207 
 
 1270 
 
 1333 
 
 1396 
 
 1459 
 
 1522 
 
 1585 
 
 1648 
 
 
 
 18 
 
 19 
 
 19 
 
 9 
 
 19 
 
 20 
 
 231 
 
 1711 
 
 1774 
 
 1837 
 
 1900 
 
 1963 
 
 2026 
 
 2089 
 
 2152 
 
 2215 
 
 2278 
 
 1 
 
 19 
 
 19 
 
 20 
 
 20 
 
 20 
 
 20 
 
 232 
 
 2341 
 
 2404 
 
 2467 
 
 2530 
 
 2593 
 
 2656 
 
 2719 
 
 2782 
 
 2846 
 
 2909 
 
 2 
 
 20 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 233 
 
 2972 
 
 3035 
 
 3098 
 
 3161 
 
 3224 
 
 3287 
 
 3350 
 
 3414 
 
 3477 
 
 3540 
 
 3 
 
 20 
 
 20 
 
 21 
 
 ^1 
 
 21 
 
 22 
 
 234 
 
 3603 
 
 3666 
 
 3729 
 
 3792 
 
 3856 
 
 3919 
 
 3982 
 
 4045 
 
 4108 
 
 4172 
 
 4 
 
 21 
 
 21 
 
 21 
 
 ^2 
 
 22 
 
 22 
 
 235 
 
 4235 
 
 4298 
 
 4361 
 
 4424 
 
 4488 
 
 4551 
 
 4614 
 
 4677 
 
 4741 
 
 4804 
 
 5 
 
 21 
 
 22 
 
 22 
 
 2 
 
 23 
 
 23 
 
 236 
 
 4867 
 
 4930 
 
 4994 
 
 5057 
 
 5120 
 
 5184 
 
 5247 
 
 5310 
 
 5373 
 
 5437 
 
 6 
 
 22 
 
 22 
 
 23 
 
 3 
 
 2.3 
 
 24 
 
 237 
 
 5500 
 
 5563 
 
 5627 
 
 5690 
 
 5753 
 
 5817 
 
 5880 
 
 5943 
 
 6007 
 
 6070 
 
 7 
 
 23 
 
 23 
 
 23 
 
 4- 
 
 24 
 
 24 
 
 238 
 
 6133 
 
 6197 
 
 6260 
 
 6324 
 
 6387 
 
 6450 
 
 6514 
 
 6577 
 
 6641 
 
 6704 
 
 8 
 
 23 
 
 24 
 
 24 
 
 -t 
 
 25 
 
 25 
 
 239 
 
 6767 
 
 6831 
 
 6894 
 
 6958 
 
 7021 
 
 7085 
 
 7148 
 
 7211 
 
 7275 
 
 7338 
 
 9 
 
 24- 
 
 24- 
 
 25 
 
 5 
 
 25 
 
 ?6 
 
 240 
 
 43 7402 
 
 7465 
 
 7529 
 
 7592 
 
 7656 
 
 r '719 
 
 7783 
 
 7846 
 
 7910 
 
 7973 
 
 
 
 24 
 
 25 
 
 25 
 
 6 
 
 26 
 
 26 
 
 241 
 
 8037 
 
 8100 
 
 8164 
 
 8227 
 
 8291 
 
 8355 
 
 8418 
 
 8482 
 
 8545 
 
 8609 
 
 1 
 
 25 
 
 25 
 
 26 
 
 6 
 
 27 
 
 27 
 
 242 
 
 8672 
 
 8736 
 
 8800 
 
 8863 
 
 8927 
 
 8990 
 
 9054 
 
 9118 
 
 9181 
 
 9245 
 
 2 
 
 26 
 
 26 
 
 26 
 
 7 
 
 27 
 
 ?8 
 
 243 
 
 9308 
 
 9372 
 
 9436 
 
 9499 
 
 9563 
 
 9627 
 
 9690 
 
 9754 
 
 9818 
 
 9881 
 
 3 
 
 2G 
 
 27 
 
 27 
 
 8 
 
 28 
 
 >8 
 
 244 
 
 9945 
 
 0009 
 
 0072 
 
 o!36 
 
 0200 
 
 0264 
 
 0327 
 
 0391 
 
 0455 
 
 0518 
 
 4- 
 
 27 
 
 27 
 
 28 
 
 8 
 
 29 
 
 >9 
 
 245 
 
 44 0582 
 
 0646 
 
 0710 
 
 0773 
 
 0837 
 
 0901 
 
 0965 
 
 1029 
 
 1092 
 
 1156 
 
 5 
 
 27 
 
 28 
 
 28 
 
 9 
 
 29 
 
 30 
 
 246 
 
 1220 
 
 1284 
 
 1348 
 
 1411 
 
 1475 
 
 1539 
 
 1603 
 
 1667 
 
 1730 
 
 1794 
 
 6 
 
 28 
 
 29 
 
 29 
 
 9 
 
 30 
 
 30 
 
 247 
 
 1858 
 
 1922 
 
 1986 
 
 2050 
 
 2114 
 
 2177 
 
 2241 
 
 2305 
 
 2369 
 
 2433 
 
 7 
 
 2929 
 
 30 
 
 
 
 31 
 
 1 
 
 248 
 
 2497 
 
 2561 
 
 2625 
 
 2689 
 
 2752 
 
 2816 
 
 2880 
 
 2944 
 
 3008 
 
 3072 
 
 8 
 
 2930 
 
 30 
 
 1 
 
 31 
 
 2 
 
 249 
 
 3136 
 
 3200 
 
 3264 
 
 3328 
 
 3392 
 
 3456 
 
 3520 
 
 3584 
 
 3648 
 
 3712 
 
 9 
 
 30 
 
 30 
 
 31 
 
 _^ 
 
 32 
 
 2 
 
 
 
 
 1 
 
 2~ 
 
 3 
 
 4 
 
 5 
 
 () 
 
 7 
 
 8 
 
 9 
 
 
 3162 
 
 S3 
 
 4 
 
 35 
 
 6 
 
TABLE I. 
 
 Log 
 
 Log (1 -far) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 0-250 
 
 0-44 3776 
 
 3840 
 
 3904 
 
 3968 
 
 4032 
 
 4096 
 
 4160 
 
 4224 
 
 4288 
 
 4352 
 
 50 
 
 31 
 
 31 
 
 32 
 
 32 
 
 33 
 
 33 
 
 251 
 
 4416 
 
 4480 
 
 4544 
 
 4608 
 
 4673 
 
 4737 
 
 4801 
 
 4865 
 
 4929 
 
 4993 
 
 5l|31 
 
 32 
 
 3233 
 
 33 
 
 34 
 
 252 
 
 5057 
 
 5121 
 
 5185 
 
 5249 
 
 5314 
 
 5378 
 
 5442 
 
 5506 
 
 5570 
 
 5634 
 
 52 
 
 32 
 
 32 
 
 3333 
 
 34 
 
 34 
 
 253 
 
 5698 
 
 5763 
 
 5827 
 
 5891 
 
 5955 
 
 6019 
 
 6084 
 
 6148 
 
 6212 
 
 6276 
 
 53 
 
 32 
 
 33 
 
 3334 
 
 34 
 
 35 
 
 254 
 
 6340 
 
 6405 
 
 6469 
 
 6533 6597 
 
 6662 
 
 6726 
 
 6790 
 
 6854 
 
 6919 
 
 54 
 
 33 
 
 33 
 
 3435 
 
 35J36 
 
 255 
 
 6983 
 
 7047 
 
 7111 
 
 7176 
 
 7240 
 
 7304 
 
 7369 
 
 7433 
 
 7497 
 
 7562 
 
 55 
 
 34 
 
 34 
 
 3535 
 
 3636 
 
 256 
 
 7626 
 
 7690 
 
 7755 
 
 7819 
 
 7883 
 
 7948 
 
 8012 
 
 8076 
 
 8141 
 
 8205 
 
 56 
 
 34 
 
 35 35|36 
 
 36;37 
 
 257 
 
 8269 
 
 8334 
 
 8398 
 
 8463 8527 
 
 8591 
 
 8656 
 
 8720 
 
 8785 
 
 8849 
 
 57 
 
 35 
 
 35 
 
 3636 
 
 3738 
 
 258 
 
 8913 8978 
 
 9042 
 
 9107 
 
 9171 
 
 9236 
 
 9300 
 
 9365 
 
 9429 
 
 9493 
 
 58 
 
 35 
 
 36 
 
 37 
 
 37 
 
 3838 
 
 259 
 
 9558 
 
 9622 
 
 9687 
 
 9751 
 
 9816 
 
 9880 
 
 9945 
 
 0009 
 
 0074 
 
 0139 
 
 59 
 
 36 
 
 3 r 
 
 37 
 
 38 
 
 38 
 
 39 
 
 260 
 
 45 0203 
 
 0268 
 
 0332 
 
 0397 
 
 0461 
 
 0526 
 
 0590 
 
 0655 
 
 0719 
 
 0784 
 
 60 
 
 37 
 
 3 r- 
 
 38 
 
 38 
 
 39 
 
 40 
 
 261 
 
 0849 
 
 0913 
 
 0978 
 
 1042 
 
 1107 
 
 1172 
 
 1236 
 
 1301 
 
 1366 
 
 1430 
 
 61 
 
 37 
 
 38 
 
 38 
 
 39 
 
 4040 
 
 262 
 
 1495 
 
 1559 
 
 1624 
 
 1689 
 
 1753 
 
 1818 
 
 1883| 1947 
 
 2012 
 
 2077 
 
 62 
 
 38 
 
 38 
 
 39 
 
 40 
 
 40 
 
 41 
 
 263 
 
 2141 
 
 2206 
 
 2271 
 
 2336 
 
 2400 
 
 2465 
 
 2530 
 
 2594 
 
 2659 
 
 2724 
 
 63 
 
 38 
 
 39 
 
 40 
 
 40 
 
 41 
 
 42 
 
 264 
 
 2789 
 
 2853 
 
 2918 
 
 2983 
 
 3048 
 
 3112 
 
 3177 
 
 3242 
 
 3307 
 
 3372 
 
 64 
 
 39 
 
 40 
 
 40 
 
 41 
 
 42 
 
 42 
 
 265 
 
 3436 
 
 3501 
 
 3566 
 
 3631 
 
 3696 
 
 3760 
 
 3825 
 
 3890 
 
 3955 
 
 4020 
 
 65 
 
 40 
 
 40 
 
 41 
 
 42 
 
 42 
 
 43 
 
 266 
 
 4085 
 
 4149 
 
 4214 
 
 4279 
 
 4344 
 
 4409 
 
 4474 
 
 4539 
 
 4604 
 
 4668 
 
 66 
 
 40 
 
 41 
 
 42 
 
 42 
 
 43 
 
 44 
 
 267 
 
 4733 
 
 4798 
 
 4863 
 
 4928 
 
 4993 
 
 5058 
 
 5123 
 
 5188 
 
 5253 
 
 5318 
 
 67 
 
 41 
 
 42 
 
 42 
 
 43 
 
 44 
 
 44 
 
 268 
 
 5383 
 
 5448 
 
 5513 
 
 5578 
 
 5643 
 
 5708 
 
 5773 
 
 5838 
 
 5903 
 
 5968 
 
 68 
 
 41 
 
 42 
 
 43 
 
 44 
 
 44 
 
 45 
 
 269 
 
 6033 
 
 6098 
 
 6163 
 
 6228 
 
 6293 
 
 6358 
 
 6423 
 
 6488 
 
 6553 
 
 6618 
 
 69 
 
 42 
 
 43 
 
 43 
 
 44 
 
 45 
 
 46 
 
 270 
 
 45 6683 
 
 6748 
 
 6813 
 
 6878 
 
 6943 
 
 7008 
 
 7073 
 
 7138 
 
 7204 
 
 7269 
 
 70 
 
 43 
 
 43 
 
 44 
 
 45 
 
 45 
 
 46 
 
 271 
 
 7334 
 
 7399 
 
 7464 
 
 7529 
 
 7594 
 
 7659 
 
 7724 
 
 7790 
 
 7855 
 
 7920 
 
 71 
 
 43 
 
 44 
 
 45 
 
 45 
 
 46 
 
 47 
 
 272 
 
 7985 
 
 8050 
 
 8115 
 
 8181 
 
 8246 
 
 8311 
 
 8376 
 
 8441 
 
 8507 
 
 8572 
 
 72 
 
 44 
 
 45 
 
 45 
 
 46 
 
 47 
 
 48 
 
 273 
 
 8637 
 
 8702 
 
 8767 
 
 8833 
 
 8898 
 
 8963 
 
 9028 
 
 9094 
 
 9159 
 
 9224 
 
 73 
 
 45 
 
 45 
 
 46 
 
 47 
 
 47 
 
 48 
 
 274 
 
 9289 
 
 9355 
 
 9420 
 
 9485 
 
 9551 
 
 9616 
 
 9681 
 
 9746 
 
 9812 
 
 9877 
 
 74 
 
 45 
 
 46 
 
 47 
 
 47 
 
 48 
 
 49 
 
 275 
 
 9942 
 
 0008 
 
 0073 
 
 0138 
 
 0204 
 
 0269 
 
 0334 
 
 0400 
 
 0465 
 
 0531 
 
 75 
 
 46 
 
 46 
 
 47 
 
 48 
 
 49 
 
 49 
 
 276 
 
 46 0596 
 
 0661 
 
 0727 
 
 0792 
 
 0857 
 
 0923 
 
 0988 
 
 1054 
 
 1119 
 
 1184 
 
 76 
 
 46 
 
 47 
 
 48 
 
 49 
 
 49 
 
 50 
 
 277 
 
 1250 
 
 1315 
 
 1381 
 
 1446 
 
 1512 
 
 1577 
 
 1643 
 
 1708 
 
 1773 
 
 1839 
 
 77 
 
 47 
 
 48 
 
 49 
 
 49 
 
 50 
 
 51 
 
 278 
 
 1904 
 
 1970 
 
 2035 
 
 2101 
 
 2166 
 
 2232 
 
 2297 
 
 2363 
 
 2428 
 
 2494 
 
 78 
 
 48 
 
 48 
 
 49 
 
 50 
 
 51 
 
 51 
 
 279 
 
 2559 
 
 2625 
 
 2691 
 
 2756 
 
 2822 
 
 2887 
 
 2953 
 
 3018 
 
 3084 
 
 3149 
 
 79 
 
 48 
 
 49 
 
 50 
 
 51 
 
 51 
 
 52 
 
 280 
 
 463215 
 
 3281 
 
 3346 
 
 3412 
 
 3477 
 
 3543 
 
 3609 
 
 3674 
 
 3740 
 
 3805 
 
 80 
 
 49 
 
 50 
 
 50 
 
 51 
 
 52 
 
 53 
 
 281 
 
 3871 
 
 3937 
 
 4002 
 
 4068 
 
 4134 
 
 4199 
 
 4265 
 
 4331 
 
 4396 
 
 4462 
 
 81 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 53 
 
 282 
 
 4528 
 
 4593 
 
 4659 
 
 4725 
 
 4790 
 
 4856 
 
 4922 
 
 4988 
 
 5053 
 
 5119 
 
 82 
 
 50 
 
 51 
 
 52 
 
 52 
 
 53 
 
 54 
 
 283 
 
 5185 
 
 5251 
 
 5316 
 
 5382 
 
 5448 
 
 5514 
 
 5579 
 
 5645 
 
 5711 
 
 5777 
 
 83 
 
 51 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 284 
 
 5842 
 
 5908 
 
 5974 
 
 6040 
 
 6106 
 
 6171 
 
 6237 
 
 6303 
 
 6369 
 
 6435 
 
 84 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 55 
 
 285 
 
 6501 
 
 6566 
 
 6632 
 
 6698 
 
 6764 
 
 6830 
 
 6896 
 
 6962 
 
 7027 
 
 7093 
 
 85 
 
 52 
 
 53 
 
 54 
 
 54 
 
 55 
 
 56 
 
 286 
 
 7159 
 
 7225 
 
 7291 
 
 7357 
 
 7423 
 
 7489 
 
 7555 
 
 7621 
 
 7687 
 
 7753 
 
 86 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 287 
 
 7818 
 
 7884 
 
 7950 
 
 8016 
 
 8082 
 
 8148 
 
 8214 
 
 8280 
 
 8346 
 
 8412 
 
 87 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 57 
 
 288 
 
 8478 
 
 8544 
 
 8610 
 
 8676 
 
 8742 
 
 8808 
 
 8874 
 
 8940 
 
 9006 
 
 9072 
 
 88 
 
 54 
 
 55 
 
 55 
 
 56 
 
 57 
 
 58 
 
 289 
 
 9138 
 
 9204 
 
 9271 
 
 9337 
 
 9403 
 
 9469 
 
 9535 
 
 9601 
 
 9667 
 
 9733 
 
 89 
 
 54 
 
 55 
 
 06 
 
 57 
 
 58 
 
 59 
 
 290 
 
 46 9799 
 
 9865 
 
 9931 
 
 9997 
 
 0064 
 
 0130 
 
 0196 
 
 0262 
 
 0328 
 
 0394 
 
 90 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 59 
 
 291 
 
 47 0460 
 
 0527 
 
 0593 
 
 0659 
 
 0725 
 
 0791 
 
 0857 
 
 0924 
 
 0990! 1056 
 
 91 
 
 56 
 
 56 
 
 57 
 
 58 
 
 5960 
 
 292 
 
 1122 
 
 1188 
 
 1255 
 
 1321 
 
 1387 
 
 1453 
 
 1519 
 
 1586 1652 
 
 1718 
 
 92 
 
 56 
 
 57 
 
 58 
 
 59 
 
 6061 
 
 293 
 
 1784 
 
 1851 
 
 1917 
 
 1983 
 
 2050 
 
 2116 
 
 2182 
 
 2248 
 
 2315 
 
 2381 
 
 93 
 
 57 
 
 58 
 
 59 
 
 60 
 
 60 
 
 61 
 
 294 
 
 2447 
 
 2514 
 
 2580 
 
 2646 
 
 2713 
 
 2779 
 
 2845 
 
 2912 
 
 2978 
 
 3044 
 
 94 
 
 57 
 
 58 
 
 59 
 
 60 
 
 61 
 
 62 
 
 295 
 
 3111 
 
 3177 
 
 3243 
 
 3310 
 
 3376 
 
 3442 
 
 3509 
 
 3575 
 
 3642 
 
 3708 
 
 95 
 
 58 
 
 59 
 
 60 
 
 61 
 
 62 
 
 63 
 
 296 
 
 3774 
 
 3841 
 
 3907 
 
 3974 
 
 4040 
 
 4107 
 
 4173 
 
 4239 
 
 4306 
 
 4372 
 
 96 
 
 59 
 
 60 
 
 60 
 
 61 
 
 62 
 
 63 
 
 297 
 
 4439 
 
 4505 
 
 4572 
 
 4638 
 
 4705 
 
 4771 
 
 4838 
 
 4904 
 
 4971 
 
 5037 
 
 97 
 
 59 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 298 
 
 5104 
 
 5170 
 
 5237 
 
 5303 
 
 5370 
 
 5436 
 
 5503 
 
 5569 
 
 5636 
 
 5702 
 
 98 
 
 80 
 
 61 
 
 62 
 
 S3 
 
 64 
 
 65 
 
 299 
 
 5769 
 
 5836 
 
 5902 
 
 5969 
 
 6035 
 
 6102 
 
 6168 
 
 6235 
 
 6302 
 
 6368 
 
 99 
 
 50 
 
 61 
 
 62 
 
 33 
 
 64 
 
 65 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 31 
 
 62 
 
 3 
 
 34 
 
 55 
 
 36 
 
TABLE I. 
 
 Log 
 
 
 
 Log(l- 
 
 f*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 t> 
 
 7 
 
 8 
 
 9 
 
 
 67 
 
 6 
 
 6 
 
 7( 
 
 )7 
 
 72 
 
 0-300 
 
 0-47 6435 
 
 6501 
 
 6568 
 
 6635 
 
 6701 
 
 6768 
 
 6835 
 
 6901 
 
 696 
 
 703 
 
 00 
 
 t 
 
 t 
 
 
 C 
 
 ) 
 
 
 301 
 
 7101 
 
 7168 
 
 7235 
 
 7301 
 
 7368 
 
 7435 
 
 750 
 
 7568 763^ 
 
 770 
 
 01 
 
 1 
 
 1 
 
 
 
 
 
 302 
 
 7768 
 
 7835 
 
 7902 
 
 7968 
 
 8035 
 
 8102 
 
 8169 
 
 8235 830 
 
 836 
 
 02 
 
 1 
 
 1 
 
 
 1 
 
 
 
 303 
 
 8436 
 
 8502 
 
 8569 
 
 8636 
 
 8703 
 
 8769 
 
 8836 
 
 8903 897 
 
 903 
 
 03 
 
 
 2 
 
 
 i 
 
 < 
 
 
 304 
 
 9104 
 
 9170 
 
 9237 
 
 9304 
 
 9371 
 
 9438 
 
 950o 
 
 9571 963 
 
 970o 
 
 04 
 
 t 
 
 a 
 
 
 2 
 
 3 
 
 
 305 
 
 9772 
 
 9839 
 
 9906 
 
 9973 
 
 0039 
 
 0106 
 
 0173 
 
 0240 o30 
 
 0374 
 
 05 
 
 s 
 
 a 
 
 
 4 
 
 4 
 
 
 306 
 
 48 0441 
 
 0508 
 
 0575 
 
 0642 
 
 0709 
 
 0776 
 
 0843 
 
 0909 097 
 
 1043 
 
 06 
 
 4 
 
 4 
 
 
 4 
 
 4 
 
 
 307 
 
 1110 
 
 1177 
 
 1244 
 
 1311 
 
 1378 
 
 1445 
 
 1512 
 
 157S 
 
 1646 
 
 1713 
 
 07 
 
 5 
 
 5 
 
 
 5 
 
 5 
 
 
 308 
 
 1780 
 
 1847 
 
 1914 
 
 1981 
 
 2048 
 
 2116 
 
 2183 
 
 225C 
 
 2317 
 
 2384 
 
 08 
 
 A 
 
 5 
 
 
 6 6 
 
 
 309 
 
 2451 
 
 2518 
 
 2585 
 
 2652 
 
 2719 
 
 2786 
 
 2853 
 
 292C 
 
 2988 
 
 3055 
 
 09 
 
 6 
 
 6 
 
 
 6 
 
 6 
 
 
 310 
 
 48 3122 
 
 3189 
 
 3256 
 
 3323 
 
 3390 
 
 3457 
 
 3525 
 
 3592 
 
 3659 
 
 3726 
 
 10 
 
 7 
 
 7 
 
 
 f 
 
 
 
 311 
 
 3793 
 
 3860 
 
 3928 
 
 3995 
 
 4062 
 
 4129 
 
 4196 
 
 4264 
 
 433 
 
 4398 
 
 11 
 
 7 
 
 7 
 
 
 8 
 
 8 
 
 
 312 
 
 4465 
 
 4533 
 
 4600 
 
 4667 
 
 4734 
 
 4801 
 
 4869 
 
 4936 
 
 5003 
 
 507 
 
 12 
 
 8 
 
 8 
 
 
 8 
 
 9 
 
 
 313 
 
 5138 
 
 5205 
 
 5272 
 
 5340 
 
 5407 
 
 5474 
 
 5542 
 
 5609 
 
 5676 
 
 5743 
 
 13 
 
 9 
 
 9 
 
 
 c 
 
 9 
 
 
 314 
 
 5811 
 
 5878 
 
 5945 
 
 6013 
 
 6080 
 
 6148 
 
 6215 
 
 6282 
 
 6350 
 
 6417 
 
 14 
 
 9 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 315 
 
 6484 
 
 6552 
 
 6619 
 
 6686 
 
 6754 
 
 6821 
 
 6889 
 
 6956 
 
 7024 
 
 7091 
 
 15 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 316 
 
 7158 
 
 7226 
 
 7293 
 
 7361 
 
 7428 
 
 7496 
 
 7563 
 
 7630 
 
 7698 
 
 7765 
 
 16 
 
 11 
 
 11 
 
 1 
 
 11 
 
 11 
 
 12 
 
 317 
 
 7833 
 
 7900 
 
 7968 
 
 8035 
 
 8103 
 
 8170 
 
 8238 
 
 8305 
 
 8373 
 
 8440 
 
 17 
 
 11 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 318 
 
 8508 
 
 8575 
 
 8643 
 
 8711 
 
 8778 
 
 8846 
 
 8913 
 
 8981 
 
 9048 
 
 9116 
 
 18 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 319 
 
 9183 
 
 9251 
 
 9319 
 
 9386 
 
 9454 
 
 9521 
 
 9589 
 
 9657 
 
 9724 
 
 9792 
 
 19 
 
 13 
 
 13 
 
 i: 
 
 13 
 
 13 
 
 14 
 
 320 
 
 48 9860 
 
 9927 
 
 9995 
 
 0062 
 
 0130 
 
 0198 
 
 0265 
 
 0333 
 
 0401 
 
 0468 
 
 20 
 
 13 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 321 
 
 49 0536 
 
 0604 
 
 0671 
 
 0739 
 
 0807 
 
 0875 
 
 0942 
 
 1010 
 
 1078 
 
 1145 
 
 21 
 
 14 
 
 14 
 
 14 
 
 I C 
 
 1 5 
 
 15 
 
 322 
 
 1213 
 
 1281 
 
 1349 
 
 1416 
 
 1484 
 
 1552 
 
 1620 
 
 1687 
 
 1755 
 
 1823 
 
 22 
 
 15 
 
 15 
 
 15 
 
 1 *\ 
 
 16 
 
 16 
 
 323 
 
 1891 
 
 1958 
 
 2026 
 
 2094 
 
 2162 
 
 2230 
 
 2297 
 
 2365 
 
 2433 
 
 2501 
 
 23 
 
 15 
 
 16 
 
 16 
 
 16 
 
 16 
 
 [7 
 
 324 
 
 2569 
 
 2637 
 
 2704 
 
 2772 
 
 2840 
 
 2908 
 
 2976 
 
 3044 
 
 3112 
 
 3179 
 
 24 
 
 16 
 
 16 
 
 17 
 
 17 
 
 17 
 
 1 
 
 325 
 
 3247 
 
 3315 
 
 3383 
 
 3451 
 
 3519 
 
 3587 
 
 3655 
 
 3723 
 
 3791 
 
 3858 
 
 25 
 
 17 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 326 
 
 3926 
 
 3994 
 
 4062 
 
 4130 
 
 4198 
 
 4266 
 
 4334 
 
 4402 
 
 4470 
 
 4538 
 
 26 
 
 17 
 
 18 
 
 18 
 
 18 
 
 18 
 
 19 
 
 327 
 
 4606 
 
 4674 
 
 4742 
 
 4810 
 
 4878 
 
 4946 
 
 5014 
 
 5082 
 
 5150 
 
 5218 
 
 27 
 
 18 
 
 18 
 
 J C 
 
 19 
 
 19 
 
 9 
 
 328 
 
 5286 
 
 5354 
 
 5422 
 
 5490 
 
 5558 
 
 5626 
 
 5694 
 
 5762 
 
 5830 
 
 5899 
 
 28 
 
 19 
 
 19 
 
 19 
 
 20 
 
 20 
 
 20 
 
 329 
 
 5967 
 
 6035 
 
 6103 
 
 6171 
 
 6239 
 
 6307 
 
 6375 
 
 6443 
 
 6511 
 
 6580 
 
 29 
 
 19 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 330 
 
 49 6648 
 
 6716 
 
 6784 
 
 6852 
 
 6920 
 
 6988 
 
 7057 
 
 7125 
 
 7193 
 
 7261 
 
 3020 
 
 20 
 
 21 
 
 21 
 
 21 
 
 22 
 
 331 
 
 7329 
 
 7397 
 
 7466 
 
 7534 
 
 7602 
 
 7670 
 
 7738 
 
 7807 
 
 7875 
 
 7943 
 
 31 21 
 
 2121 
 
 22 
 
 22 
 
 22 
 
 332 
 
 8011 
 
 8080 
 
 8148 
 
 8216 
 
 8284 
 
 8353 
 
 8421 
 
 8489 
 
 8557 
 
 8626 
 
 3221 
 
 22 
 
 22 
 
 22 
 
 23 
 
 2'; 
 
 333 
 
 8694 
 
 8762 
 
 8830 
 
 8899 
 
 8967 
 
 9035 
 
 9104 
 
 9172 
 
 9240 
 
 9309 
 
 33 22 
 
 22 
 
 23 
 
 23 
 
 23 
 
 24 
 
 334 
 
 9377 
 
 9445 
 
 9514 
 
 9582 
 
 9650 
 
 9719 
 
 9787 
 
 9855 
 
 9924 
 
 9992 
 
 3423 
 
 23 
 
 23 
 
 24 
 
 24 
 
 24 
 
 335 
 
 500061 
 
 0129 
 
 0197 
 
 0266 
 
 0334 
 
 0403 
 
 0471 
 
 0539 
 
 0608 
 
 0676 
 
 35'23 
 
 24 
 
 24 
 
 24 
 
 25 
 
 25 
 
 336 
 
 0745 
 
 0813 
 
 0881 
 
 0950 
 
 1018 
 
 1087 
 
 1155 
 
 1224 
 
 1292 
 
 1361 
 
 3624 
 
 2425 
 
 '5 
 
 26 
 
 ? 6 
 
 337 
 
 1429 
 
 1498 
 
 1566 
 
 1635 
 
 1703 
 
 1772 
 
 1840 
 
 1909 
 
 1977 
 
 2046 
 
 72525 
 
 26 
 
 26 
 
 26 
 
 t-7 
 
 338 
 
 2114 
 
 2183 
 
 2251 
 
 2320 
 
 2388 
 
 2457 
 
 2526 
 
 2594 
 
 2663 
 
 2731 
 
 S 25 26 
 
 26 
 
 27 
 
 27 
 
 7 
 
 339 
 
 2800 
 
 2868 
 
 2937 
 
 3006 
 
 3074 
 
 3143 
 
 3211 
 
 3280 
 
 3349 
 
 3417 
 
 9 
 
 26 
 
 27 
 
 27 
 
 27 
 
 28 
 
 28 
 
 340 
 
 50 3486 
 
 3554 
 
 3623 
 
 3692 
 
 3760 
 
 3829 
 
 3898 
 
 3966 
 
 4035 
 
 4104 
 
 40 
 
 27 
 
 27 
 
 28 
 
 28 
 
 28 
 
 29 
 
 341 
 
 4172 
 
 4241 
 
 4310 
 
 4378 
 
 4447 
 
 4516 
 
 4585 
 
 4653 
 
 4722 
 
 4791 
 
 127 
 
 2828 
 
 99 
 
 29 
 
 30 
 
 342 
 
 4859 
 
 4928 
 
 4997 
 
 5066 
 
 5134 
 
 5203 
 
 5272 
 
 5341 
 
 5409 
 
 5478 
 
 2* 28 
 
 2929 
 
 9 
 
 30 
 
 30 
 
 343 
 
 5547 
 
 5616 
 
 5685 
 
 5753 
 
 5822 
 
 5891 
 
 5960 
 
 6029 
 
 6097 
 
 6166 
 
 329 
 
 29|30 
 
 
 
 31 
 
 31 
 
 344 
 
 6235 
 
 6304 
 
 6373 
 
 6442 
 
 6510 
 
 6579 
 
 6648 
 
 6717 
 
 6786 
 
 6855 
 
 44 
 
 29 
 
 30|30 
 
 1 
 
 31 
 
 32 
 
 345 
 
 6924 
 
 6992 
 
 7061 
 
 7130 
 
 7199 
 
 7268 
 
 7337 
 
 7406 
 
 7475 
 
 7544 
 
 5. 
 
 30 
 
 31(31 
 
 2 
 
 32 
 
 32 
 
 346 
 
 7613 
 
 7681 
 
 7750 
 
 7819 
 
 7888 
 
 7957 
 
 8026 
 
 8095 
 
 8164 
 
 8233 
 
 631 
 
 3132 
 
 2 
 
 33 
 
 33 
 
 347 
 
 8302 
 
 8371 
 
 8440 
 
 8509 
 
 8578 
 
 8647 
 
 8716 
 
 8785 
 
 8854 
 
 8923 
 
 7313232 
 
 3 
 
 33 
 
 34 
 
 348 
 
 8992 
 
 9061 
 
 9130 
 
 9199 
 
 9268 
 
 9337 
 
 9406 
 
 9475 
 
 9544 
 
 9613 
 
 832, 
 
 33, 
 
 33 
 
 4 
 
 34 
 
 35 
 
 349 
 
 9683 
 
 9752 
 
 9821 
 
 9890 
 
 9959 
 
 0028 
 
 0097 
 
 0166 
 
 0235 
 
 0304 
 
 933, 
 
 33, 
 
 34 
 
 4 
 
 35 
 
 J5 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ( 
 
 )?< 
 
 38 ( 
 
 39 
 
 
 
 71 
 
 "2 
 
TABLE I. 
 
 Log 
 
 Log(l+ar) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 676869 
 
 70 
 
 71 72 
 
 0350 
 
 0-51 0374 
 
 0443 
 
 0512 
 
 0581 
 
 0650 
 
 0719 
 
 0788 
 
 0858 
 
 0927 
 
 0996 
 
 5034343535 
 
 3636 
 
 351 
 
 1065 
 
 1134 
 
 1203 1273 
 
 1342 
 
 1411 
 
 1480 
 
 1549 
 
 1619 
 
 1688 
 
 51 
 
 34353536:36:37 
 
 352 
 
 1757 
 
 1826 
 
 1895 1965 
 
 2034 
 
 2103 
 
 2172 2242 
 
 2311 
 
 2380 
 
 5235353636-3737 
 
 353 
 
 2449 
 
 2519 
 
 2588! 2657 
 
 2727 
 
 2796 2865 
 
 2934 
 
 3004 
 
 3073 
 
 53363637 
 
 37 38 38 
 
 354 
 
 3142 
 
 3212 
 
 3281 3350 
 
 3420 
 
 3489 3558 
 
 3628 
 
 3697 
 
 3767 
 
 54 36 37j37 38 
 
 38J39 
 
 355 
 
 3836 
 
 3905 
 
 3975 4044 
 
 4113 
 
 4183; 4252 
 
 4322 
 
 4391 
 
 4460 
 
 55373713838 
 
 3940 
 
 356 
 
 4530 
 
 4599 
 
 4669! 4738 
 
 4808 
 
 4877! 4946 
 
 5016 5085 
 
 5155 
 
 56383839394040 
 
 357 
 
 5224 
 
 5294 
 
 5363' 5433 
 
 5502 
 
 5572 
 
 5641 
 
 5711 
 
 5780 5850 
 
 573839394040 ! 41 
 
 358 
 
 5919 
 
 5989 
 
 6058 6128 
 
 6197 
 
 6267 
 
 6336 
 
 6406 
 
 6475 
 
 6545 
 
 58393940 
 
 41 
 
 4142 
 
 359 
 
 6615 
 
 6684 
 
 6754 6823 
 
 6893 
 
 6962 7032 
 
 7102 
 
 7171 
 
 7241 
 
 59404041 
 
 41 
 
 4242 
 
 360 
 
 51 7310 
 
 7380 
 
 7450 
 
 7519 
 
 7589 
 
 7659 
 
 7728 
 
 7798 
 
 7867 7937 
 
 60404141 
 
 424343 
 
 361 
 
 8007 
 
 8076 
 
 8146 
 
 8216 
 
 8285 
 
 8355 
 
 8425 
 
 8495 
 
 8564 
 
 8634 
 
 61 
 
 41 41 42 
 
 434344 
 
 362 
 
 8704 
 
 8773 
 
 8843 
 
 8913 
 
 8983 
 
 9052 9122 
 
 9192 
 
 9261 
 
 9331 
 
 62424243|43,444o 
 
 363 
 
 9401 
 
 9471 
 
 9541 
 
 9610 
 
 9680 
 
 9750 
 
 9820 
 
 9889 
 
 9959 
 
 0029 
 
 63 42 43 43|44 
 
 4545 
 
 364 
 
 52 0099 
 
 0169 
 
 0238 
 
 0308 
 
 0378 
 
 0448 
 
 0518 
 
 0588 
 
 0657 
 
 0727 
 
 644344 
 
 44 
 
 45 
 
 4546 
 
 365 
 
 0797 
 
 0867 
 
 0937 
 
 1007 
 
 1077 
 
 1146 
 
 1216 
 
 1286 
 
 1356 
 
 1426 
 
 G544!44 
 
 45 
 
 46 
 
 46 47 
 
 366 
 
 1496 
 
 1566 
 
 1636 
 
 1706 
 
 1776 
 
 1846 
 
 1915 
 
 1985 
 
 2055 
 
 2125 
 
 66 44 45;46;46J47 48 
 
 367 
 
 2195 
 
 2265 
 
 2335 
 
 2405 
 
 2475 
 
 2545 
 
 2615 
 
 2685 
 
 2755 
 
 2825 
 
 67 >4546 
 
 46147 
 
 48 4b 
 
 368 
 369 
 
 2895 
 3595 
 
 2965 
 3665 
 
 3035 
 3735 
 
 3105 
 
 3805 
 
 3175 
 
 3875 
 
 3245 
 3946 
 
 3315 
 4016 
 
 3385 
 4086 
 
 3455 
 4156 
 
 3525 
 4226 
 
 68 
 
 69 
 
 46 
 46 
 
 46 
 47 
 
 47 
 
 48 
 
 48 
 48 
 
 48 
 49 
 
 k9 
 50 
 
 370 
 
 52 4296 
 
 4366 
 
 4436 
 
 4506 
 
 4576 
 
 4647 
 
 4717 
 
 4787 
 
 4857 
 
 4927 
 
 70 
 
 47 
 
 48 
 
 48 
 
 49 
 
 50 
 
 50 
 
 371 
 
 4997 
 
 5067 
 
 5138 
 
 5208 
 
 5278 
 
 5348 
 
 5418 
 
 5488 
 
 5559 
 
 5629 
 
 71 
 
 48 
 
 48 
 
 49 
 
 50 
 
 50 
 
 51 
 
 372 
 
 5699 
 
 5769 
 
 5839 
 
 5910 
 
 5980 
 
 6050 
 
 6120 
 
 6190 
 
 6261 
 
 6331 
 
 72 
 
 48 
 
 49 
 
 50 
 
 50 
 
 51 
 
 52 
 
 373 
 
 6401 
 
 6471 
 
 6542 
 
 6612 
 
 6682 
 
 6752 
 
 6823 
 
 6893 
 
 6963 
 
 7033 
 
 73 
 
 49 
 
 50 
 
 50 
 
 51 
 
 52 
 
 53 
 
 374 
 
 7104 
 
 7174 
 
 7244 
 
 7315 
 
 7385 
 
 7455 
 
 7526 
 
 7596 
 
 7666 
 
 7737 
 
 74 
 
 50 
 
 50 
 
 5152 
 
 53 
 
 53 
 
 375 
 
 7807 
 
 7877 
 
 7948 
 
 8018 
 
 8088 
 
 8159 
 
 8229 
 
 8299 
 
 8370 
 
 8440 
 
 75 
 
 50 
 
 51 
 
 5-2 52 53J54 
 
 376 
 
 8511 
 
 8581 
 
 8651 
 
 8722 
 
 8792 
 
 8863 
 
 8933 
 
 9003 
 
 9074 
 
 9144 
 
 76 
 
 51 
 
 52 
 
 52 53 54 55 
 
 377 
 
 9215 
 
 9285 
 
 9356 
 
 9426 
 
 9496 
 
 9567 
 
 9637 
 
 9708 
 
 9778 
 
 9849 
 
 77 
 
 52 52 53J54I55J55 
 
 378 
 
 9919 
 
 9990 
 
 0060 
 
 0131 
 
 0201 
 
 0272 
 
 0342 
 
 0413 
 
 0483 
 
 0554 
 
 78J52 
 
 53 54|55|55|56 
 
 379 
 
 53 0624 
 
 0695 
 
 0765 
 
 0836 
 
 0906 
 
 0977 
 
 1048 
 
 1118 
 
 1189 
 
 1259 
 
 7953 
 
 54 
 
 55 
 
 5556 
 
 57 
 
 380 
 
 53 1330 
 
 1400 
 
 1471 
 
 1542 
 
 1612 
 
 1683 
 
 1753 
 
 1824 
 
 1895 
 
 1965 
 
 8054 
 
 54 
 
 55 
 
 56157 
 
 58 
 
 '381 
 
 2036 
 
 2106 
 
 2177 
 
 2248 
 
 2318 
 
 2389 
 
 2460 
 
 2530 
 
 2601 
 
 2672 
 
 81 
 
 54 
 
 55 
 
 56;57 
 
 58 
 
 58 
 
 382 
 
 2742 
 
 2813 
 
 2884 
 
 2954 
 
 3025 
 
 3096 
 
 3166 
 
 3237 
 
 3308 
 
 3379 
 
 8255 
 
 56 
 
 5757 
 
 58 
 
 59 
 
 383 
 
 3449 
 
 3520 
 
 3591 
 
 3661 
 
 3732 
 
 3803 
 
 3874 
 
 3944 
 
 4015 
 
 4086 
 
 83156 56 
 
 5758 
 
 59 
 
 60 
 
 384 
 
 4157 
 
 4228 
 
 4298 
 
 4369 
 
 4440 
 
 4511 
 
 4581 
 
 4652 
 
 4723 
 
 4794 
 
 845657 
 
 58J59 
 
 60 
 
 60 
 
 385 
 
 4865 
 
 4936 
 
 5006 
 
 5077 
 
 5148 
 
 5219 
 
 5290 
 
 5361 
 
 5431 
 
 5502 
 
 8557 
 
 58 59 60 
 
 eoiei 
 
 386 
 
 5573 
 
 5644 
 
 5715 
 
 5786 
 
 5857 
 
 5927 
 
 5998 
 
 6069 
 
 6140 
 
 6211 
 
 86 
 
 5858596061 
 
 62 
 
 387 
 
 6282 
 
 6353 
 
 6424 
 
 6495 
 
 6566 
 
 6637 
 
 6708 
 
 6778 
 
 6849 
 
 6920 
 
 87 
 
 58!59!60j61 
 
 62 
 
 63 
 
 388 
 
 6991 
 
 7062 
 
 7133 
 
 7204 
 
 7275 
 
 7346 
 
 7417 
 
 7488 
 
 7559 
 
 7630 
 
 88|59|60'61 
 
 6262 
 
 63 
 
 389 
 
 7701 
 
 7772 
 
 7843 
 
 7914 
 
 7985 
 
 8056 
 
 8127 
 
 8198 
 
 8269 
 
 8340 
 
 89 
 
 6061 
 
 61 
 
 62 
 
 63 
 
 64 
 
 390 
 
 538411 
 
 8483 
 
 8554 
 
 8625 
 
 8696 
 
 8767 
 
 8838 
 
 8909 
 
 8980 
 
 9051 
 
 90 
 
 6061 
 
 6263 
 
 64 
 
 65 
 
 391 
 
 9122 
 
 9193 
 
 9264 
 
 9336 
 
 9407 
 
 9478 9549 
 
 9620 
 
 9691 
 
 9762 
 
 91 
 
 6162 
 
 636465 
 
 66 
 
 392 
 
 9833 
 
 9905 
 
 9976 
 
 0047 
 
 0118 
 
 0189 
 
 0260 
 
 0332 
 
 o403 
 
 o474 
 
 92 
 
 6263636465 
 
 66 
 
 393 
 
 54 0545 
 
 0616 
 
 0688 
 
 0759 
 
 0830 
 
 0901 
 
 0972 
 
 1044 
 
 1115 
 
 1186 
 
 93 
 
 6263 
 
 646566 
 
 67 
 
 394 
 
 1257 
 
 1329 
 
 1400 
 
 1471 
 
 1542 
 
 1614 
 
 1685 
 
 1756 
 
 1828 
 
 1899 
 
 94 
 
 6364 
 
 65 66 67 
 
 68 
 
 395 
 
 1970 
 
 2041 
 
 2113 
 
 2164 
 
 2255 
 
 2327 
 
 2398 
 
 2469 
 
 2541 
 
 2612 
 
 95 
 
 64 65 ee'ee 67 
 
 68 
 
 396 
 
 2683 
 
 2755 
 
 2826 
 
 2897 
 
 2969 
 
 3040 
 
 3111 
 
 3183 
 
 3254 
 
 3325 
 
 96 
 
 64i6566'67! 
 
 68 
 
 69 
 
 397 
 
 3397 
 
 3468 
 
 3540 
 
 3611 
 
 3682 
 
 3754 
 
 3825 
 
 3897 
 
 3968 
 
 4039 
 
 976566!67;6869 
 
 70 
 
 398 
 
 4111 
 
 4182 
 
 4254 
 
 4325 
 
 4397 
 
 4468 
 
 4540 
 
 4611 
 
 4682 
 
 4754 
 
 98666768,69170 
 
 71 
 
 399 
 
 4825 
 
 4897 
 
 4968 
 
 5040 
 
 5111 
 
 5183 
 
 5254 
 
 5326 
 
 5397 
 
 5469 
 
 99 
 
 66 
 
 67 68 
 
 39 
 
 70 
 
 71 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 6768 
 
 69 
 
 70 
 
 71 
 
 72 
 
 
 
 
 
 
 
 
 
 
 
 
 
TABLE I. 
 
 Log 
 
 Log(l-fx) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 0-4-00 
 
 54 5540 
 
 5612 
 
 5684 
 
 5755 
 
 5827 
 
 5898 
 
 5970 
 
 6041 
 
 6113 
 
 6184 
 
 00 
 
 
 
 
 
 
 
 ojo 
 
 401 
 
 6256 
 
 6328 
 
 6399 
 
 6471 
 
 6542 
 
 66 14| 6685 
 
 6757 
 
 6829 
 
 6900 
 
 01 1 
 
 1 
 
 1 
 
 1 
 
 1 1 
 
 402 
 
 6972 
 
 7044 
 
 7115 
 
 7187 
 
 7258 
 
 7330 7402 
 
 7473 
 
 7545 
 
 7617 
 
 02 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 
 403 
 
 7688 
 
 7760 
 
 7832 
 
 7903 
 
 7975 
 
 8047 8118 
 
 8190 
 
 8262 
 
 8334 
 
 03 
 
 2 2 
 
 2 
 
 2 
 
 2 
 
 
 404 
 
 8405 
 
 8477 
 
 8549 
 
 8620 
 
 8692 
 
 8764 8836 
 
 8907 
 
 8979 
 
 9051 
 
 04 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 
 405 
 
 9123 
 
 9194 
 
 9266 
 
 9338 
 
 9410 
 
 9481 
 
 9553 
 
 9625 
 
 9697 
 
 9769 
 
 05 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 
 406 
 
 9840 
 
 9912 
 
 9984 
 
 0056 
 
 o!28 
 
 o!99 
 
 0271 
 
 0343 
 
 0415 
 
 o487 
 
 06 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 
 407 
 
 55 0559 
 
 0631 
 
 0702 
 
 0774 
 
 0846 
 
 0918 
 
 099U 
 
 1062 
 
 1134 
 
 1206 
 
 07 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 
 408 
 
 1277 
 
 1349 
 
 1421 ! 1493 
 
 1565 
 
 1637 
 
 1709 
 
 1781 
 
 1853 
 
 1925 
 
 08 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 
 409 
 
 1997 
 
 2069 
 
 2141 
 
 2213 
 
 2284 
 
 2356 
 
 2428 
 
 2500 
 
 2572 
 
 2644 
 
 09 
 
 6 
 
 6 
 
 7 
 
 7 
 
 7 
 
 
 410 
 
 552716 
 
 2788 
 
 2860 
 
 2932 
 
 3004 
 
 3076 
 
 3148 
 
 3220 
 
 3292 
 
 3364 
 
 10 
 
 7 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 411 
 
 3437 
 
 3509 
 
 3581 
 
 3653 
 
 3725 
 
 3797 
 
 3869 
 
 3941 
 
 4013 
 
 4085 
 
 11 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 412 
 
 4157 
 
 4229 
 
 4301 
 
 4373 
 
 4446 
 
 4518 
 
 4590 
 
 4662 
 
 4734 
 
 4806 
 
 12 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 413 
 
 4878 
 
 4950 
 
 5022 
 
 5095 
 
 5167 
 
 5239 
 
 5311 
 
 5383 
 
 5455 
 
 5528 
 
 13 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 414 
 
 5600 
 
 5672 
 
 5744 
 
 5816 
 
 5888 
 
 5961 
 
 6033 
 
 6105 
 
 6177 
 
 6250 
 
 14 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 415 
 
 6322 
 
 6394 
 
 6466 
 
 6538 
 
 6611 
 
 6683 
 
 6755 
 
 6827 
 
 6900 
 
 6972 
 
 15 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 416 
 
 7044 
 
 7116 
 
 7189 
 
 7261 
 
 7333 
 
 7406 
 
 7478 
 
 7550 
 
 7623 
 
 7695 
 
 16 
 
 11 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 417 
 
 7767 
 
 7839 
 
 7912 
 
 7984 
 
 8056 
 
 8129 
 
 8201 
 
 8273 
 
 8346 
 
 8418 
 
 17 
 
 12|12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 418 
 
 8491 
 
 8563 
 
 8635 
 
 8708 
 
 8780 
 
 8852 
 
 8925 
 
 8997 
 
 9070 
 
 9142 
 
 18 
 
 13 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 419 
 
 9214 
 
 9287 
 
 9359 
 
 9432 
 
 9504 
 
 9576 
 
 9649 
 
 9721 
 
 9794 
 
 9866 
 
 19 
 
 13 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 420 
 
 55 9939 
 
 oOll 
 
 0084 
 
 0156 
 
 0229 
 
 0301 
 
 0373 
 
 0446 
 
 0518 
 
 0591 
 
 20 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 15 
 
 421 
 
 56 0663 
 
 0736 
 
 0808 
 
 0881 
 
 0953 
 
 1026 
 
 1099 
 
 1171 
 
 1244 
 
 1316 
 
 21 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 422 
 
 1389 
 
 1461 
 
 1534 
 
 1606 
 
 1679 
 
 1751 
 
 1824 
 
 1897 
 
 1969 
 
 2042 
 
 22 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 17 
 
 423 
 
 2114 
 
 2187 
 
 2260 
 
 2332 
 
 2405 
 
 2477 
 
 2550 
 
 2623 
 
 2695 
 
 2768 
 
 23 
 
 16 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 424 
 
 2841 
 
 2913 
 
 2986 
 
 3058 
 
 3131 
 
 3204 
 
 3276 
 
 3349 
 
 3422 
 
 3494 
 
 24 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 18 
 
 425 
 
 3567 
 
 3640 
 
 3712 
 
 3785 
 
 3858 
 
 3931 
 
 4003 
 
 4076 
 
 4149 
 
 4221 
 
 25 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 19 
 
 426 
 
 4294 
 
 4367 
 
 4440 
 
 4512 
 
 4585 
 
 4658 
 
 4731 
 
 4803 
 
 4876 
 
 4949 
 
 26 
 
 18 
 
 19 
 
 19 
 
 19 
 
 20 
 
 20 
 
 427 
 
 5022 
 
 5094 
 
 5167 
 
 5240 
 
 5313 
 
 5386 
 
 5458 
 
 5531 
 
 5604 
 
 5677 
 
 27 
 
 19192020 
 
 20 
 
 21 
 
 428 
 
 5750 
 
 5822 
 
 5895 
 
 5968 
 
 6041 
 
 6114 
 
 6187 
 
 6260 
 
 6332 
 
 6405 
 
 28 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 21 
 
 429 
 
 6478 
 
 6551 
 
 6624 
 
 6697 
 
 6770 
 
 6842 
 
 6915 
 
 6988 
 
 7061 
 
 7134 
 
 29J21 
 
 21 
 
 21 
 
 21 
 
 22 
 
 22 
 
 430 
 
 56 7207 
 
 7280 
 
 7353 
 
 7426 
 
 7499 
 
 7572 
 
 7645 
 
 7717 
 
 7790 
 
 7863 
 
 3021 
 
 22 22 
 
 22 
 
 22 
 
 23 
 
 431 
 
 7936 
 
 8009 
 
 8082 
 
 8155 
 
 8228 
 
 8301 
 
 8374 
 
 8447 
 
 852( 
 
 8593 
 
 3 1 22 22 23 
 
 23 
 
 23 
 
 24 
 
 432 
 433 
 
 8666 
 9396 
 
 8739 
 9469 
 
 8812 
 9542 
 
 8885 
 9616 
 
 8958 
 9689 
 
 9031 
 9762 
 
 9104 
 9835 
 
 9177 
 9908 
 
 9250 
 9981 
 
 9323 
 0054 
 
 32 23 23 23 24 
 
 3:} 23 24I24J24 
 
 24 
 25 
 
 24 
 
 25 
 
 434 
 
 57 0127 
 
 0200 
 
 0273 
 
 0346 
 
 0419 
 
 0493 
 
 0566 
 
 0639 
 
 0712 
 
 0785 
 
 34 
 
 24 24 25 
 
 25 
 
 26 
 
 26 
 
 435 
 
 0858 
 
 0931 
 
 1004 
 
 1078 
 
 1151 
 
 1224 
 
 1297 
 
 1370 
 
 1443 
 
 1517 
 
 35 
 
 25 25 26 
 
 26 
 
 26 
 
 27 
 
 436 
 
 1590 
 
 1663 
 
 1736 
 
 1809 
 
 1883 
 
 1956 
 
 2029 
 
 2102 
 
 2175 
 
 2249 
 
 36 
 
 26 26 26 
 
 27 
 
 27 
 
 27 
 
 437 
 
 2322 
 
 2395 
 
 2468 
 
 2542 
 
 2615 
 
 2688 
 
 2761 
 
 2835 
 
 2908 
 
 2981 
 
 37 
 
 26 27 27 
 
 27 
 
 28 
 
 28 
 
 438 
 
 3054 
 
 3128 
 
 3201 
 
 3274 
 
 3347 
 
 3421 
 
 3494 
 
 3567 
 
 3641 
 
 3714 
 
 38 
 
 27J2728 
 
 28 
 
 28 
 
 29 
 
 439 
 
 3787 
 
 3861 
 
 3934 
 
 4007 
 
 4081 
 
 4154 
 
 4227 
 
 4301 
 
 4374 
 
 4447 
 
 39 
 
 28J2828 
 
 29 
 
 29 
 
 30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 440 
 
 574521 
 
 4594 
 
 4667 
 
 4741 
 
 4814 
 
 4888 
 
 4961 
 
 5034 
 
 5108 
 
 5181 
 
 40 
 
 28 29 29 
 
 30 
 
 30 
 
 30 
 
 441 
 
 5255 
 
 5328 
 
 5401 
 
 5475 
 
 5548 
 
 5622 
 
 5695 
 
 5769 
 
 5842 
 
 5915 
 
 41 
 
 29 30 30 
 
 30 
 
 31 
 
 31 
 
 442 
 
 5989 
 
 6062 
 
 6136 
 
 6209 
 
 6283 
 
 6356 
 
 6430 
 
 6503 
 
 6577 
 
 6650 
 
 42 
 
 30 
 
 30 
 
 31 
 
 31 
 
 32 
 
 32 
 
 44:3 
 
 6724 
 
 6797 
 
 6871 
 
 6944 
 
 7018 
 
 7091 
 
 7165 
 
 7238 
 
 7312 
 
 7385 
 
 43 
 
 31 
 
 31 
 
 31 
 
 32 
 
 32 
 
 33 
 
 444 
 
 7459 
 
 7532 
 
 7606 
 
 7680 
 
 7753 
 
 7827 
 
 7900 
 
 7974 
 
 8047 
 
 8121 
 
 44 
 
 313232 
 
 33 
 
 33 
 
 33 
 
 445 
 
 8195 
 
 8268 
 
 8342 
 
 8415 
 
 8489 
 
 8563 
 
 8636 
 
 8710 
 
 8783 
 
 8857 
 
 45 
 
 32 32 33 
 
 '33 
 
 34 
 
 34 
 
 446 
 
 8931 
 
 9004 
 
 9078 
 
 9152 
 
 9225 
 
 9299 
 
 9372 
 
 9446 
 
 9520 
 
 9593 
 
 46 
 
 333334 
 
 34 
 
 34 
 
 35 
 
 447 
 
 9667 
 
 9741 
 
 9815 
 
 9888 
 
 9962 
 
 0036 
 
 o!09 
 
 0183 
 
 0257 
 
 0330 
 
 47 
 
 33 34 34 
 
 35 
 
 35 
 
 36 
 
 448 
 
 58 0404 
 
 0478 
 
 0552 
 
 0625 
 
 0699 
 
 0773 
 
 0847 
 
 0920 
 
 0994 
 
 1068 
 
 48 
 
 31 35 35 
 
 36 
 
 36 
 
 36 
 
 449 
 
 1142 
 
 1215 
 
 1289 
 
 1363 
 
 1437 
 
 1510 
 
 1584 
 
 1658 
 
 1732 
 
 1806 
 
 49 
 
 35 
 
 35 
 
 36 
 
 36 
 
 37 
 
 37 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 7172 
 
 73 
 
 74 
 
 75 
 
 76 
 
TABLE I. 
 
 Log 
 
 LogO-HO 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 71 
 
 7S 
 
 '7374 
 
 -75J76 
 
 0-450 
 
 58 1875 
 
 195c 
 
 $ 2027 
 
 ' 210 
 
 217 
 
 224 
 
 232$ 
 
 > 2396 
 
 > 2470 
 
 2544 
 
 50 36 36 37 
 
 37 
 
 3838 
 
 451 
 
 2616 
 
 269$ 
 
 \ 2766 
 
 283 
 
 291 
 
 2987 
 
 3061 313* 
 
 3209 
 
 3283 
 
 51J363737 
 
 38 
 
 3839 
 
 452 
 
 3357 
 
 3430 3504 
 
 357 
 
 365 
 
 3726 
 
 3800 3874 
 
 < 3948 
 
 4022 
 
 52 37,37 3^ 
 
 38 39 40 
 
 453 
 
 4096 
 
 4170 4244 
 
 431 
 
 439 
 
 4466 
 
 454C 
 
 ) 4614 
 
 < 4688 
 
 4761 
 
 53 38,38 3 
 
 3940 
 
 40 
 
 454 
 
 4835 
 
 4909 
 
 4983 
 
 505 
 
 513 
 
 5205 
 
 527< 
 
 > 5354 
 
 5428 
 
 5502 
 
 54 38 39 3 
 
 4C 
 
 40 
 
 41 
 
 455 
 
 5576 
 
 565C 
 
 5724 
 
 579 
 
 587 
 
 5946 
 
 602C 
 
 > 6094 
 
 6168 
 
 6242 
 
 3.5 39 40 4C 
 
 41 
 
 41 
 
 42 
 
 456 
 
 6316 
 
 6390 
 
 6464 
 
 653 
 
 661 
 
 6687 
 
 6761 
 
 6835 
 
 6909 
 
 6983 
 
 56404014141 
 
 42 
 
 43 
 
 457 
 
 7057 
 
 713 
 
 7205 
 
 728 
 
 735 
 
 7428 
 
 750$ 
 
 ' 7576 
 
 7650 
 
 7724 
 
 5740414242 
 
 43 
 
 43 
 
 458 
 
 7799 
 
 7873 
 
 7947 
 
 802 
 
 809 
 
 8169 
 
 8244 
 
 < 8318 
 
 8392 
 
 8466 
 
 5841424$ 
 
 45 
 
 44 
 
 44 
 
 459 
 
 8540 
 
 8615 
 
 8689 
 
 8763 
 
 883 
 
 8912 
 
 8986 
 
 9060 
 
 9134 
 
 9208 
 
 59 42 42 4: 
 
 44 
 
 44 
 
 45 
 
 
 
 
 
 
 
 
 
 
 
 
 !-- 
 60 43|43|44 
 
 44 
 
 45 
 
 46 
 
 460 
 
 58 9283 
 
 9357 
 
 9431 
 
 9506 
 
 9580 
 
 9654 
 
 9728 
 
 9803 
 
 9877 
 
 9951 
 
 461 
 
 59 0025 
 
 0100 
 
 0174 
 
 0248 
 
 0323 
 
 0397 
 
 0471 
 
 0546 0620 
 
 0694 
 
 6143'444^ 
 
 45 
 
 46 
 
 46 
 
 462 
 
 0769 
 
 0843 
 
 0917 
 
 0992 
 
 1066 
 
 1140 
 
 1215 
 
 1289 1364 
 
 1438 
 
 62 44145 45<46|46 
 
 47 
 
 463 
 
 1512 
 
 1587 
 
 1661 
 
 1736 
 
 1810 
 
 1884 
 
 1959 2033 2108 
 
 2182 
 
 6345454647 
 
 4-7 
 
 48 
 
 464 
 
 2256 
 
 2331 
 
 2405 
 
 2480 
 
 2554 
 
 2629 
 
 2703 
 
 2778 
 
 2852 
 
 2926 
 
 64 45 46 
 
 47 
 
 47 
 
 4b 
 
 49 
 
 465 
 
 3001 
 
 3075 
 
 3150 
 
 3224 
 
 3299 
 
 3373 
 
 3448 
 
 3522 
 
 3597 
 
 3671 
 
 65 46 47 
 
 47 
 
 48 
 
 49 
 
 49 
 
 466 
 
 3746 
 
 3820 
 
 3895 
 
 3969 
 
 4044 
 
 4119 4193 
 
 4268 4342 
 
 4417 
 
 36 47 48 
 
 4S 
 
 4f 
 
 50 
 
 50 
 
 467 
 
 4491 
 
 4566 
 
 4640 
 
 4715 
 
 4790 
 
 4864 
 
 4939 
 
 5013 5088 
 
 5162 
 
 57 48 48 4S 
 
 50 
 
 50 
 
 5 
 
 468 
 
 5237 
 
 5312 
 
 5386 
 
 5461 
 
 5536 
 
 5610 
 
 5685 
 
 5759 5834 
 
 5909 
 
 3848 
 
 495 
 
 50 
 
 51 
 
 52 
 
 469 
 
 5983 
 
 6058 
 
 6133 
 
 6207 
 
 6282 
 
 6357 
 
 6431 
 
 6506 
 
 6581 
 
 6655 
 
 5949 
 
 50 
 
 5 
 
 51 
 
 52 
 
 52 
 
 470 
 
 59 6730 
 
 6805 
 
 6879 
 
 6954 
 
 7029 
 
 7104 
 
 7178 
 
 7253 
 
 7328 
 
 7402 
 
 7050 
 
 505 
 
 52 
 
 52 
 
 53 
 
 471 
 
 7477 
 
 7552 
 
 7627 
 
 7701 
 
 7776 
 
 7851 
 
 7926 
 
 8000 
 
 8075 
 
 8150 
 
 7150 
 
 51|5 
 
 53 
 
 53 
 
 54 
 
 472 
 
 8225 
 
 8300 
 
 8374 
 
 8449 
 
 8524 
 
 8599 
 
 8673 
 
 8748 
 
 8823 
 
 8898 
 
 7251 
 
 52 
 
 5 
 
 53 
 
 54 
 
 55 
 
 473 
 
 8973 
 
 9048 
 
 9122 
 
 9197 
 
 9272 
 
 9347 
 
 9422 
 
 9497 
 
 9571 
 
 9646 
 
 7352 
 
 53 
 
 5 
 
 5 
 
 55|55 
 
 474 
 
 9721 
 
 9796 
 
 9871 
 
 9946 
 
 0021 
 
 0096 
 
 0170 
 
 0245 
 
 0320 
 
 0395 
 
 7453 
 
 53 
 
 5 
 
 5 
 
 56 
 
 56 
 
 475 
 
 60 0470 
 
 0545 
 
 0620 
 
 0695 
 
 0770 
 
 0845 
 
 0920 
 
 0994 
 
 1069 
 
 1144 
 
 7553 
 
 54 
 
 5 
 
 5 
 
 56 
 
 57 
 
 476 
 
 1219 
 
 1294 
 
 1369 
 
 1444 
 
 1519 
 
 1594 
 
 1669 
 
 1744 
 
 1819 
 
 1894 
 
 7654 
 
 55 
 
 5 
 
 5 
 
 57 
 
 58 
 
 477 
 
 1969 
 
 2044 
 
 2119 
 
 2194 
 
 2269 
 
 2344 
 
 2419 
 
 2494 
 
 2569 
 
 2644 
 
 77 55 
 
 55 
 
 5 
 
 5 
 
 58 
 
 59 
 
 478 
 
 2719 
 
 2794 
 
 2869 
 
 2944 
 
 3019 
 
 3094 
 
 3170 
 
 3245 
 
 3320 
 
 3395 
 
 r855 
 
 56 
 
 5 
 
 5 
 
 58 
 
 59 
 
 479 
 
 3470 
 
 3545 
 
 3620 
 
 3695 
 
 3770 
 
 3845 
 
 3920 
 
 3995 
 
 4071 
 
 4146 
 
 r956 
 
 57 
 
 5 
 
 5 
 
 59 
 
 60 
 
 480 
 
 604221 
 
 4296 
 
 4371 
 
 4446 
 
 4521 
 
 4597 
 
 4672 
 
 4747 
 
 4822 
 
 4897 
 
 JO 57 
 
 58 
 
 58 
 
 5 
 
 60 
 
 61 
 
 481 
 
 4972 
 
 5047 
 
 5123 
 
 5198 
 
 5273 
 
 5348 5423 
 
 5499 
 
 5574 
 
 5649 
 
 $1585859 
 
 60 
 
 61 
 
 62 
 
 482 
 
 5724 
 
 5799 
 
 5875 
 
 5950 
 
 6025 
 
 6100 6176 
 
 6251 
 
 6326 
 
 6401 
 
 *2 58 59 60 
 
 6 
 
 6262 
 
 483 
 
 6476 
 
 6552 
 
 6627 
 
 6702 
 
 6778 
 
 6853 
 
 6928 
 
 7003 
 
 7079 
 
 7154 
 
 !3 59 60 61 
 
 6 
 
 6263 
 
 484 
 
 7229 
 
 7305 
 
 7380 
 
 7455 
 
 7530 
 
 7606 7681 
 
 7756 
 
 7832 
 
 7907 
 
 46060 
 
 61 
 
 62 
 
 63 
 
 64 
 
 485 
 
 7982 
 
 8058 
 
 8133 
 
 8208 
 
 8284 
 
 8359 8435 
 
 8510 
 
 8585 
 
 8661 
 
 5 60 61 
 
 62 
 
 5c 
 
 64 
 
 65 
 
 486 
 
 8736 8811 
 
 8887 
 
 8962 
 
 9038 
 
 9113 9188 
 
 9264 
 
 9339 
 
 9415 
 
 66162:63 
 
 j- 
 
 64 
 
 65 
 
 487 
 
 9490 9565 
 
 9641 
 
 9716 
 
 9792 
 
 9867 9943 
 
 0018 
 
 0094 
 
 0169 
 
 7|62 63 64 
 
 >^ 
 
 65 
 
 66 
 
 488 
 
 61 0244 
 
 0320 
 
 0395 
 
 0471 
 
 0546 
 
 0622 0697 0773 
 
 0848 
 
 0924 
 
 8626364 
 
 j^ 
 
 66 
 
 67 
 
 489 
 
 0999 
 
 1075 
 
 1150 
 
 1226 
 
 1301 
 
 377 
 
 1453 
 
 1528 
 
 1604 
 
 1679 
 
 9 63^4:65 
 
 66 
 
 67 
 
 58 
 
 490 
 
 61 1755 
 
 1830 
 
 1906 
 
 1981 
 
 2057 
 
 2132 
 
 2208 
 
 22841 2359 
 
 2435 
 
 64J65 66 
 
 67 
 
 68 
 
 38 
 
 491 
 
 2510 
 
 2586 
 
 2662 
 
 2737 
 
 2813 
 
 888 
 
 2964 
 
 3040 3115 
 
 3191 
 
 1 65 ee'ee 
 
 T 
 
 68 
 
 39 
 
 492 
 
 3267 
 
 3342 
 
 3418 
 
 3493 
 
 3569 
 
 645 
 
 3720 
 
 3796 3872 
 
 3947 
 
 2 6566,67 
 
 8 
 
 59 
 
 70 
 
 493 
 
 4023 
 
 4099 
 
 4175 
 
 4250 
 
 4326 
 
 402 
 
 4477 
 
 4553 
 
 4629 
 
 4704 
 
 3 66:67 68 
 
 9 
 
 
 
 n 
 
 494 
 
 4780 
 
 4856 
 
 4932 
 
 5007 
 
 5083 
 
 159 
 
 5235 5310 
 
 5386 
 
 5462 
 
 46768J69 
 
 
 
 
 
 n 
 
 495 
 
 5538 
 
 5613 
 
 5689 
 
 5765 
 
 5841 
 
 916 
 
 5992 
 
 6068 
 
 6144 
 
 6220 
 
 567( 
 
 )8'69 
 
 
 
 1 
 
 >2 
 
 496 
 
 6295 
 
 6371 
 
 6447 
 
 6523 
 
 6599 
 
 675 
 
 6750 
 
 6826 
 
 6902 
 
 6978 
 
 6 68 69 70 
 
 1 
 
 2 
 
 '3 
 
 497 
 
 7054 
 
 7130 
 
 7205 
 
 7281 
 
 357 
 
 433 
 
 7509 
 
 7585 
 
 7661 
 
 7737 
 
 7 69 70 71 
 
 2 
 
 374 
 
 498 
 
 7812 
 
 7888 
 
 7964 
 
 8040 
 
 8116 
 
 192 
 
 8268 
 
 8344 
 
 8420 
 
 8496 
 
 37071' 
 
 ?2 
 
 3 
 
 474 
 
 499 
 
 8572 
 
 8647 
 
 8723 
 
 8799 
 
 8875 
 
 951 
 
 9027 
 
 9103 
 
 9179 
 
 9255 
 
 37071 ' 
 
 i2 
 
 3 
 
 475 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 6 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 , 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 71 72 f 
 
 3 
 
 4 
 
 $ 
 
 11 
 
TABLE I. 
 
 Lo 
 
 Log (1+07) 
 
 Pro. Par 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 76 
 
 77 
 
 78 
 
 0-500 
 
 0-61 9331 
 
 9407 
 
 9483 
 
 9559 
 
 9635 
 
 9711 
 
 9787 
 
 9863 
 
 9939 
 
 0015 
 
 00 
 
 
 
 
 
 
 
 501 
 
 620091 
 
 0167 
 
 0243 
 
 0319 
 
 0395 
 
 0471 
 
 0547 
 
 0623 
 
 0699 
 
 0775 
 
 01 
 
 1 
 
 1 
 
 1 
 
 502 
 
 0851 
 
 0927 
 
 1004 
 
 1080 
 
 1156 
 
 1232 
 
 1308 
 
 1384 
 
 1460 
 
 1536 
 
 02 
 
 2 
 
 2 
 
 2 
 
 503 
 
 1612 
 
 1688 
 
 1764 
 
 1840 
 
 1917 
 
 1993 
 
 2069 
 
 2145 
 
 2221 
 
 2297 
 
 03 
 
 2 
 
 2 
 
 2 
 
 504 
 
 2373 
 
 2450 
 
 2526 
 
 2602 
 
 2678 
 
 2754 
 
 2830 
 
 2906 
 
 2983 
 
 3059 
 
 04 
 
 3 
 
 3 
 
 3 
 
 505 
 
 3135 
 
 3211 
 
 3287 
 
 3364 
 
 3440 
 
 3516 
 
 3592 
 
 3668 
 
 3745 
 
 3821 
 
 05 
 
 4 
 
 4 
 
 4 
 
 506 
 
 3897 
 
 3973 
 
 4050 
 
 4126 
 
 4202 
 
 4278 
 
 4355 
 
 4431 
 
 4507 
 
 4583 
 
 06 
 
 5 
 
 5 
 
 
 o 
 
 507 
 
 4660 
 
 4736 
 
 4812 
 
 4888 
 
 4965 
 
 5041 
 
 5117 
 
 5194 
 
 5270 
 
 5346 
 
 07 
 
 5 
 
 5 
 
 K 
 
 508 
 
 5422 
 
 5499 
 
 5575 
 
 5651 
 
 5728 
 
 5804 
 
 5880 
 
 5957 
 
 6033 
 
 6109 
 
 08 
 
 6 
 
 6 
 
 6 
 
 509 
 
 6186 
 
 6262 
 
 6338 
 
 6415 
 
 6491 
 
 6568 
 
 6644 
 
 6720 
 
 6797 
 
 6873 
 
 09 
 
 7 
 
 7 
 
 7 
 
 510 
 
 62 6949 
 
 7026 
 
 7102 
 
 7179 
 
 7255 
 
 7331 
 
 7408 
 
 7484 
 
 7561 
 
 7637 
 
 10 
 
 8 
 
 8 
 
 8 
 
 511 
 
 7714 
 
 7790 
 
 7866 
 
 7943 
 
 8019 
 
 8096 
 
 8172 
 
 8249 
 
 8325 
 
 8402 
 
 11 
 
 8 
 
 8 
 
 8 
 
 512 
 
 8478 
 
 8555 
 
 8631 
 
 8708 
 
 8784 
 
 8861 
 
 8937 
 
 9014 
 
 9090 
 
 9167 
 
 12 
 
 9 
 
 9 
 
 9 
 
 513 
 
 9243 
 
 9320 
 
 9396 
 
 9473 
 
 9549 
 
 9626 
 
 9702 
 
 9779 
 
 9855 
 
 9932 
 
 13 
 
 10 
 
 10 
 
 10 
 
 514 
 
 63 0008 
 
 0085 
 
 0162 
 
 0238 
 
 0315 
 
 0391 
 
 0468 
 
 0544 
 
 0621 
 
 0698 
 
 14 
 
 11 
 
 11 
 
 11 
 
 515 
 
 0774 
 
 0851 
 
 0927 
 
 1004 
 
 1081 
 
 1157 
 
 1234 
 
 1311 
 
 1387 
 
 1464 
 
 15 
 
 11 
 
 12 
 
 12 
 
 516 
 
 1540 
 
 1617 
 
 1694 
 
 1770 
 
 1847 
 
 1924 
 
 2000 
 
 2077 
 
 2154 
 
 2230 
 
 16 
 
 12 
 
 12 
 
 12 
 
 517 
 
 2307 
 
 2384 
 
 2460 
 
 2537 
 
 2614 
 
 2691 
 
 2767 
 
 2844 
 
 2921 
 
 2997 
 
 17 
 
 13 
 
 13 
 
 13 
 
 518 
 
 3074 
 
 3151 
 
 3228 
 
 3304 
 
 3381 
 
 3458 
 
 3535 
 
 3611 
 
 3688 
 
 3765 
 
 18 
 
 14 
 
 14 
 
 14 
 
 519 
 
 3842 
 
 3918 
 
 3995 
 
 4072 
 
 4149 
 
 4225 
 
 4302 
 
 4379 
 
 4456 
 
 4533 
 
 19 
 
 14 
 
 15 
 
 15 
 
 520 
 
 63 4609 
 
 4686 
 
 4763 
 
 4840 
 
 4917 
 
 4993 
 
 5070 
 
 5147 
 
 5224 
 
 5301 
 
 20 
 
 15 
 
 15 
 
 16 
 
 521 
 
 5378 
 
 5454 
 
 5531 
 
 5608 
 
 5685 
 
 5762 
 
 5839 
 
 5916 
 
 5993 
 
 6069 
 
 21 
 
 16 
 
 16 
 
 16 
 
 -522 
 
 6146 
 
 6223 
 
 6300 
 
 6377 
 
 6454 
 
 6531 
 
 6608 
 
 6685 
 
 6762 
 
 6838 
 
 22 
 
 17 
 
 17 
 
 17 
 
 523 
 
 6915 
 
 6992 
 
 7069 
 
 7146 
 
 7223 
 
 7300 
 
 7377 
 
 7454 
 
 7531 
 
 7608 
 
 23 
 
 17 
 
 18 
 
 18 
 
 524 
 
 7685 
 
 7762 
 
 7839 
 
 7916 
 
 7993 
 
 8070 
 
 8147 
 
 8224 
 
 8301 
 
 8378 
 
 24 
 
 18 
 
 18 
 
 19 
 
 525 
 
 8455 
 
 8532 
 
 8609 
 
 8686 
 
 8763 
 
 8840 
 
 8917 
 
 8994 
 
 9071 
 
 9148 
 
 25 
 
 19 
 
 19 
 
 20 
 
 526 
 
 9225 
 
 9302 
 
 9379 
 
 9456 
 
 9533 
 
 9610 
 
 9687 
 
 9765 
 
 9842 
 
 9919 
 
 26 
 
 20 
 
 20 
 
 20 
 
 527 
 
 9996 
 
 0073 
 
 0150 
 
 0227 
 
 0304 
 
 0381 
 
 0458 
 
 0535 
 
 0613 
 
 0690 
 
 27 
 
 21 
 
 21 
 
 21 
 
 528 
 
 64 0767 
 
 0844 
 
 0921 
 
 0998 
 
 1075 
 
 1153 
 
 1230 
 
 1307 
 
 1384 
 
 1461 
 
 28 
 
 21 
 
 22 
 
 22 
 
 529 
 
 1538 
 
 1616 
 
 1693 
 
 1770 
 
 1847 
 
 1924 
 
 2002 
 
 2079 
 
 2156 
 
 2233 
 
 29 
 
 22 
 
 22 
 
 23 
 
 530 
 
 642310 
 
 2388 
 
 2465 
 
 2542 
 
 2619 
 
 2696 
 
 2774 
 
 2851 
 
 2928 
 
 3005 
 
 30 
 
 23 
 
 23 
 
 23 
 
 531 
 
 3083 
 
 3160 
 
 3237 
 
 3314 
 
 3392 
 
 3469 
 
 3546 
 
 3624 
 
 3701 
 
 3778 
 
 31 
 
 24 
 
 24 
 
 24 
 
 532 
 
 3855 
 
 3933 
 
 4010 
 
 4087 
 
 4165 
 
 4242 
 
 4319 
 
 4397 
 
 4474 
 
 4551 
 
 32 
 
 24 
 
 25 
 
 25 
 
 533 
 
 4629 
 
 4706 
 
 4783 
 
 4861 
 
 4938 
 
 5015 
 
 5093 
 
 5170 
 
 5247 
 
 5325 
 
 33 
 
 25 
 
 25 
 
 26 
 
 534 
 
 5402 
 
 5479 
 
 5557 
 
 5634 
 
 5712 
 
 5789 
 
 5866 
 
 5944 
 
 6021 
 
 6099 
 
 34 
 
 26 
 
 26 
 
 27 
 
 535 
 
 6176 
 
 6253 
 
 6331 
 
 6408 
 
 6486 
 
 6563 
 
 6641 
 
 6718 
 
 6795 
 
 6873 
 
 35 
 
 27 
 
 27 
 
 27 
 
 536 
 
 6950 
 
 7028 
 
 7105 
 
 7183 
 
 7260 
 
 7338 
 
 7415 
 
 7493 
 
 7570 
 
 7648 
 
 36 
 
 27 
 
 28 
 
 28 
 
 537 
 
 7725 
 
 7803 
 
 7880 
 
 7958 
 
 8035 
 
 8113 
 
 8190 
 
 8268 
 
 8345 
 
 8423 
 
 37 
 
 28 
 
 28 
 
 29 
 
 538 
 
 8500 
 
 8578 
 
 8655 
 
 8733 
 
 8810 
 
 8888 
 
 8966 
 
 9043 
 
 9121 
 
 9198 
 
 38 
 
 29 
 
 29 
 
 30 
 
 539 
 
 9276 
 
 9353 
 
 9431 
 
 9509 
 
 9586 
 
 9664 
 
 9741 
 
 9819 
 
 9897 
 
 9974 
 
 39 
 
 30 
 
 30 
 
 30 
 
 540 
 
 65 0052 
 
 0129 
 
 0207 
 
 0285 
 
 0362 
 
 0440 
 
 0518 
 
 0595 
 
 0673 
 
 0750 
 
 40 
 
 30 
 
 31 
 
 31 
 
 541 
 
 0828 
 
 0906 
 
 0983 
 
 1061 
 
 1139 
 
 1216 
 
 1294 
 
 1372 
 
 1450 
 
 1527 
 
 41131 
 
 32 
 
 32 
 
 542 
 
 1605 
 
 1683 
 
 1760 
 
 1838 
 
 1916 
 
 1993 
 
 2071 
 
 2149 
 
 2227 
 
 2304 
 
 4232 
 
 32 
 
 33 
 
 543 
 
 2382 
 
 2460 
 
 2538 
 
 2615 
 
 2693 
 
 2771 
 
 2849 
 
 2926 
 
 3004 
 
 3082 
 
 4333 
 
 33 
 
 34 
 
 544 
 
 3160 
 
 3237 
 
 3315 
 
 3393 
 
 3471 
 
 3549 
 
 3626 
 
 3704 
 
 3782 
 
 3860 
 
 4433 
 
 34 
 
 34 
 
 545 
 
 3938 
 
 4015 
 
 4093 
 
 4171 4249 
 
 4327 
 
 4405 
 
 4482 
 
 4560 
 
 4638 
 
 453435 
 
 35 
 
 548 
 
 4716 
 
 4794 
 
 4872 
 
 4949 
 
 5027 
 
 5105 
 
 5183 
 
 5261 
 
 5339 
 
 5417 
 
 463535 
 
 36 
 
 547 
 
 5495 
 
 5573 
 
 5650 
 
 5728 
 
 5806 
 
 5884 
 
 5962 
 
 6040 
 
 6118 
 
 6196 
 
 47 36 36 
 
 37 
 
 548 
 
 6274 
 
 6352 
 
 6430 
 
 6508 
 
 6586 
 
 6663 
 
 6741 
 
 6819 
 
 6897 
 
 6975 
 
 483637 
 
 37 
 
 549 
 
 7053 
 
 7131 
 
 7209 
 
 7287 
 
 7365 
 
 7443 
 
 7521 
 
 7599 
 
 7677 
 
 7755 
 
 493738 
 
 38 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 76 
 
 77 
 
 78 
 
 12 
 
TABLE I. 
 
 I Lo^ 
 
 ^^^^^^^^^I^^H 
 
 Log (1+s) 
 
 m^*mmm^mnr*maa*~' 
 
 Pro. Parts. 
 
 * 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 76 
 
 77 
 
 78 
 
 79 
 
 80 
 
 IO-550 
 
 0-65 7833 
 
 7911 
 
 7989 
 
 8067 
 
 8145 
 
 8223 
 
 8301 
 
 8379 
 
 8457 
 
 8536 
 
 50 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 1 '551 
 
 8614 
 
 8692 
 
 8770 
 
 8848 
 
 8926 
 
 9004 
 
 9082 
 
 9160 
 
 9238 
 
 9326 
 
 51 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 1 '552 
 
 9394 
 
 9472 
 
 9550 
 
 9629 
 
 9707 
 
 9785 
 
 9863 
 
 9941 
 
 0019 
 
 0097 
 
 52 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 1 '553 
 
 66 0175 
 
 0254 
 
 0332 
 
 0410 
 
 0488 
 
 0566 
 
 0644 
 
 0722 
 
 0801 
 
 0879 
 
 53 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 1 '554 
 
 0957 
 
 1035 
 
 1113 
 
 1191 
 
 1270 
 
 1348 
 
 1426 
 
 1504 
 
 1582 
 
 1661 
 
 54 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 1 '555 
 
 1739 
 
 1817 
 
 1895 
 
 1973 
 
 2052 
 
 2130 
 
 2208 
 
 2286 
 
 2365 
 
 2443 
 
 55 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 1 '556 
 
 2521 
 
 2599 
 
 2678 
 
 2756 
 
 2834 
 
 2912 
 
 2991 
 
 3069 
 
 3147 
 
 3226 
 
 56 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 1 '557 
 
 3304 
 
 3382 
 
 3460 
 
 3539 
 
 3617 
 
 3695 
 
 3774 
 
 3852 
 
 3930 
 
 4009 
 
 57 
 
 43 
 
 44 
 
 44 
 
 45 
 
 46 
 
 1 '558 
 
 4087 
 
 4165 
 
 4244 
 
 4322 
 
 4400 
 
 4479 
 
 4557 
 
 4635 
 
 4714 
 
 4792 
 
 58 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 1 '559 
 
 4870 
 
 4949 
 
 5027 
 
 5105 
 
 5184 
 
 5262 
 
 5341 
 
 5419 
 
 5497 
 
 5576 
 
 59 
 
 45 
 
 45 
 
 46 
 
 47 
 
 47 
 
 1 -560 
 
 66 5654 
 
 5733 
 
 5811 
 
 5889 
 
 5968 
 
 6046 
 
 6125 
 
 6203 
 
 6282 
 
 6360 
 
 60 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 1 -561 
 
 6438 
 
 6517 
 
 6595 
 
 6674 
 
 6752 
 
 6831 
 
 6909 
 
 6988 
 
 7066 
 
 7145 
 
 61 
 
 46 
 
 47 
 
 48 
 
 48 
 
 49 
 
 1 -562 
 
 7223 7302 
 
 7380 
 
 7459 
 
 7537 
 
 7616 
 
 7694 
 
 7773 
 
 7851 
 
 7930 
 
 62 
 
 47 
 
 48 
 
 48 
 
 49 
 
 50 
 
 1 -563 
 
 8008 
 
 8087 
 
 8165 
 
 8244 
 
 8322 
 
 8401 
 
 8479 
 
 8558 
 
 8636 
 
 8715 
 
 63 
 
 48 
 
 49 
 
 49 
 
 50 
 
 50 
 
 564 
 
 8794 
 
 8872 
 
 8951 
 
 9029 
 
 9108 
 
 9186 
 
 9265 
 
 9344 
 
 9422 
 
 9501 
 
 64 
 
 49 
 
 49 
 
 50 
 
 51 
 
 51 
 
 565 
 
 9579 
 
 9658 
 
 9737 
 
 9815 
 
 9894 
 
 9972 
 
 0051 
 
 0130 
 
 0208 
 
 0287 
 
 65 
 
 49 
 
 50 
 
 51 
 
 51 
 
 52 
 
 1 -566 
 
 67 0366 
 
 0444 
 
 0523 
 
 0601 
 
 0680 
 
 0759 
 
 0837 
 
 0916 
 
 0995 
 
 1073 
 
 66 
 
 50 
 
 51 
 
 51 
 
 52 
 
 53 
 
 567 
 
 1152 
 
 1231 
 
 1309 
 
 1388 
 
 1467 
 
 1546 
 
 1624 
 
 1703 
 
 1782 
 
 1860 
 
 67 
 
 51 
 
 52 
 
 52 
 
 53 
 
 54 
 
 1 '568 
 
 1939 
 
 2018 
 
 2097 
 
 2175 
 
 2254 
 
 2333 
 
 2411 
 
 2490 
 
 2569 
 
 2648 
 
 68 
 
 52 
 
 52 
 
 53 
 
 54 
 
 54 
 
 569 
 
 2726 
 
 2805 
 
 2884 
 
 2963 
 
 3041 
 
 3120 
 
 3199 
 
 3278 
 
 3357 
 
 3435 
 
 69 
 
 52 
 
 53 
 
 54 
 
 55 
 
 55 
 
 570 
 
 67 3514 
 
 3593 
 
 3672 
 
 3751 
 
 3829 
 
 3908 
 
 3987 
 
 4066 
 
 4145 
 
 4223 
 
 70 
 
 53 
 
 54 
 
 55 
 
 55 
 
 56 
 
 571 
 
 4302 
 
 4381 
 
 4460 
 
 4539 
 
 4618 
 
 4696 
 
 4775 
 
 4854 
 
 4933 
 
 5012 
 
 71 
 
 54 
 
 55 
 
 55 
 
 56 
 
 57 
 
 572 
 
 5091 
 
 5170 
 
 5249 
 
 5327 
 
 5406 
 
 5485 
 
 5564 
 
 5643 
 
 5722 
 
 5801 
 
 72 
 
 55 
 
 55 
 
 56 
 
 57 
 
 58 
 
 573 
 
 5880 
 
 5959 
 
 6037 
 
 6116 
 
 6195 
 
 6274 
 
 6353 
 
 6432 
 
 6511 
 
 6590 
 
 73 
 
 55 
 
 56 
 
 57 
 
 58 
 
 58 
 
 574 
 
 6669 
 
 6748 
 
 6827! 6906 
 
 6985 
 
 7064 
 
 7143 
 
 7222 
 
 7301 
 
 7380 
 
 74 
 
 56 
 
 57 
 
 58 
 
 58 
 
 59 
 
 575 
 
 7459 
 
 7538 
 
 7617 
 
 7696 
 
 7775 
 
 7854 
 
 7933 
 
 8012 
 
 8091 
 
 8170 
 
 75 
 
 57 
 
 58 
 
 58 
 
 59 
 
 60 
 
 576 
 
 8249 
 
 8328 
 
 8407 
 
 8486 
 
 8565 
 
 8644 
 
 8723 
 
 8802 
 
 8881 
 
 8960 
 
 76 
 
 58 
 
 59 
 
 59 
 
 60 
 
 61 
 
 577 
 
 9039 
 
 9118 
 
 9197 
 
 9276 
 
 9355 
 
 9434 
 
 9513 
 
 9593 
 
 9672 
 
 9751 
 
 77 
 
 59 
 
 59 
 
 60 
 
 61 
 
 62 
 
 578 
 
 9830 
 
 9909 
 
 9988 
 
 0067 
 
 0146 
 
 0225 
 
 0304 
 
 0384 
 
 0463 
 
 0542 
 
 78 
 
 59 
 
 60 
 
 61 
 
 62 
 
 62 
 
 579 
 
 68 0621 
 
 0700 
 
 0779 
 
 0858 
 
 0938 
 
 1017 
 
 1096 
 
 1175 
 
 1254 
 
 1333 
 
 79 
 
 60 
 
 61 
 
 62 
 
 62 
 
 63 
 
 580 
 
 68 1413 
 
 1492 
 
 1571 
 
 1650 
 
 1729 
 
 1808 
 
 1888 
 
 1967 
 
 2046 
 
 2125 
 
 80 
 
 61 
 
 62 
 
 62 
 
 63 
 
 64 
 
 581 
 
 2205 
 
 2284 
 
 2363 
 
 2442 
 
 2521 
 
 2601 
 
 2680 
 
 2759 
 
 2838 
 
 2918 
 
 81 
 
 62 
 
 62 
 
 63 
 
 64 
 
 65 
 
 1 '582 
 
 2997 
 
 3076 
 
 3155 
 
 3235 
 
 3314 
 
 3393 
 
 3472 
 
 3552 
 
 3631 
 
 3710 
 
 82 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 583 
 
 3790 
 
 3869 
 
 3948 
 
 4027 
 
 4107 
 
 4186 
 
 4265 
 
 4345 
 
 4424 
 
 4503 
 
 83 
 
 63 
 
 64 
 
 65 
 
 66 
 
 66 
 
 584 
 
 4583 
 
 4662 
 
 4741 
 
 4821 
 
 4900 
 
 4979 
 
 5059 
 
 5138 
 
 5217 
 
 5297 
 
 84 
 
 64 
 
 65 
 
 66 
 
 66 
 
 67 
 
 585 
 
 5376 
 
 5455 
 
 5535 
 
 5614 
 
 5694 
 
 5773 
 
 5852 
 
 5932 
 
 6011 
 
 6090 
 
 85 
 
 65 
 
 65 
 
 66 
 
 67 
 
 68 
 
 586 
 
 6170 
 
 6250 
 
 6329 
 
 6408 
 
 6488 
 
 6567 
 
 6646 
 
 6726 
 
 6805 
 
 6885 
 
 86 
 
 65 
 
 66 
 
 67 
 
 68 
 
 69 
 
 587 
 
 6964 
 
 7044 
 
 7123 
 
 7202 
 
 7282 
 
 7361 
 
 7441 
 
 7520 
 
 7600 
 
 7679 
 
 87 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 588 
 
 7759 
 
 7838 
 
 7918 
 
 7997 
 
 8077 
 
 8156 
 
 8236 
 
 8315 
 
 8395 
 
 8474 
 
 88 
 
 67 
 
 68 
 
 69 
 
 70 
 
 70 
 
 589 
 
 8554 
 
 8633 
 
 8713 
 
 8792 
 
 8872 
 
 8951 
 
 9031 
 
 9110 
 
 9190 
 
 9269 
 
 89 
 
 68 
 
 69 
 
 69 
 
 70 
 
 71 
 
 590 
 
 68 9349 
 
 9429 
 
 9508 
 
 9588 
 
 9667 
 
 9747 
 
 9826 
 
 9906 
 
 9985 
 
 0065 
 
 90 
 
 6869 
 
 70 
 
 71 
 
 72 
 
 591 
 
 690145 
 
 0224 
 
 0304 
 
 0383 
 
 0463 
 
 0543 
 
 0622 
 
 0702 
 
 0781 
 
 0861 
 
 91 
 
 6970 
 
 71 
 
 72 
 
 73 
 
 592 
 
 0941 
 
 1020 
 
 1100 
 
 1180 
 
 1259 
 
 1339 
 
 1419 
 
 1498 
 
 1578 
 
 1658 
 
 92 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 593 
 
 1737 
 
 1817 
 
 1897 
 
 1976 
 
 2056 
 
 2136 
 
 2215 
 
 2295 
 
 2375 
 
 2454 
 
 93 
 
 71 
 
 72 
 
 73 
 
 73 
 
 74 
 
 594 
 
 2534 
 
 2614 
 
 2693 
 
 2773 
 
 2853 
 
 2933 
 
 3012 
 
 3092 
 
 3172 
 
 3251 
 
 94 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 1 -595 
 
 3331 
 
 3411 
 
 3491 
 
 3570 
 
 3650 
 
 3730 
 
 3810 
 
 3889 
 
 3969 
 
 4049 
 
 95 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 1 '596 
 
 4129 
 
 4209 
 
 4288 
 
 4368 
 
 4448 
 
 4528 
 
 4fiOft 
 
 4687 
 
 47fi7 
 
 4847 
 
 96 
 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 1 -597 
 
 4927 
 
 5007 
 
 5086 
 
 5166 
 
 5246 
 
 5326 
 
 5406 
 
 5486 
 
 T I U i 
 
 5565 
 
 TFOT l 
 
 5645 
 
 97 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 1 -598 
 
 5725 
 
 5805 
 
 5885 
 
 5965 
 
 6044 
 
 6124 
 
 6204 
 
 6284 
 
 6364 
 
 6444 
 
 98 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 1 -599 
 
 6524 
 
 6604 
 
 6684 
 
 6763 
 
 6843 
 
 6923 
 
 7003 
 
 7083 
 
 7163 
 
 7243 
 
 99 
 
 75 
 
 76 
 
 77 
 
 78 
 
 79 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 76 
 
 77 
 
 78 
 
 79 
 
 90 
 
 13 
 
TABLE I. 
 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 Log 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 8( 
 
 81 
 
 82 
 
 8384 
 
 0600 
 
 0-69 7323 
 
 7403 
 
 7483 
 
 7563 
 
 7643 
 
 7722 
 
 7802 
 
 7882 
 
 7962 
 
 8042 
 
 00 
 
 
 
 
 
 
 
 
 
 
 
 601 
 
 812218202 
 
 8282 
 
 8362 
 
 8442 
 
 8522 
 
 8602 
 
 8682 
 
 8762 
 
 8842 
 
 01 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 602 
 
 8922 9002 
 
 9082 
 
 9162 
 
 9242 
 
 9322 
 
 9402 
 
 9482 
 
 9562 
 
 9642 
 
 02 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 603 
 
 9722 
 
 9802 
 
 9882 
 
 9962 
 
 0042 
 
 0122 
 
 0202 
 
 0283 
 
 0363 
 
 0443 
 
 03 
 
 2 
 
 2 
 
 2 
 
 2 
 
 : 
 
 604 
 
 700523 
 
 0603 
 
 0683 
 
 0763 
 
 0843 
 
 0923 
 
 1003 
 
 1083 
 
 1163 
 
 1243 
 
 04 
 
 
 
 3 
 
 3 
 
 3 
 
 c 
 
 605 
 
 1324 
 
 1404 
 
 1484 
 
 1564 
 
 1644 
 
 1724 
 
 1804 
 
 1884 
 
 1965 
 
 2045 
 
 05 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 606 
 
 2125 
 
 2205 
 
 2285 
 
 2365 
 
 2445 
 
 2526 
 
 2606 
 
 2686 
 
 2766 
 
 2846 
 
 06 
 
 i 
 
 5 
 
 K 
 
 5 
 
 5 
 
 607 
 
 2926 
 
 3007 
 
 3087 
 
 3167 
 
 3247 
 
 3327 
 
 3408 
 
 3488 
 
 3568 
 
 3648 
 
 07 
 
 ( 
 
 6 
 
 6 
 
 6 
 
 6 
 
 608 
 
 3728 
 
 3809 
 
 3889 
 
 3969 
 
 4049 
 
 4130 
 
 4210 
 
 4290 
 
 4370 
 
 4451 
 
 08 
 
 ( 
 
 6 
 
 7 
 
 7 
 
 7 
 
 609 
 
 4531 
 
 4611 
 
 4691 
 
 4772 
 
 4852 
 
 4932 
 
 5012 
 
 5093 
 
 5173 
 
 5253 
 
 09 
 
 7 
 
 7 
 
 i 
 
 7 
 
 8 
 
 610 
 
 70 5334 
 
 5414 
 
 5494 
 
 5574 
 
 5655 
 
 5735 
 
 5815 
 
 5896 
 
 5976 
 
 6056 
 
 10 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 611 
 
 6137 
 
 6217 
 
 6297 
 
 6378 
 
 6458 
 
 6538 
 
 6619 
 
 6699 
 
 6779 
 
 6860 
 
 11 
 
 9 
 
 9 
 
 f 
 
 9 
 
 < 
 
 612 
 
 6940 
 
 7020 
 
 7101 
 
 7181 
 
 7262 
 
 7342 
 
 7422 
 
 7503 
 
 7583 
 
 7664 
 
 12 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 613 
 
 7744 
 
 7824 
 
 7905 
 
 7985 
 
 8066 
 
 8146 
 
 8226 
 
 8307 
 
 8387 
 
 8468 
 
 13 
 
 10 
 
 11 
 
 11 
 
 11 
 
 i 
 
 614 
 
 8548 
 
 8629 
 
 8709 
 
 8789 
 
 8870 
 
 8950 
 
 9031 
 
 9111 
 
 9192 
 
 9272 
 
 14 
 
 11 
 
 11 
 
 11 
 
 12 
 
 2 
 
 615 
 
 9353 
 
 9433 
 
 9514 
 
 9594 
 
 9675 
 
 9755 
 
 9836 
 
 9916 
 
 9997 
 
 0077 
 
 15 
 
 g 
 
 12 
 
 12 
 
 12 
 
 ;$ 
 
 616 
 
 710158 
 
 0238 
 
 0319 
 
 0399 
 
 0480 
 
 0560 
 
 0641 
 
 0721 
 
 0802 
 
 0882 
 
 U 
 
 
 
 13 
 
 13 
 
 13 
 
 j 
 
 617 
 
 0963 
 
 1043 
 
 1124 
 
 1204 
 
 1285 
 
 1366 
 
 1446 
 
 1527 
 
 1607 
 
 1688 
 
 17 
 
 i 
 
 14 
 
 14 
 
 14 
 
 i i 
 
 1 
 
 618 
 
 1768 
 
 1849 
 
 1930 
 
 2010 
 
 2091 
 
 2171 
 
 2252 
 
 2333 
 
 2413 
 
 2494 
 
 18 
 
 i 
 
 15 
 
 15 
 
 15 
 
 15 
 
 619 
 
 2574 
 
 2655 
 
 2736 
 
 2816 
 
 2897 
 
 2978 
 
 3058 
 
 3139 
 
 3220 
 
 3300 
 
 If 
 
 t 
 
 15 
 
 16 
 
 16 
 
 16 
 
 620 
 
 71 3381 
 
 3461 
 
 3542 
 
 3623 
 
 3703 
 
 3784 
 
 3865 
 
 3945 
 
 4026 
 
 4107 
 
 20 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 621 
 
 4188 
 
 4268 
 
 4349 
 
 4430 
 
 4510 
 
 4591 
 
 4672 
 
 4752 
 
 4833 
 
 4914 
 
 21 
 
 17 
 
 17 
 
 17 
 
 17 
 
 18 
 
 622 
 
 4995 
 
 5075 
 
 5156 
 
 5237 
 
 5318 
 
 5398 
 
 5479 
 
 5560 
 
 5640 
 
 5721 
 
 22 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 623 
 
 5802 
 
 5883 
 
 5964 
 
 6044 
 
 6125 
 
 6206 
 
 6287 
 
 6367 
 
 6448 
 
 6529 
 
 23 
 
 18 
 
 If) If 
 
 19 
 
 19 
 
 624 
 
 6610 
 
 6691 
 
 6771 
 
 6852 
 
 6933 
 
 7014 
 
 7095 
 
 7175 
 
 7256 
 
 7337 
 
 24 
 
 19 
 
 19 
 
 20 
 
 20 
 
 20 
 
 625 
 
 7418 
 
 7499 
 
 7580 
 
 7660 
 
 7741 
 
 7822 
 
 7903 
 
 7984 
 
 8065 
 
 8146 
 
 25 
 
 20 
 
 20 
 
 21 
 
 '21 
 
 21 
 
 626 
 
 8226 
 
 8307 
 
 8388 
 
 8469 
 
 8550 
 
 8631 
 
 8712 
 
 8793 
 
 8873 
 
 8954 
 
 26 
 
 21 
 
 21 21 
 
 22 
 
 22 
 
 627 
 
 9035 
 
 9116 
 
 9197 
 
 9278 
 
 9359 
 
 9440 
 
 9521 
 
 9602 
 
 9683 
 
 9764 
 
 2? 
 
 22 
 
 2222 
 
 2223 
 
 628 
 
 9844 
 
 9925 
 
 0006 
 
 0087 
 
 o!68 
 
 0249 
 
 0330 
 
 0411 
 
 0492 
 
 0573 
 
 28 
 
 22 
 
 23 
 
 23 
 
 2324 
 
 629 
 
 72 0654 
 
 0735 
 
 0816 
 
 0897 
 
 0978 
 
 1059 
 
 1140 
 
 1221 
 
 1302 
 
 1383 
 
 29 
 
 23 
 
 23 
 
 24 
 
 24 
 
 24 
 
 630 
 
 72 1464 
 
 1545 
 
 1626 
 
 1707 
 
 1788 
 
 1869 
 
 1950 
 
 2031 
 
 2112 
 
 2193 
 
 30 
 
 24 
 
 2425 
 
 25 
 
 25 
 
 631 
 
 2274 
 
 2355 
 
 2436 
 
 2517 
 
 2598 
 
 2680 
 
 2761 
 
 2842 
 
 2923 
 
 3004 
 
 31 
 
 25 
 
 2525 
 
 26:26 
 
 632 
 
 3085 
 
 3166 
 
 3247 
 
 3328 
 
 3409 
 
 3490 
 
 3571 
 
 3653 
 
 3734 
 
 3815 
 
 32 
 
 26 
 
 2626 
 
 27 
 
 27 
 
 633 
 
 3896 
 
 3977 
 
 4058 
 
 4139 
 
 4220 
 
 4301 
 
 4383 
 
 4464 
 
 4545 
 
 4626 
 
 33 
 
 26 
 
 27:27 
 
 27 
 
 28 
 
 634 
 
 4707 
 
 4788 
 
 4869 
 
 4951 
 
 5032 
 
 5113 
 
 5194 
 
 5275 
 
 5356 
 
 5438 
 
 34 
 
 27 
 
 28(28 
 
 28 
 
 29 
 
 635 
 
 5519 
 
 5600 
 
 5681 
 
 5762 
 
 5844 
 
 5925 
 
 6006 
 
 6087 
 
 6168 
 
 6250 
 
 35 
 
 28 
 
 2829 
 
 29 
 
 29 
 
 636 
 
 6331 
 
 6412 
 
 6493 
 
 6575 
 
 6656 
 
 6737 
 
 6818 
 
 6900 
 
 6981 
 
 7062 
 
 36 
 
 29 
 
 29 30 
 
 30 
 
 30 
 
 637 
 
 7143 
 
 7225 
 
 7306 
 
 7387 
 
 7468 
 
 7550 
 
 7631 
 
 7712 
 
 7793 
 
 7875 
 
 37 
 
 30 
 
 3030 
 
 31 
 
 31 
 
 638 
 
 7956 
 
 8037 
 
 8119 
 
 8200 
 
 8281 
 
 8363 
 
 8444 
 
 8525 
 
 8606 
 
 8688 
 
 38 
 
 30 
 
 3131 
 
 32J32 
 
 639 
 
 8769 
 
 8850 
 
 8932 
 
 9013 
 
 9094 
 
 9176 
 
 9257 
 
 9338 
 
 9420 
 
 9501 
 
 39 
 
 31 
 
 32 
 
 32 
 
 32J33 
 
 640 
 
 72 9583 
 
 9664 
 
 9745 
 
 9827 
 
 9908 
 
 9989 
 
 0071 
 
 0152 
 
 0234 
 
 0315 
 
 40 
 
 32 
 
 32 
 
 33 
 
 33 
 
 34 
 
 641 
 
 73 0396 
 
 0478 
 
 0559 
 
 0641 
 
 0722 
 
 0803 
 
 0885 
 
 0966 
 
 1048 
 
 1129 
 
 41 
 
 33 
 
 33 34 34 34 
 
 642 
 
 1210 
 
 1292 
 
 1373 
 
 1455 
 
 1536 
 
 1618 
 
 1699 
 
 1781 
 
 1862 
 
 1943 
 
 42 
 
 34 
 
 343435 
 
 35 
 
 643 
 
 2025 
 
 2106 
 
 2188 
 
 2269 
 
 2351 
 
 2432 
 
 2514 
 
 2595 
 
 2677 
 
 2758 
 
 43 
 
 34 
 
 3535 
 
 36 
 
 36 
 
 644 
 
 2840 
 
 2921 
 
 3003 
 
 3084 
 
 3166 
 
 3247 
 
 3329 
 
 3410 
 
 3492 
 
 3573 
 
 44 
 
 35 
 
 36 
 
 36 
 
 37 
 
 37 
 
 645 
 
 3655 
 
 3736 
 
 3818 
 
 3900 
 
 3981 
 
 4063 
 
 4144 
 
 4226 
 
 4307 
 
 4389 
 
 45 
 
 36 
 
 36 
 
 37 
 
 37 
 
 38 
 
 646 
 
 4470 
 
 4552 
 
 4634 
 
 4715 
 
 4797 
 
 4878 
 
 4960 
 
 5042 
 
 5123 
 
 5205 
 
 46 
 
 37 
 
 37 
 
 38 
 
 38 
 
 39 
 
 647 
 
 5286 
 
 5368 
 
 5450 
 
 5531 
 
 5613 
 
 5694 
 
 5776 
 
 5858 
 
 5939 
 
 6021 
 
 47 
 
 38 
 
 38 
 
 39 
 
 39 
 
 39 
 
 648 
 
 6103 
 
 6184 
 
 6266 
 
 6348 
 
 6429 
 
 6511 
 
 6592 
 
 6674 
 
 6756 
 
 6837 
 
 48 
 
 38 
 
 3939 
 
 40 
 
 40 
 
 649 
 
 6919 
 
 7001 
 
 7082 
 
 7164 
 
 7246 
 
 7328 
 
 7409 
 
 7491 
 
 7573 
 
 7654 
 
 49 
 
 39 
 
 40 
 
 10 
 
 41 
 
 41 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 ^0 
 
 81 
 
 S2 
 
 S3 
 
 34 
 
 14 
 
TABLE I. 
 
 Log 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 0-650 
 
 0-73 7736 
 
 7818 
 
 7899 
 
 7981 
 
 8063 
 
 8145 
 
 8226 
 
 8308 
 
 8390 
 
 8472 
 
 50 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 651 
 
 8553 
 
 8635 
 
 8717 
 
 8799 
 
 8880 
 
 8962 9044 
 
 9126 
 
 9207 
 
 9289 
 
 51 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 652 
 
 9371 
 
 9453 9534 
 
 9616 
 
 9698 
 
 9780 9862 
 
 9943 
 
 0025 
 
 0107 
 
 52 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 653 
 
 740189 
 
 0271 
 
 0352 
 
 0434 
 
 0516 
 
 0598 
 
 0680 
 
 0762 
 
 0843 
 
 0925 
 
 53 
 
 42 
 
 43 
 
 43 
 
 44 
 
 45 
 
 654 
 
 1007 
 
 1089 
 
 1171 
 
 1253 
 
 1335 
 
 1416 
 
 1498 
 
 1580 
 
 1662 
 
 1744 
 
 54 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 655 
 
 1826 
 
 1908 
 
 1989 
 
 2071 
 
 2153 
 
 2235 
 
 2317 
 
 2399 
 
 2481 
 
 2563 
 
 55 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 656 
 
 2645 
 
 2727 
 
 2809 
 
 2890 
 
 2972 
 
 3054 
 
 3136 
 
 3218 
 
 3300 
 
 3382 
 
 56 
 
 45 
 
 45 
 
 46 
 
 46 
 
 47 
 
 657 
 
 3464 
 
 3546 
 
 3628 
 
 3710 
 
 3792 
 
 3874 
 
 3956 
 
 4038 
 
 4120 
 
 4202 
 
 57 
 
 46 
 
 46 
 
 4" 
 
 47 
 
 48 
 
 .fiKQ 
 
 49ft4 
 
 4Sfifi 
 
 444ft 
 
 4^0 
 
 4fil9 
 
 4fiQ<: 
 
 14,77ft 
 
 40 eo 
 
 4.OJ.I 
 
 KAgo 
 
 ~<_ 
 
 4r 
 
 4? 
 
 4ft 
 
 4ft 
 
 40 
 
 DOO 
 
 659 
 
 T?4t5l 
 
 5104 
 
 T?OOv 
 
 5186 
 
 TTTO 
 
 5268 
 
 TOOL 
 
 5350 
 
 i*\j 1 
 
 5432 
 
 T?Oc/ . 
 
 5514 
 
 ~t 1 1 O 
 
 5596 
 
 TOOC 
 
 5678 
 
 ty'tf 
 5760 
 
 <J\}4:lC 
 
 5842 
 
 Jo 
 
 59 
 
 -tu 
 
 4? 
 
 / 
 
 48 
 
 TO 
 
 48 
 
 TO 
 
 49 
 
 <*> 
 
 50 
 
 660 
 
 74 5924 
 
 6006 
 
 6088 
 
 6170 
 
 6252 
 
 6334 
 
 6416 
 
 6498 
 
 6580 
 
 6663 
 
 60 
 
 48 
 
 49 
 
 49 
 
 50 
 
 50 
 
 661 
 
 6745 
 
 6827 
 
 6909 
 
 6991 
 
 7073 
 
 7155 
 
 7237 
 
 7319 
 
 7401 
 
 7484 
 
 6] 
 
 49 
 
 49 
 
 50 
 
 5] 
 
 51 
 
 662 
 
 7566 
 
 7648 
 
 7730 
 
 7812 
 
 7894 
 
 7976 
 
 8058 
 
 8141 
 
 8223 
 
 8305 
 
 62 
 
 50 
 
 50 
 
 51 
 
 51 
 
 52 
 
 663 
 
 8387 
 
 8469 
 
 8551 
 
 8633 
 
 8716 
 
 8798 
 
 8880 
 
 8962 
 
 9044 
 
 9126 
 
 63 
 
 50 
 
 51 
 
 52 
 
 52 
 
 53 
 
 664 
 
 9209 
 
 9291 
 
 9373 
 
 9455 
 
 9537 
 
 9620 
 
 9702 
 
 9784 
 
 9866 
 
 9948 
 
 64 
 
 5] 
 
 52 
 
 52 
 
 53 
 
 54 
 
 665 
 
 75 0031 
 
 0113 
 
 0195 
 
 0277 
 
 0360 
 
 0442 
 
 0524 
 
 0606 
 
 0689 
 
 0771 
 
 65 
 
 52 
 
 53 
 
 53 
 
 54 
 
 55 
 
 666 
 
 0853 
 
 0935 
 
 1018 
 
 1100 
 
 1182 
 
 1264 
 
 1347 
 
 1429 
 
 1511 
 
 1593 
 
 66 
 
 53 
 
 53 
 
 54 
 
 55 
 
 55 
 
 667 
 
 1676 
 
 1758 
 
 1840 
 
 1923 
 
 2005 
 
 2087 
 
 2169 
 
 2252 
 
 2334 
 
 2416 
 
 67 
 
 o-J 
 
 54 
 
 55 
 
 56 
 
 56 
 
 668 
 
 2499 
 
 2581 
 
 2663 
 
 2746 
 
 2828 
 
 2910 
 
 2993 
 
 3075 
 
 3157 
 
 3240 
 
 68 
 
 5-J 
 
 55 
 
 56 
 
 56 
 
 57 
 
 669 
 
 3322 
 
 3404 
 
 3487 
 
 3569 
 
 3652 
 
 3734 
 
 3816 
 
 3899 
 
 3981 
 
 4063 
 
 69 
 
 55 
 
 56 
 
 57 
 
 57 
 
 58 
 
 670 
 
 754146 
 
 4228 
 
 4311 
 
 4393 
 
 4475 
 
 4558 
 
 4640 
 
 4723 
 
 4805 
 
 4887 
 
 70 
 
 56 
 
 57 
 
 57 
 
 58 
 
 59 
 
 671 
 
 4970 
 
 5052 
 
 5135 
 
 5217 
 
 5300 
 
 5382 
 
 5464 
 
 5547 
 
 5629 
 
 5712 
 
 71 
 
 57 
 
 58 
 
 58 
 
 59 
 
 60 
 
 672 
 
 5794 
 
 5877 
 
 5959 
 
 6042 
 
 6124 
 
 6206 
 
 6289 
 
 6371 
 
 6454 
 
 6536 
 
 72 
 
 58 
 
 58 
 
 59 
 
 60 
 
 60 
 
 673 
 
 6619 
 
 6701 
 
 6784 
 
 6866 
 
 6949 
 
 7031 
 
 7114 
 
 7196 
 
 7279 
 
 7361 
 
 73 
 
 58 
 
 59 
 
 60 
 
 61 
 
 61 
 
 674 
 
 7444 
 
 7526 
 
 7609 
 
 7691 
 
 7774 
 
 7857 
 
 7939 
 
 8022 
 
 8104 
 
 8187 
 
 74 
 
 59 
 
 60 
 
 61 
 
 61 
 
 62 
 
 675 
 
 8269 
 
 8352 
 
 8434 
 
 8517 
 
 8599 
 
 8682 
 
 8765 
 
 8847 
 
 8930 
 
 9012 
 
 75 
 
 60 
 
 61 
 
 61 
 
 62 
 
 63 
 
 676 
 
 9095 
 
 9178 
 
 9260 
 
 9343 
 
 9425 
 
 9508 
 
 9591 
 
 9673 
 
 9756 
 
 9838 
 
 76 
 
 61 
 
 62 
 
 62 
 
 63 
 
 64 
 
 677 
 
 9921 
 
 0004 
 
 0086 
 
 0169 
 
 0251 
 
 0334 
 
 0417 
 
 0499 
 
 0582 
 
 0665 
 
 77 
 
 62 
 
 62 
 
 63 
 
 64 
 
 65 
 
 678 
 
 76 0747 
 
 0830 
 
 0913 
 
 0995 
 
 1078 
 
 1161 
 
 1243 
 
 1326 
 
 1409 
 
 1491 
 
 78 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 679 
 
 1574 
 
 1657 
 
 1739 
 
 1822 
 
 1905 
 
 1987 
 
 2070 
 
 2153 
 
 2236 
 
 2318 
 
 79 
 
 63 
 
 64 
 
 65 
 
 66 
 
 66 
 
 680 
 
 762401 
 
 2484 
 
 2566 
 
 2649 
 
 2732 
 
 2815 
 
 2897 
 
 2980 
 
 3063 
 
 3146 
 
 80 
 
 64 
 
 65 
 
 66 
 
 66 
 
 67 
 
 681 
 
 3228 
 
 3311 
 
 3394 
 
 3477 
 
 3559 
 
 3642 
 
 3725 
 
 3808 
 
 3890 
 
 3973 
 
 81 
 
 65 
 
 66 
 
 66 
 
 >7 
 
 68 
 
 682 
 
 4056 
 
 4139 
 
 4222 
 
 4304 
 
 4387 
 
 4470 
 
 4553 
 
 4636 
 
 4718 
 
 4801 
 
 82 
 
 66 
 
 66 
 
 67 
 
 68 
 
 69 
 
 683 
 
 4884 
 
 4967 
 
 5050 
 
 5132 
 
 5215 
 
 5298 
 
 5381 
 
 5464 
 
 5547 
 
 5630 
 
 83 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 684 
 
 5712 
 
 5795 
 
 5878 
 
 5961 
 
 6044 
 
 6127 
 
 6210 
 
 6292 
 
 6375 
 
 6458 
 
 84 
 
 )7 
 
 68 
 
 69 
 
 70 
 
 71 
 
 685 
 
 6541 
 
 6624 
 
 6707 
 
 6790 
 
 6873 
 
 6955 
 
 7038 
 
 7121 
 
 7204 
 
 7287 
 
 85 
 
 68 
 
 69 
 
 70 
 
 71 
 
 71 
 
 686 
 
 7370 
 
 7453 
 
 7536 
 
 7619 
 
 7702 
 
 7785 
 
 7868 
 
 7950 
 
 8033 
 
 8116 
 
 86 
 
 69 
 
 70 
 
 71 
 
 71 
 
 72 
 
 687 
 
 8199 
 
 8282 
 
 8365 
 
 8448 
 
 8531 
 
 8614 
 
 8697 
 
 8780 
 
 8863 
 
 8946 
 
 87 
 
 70 
 
 70 
 
 71 
 
 72 
 
 73 
 
 688 
 
 9029 
 
 9112 
 
 9195 
 
 9278 
 
 9361 
 
 9444 
 
 9527 
 
 9610 
 
 9693 
 
 9776 
 
 88 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 689 
 
 9859 
 
 9942 
 
 0025 
 
 0108 
 
 0191 
 
 0274 
 
 0357 
 
 0440 
 
 0523 
 
 0606 
 
 89 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 690 
 
 77 0689 
 
 0772 
 
 0855 
 
 0938 
 
 1021 
 
 1104 
 
 1187 
 
 1271 
 
 1354 
 
 1437 
 
 90 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 691 
 
 1520 
 
 1603 
 
 1686 
 
 1769 
 
 1852 
 
 1935 
 
 2018 
 
 2101 
 
 2184 
 
 2268 
 
 91 
 
 73 
 
 74 
 
 75 
 
 76 
 
 76 
 
 692 
 
 2351 
 
 2434 
 
 2517 
 
 2600 
 
 2683 
 
 2766 
 
 2849 
 
 2933 
 
 3016 
 
 3099 
 
 92 
 
 74 
 
 75 
 
 75 
 
 76 
 
 77 
 
 693 
 
 3182 
 
 3265 
 
 3348 
 
 3431 
 
 3515 
 
 3598 
 
 3681 
 
 3764 
 
 3847 
 
 3930 
 
 93 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 694 
 
 4014 
 
 4097 
 
 4180 
 
 4263 
 
 4346 
 
 4429 
 
 4513 
 
 4596 
 
 4679 
 
 4762 
 
 94 
 
 75 
 
 76 
 
 77 
 
 78 
 
 79 
 
 695 
 
 4845 
 
 4929 
 
 5012 
 
 5095 
 
 5178 
 
 5262 
 
 5345 
 
 5428 
 
 5511 
 
 5594 
 
 95 
 
 '6 
 
 77 
 
 78 
 
 79 
 
 SO 
 
 696 
 
 5678 
 
 5761 
 
 5844 
 
 5927 
 
 6011 
 
 6094 
 
 6177 
 
 6260 
 
 6344 
 
 6427 
 
 96 
 
 77 
 
 78 
 
 79 
 
 80 
 
 SI 
 
 697 
 
 6510 
 
 6593 
 
 6677 
 
 6760 
 
 6843 
 
 6927 
 
 7010 
 
 7093 
 
 7176 
 
 7260 
 
 97 
 
 "8 
 
 79 
 
 80 
 
 81 
 
 31 
 
 698 
 
 7343 
 
 7426 
 
 7510 
 
 7593 
 
 7676 
 
 7760 
 
 7843 
 
 7926 
 
 8010 
 
 8093 
 
 98 
 
 "8 
 
 79 
 
 80 
 
 81 
 
 32 
 
 699 
 
 8176 
 
 8260 
 
 8343 
 
 8426 
 
 8510 
 
 8593 
 
 8676 
 
 8760 
 
 8843 
 
 8926 
 
 99 
 
 79 
 
 80 
 
 81 
 
 32 
 
 33 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 30 
 
 SI 
 
 32 
 
 33 
 
 34 
 
 15 
 
TABLE I. 
 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 Log 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 8384 
 
 85'86 
 
 87 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0-700 
 
 0-77 9010 
 
 9093 
 
 9176 
 
 9260 
 
 9343 
 
 9427 
 
 9510 
 
 9593 
 
 9677 
 
 9760 
 
 00 
 
 
 
 
 
 
 
 
 
 701 
 
 9844 
 
 9927 
 
 0010 
 
 0094 
 
 0177 
 
 0261 
 
 0344 
 
 0427 
 
 0511 
 
 0594 
 
 01 
 
 1 
 
 1 
 
 1 1 
 
 1 
 
 702 
 
 78 0678 
 
 0761 
 
 0845 
 
 0928 
 
 1011 
 
 1095 
 
 1178 
 
 1262 
 
 1345 
 
 1429 
 
 02 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 703 
 
 1512 
 
 1596 
 
 1679 
 
 1763 
 
 1846 
 
 1930 
 
 2013 
 
 2096 
 
 2180 
 
 2263 
 
 03 
 
 2 
 
 3 
 
 3 3 
 
 3 
 
 704 
 
 2347 
 
 2430 
 
 2514 
 
 2597 
 
 2681 
 
 2764 
 
 2848 
 
 2931 
 
 3015 
 
 3099 
 
 04 
 
 3 
 
 3 
 
 3 3 
 
 3 
 
 705 
 
 3182 
 
 3266 
 
 3349 
 
 3433 
 
 3516 
 
 3600 
 
 3683 
 
 3767 
 
 3850 3934 
 
 05 
 
 4 
 
 4 
 
 *N 
 
 4 
 
 706 
 
 4017 
 
 4101 
 
 4185 
 
 4268 
 
 4352 
 
 4435 
 
 4519 
 
 4602 
 
 4686 4770 
 
 06 
 
 5 
 
 5 
 
 5 5 
 
 5 
 
 707 
 
 4853 
 
 4937 
 
 5020 
 
 5104 
 
 5188 
 
 5271 
 
 5355 
 
 5438 
 
 5522 5606 
 
 07 
 
 6 
 
 6 
 
 6 6 
 
 6 
 
 708 
 
 5689 
 
 5773 
 
 5856 
 
 5940 
 
 6024 
 
 6107 
 
 6191 
 
 6275 
 
 6358 6442 
 
 08 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 709 
 
 6526 
 
 6609 
 
 6693 
 
 6777 
 
 6860 
 
 6944 
 
 7028 
 
 7111 
 
 7195 
 
 7279 
 
 09 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 710 
 
 78 7362 
 
 7446 
 
 7530 
 
 7613 
 
 7697 
 
 7781 
 
 7864 
 
 7948 
 
 8032 
 
 8116 
 
 10 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 711 
 
 8199 
 
 8283 
 
 8367 
 
 8450 
 
 8534 
 
 8618 8702 
 
 8785 
 
 8869 
 
 8953 
 
 11 
 
 9 
 
 9 
 
 9 
 
 9 
 
 10 
 
 712 
 
 9037 
 
 9120 
 
 9204 
 
 9288 
 
 9372 
 
 9455 
 
 9539 
 
 9623 
 
 9707 
 
 9790 
 
 12 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 713 
 
 9874 
 
 9958 
 
 0042 
 
 0126 
 
 0209 
 
 0293 
 
 0377 
 
 o461 
 
 0544 
 
 0628 
 
 13 
 
 11 
 
 N 
 
 11 
 
 11 
 
 11 
 
 714 
 
 790712 
 
 0796 
 
 0880 
 
 0964 
 
 1047 
 
 1131 
 
 1215 
 
 1299 
 
 1383 
 
 1466 
 
 14 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 715 
 
 1550 
 
 1634 
 
 1718 
 
 1802 
 
 1886 
 
 1970 
 
 2053 
 
 2137 
 
 2221 
 
 2305 
 
 15 
 
 12 
 
 13 
 
 13 
 
 13 
 
 13 
 
 716 
 
 2389 
 
 2473 
 
 2557 
 
 2641 
 
 2724 
 
 2808 
 
 2892 
 
 2976 
 
 3060 
 
 3144 
 
 16 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 717 
 
 3228 
 
 3312 
 
 3396 
 
 3479 
 
 3563 
 
 3647 
 
 3731 
 
 3815 
 
 3899 
 
 3983 
 
 17 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 718 
 
 4067 
 
 4151 
 
 4235 
 
 4319 
 
 4403 
 
 4487 
 
 4571 
 
 4655 
 
 4738 
 
 4822 
 
 18 
 
 15 
 
 15 
 
 15 
 
 15 
 
 16 
 
 719 
 
 4906 
 
 4990 
 
 5074 
 
 5158 
 
 5242 
 
 5326 
 
 5410 
 
 5494 
 
 5578 
 
 5662 
 
 19 
 
 16 
 
 16 
 
 16 
 
 16 
 
 17 
 
 720 
 
 79 5746 
 
 5830 
 
 5914 
 
 5998 
 
 6082 
 
 6166 
 
 6250 
 
 6334 
 
 6418 
 
 6502 
 
 20 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 721 
 
 6586 
 
 6670 
 
 6754 
 
 6838 
 
 6922 
 
 7006 
 
 7091 
 
 7175 
 
 7259 
 
 7343 
 
 21 
 
 17 
 
 18 
 
 1818 
 
 18 
 
 722 
 
 7427 
 
 7511 
 
 7595 
 
 7679 
 
 7763 
 
 7847 
 
 7931 
 
 8015 
 
 8099 
 
 8183 
 
 22 
 
 18 
 
 18 
 
 19|19 
 
 19 
 
 723 
 
 8267 
 
 8352 
 
 8436 
 
 8520 
 
 8604 
 
 8688 
 
 8772 
 
 8856 
 
 8940 
 
 9024 
 
 23 
 
 19 
 
 19 
 
 20 20 
 
 20 
 
 724 
 
 9108 
 
 9193 
 
 9277 
 
 9361 
 
 9445 
 
 9529 
 
 9613 
 
 9697 
 
 9782 
 
 9866 
 
 24 
 
 80 
 
 20 
 
 20 
 
 21 
 
 21 
 
 725 
 
 9950 
 
 0034 
 
 0118 
 
 0202 
 
 0286 
 
 0371 
 
 0455 
 
 0539 
 
 0623 
 
 o707 
 
 25 
 
 21 
 
 21 
 
 21 
 
 22 
 
 22 
 
 726 
 
 80 0791 
 
 0876 
 
 0960 
 
 104* 
 
 1128 
 
 1212 
 
 1297 
 
 1381 
 
 1465 
 
 1549 
 
 86 
 
 22 
 
 22 
 
 22 
 
 22 
 
 23 
 
 727 
 
 1633 
 
 1718 
 
 1802 
 
 1886 
 
 1970 
 
 2055 
 
 2139 
 
 2223 
 
 2307 
 
 2391 
 
 27 
 
 22 
 
 23 
 
 23 
 
 23 
 
 23 
 
 728 
 
 2476 
 
 2560 
 
 2644 
 
 2728 
 
 2813 
 
 2897 
 
 2981 
 
 3065 
 
 3150 
 
 3234 
 
 88 
 
 23 
 
 24 
 
 24 24 
 
 24 
 
 729 
 
 3318 
 
 3403 
 
 3487 
 
 3571 
 
 3655 
 
 3740 
 
 3824 
 
 3908 
 
 3993 
 
 4077 
 
 89 
 
 24 
 
 24 
 
 2525 
 
 25 
 
 730 
 
 804161 
 
 4245 
 
 4330 
 
 4414 
 
 4498 
 
 4583 
 
 4667 
 
 4751 
 
 4836 
 
 4920 
 
 30 
 
 25 
 
 25 
 
 2526 
 
 26 
 
 731 
 
 5004 
 
 5089 
 
 5173 
 
 5257 
 
 5342 
 
 5426 
 
 5510 
 
 5595 
 
 5679 
 
 5763 
 
 31 
 
 26 
 
 26 
 
 2627 
 
 27 
 
 732 
 
 5848 
 
 5932 
 
 6016 
 
 6101 
 
 6185 
 
 6270 
 
 6354 
 
 6438 
 
 6523 
 
 6607 
 
 32 
 
 27 
 
 27 
 
 27;28 
 
 28 
 
 733 
 
 6692 
 
 6776 
 
 6860 
 
 6945 
 
 7029 
 
 7114 
 
 7198 
 
 7282 
 
 7367 
 
 7451 
 
 33 
 
 27 
 
 28 
 
 28128 
 
 29 
 
 734 
 
 7536 
 
 7620 
 
 7704 
 
 7789 
 
 7873 
 
 7958 
 
 8042 
 
 8127 
 
 8211 
 
 8296 
 
 34 
 
 28 
 
 29 
 
 29 29 
 
 30 
 
 735 
 
 8380 
 
 8464 
 
 8549 
 
 8633 
 
 8718 
 
 8802 
 
 8887 
 
 8971 
 
 9056 
 
 9140 
 
 35 
 
 29 
 
 29 
 
 30 
 
 30 
 
 30 
 
 736 
 
 9225 
 
 9309 
 
 9394 
 
 9478 
 
 9563 
 
 9647 
 
 9732 
 
 9816 
 
 9901 
 
 9985 
 
 3630 
 
 30 
 
 3131 
 
 31 
 
 737 
 
 81 0070 
 
 0154 
 
 0239 
 
 0323 
 
 0408 
 
 0492 
 
 0577 
 
 0661 
 
 0746 
 
 0830 
 
 3731 
 
 313l|32 
 
 
 738 
 
 0915 
 
 1000 
 
 1084 
 
 1169 
 
 1253 
 
 1338 
 
 1422 
 
 1507 
 
 1591 
 
 1676 
 
 38J32 
 
 32 32 33 
 
 33 
 
 739 
 740 
 
 1761 
 
 1845 
 
 1930 
 
 2014 
 
 2099 
 
 2184 
 
 2268 
 
 2353 
 
 2437 
 
 2522 
 
 39 
 
 32 
 33 
 
 33,33 
 
 34134 
 
 .-* 
 
 34 
 
 81 2606 
 
 2691 
 
 2776 
 
 2860 
 
 2945 
 
 3030 
 
 3114 
 
 3199 
 
 3283 
 
 3368 
 
 40 
 
 : 
 
 35 
 
 741 
 
 3453 
 
 3537 
 
 3622 
 
 3707 
 
 3791 
 
 3876 
 
 3961 
 
 4045 
 
 4130 
 
 4215 
 
 4134 
 
 34 
 
 35 35 36 
 
 742 
 
 4299 4384 
 
 4469 
 
 4553 
 
 4638 
 
 4723 
 
 4807 
 
 4892 
 
 4977 
 
 5061 
 
 42 ! 35 
 
 35 
 
 36 36 37 
 
 743 
 
 5146 5231 
 
 5315 
 
 5400 
 
 5485 
 
 5569 5654 
 
 5739 
 
 5824 
 
 5908 
 
 4336 
 
 36 
 
 37)3737 
 
 744 
 
 5993 6078 
 
 6163 
 
 6247 
 
 6332 
 
 6417 
 
 6501 
 
 6586 
 
 6671 
 
 6756 
 
 4437 
 
 37 
 
 37 38(38 
 
 745 
 
 6840 6925 
 
 7010 
 
 7095 
 
 7179 
 
 7264 
 
 7349 
 
 7434 
 
 7519 
 
 7603 
 
 4537 
 
 38 
 
 38 
 
 39 
 
 39 
 
 746 
 
 7688 7773 
 
 7858 
 
 7943 
 
 8027 
 
 8112 
 
 8197 
 
 8282 
 
 8367 
 
 8451 
 
 4638 
 
 39 
 
 39140 
 
 40 
 
 747 
 
 8536 8621 
 
 8706 
 
 8791 
 
 8875 
 
 8960 
 
 9045 
 
 9130 
 
 9215 
 
 9300 
 
 47 
 
 39 
 
 394040 
 
 41 
 
 748 
 
 9384 
 
 9469 
 
 9554 
 
 9639 
 
 9724 
 
 9809 
 
 9894 
 
 9978 
 
 0063 
 
 0148 
 
 48 
 
 40 
 
 4041 
 
 41 
 
 42 
 
 749 
 
 82 0233 
 
 0318 
 
 0403 
 
 0488 
 
 0572 
 
 0657 
 
 0742 
 
 0827 
 
 0912 
 
 0997 
 
 49 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 83 
 
 84 
 
 8586 
 
 87 
 
 10 
 
TABLE I. 
 
 1, 
 
 Log(l+#) 
 
 Pro. Parts. 
 
 Log 
 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 83 
 
 84|85 
 
 se 
 
 87 
 
 JO-750 
 
 0'82 1082 
 
 1167 
 
 1252 
 
 1337 
 
 1421 
 
 1506 
 
 1591 
 
 1676 
 
 1761 
 
 1846 
 
 50 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 || '751 
 
 1931 
 
 2016 
 
 2101 
 
 2186 
 
 2271 
 
 2356 
 
 2441 
 
 2526 
 
 2611 
 
 2696 
 
 51 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 1 -752 
 
 2780 
 
 2865 
 
 2950 
 
 3035 
 
 3120 
 
 3205 
 
 3290 
 
 3375 
 
 3460 
 
 3545 
 
 52 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 I '753 
 
 3630 
 
 3715 
 
 3800 
 
 3885 
 
 3970 
 
 4055 
 
 4140 
 
 4225 
 
 4310 
 
 4395 
 
 53 
 
 44 
 
 45 
 
 45 
 
 46J46 
 
 1 '754 
 
 4480 
 
 4565 
 
 4650 
 
 4735 
 
 4820 
 
 4905 
 
 4990 
 
 5076 
 
 5161 
 
 5246 
 
 54 
 
 45 
 
 45 
 
 46 
 
 46 
 
 47 
 
 1 ' 755 
 
 5331 
 
 5416 
 
 5501 
 
 5586 
 
 5671 
 
 5756 
 
 5841 
 
 5926 
 
 6011 
 
 6096 
 
 55 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 1 -756 
 
 6181 
 
 6266 
 
 6351 
 
 6437 
 
 6522 
 
 6607 
 
 6692 
 
 6777 
 
 6862 
 
 6947 
 
 56 
 
 46 
 
 47 
 
 48 
 
 48 
 
 49 
 
 1 '757 
 
 7032 
 
 7117 
 
 7202 
 
 7288 
 
 7373 
 
 7458 
 
 7543 
 
 7628 
 
 7713 
 
 7798 
 
 57 
 
 47 
 
 48|48 
 
 49 
 
 50 
 
 1 -758 
 
 7883 
 
 7969 
 
 8054 
 
 8139 
 
 8224 
 
 8309 
 
 8394 
 
 8479 
 
 8565 
 
 8650 
 
 58 
 
 48 
 
 49 
 
 49 
 
 50 
 
 50 
 
 1 ' 759 
 
 8735 
 
 8820 
 
 8905 
 
 8990 
 
 9076 
 
 9161 
 
 9246 
 
 9331 
 
 9416 
 
 9502 
 
 59 
 
 49 
 
 50 
 
 50 
 
 51 
 
 51 
 
 1 -760 
 
 82 9587 
 
 9672 
 
 9757 
 
 9842 
 
 9928 
 
 0013 
 
 0098 
 
 0183 
 
 0268 
 
 0354 
 
 60 
 
 50 
 
 50 
 
 51 
 
 52 
 
 52 
 
 1 -761 
 
 83 0439 
 
 0524 
 
 0609 
 
 0695 
 
 0780 
 
 0865 
 
 0950 
 
 1035 
 
 1121 
 
 1206 
 
 61 
 
 51 
 
 51 
 
 52 
 
 52 
 
 53 
 
 1 -762 
 
 1291 
 
 1376 
 
 1462 
 
 1547 
 
 1632 
 
 1718 
 
 1803 
 
 1888 
 
 1973 
 
 2059 
 
 62 
 
 51 
 
 52 
 
 53 
 
 53 
 
 54 
 
 1 -763 
 
 2144 
 
 2229 
 
 2314 
 
 2400 
 
 2485 
 
 2570 
 
 2656 
 
 2741 
 
 2826 
 
 2912 
 
 63 
 
 52 
 
 53 
 
 54 
 
 54 
 
 55 
 
 1 -764 
 
 2997 
 
 3082 
 
 3167 
 
 3253 
 
 3338 
 
 3423 
 
 3509 
 
 3594 
 
 3679 
 
 3765 
 
 64 
 
 53 
 
 54 
 
 54 
 
 55 
 
 56 
 
 1 -765 
 
 3850 
 
 3935 
 
 4021 
 
 4106 
 
 4191 
 
 4277 
 
 4362 
 
 4448 
 
 4533 
 
 4618 
 
 65 
 
 54 
 
 55 
 
 55 
 
 56 
 
 57 
 
 1 -766 
 
 4704 
 
 4789 
 
 4874 
 
 4960 
 
 5045 
 
 5131 
 
 5216 
 
 5301 
 
 5387 
 
 5472 
 
 66 
 
 55 
 
 55 
 
 56 
 
 57 
 
 57 
 
 1 -767 
 
 5557 
 
 5643 
 
 5728 
 
 5814 
 
 5899 
 
 5984 
 
 6070 
 
 6155 
 
 6241 
 
 6326 
 
 67 
 
 56 
 
 56 
 
 57 
 
 58 
 
 58 
 
 1 -768 
 
 6412 
 
 6497 
 
 6582 
 
 6668 
 
 6753 
 
 6839 
 
 6924 
 
 7010 
 
 7095 
 
 7181 
 
 68 
 
 56 
 
 57 
 
 58 
 
 58 
 
 59 
 
 1 -769 
 
 7266 
 
 7351 
 
 7437 
 
 7522 
 
 7608 
 
 7693 
 
 7779 
 
 7864 
 
 7950 
 
 8035 
 
 69 
 
 57 
 
 58 
 
 59 
 
 59 
 
 60 
 
 1 -770 
 
 838121 
 
 8206 
 
 8292 
 
 8377 
 
 8463 
 
 8548 
 
 8634 
 
 8719 
 
 8805 
 
 8890 
 
 70 
 
 58 
 
 59 
 
 59 
 
 60 
 
 61 
 
 1 -771 
 
 8976 
 
 9061 
 
 9147 
 
 9232 
 
 9318 
 
 9403 
 
 9489 
 
 9574 
 
 9660 
 
 9745 
 
 71 
 
 59 
 
 60 
 
 60 
 
 61 
 
 62 
 
 1 -772 
 
 9831 
 
 9916 
 
 0002 
 
 0088 
 
 0173 
 
 0259 
 
 0344 
 
 0430 
 
 0515 
 
 0601 
 
 72 
 
 60 
 
 60 
 
 61 
 
 62 
 
 63 
 
 1 -773 
 
 84 0686 
 
 0772 
 
 0858 
 
 0943 
 
 1029 
 
 1114 
 
 1200 
 
 1285 
 
 1371 
 
 1457 
 
 73 
 
 j 1 
 
 61 
 
 62 
 
 63 
 
 64 
 
 1 -774 
 
 1542 
 
 1628 
 
 1713 
 
 1799 
 
 1885 
 
 1970 
 
 2056 
 
 2142 
 
 2227 
 
 2313 
 
 74 
 
 61 
 
 62 
 
 63 
 
 64 
 
 64 
 
 775 
 
 2398 
 
 2484 
 
 2570 
 
 2655 
 
 2741 
 
 2827 
 
 2912 
 
 2998 
 
 3083 
 
 3169 
 
 75 
 
 62 
 
 63 
 
 64 
 
 64 
 
 65 
 
 776 
 
 3255 
 
 3340 
 
 3426 
 
 3512 
 
 3597 
 
 3683 
 
 3769 
 
 3854 
 
 3940 
 
 4026 
 
 76 
 
 63 
 
 64 
 
 65 
 
 65 
 
 66 
 
 1 -777 
 
 4111 
 
 4197 
 
 4283 
 
 4368 
 
 4454 
 
 4540 
 
 4626 
 
 4711 
 
 4797 
 
 4883 
 
 77 
 
 64 
 
 65 
 
 65 
 
 66 
 
 67 
 
 1 -778 
 
 4968 
 
 5054 
 
 5140 
 
 5226 
 
 5311 
 
 5397 
 
 5483 
 
 5568 
 
 5654 
 
 5740 
 
 78 
 
 65 
 
 66 
 
 66 
 
 67 
 
 68 
 
 I -779 
 
 5826 
 
 5911 
 
 5997 
 
 6083 
 
 6169 
 
 6254 
 
 6340 
 
 6426 
 
 6512 
 
 6597 
 
 79 
 
 66 
 
 66 
 
 67 
 
 68 
 
 69 
 
 1 -780 
 
 84 6683 
 
 6769 
 
 6855 
 
 6940 
 
 7026 
 
 7112 
 
 7198 
 
 7284 
 
 7369 
 
 7455 
 
 80 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 1 -781 
 
 7541 
 
 7627 
 
 7713 
 
 7798 
 
 7884 
 
 7970 
 
 8056 
 
 8142 
 
 8227 
 
 8313 
 
 81 
 
 67 
 
 68 
 
 69 
 
 70 
 
 70 
 
 1 '782 
 
 8399 
 
 8485 
 
 8571 
 
 8657 
 
 8742 
 
 8828 
 
 8914 
 
 9000 
 
 9086 
 
 9172 
 
 82 
 
 68 
 
 69 
 
 70 
 
 71 
 
 71 
 
 1 -783 
 
 9257 
 
 9343 
 
 9429 
 
 9515 
 
 9601 
 
 9687 
 
 9773 
 
 9858 
 
 9944 
 
 0030 
 
 83 
 
 69 
 
 70 
 
 71 
 
 71 
 
 72 
 
 784 
 
 850116 
 
 0202 
 
 0288 
 
 0374 
 
 0460 
 
 0545 
 
 0631 
 
 0717 
 
 0803 
 
 0889 
 
 84 
 
 70 
 
 71 
 
 71 
 
 72 
 
 73 
 
 785 
 
 0975 
 
 1061 
 
 1147 
 
 1233 
 
 1319 
 
 1405 
 
 1490 
 
 1576 
 
 1662 
 
 1748 
 
 85 
 
 71 
 
 71 
 
 72 
 
 73 
 
 74 
 
 786 
 
 1834 
 
 1920 
 
 2006 
 
 2092 
 
 2178 
 
 2264 
 
 2350 
 
 2436 
 
 2522 
 
 2608 
 
 86 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 787 
 
 2694 
 
 2780 
 
 2866 
 
 2952 
 
 3038 
 
 3124 
 
 3209 
 
 3295 
 
 3381 
 
 3467 
 
 87 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 1 -788 
 
 3553 
 
 3639 
 
 3725 
 
 3811 
 
 3897 
 
 3983 
 
 4069 
 
 4155 
 
 4241 
 
 4327 
 
 88 
 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 I -789 
 
 4413 
 
 4499 
 
 4586 
 
 4672 
 
 4758 
 
 4844 
 
 4930 
 
 5016 
 
 5102 
 
 5188 
 
 89 
 
 74 
 
 75 
 
 76 
 
 77 
 
 77 
 
 1 -790 
 
 85 5274 
 
 5360 
 
 5446 
 
 5532 
 
 5618 
 
 5704 
 
 5790 
 
 5876 
 
 5962 
 
 6048 
 
 90 
 
 75 
 
 76 
 
 77 
 
 77 
 
 78 
 
 791 
 
 6134 
 
 6220 
 
 6307 
 
 6393 
 
 6479 
 
 6565 
 
 6651 
 
 6737 
 
 6823 
 
 6909 
 
 91 
 
 76 
 
 76 
 
 77 
 
 78 
 
 79 
 
 792 
 
 6995 
 
 7081 
 
 7167 
 
 7254 
 
 7340 
 
 7426 
 
 7512 
 
 7598 
 
 7684 
 
 7770 
 
 92 
 
 "6 
 
 77 
 
 78 
 
 79 
 
 80 
 
 793 
 
 78561 7942 
 
 8029 
 
 8115 
 
 8201 
 
 8287 
 
 8373 8459 
 
 8545 
 
 8632 
 
 93 
 
 ~7 
 
 78 
 
 79 
 
 80 
 
 81 
 
 794 
 
 8718 
 
 8804 
 
 8890 
 
 8976 
 
 9062 
 
 9149 
 
 9235 
 
 9321 
 
 9407 
 
 9493 
 
 94 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 795 
 
 9579 
 
 9666 
 
 9752 
 
 9838 
 
 9924 
 
 0010 
 
 0097 
 
 0183 
 
 0269 
 
 0355 
 
 95 
 
 /9 
 
 80 
 
 81 
 
 82 
 
 83 
 
 796 
 
 86 0441 
 
 0528 
 
 0614 
 
 0700 
 
 0786 
 
 0873 
 
 0959 
 
 1045 
 
 1131 
 
 1217 
 
 96 
 
 30 
 
 8182 
 
 S3 
 
 84 
 
 797 
 
 1304 
 
 1390 
 
 1476 
 
 1562 
 
 1649 
 
 1735 
 
 1821 
 
 1907 
 
 1994 
 
 2080 
 
 97 
 
 31 
 
 8182 
 
 33 
 
 84 
 
 798 
 
 2166 
 
 2252 
 
 2339 
 
 2425 
 
 2511 
 
 2598 
 
 2684 
 
 2770 
 
 2856 
 
 2943 
 
 98 
 
 31 
 
 8283 
 
 34 
 
 85 
 
 1 -799 
 
 3029 
 
 3115 
 
 3202 
 
 3288 
 
 3374 
 
 3460 
 
 3547 
 
 3633 
 
 3719 
 
 3806 
 
 99 
 
 32 
 
 83 
 
 34 
 
 35 
 
 86 
 
 t 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 33 
 
 84 
 
 85 
 
 36 
 
 S7 
 
 17 
 
TABLE I. 
 
 Log 
 
 Log(l+a) 
 
 Pro. Parts 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 86 
 
 87 
 
 88 
 
 0-800 
 
 0-86 3892 
 
 3978 
 
 4065 
 
 4151 
 
 4237 
 
 4324 
 
 4410 
 
 4496 
 
 4583 
 
 4669 
 
 00 
 
 
 
 
 
 
 
 801 
 
 4755 
 
 4842 
 
 4928 
 
 5014 
 
 5101 
 
 5187 
 
 5273 
 
 5360 
 
 5446 
 
 5533 
 
 01 
 
 1 
 
 1 
 
 1 
 
 802 
 
 5619 
 
 5705 
 
 5792 
 
 5878 
 
 5964 
 
 6051 
 
 6137 
 
 6224 
 
 6310 
 
 6396 
 
 02 
 
 2 
 
 2 
 
 2 
 
 803 
 
 6483 
 
 6569 
 
 6656 
 
 6742 
 
 6828 
 
 6915 
 
 7001 
 
 7088 
 
 7174 
 
 7261 
 
 03 
 
 3 
 
 3 
 
 3 
 
 804 
 
 7347 
 
 7433 
 
 7520 
 
 7606 
 
 7693 
 
 7779 
 
 7866 
 
 7952 
 
 8038 
 
 8125 
 
 04 
 
 3 
 
 3 
 
 4 
 
 805 
 
 8211 
 
 8298 
 
 8384 
 
 8471 
 
 8557 
 
 8644 
 
 8730 
 
 8817 
 
 8903 
 
 8990 
 
 05 
 
 4 
 
 4 
 
 4 
 
 806 
 
 9076 
 
 9163 
 
 9249 
 
 9336 
 
 9422 
 
 9509 
 
 9595 
 
 9682 
 
 9768 
 
 9855 
 
 06 
 
 5 
 
 5 
 
 5 
 
 807 
 
 9941 
 
 0028 
 
 0114 
 
 0201 
 
 0287 
 
 0374 
 
 0460 
 
 0547 
 
 0633 
 
 0720 
 
 07 
 
 6 
 
 6 
 
 6 
 
 808 
 
 87 0806 
 
 0893 
 
 0979 
 
 1066 
 
 1152 
 
 1239 
 
 1326 
 
 1412 
 
 1499 
 
 1585 
 
 08 
 
 7 
 
 7 
 
 7 
 
 809 
 
 1672 
 
 1758 
 
 1845 
 
 1931 
 
 2018 
 
 2105 
 
 2191 
 
 2278 
 
 2364 
 
 2451 
 
 09 
 
 8 
 
 8 
 
 8 
 
 810 
 
 87 2537 
 
 2624 
 
 2711 
 
 2797 
 
 2884 
 
 2970 
 
 3057 
 
 3144 
 
 3230 
 
 3317 
 
 10 
 
 9 
 
 9 
 
 9 
 
 811 
 
 3404 
 
 3490 
 
 3577 
 
 3663 
 
 3750 
 
 3837 
 
 3923 
 
 4010 
 
 4097 
 
 4183 
 
 11 
 
 9 
 
 10 
 
 10 
 
 812 
 
 4270 
 
 4356 
 
 4443 
 
 4530 
 
 4616 
 
 4703 
 
 4790 
 
 4876 
 
 4963 
 
 5050 
 
 12 
 
 10 
 
 10 
 
 11 
 
 813 
 
 5136 
 
 5223 
 
 5310 
 
 5396 
 
 5483 
 
 5570 
 
 5656 
 
 5743 
 
 5830 
 
 5916 
 
 13 
 
 11 
 
 11 
 
 11 
 
 814 
 
 6003 
 
 6090 
 
 6177 
 
 6263 
 
 6350 
 
 6437 
 
 6523 
 
 6610 
 
 6697 
 
 6784 
 
 14 
 
 12 
 
 12 
 
 12 
 
 815 
 
 6870 
 
 6957 
 
 7044 
 
 7130 
 
 7217 
 
 7304 
 
 7391 
 
 7477 
 
 7564 
 
 7651 
 
 15 
 
 13 
 
 13 
 
 13 
 
 816 
 
 7738 
 
 7824 
 
 7911 
 
 7998 
 
 8085 
 
 8171 
 
 8258 
 
 8345 
 
 8432 
 
 8518 
 
 16 
 
 14 
 
 14 
 
 14 
 
 817 
 
 8605 
 
 8692 
 
 8779 
 
 8866 
 
 8952 
 
 9039 
 
 9126 
 
 9213 
 
 9300 
 
 9386 
 
 17 
 
 15 
 
 15 
 
 15 
 
 818 
 
 9473 
 
 9560 
 
 9647 
 
 9734 
 
 9820 
 
 9907 
 
 9994 
 
 0081 
 
 o!68 
 
 0254 
 
 18 
 
 15 
 
 16 
 
 16 
 
 819 
 
 88 0341 
 
 0428 
 
 0515 
 
 0602 
 
 0689 
 
 0775 
 
 0862 
 
 0949 
 
 1036 
 
 1123 
 
 19 
 
 16 
 
 17 
 
 17 
 
 820 
 
 88 1210 
 
 1297 
 
 1383 
 
 1470 
 
 1557 
 
 1644 
 
 1731 
 
 1818 
 
 1905 
 
 1991 
 
 20 
 
 17 
 
 17 
 
 18 
 
 821 
 
 2078 
 
 2165 
 
 2252 
 
 2339 
 
 2426 
 
 2513 
 
 2600 
 
 2687 
 
 2773 
 
 2860 
 
 21 
 
 18 
 
 18 
 
 18 
 
 822 
 
 2947 
 
 3034 
 
 3121 
 
 3208 
 
 3295 
 
 3382 
 
 3469 
 
 3556 
 
 3643 
 
 3730 
 
 22 
 
 19 
 
 19 
 
 19 
 
 823 
 
 3816 
 
 3903 
 
 3990 
 
 4077 
 
 4164 
 
 4251 
 
 4338 
 
 4425 
 
 4512 
 
 4599 
 
 23 
 
 20 
 
 20 
 
 20 
 
 824 
 
 4686 
 
 4773 
 
 4860 
 
 4947 
 
 5034 
 
 5121 
 
 5208 
 
 5295 
 
 5382 
 
 5469 
 
 24 
 
 21 
 
 21 
 
 21 
 
 825 
 
 5556 
 
 5643 
 
 5730 
 
 5817 
 
 5904 
 
 5991 
 
 6078 
 
 6165 
 
 6252 
 
 6339 
 
 25 
 
 22 
 
 22 
 
 22 
 
 826 
 
 6426 
 
 6513 
 
 6600 
 
 6687 
 
 6774 
 
 6861 
 
 6948 
 
 7035 
 
 7122 
 
 7209 
 
 26 
 
 22 
 
 23 
 
 23 
 
 827 
 
 7296 
 
 7383 
 
 7470 
 
 7557 
 
 7644 
 
 7731 
 
 7818 
 
 7905 
 
 7992 
 
 8079 
 
 27 
 
 23 
 
 23 
 
 24 
 
 828 
 
 8166 
 
 8253 
 
 8341 
 
 8428 
 
 8515 
 
 8602 
 
 8689 
 
 8776 
 
 8863 
 
 8950 
 
 28 
 
 24 
 
 24 
 
 25 
 
 829 
 
 9037 
 
 9124 
 
 9211 
 
 9298 
 
 9386 
 
 9473 
 
 9560 
 
 9647 
 
 9734 
 
 9821 
 
 29 
 
 25 
 
 25 
 
 26 
 
 830 
 
 88 9908 
 
 9995 
 
 0082 
 
 0170 
 
 0257 
 
 0344 
 
 0431 
 
 0518 
 
 0605 
 
 0692 
 
 30 
 
 26 
 
 26 
 
 26 
 
 831 
 
 89 0779 
 
 0867 
 
 0954 
 
 1041 
 
 1128 
 
 1215 
 
 1302 
 
 1389 
 
 1477 
 
 1564 
 
 31 
 
 27 
 
 27 
 
 27 
 
 832 
 
 1651 
 
 1738 
 
 1825 
 
 1912 
 
 2000 
 
 2087 
 
 2174 
 
 2261 
 
 2348 
 
 2436 
 
 32 
 
 28 
 
 28 
 
 28 
 
 833 
 
 2523 
 
 2610 
 
 2697 
 
 2784 
 
 2872 
 
 2959 
 
 3046 
 
 3133 
 
 3220 
 
 3308 
 
 33 
 
 -28 
 
 29 
 
 29 
 
 834 
 
 3395 
 
 3482 
 
 3569 
 
 3656 
 
 3744 
 
 3831 
 
 3918 
 
 4005 
 
 4093 
 
 4180 
 
 34 
 
 29 
 
 30 
 
 30 
 
 835 
 
 4267 
 
 4354 
 
 4442 
 
 4529 
 
 4616 
 
 4703 
 
 4790 
 
 4878 
 
 4965 
 
 5052 
 
 35 
 
 30 
 
 30 
 
 31 
 
 836 
 
 5140 
 
 5227 
 
 5314 
 
 5401 
 
 5489 
 
 5576 
 
 5663 
 
 5751 
 
 5838 
 
 5925 
 
 36 
 
 31 
 
 31 
 
 32 
 
 837 
 
 6012 
 
 6100 
 
 6187 
 
 6274 
 
 6362 
 
 6449 
 
 6536 
 
 6624 
 
 6711 
 
 6798 
 
 37 
 
 3-2 
 
 32 
 
 33 
 
 838 
 
 6886 
 
 6973 
 
 7060 
 
 7148 
 
 7235 
 
 7322 
 
 7410 
 
 7497 
 
 7584 
 
 7672 
 
 38 
 
 .33 
 
 33 
 
 33 
 
 839 
 
 7759 
 
 7846 
 
 7934 
 
 8021 
 
 8108 
 
 8196 
 
 8283 
 
 8370 
 
 8458 
 
 8545 
 
 39 
 
 34 
 
 34 
 
 34 
 
 840 
 
 898632 
 
 8720 
 
 8807 
 
 8895 
 
 8982 
 
 9069 
 
 9157 
 
 9244 
 
 9332 
 
 9419 
 
 40 
 
 34 
 
 35 
 
 35 
 
 841 
 
 9506 
 
 9594 
 
 9681 
 
 9769 
 
 9856 
 
 9943 
 
 0031 
 
 0118 
 
 0206 
 
 0293 
 
 41 
 
 35 
 
 36 
 
 36 
 
 842 
 
 90 0380 
 
 0468 
 
 0555 
 
 0643 
 
 0730 
 
 0818 
 
 0905 
 
 0992 
 
 1080 
 
 1167 
 
 42 
 
 36 
 
 37 
 
 37 
 
 843 
 
 1255 
 
 1342 
 
 1430 
 
 1517 
 
 1605 
 
 1692 
 
 1779 
 
 1867 
 
 1954 
 
 2042 
 
 43 
 
 37 
 
 37 
 
 38 
 
 844 
 
 2129 
 
 2217 
 
 2304 
 
 2392 
 
 2479 
 
 2567 
 
 2654 
 
 2742 
 
 2829 
 
 2917 
 
 44 
 
 38 
 
 38 
 
 39 
 
 845 
 
 3004 
 
 3092 
 
 3179 
 
 3267 
 
 3354 
 
 3442 
 
 3529 
 
 3617 
 
 3704 
 
 3792 
 
 45 
 
 39 
 
 39 
 
 40 
 
 846 
 
 3879 
 
 3967 
 
 4054 
 
 4142 
 
 4229 
 
 4317 
 
 4404 
 
 4492 
 
 4580 
 
 4667 
 
 46 
 
 40 
 
 40 
 
 40 
 
 847 
 
 4755 
 
 4842 
 
 4930 
 
 5017 
 
 5105 
 
 5192 
 
 5280 
 
 5368 
 
 5455 
 
 5543 
 
 4? 
 
 40 
 
 41 
 
 41 
 
 848 
 
 5630 
 
 5718 
 
 5805 
 
 5893 
 
 5981 
 
 6068 
 
 6156 
 
 6243 
 
 6331 
 
 6419 
 
 48 
 
 41 
 
 42 
 
 42 
 
 849 
 
 6506 
 
 6594 
 
 6681 
 
 6769 
 
 6857 
 
 6944 
 
 7032 
 
 7119 
 
 7207 
 
 7295 
 
 49 
 
 42 
 
 43 
 
 43 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 86 
 
 87 
 
 88 
 
 18 
 
TABLE I. 
 
 Log 
 
 Log(l+aO 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 86 
 
 87 
 
 88 
 
 89 
 
 0-850 
 
 D'QO 7382 
 
 7470 
 
 7557 
 
 7645 
 
 77QQ 
 
 7820 
 
 7QOft 
 
 7QQfi 
 
 QHQQ 
 
 8171 
 
 50 
 
 4Q 
 
 1 1 
 
 11 
 
 45 
 
 851 
 
 \J *7\J i OO*& 
 
 8259 
 
 1 TT I L 
 
 8346 
 
 1 tyJ 1 
 
 8434 
 
 t UTJ 
 
 8522 
 
 * 1 OO 
 
 8609 
 
 1 OU 
 
 8697 
 
 I I/UO 
 
 8785 
 
 * 7t7\j 
 
 8872 
 
 ouoo 
 
 8960 
 
 1 * 1 
 
 9048 
 
 ou 
 51 
 
 T3 
 
 44 
 
 T?TC 
 
 44 
 
 TT 
 
 45 
 
 *rO 
 
 45 
 
 852 
 
 9135 
 
 9223 
 
 9311 
 
 9398 
 
 9486 
 
 9574 
 
 9661 
 
 9749 
 
 9837 
 
 9924 
 
 52 
 
 45 
 
 45 
 
 46 
 
 46 
 
 853 
 
 91 0012 
 
 0100 
 
 0187 
 
 0275 
 
 0363 
 
 0451 
 
 0538 
 
 0626 
 
 0714 
 
 0801 
 
 53 
 
 46 
 
 46 
 
 47 
 
 47 
 
 854 
 
 0889 
 
 0977 
 
 1065 
 
 1152 
 
 1240 
 
 1328 
 
 1416 
 
 1503 
 
 1591 
 
 1679 
 
 54 
 
 46 
 
 47 
 
 48 
 
 48 
 
 855 
 
 1766 
 
 1854 
 
 1942 
 
 2030 
 
 2117 
 
 2205 
 
 2293 
 
 2381 
 
 2469 
 
 2556 
 
 55 
 
 47 
 
 48 
 
 48 
 
 49 
 
 856 
 
 2644 
 
 2732 
 
 2820 
 
 2907 
 
 2995 
 
 3083 
 
 3171 
 
 3259 
 
 3346 
 
 3434 
 
 56 
 
 48 
 
 49 
 
 49 
 
 50 
 
 857 
 
 3522 
 
 3610 
 
 3698 
 
 3785 
 
 3873 
 
 3961 
 
 4049 
 
 4137 
 
 4224 
 
 4312 
 
 57 
 
 49 
 
 50 
 
 50 
 
 51 
 
 858 
 
 4400 
 
 4488 
 
 4576 
 
 4663 
 
 4751 
 
 4839 
 
 4927 
 
 5015 
 
 5103 
 
 5190 
 
 58 
 
 50 
 
 50 
 
 51 
 
 52 
 
 859 
 
 5278 
 
 5366 
 
 5454 
 
 5542 
 
 5630 
 
 5718 
 
 5805 
 
 5893 
 
 5981 
 
 6069 
 
 59 
 
 51 
 
 51 
 
 52 
 
 53 
 
 860 
 
 91 6157 
 
 6245 
 
 6333 
 
 6421 
 
 6508 
 
 6596 
 
 6684 
 
 6772 
 
 6860 
 
 6948 
 
 60 
 
 52 
 
 52 
 
 53 
 
 53 
 
 861 
 
 7036 
 
 7124 
 
 7212 
 
 7299 
 
 7387 
 
 7475 
 
 7563 
 
 7651 
 
 7739 
 
 7827 
 
 61 
 
 52 
 
 53 
 
 54 
 
 54 
 
 862 
 
 7915 
 
 8003 
 
 8091 
 
 8179 
 
 8267 
 
 8354 
 
 8442 
 
 8530 
 
 8618 
 
 8706 
 
 62 
 
 53 
 
 54 
 
 55 
 
 55 
 
 863 
 
 8794 
 
 8882 
 
 8970 
 
 9058 
 
 9146 
 
 9234 
 
 9322 
 
 9410 
 
 9498 
 
 9586 
 
 63 
 
 54 
 
 55 
 
 55 
 
 56 
 
 864 
 
 9674 
 
 9762 
 
 9850 
 
 9938 
 
 0026 
 
 0114 
 
 0202 
 
 0290 
 
 0378 
 
 0466 
 
 64 
 
 55 
 
 56 
 
 56 
 
 57 
 
 865 
 
 92 0554 
 
 0642 
 
 0730 
 
 0817 
 
 0906 
 
 0994 
 
 1082 
 
 1170 
 
 1258 
 
 1346 
 
 65 
 
 56 
 
 57 
 
 57 
 
 58 
 
 866 
 
 1434 
 
 1522 
 
 1610 
 
 1698 
 
 1786 
 
 1874 
 
 1962 
 
 2050 
 
 2138 
 
 2226 
 
 66 
 
 57 
 
 57 
 
 58 
 
 59 
 
 867 
 
 2314 
 
 2402 
 
 2490 
 
 2578 
 
 2666 
 
 2754 
 
 2842 
 
 2930 
 
 3018 
 
 3106 
 
 67 
 
 58 
 
 58 
 
 59 
 
 60 
 
 868 
 
 3194 
 
 3282 
 
 3371 
 
 3459 
 
 3547 
 
 3635 
 
 3723 
 
 3811 
 
 3899 
 
 3987 
 
 68 
 
 58 
 
 59 
 
 60 
 
 61 
 
 869 
 
 4075 
 
 4163 
 
 4251 
 
 4339 
 
 4428 
 
 4516 
 
 4604 
 
 4692 
 
 4780 
 
 4868 
 
 69 
 
 59 
 
 60 
 
 61 
 
 61 
 
 870 
 
 92 4956 
 
 5044 
 
 5132 
 
 5221 
 
 5309 
 
 5397 
 
 5485 
 
 5573 
 
 5661 
 
 5749 
 
 70 
 
 60 
 
 61 
 
 62 
 
 62 
 
 871 
 
 5837 
 
 5926 
 
 6014 
 
 6102 
 
 6190 
 
 6278 
 
 6366 
 
 6454 
 
 6543 
 
 6631 
 
 71 
 
 61 
 
 62 
 
 62 
 
 63 
 
 872 
 
 6719 
 
 6807 
 
 6895 
 
 6983 
 
 7072 
 
 7160 
 
 7248 
 
 7336 
 
 7424 
 
 7512 
 
 72 
 
 62 
 
 63 
 
 63 
 
 64 
 
 873 
 
 7601 
 
 7689 
 
 7777 
 
 7865 
 
 7953 
 
 8042 
 
 8130 
 
 8218 
 
 8306 
 
 8394 
 
 73 
 
 63 
 
 64 
 
 64 
 
 65 
 
 874 
 
 8483 
 
 8571 
 
 8659 
 
 8747 
 
 8836 
 
 8924 
 
 9012 
 
 9100 
 
 9188 
 
 9277 
 
 74 
 
 64 
 
 64 
 
 65 
 
 66 
 
 875 
 
 9365 
 
 9453 
 
 9541 
 
 9630 
 
 9718 
 
 9806 
 
 9894 
 
 9983 
 
 0071 
 
 0159 
 
 75 
 
 64 
 
 65 
 
 66 
 
 67 
 
 876 
 
 93 0247 
 
 0336 
 
 0424 
 
 0512 
 
 0600 
 
 0689 
 
 0777 
 
 0865 
 
 0953 
 
 1042 
 
 76 
 
 65 
 
 66 
 
 67 
 
 68 
 
 877 
 
 1130 
 
 1218 
 
 1307 
 
 1395 
 
 1483 
 
 1571 
 
 1660 
 
 1748 
 
 1836 
 
 1925 
 
 77 
 
 66 
 
 67 
 
 68 
 
 69 
 
 878 
 
 2013 
 
 2101 
 
 2190 
 
 2278 2366 
 
 2455 
 
 2543 
 
 2631 
 
 2719 
 
 2808 
 
 78 
 
 67 
 
 68 
 
 69 
 
 69 
 
 879 
 
 2896 
 
 2984 
 
 3073 
 
 3161 
 
 3249 
 
 3338 
 
 3426 
 
 3514 
 
 3603 
 
 3691 
 
 79 
 
 68 
 
 69 
 
 70 
 
 70 
 
 880 
 
 93 3780 
 
 3868 
 
 3956 
 
 4045 
 
 4133 
 
 4221 
 
 4310 
 
 4398 
 
 4486 
 
 4575 
 
 80 
 
 69 
 
 70 
 
 70 
 
 71 
 
 881 
 
 4663 
 
 4752 
 
 4840 
 
 4928 
 
 5017 
 
 5105 
 
 5193 
 
 5282 
 
 5370 
 
 5459 
 
 81 
 
 70 
 
 70 
 
 71 
 
 72 
 
 882 
 
 5547 
 
 5635 
 
 5724 
 
 5812 
 
 5901 
 
 5989 
 
 6078 
 
 6166 
 
 6254 
 
 6343 
 
 82 
 
 71 
 
 71 
 
 72 
 
 73 
 
 883 
 
 6431 
 
 6520 
 
 6608 
 
 6696 
 
 6785 
 
 6873 
 
 6962 
 
 7050 
 
 7139 
 
 7227 
 
 83 
 
 71 
 
 72 
 
 73 
 
 74 
 
 884 
 
 7316 
 
 7404 
 
 7492 
 
 7581 
 
 7669 
 
 7758 
 
 7846 
 
 7935 
 
 8023 
 
 8112 
 
 84 
 
 72 
 
 73 
 
 74 
 
 75 
 
 885 
 
 8200 
 
 8289 
 
 8377 
 
 8466 
 
 8554 
 
 8643 
 
 8731 
 
 8819 
 
 8908 
 
 8996 
 
 85 
 
 73 
 
 74 
 
 75 
 
 76 
 
 886 
 
 9085 
 
 9173 
 
 9262 
 
 9350 
 
 9439 
 
 9527 
 
 9616 
 
 9704 
 
 9793 
 
 9882 
 
 86 
 
 74|75 
 
 76 
 
 77 
 
 887 
 
 9970 
 
 0059 
 
 0147 
 
 0236 
 
 0324 
 
 o413 
 
 0501 
 
 0590 
 
 0678 
 
 0767 
 
 87 
 
 75 
 
 76 
 
 77 
 
 77 
 
 888 
 
 94 0855 
 
 0944 
 
 1032 
 
 1121 
 
 1209 
 
 1298 
 
 1387 
 
 1475 
 
 1564 
 
 1652 
 
 88 
 
 76 
 
 77 
 
 77 
 
 78 
 
 889 
 
 1741 
 
 1829 
 
 1918 
 
 2007 
 
 2095 
 
 2184 
 
 2272 
 
 2361 
 
 2449 
 
 2538 
 
 89 
 
 77 
 
 77 
 
 78 
 
 79 
 
 890 
 
 94 2627 
 
 2715 
 
 2804 
 
 2892 
 
 2981 
 
 3070 
 
 3158 
 
 3247 
 
 3335 
 
 3424 
 
 90 
 
 77 
 
 78 
 
 79 
 
 80 
 
 891 
 
 3513 
 
 3601 
 
 3690J 3778 
 
 3867 
 
 3956 
 
 4044 
 
 4133 
 
 4222 
 
 4310 
 
 91 
 
 78 
 
 79 
 
 80 
 
 81 
 
 892 
 
 4399 
 
 4487 
 
 4576 
 
 4665 
 
 4753 
 
 4842 
 
 4931 
 
 5019 
 
 5108 
 
 5197 
 
 & 
 
 79 
 
 80 
 
 81 
 
 82 
 
 893 
 
 5285 
 
 5374 
 
 5463 
 
 5551 
 
 5640 
 
 5729 
 
 5817 
 
 5906 
 
 5995 
 
 6083 
 
 93 
 
 80 
 
 81 
 
 82 
 
 83 
 
 894 
 
 6172 
 
 6261 
 
 6349 6438 1 6527 
 
 6615 
 
 6704 
 
 6793 
 
 6881 
 
 6970 
 
 94 
 
 81 
 
 82 
 
 83 
 
 84 
 
 895 
 
 7059 
 
 7148 
 
 7236 
 
 7325 7414 
 
 7502 
 
 7591 
 
 7680 
 
 7769 
 
 7857 
 
 95 
 
 82 
 
 83 
 
 84 
 
 85 
 
 896 
 897 
 
 7946 
 8833 
 
 8035 
 8922 
 
 8123 8212 8301 
 9011 9100 9188 
 
 8390 
 9277 
 
 8478 
 9366 
 
 8567 
 9455 
 
 8656 
 9543 
 
 8745 
 9632 
 
 96 
 97 
 
 83 
 83 
 
 % 
 
 84 
 
 85 
 
 85 
 86 
 
 898 
 
 9721 
 
 9810 
 
 9899 9987 
 
 0076 
 
 0165 
 
 0254 
 
 0342 
 
 0431 
 
 0520 
 
 98 
 
 84 
 
 85 
 
 86 
 
 87 
 
 899 
 
 95 0609 
 
 Ot98 
 
 0786 0875 
 
 0964 
 
 1053 
 
 1142 
 
 1230 
 
 1319 
 
 1408 
 
 99 
 
 85 
 
 86 
 
 87 
 
 88 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 86 
 
 87 
 
 88 
 
 89 
 
 19 
 
TABLE I. 
 
 Log 
 
 LogO -HO 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 88 
 
 89 
 
 90 
 
 91 
 
 0-900 
 
 0-95 1497 
 
 1586 
 
 1675 
 
 1763 
 
 1852 
 
 1941 
 
 2030 
 
 2119 
 
 2208 
 
 2296 
 
 00 
 
 
 
 
 
 
 
 
 
 901 
 
 2385 
 
 2474 
 
 2563 
 
 2652 
 
 2741 
 
 2829 
 
 2918 
 
 3007 
 
 3096 
 
 3185 
 
 01 
 
 ] 
 
 1 
 
 1 
 
 1 
 
 902 
 
 3274 
 
 3363 
 
 3451 
 
 3540 
 
 3629 
 
 3718 
 
 3807 
 
 3896 
 
 3985 
 
 4074 
 
 02 
 
 
 
 2 
 
 2 
 
 2 
 
 903 
 
 4163 
 
 4251 
 
 4340 
 
 4429 
 
 4518 
 
 4607 
 
 4696 
 
 4785 
 
 4874 
 
 4963 
 
 03 
 
 1 
 
 3 
 
 3 
 
 g^ 
 
 c 
 
 904 
 
 5052 
 
 5140 
 
 5229 
 
 5318 
 
 5407 
 
 5496 
 
 5585 
 
 5674 
 
 5763 
 
 5852 
 
 04 
 
 4 
 
 4 
 
 4 
 
 4 
 
 905 
 
 5941 
 
 6030 
 
 6119 
 
 6208 
 
 6296 
 
 6385 
 
 6474 
 
 6563 
 
 6652 
 
 6741 
 
 05 
 
 4 
 
 4 
 
 5 
 
 e 
 
 906 
 
 6830 
 
 6919 
 
 7008 
 
 7097 
 
 7186 
 
 7275 
 
 7364 
 
 7453 
 
 7542 
 
 7631 
 
 06 
 
 t 
 
 5 
 
 5 
 
 e 
 
 
 
 907 
 
 7720 
 
 7809 
 
 7898 
 
 7987 
 
 8076 
 
 8165 
 
 8254 
 
 8343 
 
 8432 
 
 8521 
 
 07 
 
 6 
 
 6 
 
 6 
 
 6 
 
 908 
 
 8610 
 
 8699 
 
 8788 
 
 8877 
 
 8966 
 
 9055 
 
 9144 
 
 9233 
 
 9322 
 
 9411 
 
 08 
 
 r 
 
 7 
 
 7 
 
 7 
 
 909 
 
 9500 
 
 9589 
 
 9678 
 
 9767 
 
 9856 
 
 9945 
 
 0034 
 
 0123 
 
 0212 
 
 0301 
 
 09 
 
 8 
 
 8 
 
 8 
 
 8 
 
 910 
 
 96 0390 
 
 0479 
 
 0568 
 
 0657 
 
 0746 
 
 0835 
 
 0924 
 
 1014 
 
 1103 
 
 1192 
 
 10 
 
 9 
 
 9 
 
 9 
 
 9 
 
 911 
 
 1281 
 
 1370 
 
 1459 
 
 1548 
 
 1637 
 
 1726 
 
 1815 
 
 1904 
 
 1993 
 
 2082 
 
 11 
 
 10 
 
 10 
 
 10 
 
 10 
 
 912 
 
 2172 
 
 2261 
 
 2350 
 
 2439 
 
 2528 
 
 2617 
 
 2706 
 
 2795 
 
 2884 
 
 2973 
 
 12 
 
 11 
 
 11 
 
 11 
 
 11 
 
 913 
 
 3063 
 
 3152 
 
 3241 
 
 3330 
 
 3419 
 
 3508 
 
 3597 
 
 3686 
 
 3775 
 
 3865 
 
 13 
 
 11 
 
 12 
 
 12 
 
 12 
 
 914 
 
 3954 
 
 4043 
 
 4132 
 
 4221 
 
 4310 
 
 4399 
 
 4489 
 
 4578 
 
 4667 
 
 4756 
 
 14 
 
 12 
 
 12 
 
 13 
 
 13 
 
 915 
 
 4845 
 
 4934 
 
 5024 
 
 5113 
 
 5202 
 
 5291 
 
 5380 
 
 5469 
 
 5559 
 
 5648 
 
 15 
 
 13 
 
 13 
 
 13 
 
 14 
 
 916 
 
 5737 
 
 5826 
 
 5915 
 
 6004 
 
 6094 
 
 6183 
 
 6272 
 
 6361 
 
 6450 
 
 6540 
 
 16 
 
 14 
 
 14 
 
 14 
 
 15 
 
 917 
 
 6629 
 
 6718 
 
 6807 
 
 6896 
 
 6986 
 
 7075 
 
 7164 
 
 7253 
 
 7342 
 
 7432 
 
 17 
 
 15 
 
 15 
 
 15 
 
 15 
 
 918 
 
 7521 
 
 7610 
 
 7699 
 
 7789 
 
 7878 
 
 7967 
 
 8056 
 
 8146 
 
 8235 
 
 8324 
 
 18 
 
 16 
 
 16 
 
 16 
 
 16 
 
 919 
 
 8413 
 
 8503 
 
 8592 
 
 8681 
 
 8770 
 
 8860 
 
 8949 
 
 9038 
 
 9127 
 
 9217 
 
 19 
 
 17 
 
 17 
 
 17 
 
 17 
 
 920 
 
 Qfi Q*?0fi 
 
 qqoK 
 
 0404 
 
 QK74 
 
 QfifiQ 
 
 9752 
 
 Q841 
 
 qoQi 
 
 o020 
 
 olOQ 
 
 20 
 
 18 
 
 18 
 
 18 
 
 18 
 
 921 
 
 7D I/3UC 
 
 970199 
 
 tfOifu 
 
 0288 
 
 j7*rOT 
 
 0377 
 
 W I T 
 
 0466 
 
 i/UUt- 
 
 0556 
 
 0645 
 
 J/OT J 
 
 0734 
 
 J/I7O 1 
 
 0824 
 
 UvJrfOV 
 
 0913 
 
 Ul Vt 
 
 1002 
 
 21 
 
 18 
 
 19 
 
 19 
 
 19 
 
 922 
 
 1092 
 
 1181 
 
 1270 
 
 1360 
 
 1449 
 
 1538 
 
 1628 
 
 1717 
 
 1806 
 
 1896 
 
 22 
 
 19 
 
 20 
 
 20 
 
 20 
 
 923 
 
 1985 
 
 2074 
 
 2164 
 
 2253 
 
 2342 
 
 2432 
 
 2521 
 
 2610 
 
 2700 
 
 2789 
 
 23 
 
 20 
 
 20 
 
 21 
 
 21 
 
 924 
 
 2878 
 
 2968 
 
 3057 
 
 3146 
 
 3236 
 
 3325 
 
 3414 
 
 3504 
 
 3593 
 
 3683 
 
 24 
 
 21 
 
 21 
 
 22 
 
 22 
 
 925 
 
 3772 
 
 3861 
 
 3951 
 
 4040 
 
 4129 
 
 4219 
 
 4308 
 
 4398 
 
 4487 
 
 4576 
 
 25 
 
 22 
 
 22 
 
 23 
 
 23 
 
 926 
 
 4666 
 
 4755 
 
 4845 
 
 4934 
 
 5023 
 
 5113 
 
 5202 
 
 5292 
 
 5381 
 
 5471 
 
 26 
 
 23 
 
 23J23 
 
 24 
 
 927 
 
 5560 
 
 5649 
 
 5739 
 
 5828 
 
 5918 
 
 6007 
 
 6097 
 
 6186 
 
 6275 
 
 6365 
 
 21 
 
 24 
 
 24J24 
 
 25 
 
 928 
 
 6454 
 
 6544 
 
 6633 
 
 6723 
 
 6812 
 
 6902 
 
 6991 
 
 7080 
 
 7170 
 
 7259 
 
 28 
 
 25 
 
 25 
 
 25 
 
 25 
 
 929 
 
 7349 
 
 7438 
 
 7528 
 
 7617 
 
 7707 
 
 7796 
 
 7886 
 
 7975 
 
 8065 
 
 8154 
 
 29 
 
 26 
 
 26 
 
 26 
 
 26 
 
 930 
 
 97 8244 
 
 8333 
 
 8423 
 
 8512 
 
 8602 
 
 8691 
 
 8781 
 
 8870 
 
 8960 
 
 9049 
 
 30 
 
 26 
 
 27 
 
 27 
 
 27 
 
 931 
 
 9139 
 
 9228 
 
 9318 
 
 9407 
 
 9497 
 
 9586 
 
 9676 
 
 9765 
 
 9855 
 
 9944 
 
 31 
 
 21 
 
 28 
 
 28 
 
 28 
 
 932 
 
 98 0034 
 
 0123 
 
 0213 
 
 0302 
 
 0392 
 
 0481 
 
 0571 
 
 0660 
 
 0750 
 
 0840 
 
 32 
 
 28 
 
 28 
 
 29 
 
 29 
 
 933 
 
 0929 
 
 1019 
 
 1108 
 
 1198 
 
 1287 
 
 1377 
 
 1466 
 
 1556 
 
 1646 
 
 1735 
 
 33 
 
 -29 
 
 2930 
 
 30 
 
 934 
 
 1825 
 
 1914 
 
 2004 
 
 2093 
 
 2183 
 
 2273 
 
 2362 
 
 2452 
 
 2541 
 
 2631 
 
 34 
 
 30 
 
 3031 
 
 31 
 
 935 
 
 2721 
 
 2810 
 
 2900 
 
 2989 3079 
 
 3169 
 
 3258 
 
 3348 
 
 3437 
 
 3527 
 
 35 
 
 31 
 
 3131 
 
 32 
 
 936 
 
 3617 
 
 3706 
 
 3796 
 
 3885 3975 
 
 4065 
 
 4154 
 
 4244 
 
 4334 
 
 4423 
 
 36 
 
 32 
 
 32 
 
 32 
 
 33 
 
 937 
 
 4513 
 
 4603 
 
 4692 
 
 4782 
 
 4871 
 
 4961 
 
 5051 
 
 5140 
 
 5230 
 
 5320 
 
 37 
 
 33 
 
 33 
 
 33 
 
 34 
 
 938 
 
 5409 
 
 5499 
 
 5589 
 
 5678 
 
 5768 
 
 5858 
 
 5947 
 
 6037 
 
 6127 
 
 6216 
 
 38 
 
 33 
 
 34 
 
 34 
 
 35 
 
 939 
 
 6306 
 
 6396 
 
 6485 
 
 6575 
 
 6665 
 
 6754 
 
 6844 
 
 6934 
 
 7024 
 
 7113 
 
 39 
 
 34 
 
 35 
 
 35 
 
 35 
 
 940 
 
 98 7203 
 
 7293 
 
 7382 
 
 7472 
 
 7562 
 
 7651 
 
 7741 
 
 7831 
 
 7921 
 
 8010 
 
 4035 
 
 36 
 
 36 
 
 36 
 
 941 
 
 8100 
 
 8190 
 
 8280 
 
 8369 
 
 8459 
 
 8549 
 
 8638 
 
 8728 
 
 8818 
 
 8908 
 
 4136 
 
 36 
 
 37 
 
 37 
 
 942 
 
 8997 
 
 9087 
 
 9177 
 
 9267 
 
 9356 
 
 9446 
 
 9536 
 
 9626 
 
 9715 
 
 9805 
 
 4237 
 
 3738 
 
 38 
 
 943 
 
 9895 
 
 9985 
 
 0074 
 
 0164 
 
 0254 
 
 0344 
 
 0434 
 
 0523 
 
 0613 
 
 0703 
 
 4338 
 
 383939 
 
 944 
 
 99 0793 
 
 0882 
 
 0972 
 
 1062 
 
 1152 
 
 1242 
 
 1331 
 
 1421 
 
 1511 
 
 1601 
 
 44 
 
 39 
 
 39 
 
 4040 
 
 945 
 
 1691 
 
 1780 
 
 1870 
 
 1960 
 
 2050 
 
 2140 
 
 2230 
 
 2319 
 
 2409 
 
 2499 
 
 45 
 
 40 
 
 40 
 
 41 ! 41 
 
 946 
 
 2589 
 
 2679 
 
 2768 
 
 2858 
 
 2948 
 
 3038 
 
 3128 
 
 3218 
 
 3308 
 
 3397 
 
 46 
 
 40 
 
 41 
 
 41 
 
 42 
 
 947 
 
 3487 
 
 3577 
 
 3667 
 
 3757 
 
 3847 
 
 3936 
 
 4026 
 
 4116 
 
 4206 
 
 4296 
 
 47 
 
 41 
 
 42 
 
 42 
 
 43 
 
 948 
 
 4386 
 
 4476 
 
 4566 
 
 4655 
 
 4745 
 
 4835 
 
 4925 
 
 5015 
 
 5105 
 
 5195 
 
 48 
 
 42 
 
 43143 
 
 44 
 
 949 
 
 5285 
 
 5374 
 
 5464 
 
 5554 
 
 5644 
 
 5734 
 
 5824 
 
 5914 
 
 6004 
 
 6094 
 
 4943 
 
 4444 
 
 *5 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 J 
 
 38 
 
 8990 
 
 n 
 
TABLE I. 
 
 
 Log (14.2-1 
 
 Pro. Parts. 
 
 Log 
 
 
 
 
 X 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 88899091 
 
 0-950 
 
 0-996184 
 
 6274 
 
 6363 
 
 6453 
 
 6543 
 
 6633 
 
 6723 
 
 6813 
 
 6903 
 
 6993 
 
 50 
 
 4445 
 
 45 
 
 46 
 
 951 
 
 7083 
 
 7173 
 
 7263 
 
 7353 
 
 7443 
 
 7533 
 
 7622 
 
 7712 7802 
 
 7892 
 
 51 
 
 4545 
 
 46 
 
 46 
 
 952 
 
 7982 
 
 8072 
 
 8162 
 
 8252 
 
 8342 
 
 8432 
 
 8522 
 
 8612 
 
 8702 
 
 8792 
 
 52 
 
 46 
 
 46 
 
 47 
 
 47 
 
 953 
 
 8882 8972 
 
 9062 
 
 9152 
 
 9242 
 
 9332 
 
 9422 
 
 9512 
 
 9602 
 
 9692 
 
 53 
 
 47 
 
 47 
 
 48 
 
 48 
 
 954 
 
 9782 
 
 9872 
 
 9962 
 
 0052 
 
 0142 
 
 0232 
 
 0322 
 
 o412 
 
 0502 
 
 0592 
 
 54 
 
 48 
 
 48 
 
 49 
 
 49 
 
 955 
 
 1-000682 
 
 0772 
 
 0862 
 
 0952 
 
 1042 
 
 1132 
 
 1222 
 
 1312 
 
 1402 
 
 1492 
 
 55 
 
 48 
 
 49 
 
 49 
 
 50 
 
 956 
 
 1582 1672 
 
 1762 
 
 1852 
 
 1942 
 
 2032 
 
 2122j22l2 
 
 2302 
 
 2392 
 
 56 
 
 49 
 
 5050 
 
 51 
 
 957 
 
 2483 2573 
 
 2663 
 
 2753 
 
 2843 
 
 2933 
 
 3023 3113 
 
 3203 
 
 3293 
 
 57 
 
 50 
 
 5151 
 
 52 
 
 958 
 
 3383 
 
 3473 
 
 3563 
 
 3653 
 
 3744 
 
 3834 
 
 3924 
 
 4014 
 
 4104 
 
 4194 
 
 58 
 
 51 
 
 5252 
 
 53 
 
 959 
 
 4284 
 
 4374 
 
 4464 
 
 4554 
 
 4644 
 
 4735 
 
 4825 
 
 4915 
 
 5005 
 
 5095 
 
 59 
 
 52 
 
 53 
 
 53 
 
 54 
 
 960 
 
 005185 
 
 5275 
 
 5365 
 
 5456 
 
 5546 
 
 5636 
 
 5726 
 
 5816 
 
 5906 
 
 5996 
 
 60 
 
 53 
 
 53 
 
 54 
 
 55 
 
 961 
 
 6086 
 
 6177 
 
 6267 
 
 6357 
 
 6447 
 
 6537 
 
 6627 
 
 6717 
 
 6808 
 
 6898 
 
 61 
 
 54 
 
 54 
 
 55 
 
 56 
 
 962 
 
 6988 7078 
 
 7168 
 
 7258 
 
 7349 
 
 7439 
 
 7529 
 
 7619 
 
 7709 
 
 7799 
 
 62 
 
 55 
 
 55 
 
 56 
 
 56 
 
 963 
 
 7890 
 
 7980 
 
 8070 
 
 8160 8250 
 
 8341 
 
 8431 
 
 8521 
 
 8611 
 
 8701 
 
 63 
 
 55 
 
 56 
 
 57 
 
 57 
 
 964 
 
 8792 
 
 8882 
 
 8972 
 
 9062 
 
 9152 
 
 9243 
 
 9333 
 
 9423 
 
 9513 
 
 9603 
 
 64 
 
 56 
 
 57 
 
 58 
 
 58 
 
 965 
 
 9694 
 
 9784 
 
 9874 
 
 9964 
 
 0055 
 
 0145 
 
 0235 
 
 0325 
 
 0415 
 
 0506 
 
 65 
 
 57 
 
 58 
 
 59 
 
 59 
 
 966 
 
 01 0596 
 
 0686 
 
 0776 
 
 0867 
 
 0957 
 
 1047 
 
 1137 
 
 1228 
 
 1318 
 
 1408 
 
 66 
 
 58 
 
 59 
 
 59 
 
 60 
 
 967 
 
 1498 
 
 1589 
 
 1679 
 
 1769 
 
 I860 
 
 1950 
 
 2040 
 
 2130 
 
 2221 
 
 2311 
 
 67 
 
 59 
 
 60 
 
 60 
 
 61 
 
 968 
 
 2401 
 
 2491 
 
 2582 
 
 2672 
 
 2762 
 
 2853 
 
 2943 
 
 3033 
 
 3124 
 
 3214 
 
 68 
 
 60 
 
 61 
 
 61 
 
 62 
 
 969 
 
 3304 
 
 3394 
 
 3485 
 
 3575 
 
 3665 
 
 3756 
 
 3846 
 
 3936 
 
 4027 
 
 4117 
 
 69 
 
 61 
 
 61 
 
 62 
 
 63 
 
 970 
 
 01 4207 
 
 4298 
 
 4388 
 
 4478 
 
 4569 
 
 4659 
 
 4749 
 
 4840 
 
 4930 
 
 5020 
 
 70 
 
 62 
 
 62 
 
 63 
 
 64 
 
 971 
 
 5111 
 
 5201 
 
 5291 
 
 5382 
 
 5472 
 
 5562 
 
 5653 
 
 5743 
 
 5833 
 
 5924 
 
 71 
 
 62 
 
 63 
 
 64 
 
 65 
 
 972 
 
 6014 
 
 6104 
 
 6195 
 
 6285 
 
 6376 
 
 6466 
 
 6556 
 
 6647 
 
 6737 
 
 6827 
 
 72 
 
 63 
 
 64 
 
 65 
 
 66 
 
 973 
 
 6918 
 
 7008 
 
 7099 
 
 7189 
 
 7279 
 
 7370 
 
 7460 
 
 7551 
 
 7641 
 
 7731 
 
 73 
 
 64 
 
 65 
 
 66 
 
 66 
 
 974 
 
 7822 
 
 7912 
 
 8003 
 
 8093 
 
 8183 
 
 8274 
 
 8364 
 
 8455 
 
 8545 
 
 8635 
 
 74 
 
 65 
 
 66 
 
 67 
 
 67 
 
 975 
 
 8726 
 
 8816 
 
 8907 
 
 8997 
 
 908S 
 
 9178 
 
 9268 
 
 9359 
 
 9449 
 
 9540 
 
 75 
 
 66 
 
 67 
 
 67 
 
 68 
 
 976 
 
 9630 
 
 9721 
 
 9811 
 
 9901 
 
 9992 
 
 0082 
 
 0173 
 
 0263 
 
 0354 
 
 0444 
 
 76 
 
 67 
 
 68 
 
 68 
 
 69 
 
 977 
 
 020535 
 
 0625 
 
 0716 
 
 0806 
 
 0897 
 
 0987 
 
 1077 
 
 1168 
 
 1258 
 
 1349 
 
 77 
 
 68 
 
 69 
 
 69 
 
 70 
 
 978 
 
 1439 
 
 1530 
 
 1620 
 
 1711 
 
 1801 
 
 1892 
 
 1982 
 
 2073 
 
 2163 
 
 2254 
 
 78 
 
 69 
 
 69 
 
 70 
 
 71 
 
 979 
 
 2344 
 
 2435 
 
 2525 
 
 2616 
 
 2706 
 
 2797 
 
 2887 
 
 2978 
 
 3068 
 
 3159 
 
 79 
 
 70 
 
 70 
 
 71 
 
 72 
 
 980 
 
 02 3249 
 
 3340 
 
 3430 
 
 3521 
 
 3612 
 
 3702 
 
 3793 
 
 3883 
 
 3974 
 
 4064 
 
 80 
 
 70 
 
 71 
 
 72 
 
 73 
 
 981 
 
 4155 
 
 4245 
 
 4336 
 
 4426 
 
 4517 
 
 4607 
 
 4698 
 
 4789 
 
 4879 
 
 4970 
 
 81 
 
 71 
 
 72 
 
 73 
 
 74 
 
 982 
 
 5060 
 
 5151 
 
 5241 
 
 5332 
 
 5422 
 
 5513 
 
 5604 
 
 5694 
 
 5785 
 
 5875 
 
 82 
 
 72 
 
 73 
 
 74 
 
 75 
 
 983 
 
 5966 
 
 6057 
 
 6147 
 
 6238 
 
 6328 
 
 6419 
 
 6509 
 
 6600 
 
 6691 
 
 6781 
 
 83 
 
 73 
 
 74 
 
 75 
 
 76 
 
 984 
 
 6872 
 
 6962 
 
 7053 
 
 7144 
 
 7234 
 
 7325 
 
 7415 
 
 7506 
 
 7597 
 
 7687 
 
 84 
 
 74 
 
 75 
 
 76 
 
 76 
 
 985 
 
 7778 
 
 7869 
 
 7959 
 
 8050 
 
 8140 
 
 8231 
 
 8322 
 
 8412 
 
 8503 
 
 8594 
 
 85 
 
 75 
 
 76 
 
 77 
 
 77 
 
 986 
 
 8684 
 
 8775 
 
 8866 
 
 8956 
 
 9047 
 
 9137 
 
 9228 
 
 9319 
 
 9409 
 
 9500 
 
 86 
 
 76 
 
 77 
 
 77 
 
 78 
 
 987 
 
 9591 
 
 9681 
 
 9772 
 
 9863 
 
 9953 
 
 0044 
 
 0135 
 
 0225 
 
 0316 
 
 o407 
 
 87 
 
 77 
 
 77 
 
 78 
 
 79 
 
 988 
 
 03 0497 
 
 0588 
 
 0679 
 
 0769 
 
 0860 
 
 0951 
 
 1042 
 
 1132 
 
 1223 
 
 1314 
 
 88 
 
 77 
 
 78 
 
 '79 
 
 80 
 
 989 
 
 1404 
 
 1495 
 
 1586 
 
 1676 
 
 1767 
 
 1858 
 
 1948 
 
 2039 
 
 2130 
 
 2221 
 
 89 
 
 78 
 
 79 
 
 80 
 
 81 
 
 990 
 
 032311 
 
 2402 
 
 2493 
 
 2584 
 
 2674 
 
 2765 
 
 2856 
 
 2946 
 
 3037 
 
 3128 
 
 90 
 
 79 
 
 80 
 
 81 
 
 82 
 
 991 
 
 3219 
 
 3309 
 
 3400 
 
 3491 
 
 3582 
 
 3672 
 
 3763 
 
 3854 
 
 3945 
 
 4035 
 
 91 
 
 80 
 
 81 
 
 82 
 
 83 
 
 992 
 
 4126 
 
 4217 
 
 4308 
 
 4398 
 
 4489 
 
 4580 
 
 4671 
 
 4761 
 
 4852 
 
 4943 
 
 92 
 
 81 
 
 82 
 
 83 
 
 84 
 
 993 
 
 5034 
 
 5125 
 
 5215 
 
 5306 
 
 5397 
 
 5488 
 
 5578 
 
 5669 
 
 5760 
 
 5851 
 
 93 
 
 82 
 
 8384 
 
 85 
 
 994 
 
 5942 
 
 6032 
 
 6123 
 
 6214 
 
 6305 
 
 6396 
 
 6486 
 
 6577 
 
 6668 
 
 6759 
 
 94 
 
 83 
 
 8485 
 
 86 
 
 995 
 
 6850 
 
 6940 
 
 7031 
 
 7122 
 
 7213 
 
 7304 
 
 7395 
 
 7485 
 
 7576 
 
 7667 
 
 95 
 
 84 
 
 85'85 
 
 86 
 
 996 
 
 7758 
 
 7849 
 
 7940 
 
 8030 
 
 8121 
 
 8212 
 
 8303 8394 
 
 8485 
 
 8575 
 
 96 84185 86 
 
 87 
 
 997 
 
 8666 
 
 8757 
 
 8848 
 
 8939 
 
 9030 
 
 9121 
 
 9211 
 
 9302 
 
 9393 
 
 9484 
 
 97 
 
 85 86 87 
 
 88 
 
 998 
 
 9575 
 
 9666 
 
 9757 
 
 9848 
 
 9938 
 
 0029 
 
 0120 
 
 0211 
 
 0302 
 
 0393 
 
 98 
 
 868788 
 
 89 
 
 999 
 
 04 0484 
 
 0575 
 
 0665 
 
 0756 
 
 0847 
 
 0938 
 
 1029 
 
 1120 
 
 1211 
 
 1302 
 
 99 
 
 87 
 
 88 
 
 89 
 
 90 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 88 
 
 89J90 
 
 91 
 
TABLE I. 
 
 Log 
 
 
 Log(l+) 
 
 Pro. Parts. 
 
 3C 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 30 
 
 31 
 
 2 
 
 93 
 
 1-000 
 
 1-04 1393 
 
 1484 
 
 1575 
 
 1665 
 
 1756 
 
 1847 
 
 1938 
 
 2029 
 
 2120 
 
 2211 
 
 00 
 
 
 
 
 
 
 
 
 
 001 
 
 2302 
 
 2393 
 
 2484 
 
 2575 
 
 2666 
 
 2757 
 
 2847 
 
 2938 
 
 3029 
 
 3120 
 
 01 
 
 1 
 
 1 
 
 1 
 
 1 
 
 002 
 
 3211 
 
 3302 
 
 3393 
 
 3484 
 
 3575 
 
 3666 
 
 3757 
 
 3848 
 
 3939 
 
 4030 
 
 02 
 
 2 
 
 2 
 
 2 
 
 2 
 
 003 
 
 4121 
 
 4212 
 
 4303 
 
 4394 
 
 4485 
 
 4576 
 
 4667 
 
 4758 
 
 4849 
 
 4940 
 
 03 
 
 
 
 3 
 
 3 
 
 3 
 
 004- 
 
 5031 
 
 5122 
 
 5213 
 
 5304 
 
 5395 
 
 5486 
 
 5577 
 
 5668 
 
 5759 
 
 5850 
 
 04 
 
 4 
 
 4 
 
 4 
 
 4 
 
 005 
 
 5941 
 
 6032 
 
 6123 
 
 6214 
 
 6305 
 
 6396 
 
 6487 
 
 6578 
 
 6669 
 
 6760 
 
 05 
 
 K 
 
 e 
 
 e 
 
 5 
 
 006 
 
 6851 
 
 6942 
 
 7033 
 
 7124 
 
 7215 
 
 7306 
 
 7397 
 
 7488 
 
 7579 
 
 7670 
 
 06 
 
 K 
 ^ 
 
 K 
 
 6 
 
 6 
 
 007 
 
 7761 
 
 7852 
 
 7943 
 
 8034 
 
 8125 
 
 8216 
 
 8307 
 
 8398 
 
 8489 
 
 8580 
 
 07 
 
 6 
 
 6 
 
 6 
 
 7 
 
 008 
 
 8671 
 
 8763 
 
 8854 
 
 8945 
 
 9036 
 
 9127 
 
 9218 
 
 9309 
 
 9400 
 
 9491 
 
 08 
 
 rv 
 1 
 
 7 
 1 
 
 r 
 
 7 
 
 009 
 
 9582 
 
 9673 
 
 9764 
 
 9855 
 
 9946 
 
 0038 
 
 0129 
 
 0220 
 
 0311 
 
 o402 
 
 09 
 
 8 
 
 8 
 
 8 
 
 8 
 
 010 
 
 05 0493 
 
 0584 
 
 0675 
 
 0766 
 
 0857 
 
 0949 
 
 1040 
 
 1131 
 
 1222 
 
 1313 
 
 10 
 
 9 
 
 9 
 
 ( 
 
 9 
 
 on 
 
 1404 
 
 1495 
 
 1586 
 
 1677 
 
 1769 
 
 1860 
 
 1951 
 
 2042 
 
 2133 
 
 2224 
 
 11 
 
 10 
 
 10 
 
 10 
 
 10 
 
 012 
 
 2315 
 
 2407 
 
 2498 
 
 2589 
 
 2680 
 
 2771 
 
 2862 
 
 2953 
 
 3045 
 
 3136 
 
 12 
 
 11 
 
 11 
 
 11 
 
 11 
 
 013 
 
 3227 
 
 3318 
 
 3409 
 
 3500 
 
 3591 
 
 3683 
 
 3774 
 
 3865 
 
 3956 
 
 4047 
 
 13 
 
 12 
 
 12 
 
 12 
 
 12 
 
 014 
 
 4138 
 
 4230 
 
 4321 
 
 4412 
 
 4503 
 
 4594 
 
 4686 
 
 4777 
 
 4868 
 
 4959 
 
 14 
 
 13 
 
 13 
 
 13 
 
 13 
 
 015 
 
 5050 
 
 5141 
 
 5233 
 
 5324 
 
 5415 
 
 5506 
 
 5597 
 
 5689 
 
 5780 
 
 5871 
 
 1.5 
 
 13 
 
 14 
 
 14 
 
 14 
 
 016 
 
 5962 
 
 6053 
 
 6145 
 
 6236 
 
 6327 
 
 6418 
 
 6510 
 
 6601 
 
 6692 
 
 6783 
 
 16 
 
 14 
 
 15 
 
 15 
 
 15 
 
 017 
 
 6874 
 
 6966 
 
 7057 
 
 7148 
 
 7239 
 
 7331 
 
 7422 
 
 7513 
 
 7604 
 
 7696 
 
 17 
 
 15 
 
 15 
 
 16 
 
 16 
 
 018 
 
 7787 
 
 7878 
 
 7969 
 
 8061 
 
 8152 
 
 8243 
 
 8334 
 
 8426 
 
 8517 
 
 8608 
 
 18 
 
 16 
 
 16 
 
 17 
 
 Iry 
 t 
 
 019 
 
 8699 
 
 8791 
 
 8882 
 
 8973 
 
 9064 
 
 9156 
 
 9247 
 
 9338 
 
 9430 
 
 9521 
 
 19 
 
 17 
 
 17 
 
 17 
 
 18 
 
 020 
 
 059612 
 
 9703 
 
 9795 
 
 9886 
 
 9977 
 
 0069 
 
 0160 
 
 0251 
 
 0342 
 
 0434 
 
 80 
 
 18 
 
 18 
 
 18 
 
 19 
 
 021 
 
 06 0525 
 
 0616 
 
 0708 
 
 0799 
 
 0890 
 
 0982 
 
 1073 
 
 1164 
 
 1255 
 
 1347 
 
 21 
 
 19 
 
 19 
 
 19 
 
 20 
 
 022 
 
 1438 
 
 1529 
 
 1621 
 
 1712 
 
 1803 
 
 1895 
 
 1986 
 
 2077 
 
 2169 
 
 2260 
 
 22 
 
 20 
 
 -20 
 
 20 
 
 20 
 
 023 
 
 2351 
 
 2443 
 
 2534 
 
 2625 
 
 2717 
 
 2808 
 
 2899 
 
 2991 
 
 3082 
 
 3173 
 
 23 
 
 2 
 
 21 
 
 2 
 
 21 
 
 024 
 
 3265 
 
 3356 
 
 3448 
 
 3539 
 
 3630 
 
 3722 
 
 3813 
 
 3904 
 
 3996 
 
 4087 
 
 24 
 
 22 
 
 22 
 
 2-- 
 
 22 
 
 025 
 
 4179 
 
 4270 
 
 4361 
 
 4453 
 
 4544 
 
 4635 
 
 4727 
 
 4818 
 
 4910 
 
 5001 
 
 2.5 
 
 23 
 
 23 
 
 23 
 
 23 
 
 026 
 
 5092 
 
 5184 
 
 5275 
 
 5367 
 
 5458 
 
 5549 
 
 5641 
 
 5732 
 
 5824 
 
 5915 
 
 26 
 
 23 
 
 24 
 
 24 
 
 24 
 
 027 
 
 6006 
 
 6098 
 
 6189 
 
 6281 
 
 6372 
 
 6463 
 
 6555 
 
 6646 
 
 6738 
 
 6829 
 
 27 
 
 24 
 
 25 
 
 2o 
 
 25 
 
 028 
 
 6921 
 
 7012 
 
 7103 
 
 7195 
 
 7286 
 
 7378 
 
 7469 
 
 7561 
 
 7652 
 
 7743 
 
 28 
 
 2o 
 
 2.5 
 
 2f 
 
 26 
 
 029 
 
 7835 
 
 7926 
 
 8018 
 
 8109 
 
 8201 
 
 8292 
 
 8384 
 
 8475 
 
 8567 
 
 8658 
 
 29 
 
 26 
 
 26 
 
 27 
 
 27 
 
 030 
 
 06 8749 
 
 8841 
 
 8932 
 
 9024 
 
 9115 
 
 9207 
 
 9298 
 
 9390 
 
 9481 
 
 9573 
 
 30 
 
 27 
 
 r 
 
 28 
 
 28 
 
 031 
 
 9664 
 
 9756 
 
 9847 
 
 9939 
 
 0030 
 
 0122 
 
 0213 
 
 0305 
 
 0396 
 
 0488 
 
 31 
 
 28 
 
 28 
 
 29 
 
 29 
 
 032 
 
 07 0579 
 
 0671 
 
 0762 
 
 0854 
 
 0945 
 
 1037 
 
 1128 
 
 1220 
 
 1311 
 
 1403 
 
 32 
 
 29 
 
 29 
 
 29 
 
 30 
 
 033 
 
 1494 
 
 1586 
 
 1677 
 
 1769 
 
 1860 
 
 1952 
 
 2043 
 
 2135 
 
 2226 
 
 2318 
 
 33 
 
 30 
 
 30 
 
 30 
 
 31 
 
 034 
 
 2409 
 
 2501 
 
 2593 
 
 2684 
 
 2776 
 
 2867 
 
 2959 
 
 3050 
 
 3142 
 
 3233 
 
 34 
 
 31 
 
 3 
 
 31 
 
 32 
 
 035 
 
 3325 
 
 3416 
 
 3508 
 
 3600 
 
 3691 
 
 3783 
 
 3874 
 
 3966 
 
 4057 
 
 4149 
 
 3.5 
 
 31 
 
 32 
 
 32 
 
 33 
 
 036 
 
 4241 
 
 4332 
 
 4424 
 
 4515 
 
 4607 
 
 4698 
 
 4790 
 
 4882 
 
 4973 
 
 5065 
 
 36 
 
 32 
 
 33 
 
 33 
 
 33 
 
 037 
 
 5156 
 
 5248 
 
 5340 
 
 5431 
 
 5523 
 
 5614 
 
 5706 
 
 5797 
 
 5889 
 
 5981 
 
 37 
 
 33 
 
 34 
 
 34 
 
 34 
 
 038 
 
 6072 
 
 6164 
 
 6256 
 
 6347 
 
 6439 
 
 6530 
 
 6622 
 
 6714 
 
 6805 
 
 6897 
 
 38 
 
 34 
 
 35 
 
 35 
 
 35 
 
 039 
 
 6988 
 
 7080 
 
 7172 
 
 7263 
 
 7355 
 
 7447 
 
 7538 
 
 7630 
 
 7722 
 
 7813 
 
 39 
 
 35 
 
 35 
 
 36 
 
 36 
 
 040 
 
 07 7905 
 
 7996 
 
 8088 
 
 8180 
 
 8271 
 
 8363 
 
 8455 
 
 8546 
 
 8638 
 
 8730 
 
 40 
 
 36 
 
 36 
 
 37 
 
 37 
 
 041 
 
 8821 
 
 8913 
 
 9005 
 
 9096 
 
 9188 
 
 9280 
 
 9371 
 
 9463 
 
 9555 
 
 9646 
 
 41 
 
 37 
 
 37 
 
 38 
 
 38 
 
 042 
 
 9738 
 
 9830 
 
 9921 
 
 0013 
 
 o!05 
 
 0196 
 
 0288 
 
 0380 
 
 0471 
 
 0563 
 
 42 
 
 38 
 
 38 
 
 39 
 
 39 
 
 043 
 
 08 0655 
 
 0747 
 
 0838 
 
 0930 
 
 1022 
 
 1113 
 
 1205 
 
 1297 
 
 1388 
 
 1480 
 
 43 
 
 39 
 
 39 
 
 40 
 
 40 
 
 044 
 
 1572 
 
 1664 
 
 1755 
 
 1847 
 
 1939 
 
 2030 
 
 2122 
 
 2214 
 
 2306 
 
 2397 
 
 44 
 
 40 
 
 40 
 
 40 
 
 41 
 
 045 
 
 2489 
 
 2581 
 
 2673 
 
 2764 
 
 2856 
 
 2948 
 
 3040 
 
 3131 
 
 3223 
 
 3315 
 
 45 
 
 41 
 
 41 
 
 41 
 
 42 
 
 016 
 
 3406 
 
 3498 
 
 3590 
 
 3682 
 
 3773 
 
 3865 
 
 3957 
 
 4049 
 
 4141 
 
 4232 
 
 46 
 
 41 
 
 42 
 
 42 
 
 43 
 
 047 
 
 4324 
 
 4416 
 
 4508 
 
 4599 
 
 4691 
 
 4783 
 
 4875 
 
 4966 
 
 5058 
 
 5150 
 
 47 
 
 4-2 
 
 43 
 
 43 
 
 44 
 
 048 
 
 5242 
 
 5334 
 
 5425 
 
 5517 
 
 5609 
 
 5701 
 
 5792 
 
 5884 
 
 5976 
 
 6068 
 
 48 
 
 43 
 
 44 
 
 44 
 
 45 
 
 049 
 
 6160 
 
 6251 
 
 6343 
 
 6435 
 
 6527 
 
 6619 
 
 6711 
 
 6802 
 
 6894 
 
 6986 
 
 49 
 
 44 
 
 45 
 
 45 
 
 46 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 ' 
 
 
 90 
 
 91 
 
 92 
 
 93 
 
 22 
 
TABLE I. 
 
 Log 
 
 Log(l+;r) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 90 
 
 91 
 
 92 
 
 93 
 
 1-050 
 
 1-08 7078 
 
 7170 
 
 7261 
 
 7353 
 
 7445 
 
 7537 
 
 7629 
 
 7721 
 
 7812 
 
 7904 
 
 50 
 
 45 
 
 46 
 
 46 
 
 47 
 
 051 
 
 7996 
 
 8088 
 
 8180 
 
 8272 
 
 8363 
 
 8455 
 
 8547 
 
 8639 
 
 8731 
 
 8823 
 
 51 
 
 46 
 
 46 
 
 47 
 
 47 
 
 052 
 
 8914 
 
 9006 
 
 9098 
 
 9190 
 
 9282 
 
 9374 
 
 9466 
 
 9557 
 
 9649 
 
 9741 
 
 52 
 
 47 
 
 47 
 
 48 
 
 48 
 
 053 
 
 9833 
 
 9925 
 
 0017 
 
 o!09 
 
 0201 
 
 0292 
 
 0384 
 
 0476 
 
 0568 
 
 0660 
 
 53 
 
 48 
 
 48 
 
 49 
 
 49 
 
 054 
 
 09 0752 
 
 0844 
 
 0936 
 
 1027 
 
 1119 
 
 1211 
 
 1303 
 
 1395 
 
 1487 
 
 1579 
 
 54 
 
 49 
 
 49 
 
 50 
 
 50 
 
 055 
 
 1671 
 
 1763 
 
 1855 
 
 1946 
 
 2038 
 
 2130 
 
 2222 
 
 2314 
 
 2406 
 
 2498 
 
 55 
 
 49 
 
 50 
 
 51 
 
 51 
 
 056 
 
 2590 
 
 2682 
 
 2774 
 
 2866 
 
 2958 
 
 3049 
 
 3141 
 
 3233 
 
 3325 
 
 3417 
 
 56 
 
 50 
 
 51 
 
 52 
 
 52 
 
 057 
 
 3509 
 
 3601 
 
 3693 
 
 3785 
 
 3877 
 
 3969 
 
 4061 
 
 4153 
 
 4245 
 
 4337 
 
 57 
 
 51 
 
 52 
 
 52 
 
 53 
 
 058 
 
 4429 
 
 4521 
 
 4613 
 
 4704 
 
 4796 
 
 4888 
 
 4980 
 
 5072 
 
 5164 
 
 5256 
 
 58 
 
 52 
 
 53 
 
 53 
 
 54 
 
 059 
 
 5348 
 
 5440 
 
 5532 
 
 5624 
 
 5716 
 
 5808 
 
 5900 
 
 5992 
 
 6084 
 
 6176 
 
 59 
 
 53 
 
 54 
 
 54 
 
 55 
 
 060 
 
 09 6268 
 
 6360 
 
 6452 
 
 6544 
 
 6636 
 
 6728 
 
 6820 
 
 6912 
 
 7004 
 
 7096 
 
 60 
 
 54 
 
 55 
 
 55 
 
 56 
 
 061 
 
 7188 
 
 7280 
 
 7372 
 
 7464 
 
 7556 
 
 7648 
 
 7740 
 
 7832 
 
 7924 
 
 8016 
 
 61 
 
 55 
 
 56 
 
 56 
 
 57 
 
 062 
 
 8108 
 
 8200 
 
 8292 
 
 8384 
 
 8476 
 
 8568 
 
 8660 
 
 8752 
 
 8844 
 
 8936 
 
 62 
 
 56 
 
 56 
 
 57 
 
 58 
 
 063 
 
 9028 
 
 9120 
 
 9213 
 
 9305 
 
 9397 
 
 9489 
 
 9581 
 
 9673 
 
 9765 
 
 9857 
 
 63 
 
 57 
 
 57 
 
 58 
 
 59 
 
 064 
 
 9949 
 
 0041 
 
 0133 
 
 0225 
 
 0317 
 
 0409 
 
 0501 
 
 0593 
 
 0685 
 
 0777 
 
 64 
 
 58 
 
 58 
 
 59 
 
 60 
 
 065 
 
 100870 
 
 0962 
 
 1054 
 
 1146 
 
 1238 
 
 1330 
 
 1422 
 
 1514 
 
 1606 
 
 1698 
 
 65 
 
 59 
 
 59 
 
 60 
 
 60 
 
 066 
 
 1790 
 
 1882 
 
 1975 
 
 2067 
 
 2159 
 
 2251 
 
 2343 
 
 2435 
 
 2527 
 
 2619 
 
 66 
 
 59 
 
 60 
 
 61 
 
 61 
 
 067 
 
 2711 
 
 2803 
 
 2896 
 
 2988 
 
 3080 
 
 3172 
 
 3264 
 
 3356 
 
 3448 
 
 3540 
 
 67 
 
 60 
 
 61 
 
 62 
 
 62 
 
 068 
 
 3632 
 
 3725 
 
 3817 
 
 3909 
 
 4001 
 
 4093 
 
 4185 
 
 4277 
 
 4370 
 
 4462 
 
 68 
 
 61 
 
 62 
 
 63 
 
 63 
 
 069 
 
 4554 
 
 4646 
 
 4738 
 
 4830 
 
 4922 
 
 5015 
 
 5107 
 
 5199 
 
 5291 
 
 5383 
 
 69 
 
 62 
 
 63 
 
 63 
 
 64 
 
 070 
 
 10 5475 
 
 5567 
 
 5660 
 
 5752 
 
 5844 
 
 5936 
 
 6028 
 
 6120 
 
 6213 
 
 6305 
 
 70 
 
 63 
 
 64 
 
 64 
 
 65 
 
 071 
 
 6397 
 
 6489 
 
 6581 
 
 6673 
 
 6766 
 
 6858 
 
 6950 
 
 7042 
 
 7134 
 
 7227 
 
 71 
 
 64 
 
 65 
 
 65 
 
 66 
 
 072 
 
 7319 
 
 7411 
 
 7503 
 
 7595 
 
 7688 
 
 7780 
 
 7872 
 
 7964 
 
 8056 
 
 8149 
 
 72 
 
 65 
 
 66 
 
 66 
 
 67 
 
 073 
 
 8241 
 
 8333 
 
 8425 
 
 8517 
 
 8610 
 
 8702 
 
 8794 
 
 8886 
 
 8978 
 
 9071 
 
 73 
 
 66 
 
 66 
 
 67 
 
 68 
 
 074 
 
 9163 
 
 9255 
 
 9347 
 
 9440 
 
 9532 
 
 9624 
 
 9716 
 
 9808 
 
 9901 
 
 9993 
 
 74 
 
 67 
 
 67 
 
 68 
 
 69 
 
 075 
 
 11 0085 
 
 0177 
 
 0270 
 
 0362 
 
 0454 
 
 0546 
 
 0639 
 
 0731 
 
 0823 
 
 0915 
 
 75 
 
 67 
 
 68 
 
 69 
 
 70 
 
 076 
 
 1008 
 
 1100 
 
 1192 
 
 1284 
 
 1377 
 
 1469 
 
 1561 
 
 1653 
 
 1746 
 
 1838 
 
 76 
 
 68 
 
 69 
 
 70 
 
 71 
 
 077 
 
 1930 
 
 2023 
 
 2115 
 
 2207 
 
 2299 
 
 2392 
 
 2484 
 
 2576 
 
 2669 
 
 2761 
 
 77 
 
 69 
 
 70 
 
 71 
 
 72 
 
 078 
 
 2853 
 
 2945 
 
 3038 
 
 3130 
 
 3222 
 
 3315 
 
 3407 
 
 3499 
 
 3591 
 
 3684 
 
 78 
 
 70 
 
 71 
 
 72 
 
 73 
 
 079 
 
 3776 
 
 3868 
 
 3961 
 
 4053 
 
 4145 
 
 4238 
 
 4330 
 
 4422 
 
 4515 
 
 4607 
 
 79 
 
 71 
 
 72 
 
 73 
 
 73 
 
 080 
 
 114699 
 
 4792 
 
 4884 
 
 4976 
 
 5068 
 
 5161 
 
 5253 
 
 5345 
 
 5438 
 
 5530 
 
 80 
 
 72 
 
 73 
 
 74 
 
 74 
 
 081 
 
 5622 
 
 5715 
 
 5807 
 
 5899 
 
 5992 
 
 6084 
 
 6177 
 
 6269 
 
 6361 
 
 6454 
 
 81 
 
 73 
 
 74 
 
 75 
 
 75 
 
 082 
 
 6546 
 
 6638 
 
 6731 
 
 6823 
 
 6915 
 
 7008 
 
 7100 
 
 7192 
 
 7285 
 
 7377 
 
 82 
 
 74 
 
 75 
 
 75 
 
 76 
 
 083 
 
 7470 
 
 7562 
 
 7654 
 
 7747 
 
 7839 
 
 7931 
 
 8024 
 
 8116 
 
 8209 
 
 8301 
 
 83 
 
 75 
 
 76 
 
 76 
 
 77 
 
 084 
 
 8393 
 
 8486 
 
 8578 
 
 8670 
 
 8763 
 
 8855 
 
 8948 
 
 9040 
 
 9132 
 
 9225 
 
 84 
 
 76 
 
 76 
 
 77 
 
 78 
 
 085 
 
 9317 
 
 9410 
 
 9502 
 
 9594 
 
 9687 
 
 9779 
 
 9872 
 
 9964 
 
 0057 
 
 0149 
 
 85 
 
 77 
 
 77 
 
 78 
 
 79 
 
 086 
 
 120241 
 
 0334 
 
 0426 
 
 0519 
 
 0611 
 
 0703 
 
 0796 
 
 0888 
 
 0981 
 
 1073 
 
 86 
 
 77 
 
 78 
 
 79 
 
 80 
 
 087 
 
 1166 
 
 1258 
 
 1351 
 
 1443 
 
 1535 
 
 1628 
 
 1720 
 
 1813 
 
 1905 
 
 1998 
 
 87 
 
 78 
 
 79 
 
 80 
 
 81 
 
 088 
 
 2090 
 
 2183 
 
 2275 
 
 2367 
 
 2460 
 
 2552 
 
 2645 
 
 2737 
 
 2830 
 
 2922 
 
 88 
 
 89 
 
 80 
 
 81 
 
 82 
 
 089 
 
 3015 
 
 3107 
 
 3200 
 
 3292 
 
 3385 
 
 3477 
 
 3569 
 
 3662 
 
 3754 
 
 3847 
 
 89 
 
 80 
 
 81 
 
 82 
 
 83 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 090 
 
 123939 
 
 4032 
 
 4124 
 
 4217 
 
 4309 
 
 4402 
 
 4494 
 
 4587 
 
 4679 
 
 4772 
 
 90 
 
 81 
 
 82 
 
 83 
 
 84 
 
 091 
 
 4864 
 
 4957 
 
 5049 
 
 5142 
 
 5234 
 
 5327 
 
 5419 
 
 5512 
 
 5604 
 
 5697 
 
 91 
 
 82 
 
 83 
 
 84 
 
 85 
 
 092 
 
 5789 
 
 5882 
 
 5974 
 
 6067 
 
 6159 
 
 6252 
 
 6344 
 
 6437 
 
 6530 
 
 6622 
 
 92 
 
 83 
 
 84 
 
 85 
 
 86 
 
 093 
 
 6715 
 
 6807 
 
 6900 
 
 6992 
 
 7085 
 
 7177 
 
 7270 
 
 7362 
 
 7455 
 
 7547 
 
 93 
 
 84 
 
 85 
 
 86 
 
 86 
 
 094 
 
 7640 
 
 7733 
 
 7825 
 
 7918 
 
 8010 
 
 8103 
 
 8195 
 
 8288 
 
 8380 
 
 8473 
 
 94 
 
 85 
 
 86 
 
 86 
 
 87 
 
 095 
 
 8566 
 
 8658 
 
 8751 
 
 8843 
 
 8936 
 
 9028 
 
 9121 
 
 9214 
 
 9306 
 
 9399 
 
 95 
 
 85 
 
 86 
 
 87 
 
 88 
 
 096 
 
 9491 
 
 9584 
 
 9676 
 
 9769 
 
 9862 
 
 9954 
 
 0047 
 
 0139 
 
 0232 
 
 0324 
 
 96 
 
 86 
 
 87 
 
 88 
 
 89 
 
 097 
 
 130417 
 
 0510 
 
 0602 
 
 0695 
 
 0787 
 
 0880 
 
 0973 
 
 1065 
 
 1158 
 
 1251 
 
 97 
 
 87 
 
 88 
 
 89 
 
 90 
 
 098 
 
 1343 
 
 1436 
 
 1528 
 
 1621 
 
 1714 
 
 1806 
 
 1899 
 
 1991 
 
 2084 
 
 2177 
 
 98 
 
 88 
 
 89 
 
 90 
 
 91 
 
 099 
 
 2269 
 
 2362 
 
 2455 
 
 2547 
 
 2640 
 
 2732 
 
 2825 
 
 2918 
 
 3010 
 
 3103 
 
 99 
 
 69 
 
 90 
 
 91 
 
 92 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 90 
 
 91 
 
 92 
 
 93 
 
TABLE I. 
 
 Log 
 
 
 
 Log (1- 
 
 h*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 92 
 
 93 
 
 94 
 
 1-100 
 
 1-133196 
 
 3288 
 
 3381 
 
 3474 
 
 3566 
 
 3659 
 
 3751 
 
 3844 
 
 3937 
 
 4029 
 
 00 
 
 
 
 
 
 
 
 101 
 
 4122 
 
 4215 
 
 4307 
 
 4400 
 
 4493 
 
 4585 
 
 4678 
 
 4771 
 
 4863 
 
 4956 
 
 
 
 1 
 
 1 
 
 ; 
 
 102 
 
 5049 
 
 5141 
 
 5234 
 
 5327 
 
 5419 
 
 5512 
 
 5605 
 
 5698 
 
 5790 
 
 5883 
 
 o-. 
 
 2 
 
 2 
 
 2 
 
 103 
 
 5976 
 
 6068 
 
 6161 
 
 6254 
 
 6346 
 
 6439 
 
 6532 
 
 6624 
 
 6717 
 
 6810 
 
 o: 
 
 3 
 
 3 
 
 < 
 
 104 
 
 6903 
 
 6995 
 
 7088 
 
 7181 
 
 7273 
 
 7366 
 
 7459 
 
 7551 
 
 7644 
 
 7737 
 
 04 
 
 4 
 
 4 
 
 4 
 
 105 
 
 7830 
 
 7922 
 
 8015 
 
 8108 
 
 8201 
 
 8293 
 
 8386 
 
 8479 
 
 8571 
 
 8664 
 
 Oo 
 
 5 
 
 5 
 
 5 
 
 106 
 
 8757 
 
 8850 
 
 8942 
 
 9035 
 
 9128 
 
 9221 
 
 9313 
 
 9406 
 
 9499 
 
 9592 
 
 06 
 
 6 
 
 6 
 
 6 
 
 107 
 
 9684 
 
 9777 
 
 9870 
 
 9963 
 
 0055 
 
 0148 
 
 0241 
 
 0334 
 
 0426 
 
 0519 
 
 97 
 
 6 
 
 7 
 
 7 
 
 108 
 
 140612 
 
 0705 
 
 0797 
 
 0890 
 
 0983 
 
 1076 
 
 1169 
 
 1261 
 
 1354 
 
 1447 
 
 08 
 
 7 
 
 7| 8 
 
 109 
 
 1540 
 
 1632 
 
 1725 
 
 1818 
 
 1911 
 
 2004 
 
 2096 
 
 2189 
 
 2282 
 
 2375 
 
 Of 
 
 8 
 
 8 
 
 8 
 
 110 
 
 142468 
 
 2560 
 
 2653 
 
 2746 
 
 2839 
 
 2932 
 
 3024 
 
 3117 
 
 3210 
 
 3303 
 
 10 
 
 9 
 
 9 
 
 9 
 
 111 
 
 3396 
 
 3488 
 
 3581 
 
 3674 
 
 3767 
 
 3860 
 
 3952 
 
 4045 
 
 4138 
 
 4231 
 
 11 
 
 10 
 
 10 
 
 10 
 
 112 
 
 4324 
 
 4417 
 
 4509 
 
 4602 
 
 4695 
 
 4788 
 
 4881 
 
 4974 
 
 5066 
 
 5159 
 
 12 
 
 11 
 
 11 
 
 11 
 
 113 
 
 5252 
 
 5345 
 
 5438 
 
 5531 
 
 5624 
 
 5716 
 
 5809 
 
 5902 
 
 5995 
 
 6088 
 
 13 
 
 1-2 
 
 12 
 
 12 
 
 114 
 
 6181 
 
 6273 
 
 6366 
 
 6459 
 
 6552 
 
 6645 
 
 6738 
 
 6831 
 
 6924 
 
 7016 
 
 14 
 
 13 
 
 13 
 
 13 
 
 115 
 
 7109 
 
 7202 
 
 7295 
 
 7388 
 
 7481 
 
 7574 
 
 7667 
 
 7759 
 
 7852 
 
 7945 
 
 lo 
 
 14 
 
 14 
 
 14 
 
 116 
 
 8038 
 
 8131 
 
 8224 
 
 8317 
 
 8410 
 
 8503 
 
 8595 
 
 8688 
 
 8781 
 
 8874 
 
 16 
 
 15 
 
 15 
 
 15 
 
 117 
 
 8967 
 
 9060 
 
 9153 
 
 9246 
 
 9339 
 
 9432 
 
 9525 
 
 9617 
 
 9710 
 
 9803 
 
 17 
 
 16 
 
 16 
 
 16 
 
 118 
 
 9896 
 
 9989 
 
 0082 
 
 0175 
 
 0268 
 
 0361 
 
 0454 
 
 0547 
 
 0640 
 
 0733 
 
 18 
 
 17 
 
 17 
 
 17 
 
 119 
 
 150825 
 
 0918 
 
 1011 
 
 1104 
 
 1197 
 
 1290 
 
 1383 
 
 1476 
 
 1569 
 
 1662 
 
 1! 
 
 17 
 
 18 
 
 18 
 
 120 
 
 15 1755 
 
 1848 
 
 1941 
 
 2034 
 
 2127 
 
 2220 
 
 2313 
 
 2406 
 
 2498 
 
 2591 
 
 20 
 
 18 
 
 19 
 
 19 
 
 121 
 
 2684 
 
 2777 
 
 2870 
 
 2963 
 
 3056 
 
 3149 
 
 3242 
 
 3335 
 
 3428 
 
 3521 
 
 y 
 
 19 
 
 20 
 
 20 
 
 122 
 
 3614 
 
 3707 
 
 3800 
 
 3893 
 
 3986 
 
 4079 
 
 4172 
 
 4265 
 
 4358 
 
 4451 
 
 22 
 
 -20 
 
 20 
 
 21 
 
 123 
 
 4544 
 
 4637 
 
 4730 
 
 4823 
 
 4916 
 
 5009 
 
 5102 
 
 5195 
 
 5288 
 
 5381 
 
 2'. 
 
 21 
 
 21 
 
 22 
 
 124 
 
 5474 
 
 5567 
 
 5660 
 
 5753 
 
 5846 
 
 5939 
 
 6032 
 
 6125 
 
 6218 
 
 6311 
 
 24 
 
 22 
 
 2-2 
 
 23 
 
 125 
 
 6404 
 
 6497 
 
 6590 
 
 6683 
 
 6776 
 
 6869 
 
 6962 
 
 7055 
 
 7148 
 
 7241 
 
 25 
 
 23 
 
 23 
 
 24 
 
 126 
 
 7335 
 
 7428 
 
 7521 
 
 7614 
 
 7707 
 
 7800 
 
 7893 
 
 7986 
 
 8079 
 
 8172 
 
 26 
 
 24 
 
 24 
 
 24 
 
 127 
 
 8265 
 
 8358 
 
 8451 
 
 8544 
 
 8637 
 
 8730 
 
 8823 
 
 8916 
 
 9009 
 
 9103 
 
 27 
 
 25 
 
 25 
 
 25 
 
 128 
 
 9196 
 
 9289 
 
 9382 
 
 9475 
 
 9568 
 
 9661 
 
 9754 
 
 9847 
 
 9940 
 
 0033 
 
 28 
 
 -26 
 
 26 
 
 26 
 
 129 
 
 160126 
 
 0219 
 
 0313 
 
 0406 
 
 0499 
 
 0592 
 
 0685 
 
 0778 
 
 0871 
 
 0964 
 
 29 
 
 27 
 
 27 
 
 27 
 
 130 
 
 16 1057 
 
 1150 
 
 1243 
 
 1337 
 
 1430 
 
 1523 
 
 1616 
 
 1709 
 
 1802 
 
 1895 
 
 30 
 
 28 
 
 28 
 
 28 
 
 131 
 
 1988 
 
 2081 
 
 2175 
 
 2268 
 
 2361 
 
 2454 
 
 2547 
 
 2640 
 
 2733 
 
 2826 
 
 31 
 
 29 
 
 29 
 
 29 
 
 132 
 
 2920 
 
 3013 
 
 3106 
 
 3199 
 
 3292 
 
 3385 
 
 3478 
 
 3571 
 
 3665 
 
 3758 
 
 32 
 
 29 
 
 30 
 
 30 
 
 133 
 
 3851 
 
 3944 
 
 4037 
 
 4130 
 
 4223 
 
 4317 
 
 4410 
 
 4503 
 
 4596 
 
 4689 
 
 33 
 
 30 
 
 31 
 
 31 
 
 134 
 
 4782 
 
 4876 
 
 4969 
 
 5062 
 
 5155 
 
 5248 
 
 5341 
 
 5435 
 
 5528 
 
 5621 
 
 31 
 
 31 
 
 32 
 
 32 
 
 135 
 
 5714 
 
 5807 
 
 5900 
 
 5994 
 
 6087 
 
 6180 
 
 6273 
 
 6366 
 
 6459 
 
 6553 
 
 35 
 
 32 
 
 33 
 
 33 
 
 136 
 
 6646 
 
 6739 
 
 6832 
 
 6925 
 
 7019 
 
 7112 
 
 7205 
 
 7298 
 
 7391 
 
 7485 
 
 36 
 
 33 
 
 33 
 
 34 
 
 137 
 
 7578 
 
 7671 
 
 7764 
 
 7857 
 
 7951 
 
 8044 
 
 8137 
 
 8230 
 
 8323 
 
 8417 
 
 37 
 
 34 
 
 34 
 
 35 
 
 138 
 
 8510 
 
 8603 
 
 8696 
 
 8790 
 
 8883 
 
 8976 
 
 9069 
 
 9162 
 
 9256 
 
 9349 
 
 38 
 
 35 
 
 35 
 
 36 
 
 139 
 
 9442 
 
 9535 
 
 9629 
 
 9722 
 
 9815 
 
 9908 
 
 0001 
 
 0095 
 
 0188 
 
 0281 
 
 39 
 
 36 
 
 36 
 
 37 
 
 140 
 
 170374 
 
 0468 
 
 0561 
 
 0654 
 
 0747 
 
 0841 
 
 0934 
 
 1027 
 
 1120 
 
 1214 
 
 40 
 
 37 
 
 3738 
 
 141 
 
 1307 
 
 1400 
 
 1494 
 
 1587 
 
 1680 
 
 1773 
 
 1867 
 
 1960 2053 
 
 2146 
 
 41 
 
 3SI38 
 
 39 
 
 142 
 
 2240 
 
 2333 
 
 2426 
 
 2519 
 
 2613 
 
 2706 
 
 2799 
 
 2893 
 
 2986 
 
 3079 
 
 4-2 
 
 39 
 
 39 
 
 39 
 
 143 
 
 3172 
 
 3266 
 
 3359 
 
 3452 
 
 3546 
 
 3639 
 
 3732 
 
 3826 
 
 3919 
 
 4012 
 
 
 40 
 
 40 
 
 40 
 
 144 
 
 4105 
 
 4199 
 
 4292 
 
 4385 
 
 4479 
 
 4572 
 
 4665 
 
 4759 4852 
 
 4945 
 
 44 
 
 40 
 
 41 
 
 41 
 
 145 
 
 5039 
 
 5132 
 
 5225 
 
 5318 
 
 5412 
 
 5505 
 
 5598 
 
 5692 5785 
 
 5878 
 
 45 
 
 41 
 
 12 
 
 42 
 
 146 
 
 5972 
 
 6065 
 
 6158 
 
 6252 
 
 6345 
 
 6438 
 
 6532 
 
 6625 
 
 6718 
 
 6812 
 
 46 
 
 42 
 
 13 
 
 13 
 
 147 
 
 6905 
 
 6998 
 
 7092 
 
 7185 
 
 7279 
 
 7372 
 
 7465 
 
 7559 
 
 7652 
 
 7745 
 
 47 
 
 1344 
 
 14 
 
 148 
 
 7839 
 
 7932 
 
 8025 
 
 8119 
 
 8212 
 
 8305 
 
 8399 
 
 8492 
 
 8586 
 
 8679 
 
 48 
 
 1445 
 
 15 
 
 149 
 
 8772 
 
 8866 
 
 8959 
 
 9052 
 
 9146 
 
 9239 
 
 9333 
 
 9426 
 
 9519 
 
 9613 
 
 49 
 
 15 < 
 
 t6 
 
 16 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 3-2 f 
 
 )3 
 
 n 
 
 24 
 
TABLE I. 
 
 Log 
 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 92 
 
 93 
 
 94 
 
 1-150 
 
 1-179706 
 
 9800 
 
 9893 
 
 9986 
 
 0080 
 
 0173 
 
 0267 
 
 0360 
 
 0453 
 
 o547 
 
 50 
 
 46 
 
 47 
 
 47 
 
 151 
 
 18 0640 
 
 0734 
 
 0827 
 
 0920 
 
 1014 
 
 1107 
 
 1201 
 
 1294 
 
 1387 
 
 1481 
 
 51 
 
 47 
 
 47 
 
 48 
 
 152 
 
 1574 
 
 1668 
 
 1761 
 
 1854 
 
 1948 
 
 2041 
 
 2135 
 
 2228 
 
 2322 
 
 2415 
 
 52 
 
 48 
 
 48 
 
 49 
 
 153 
 
 2508 
 
 2602 
 
 2695 
 
 2789 
 
 2882 
 
 2976 
 
 3069 
 
 3163 
 
 3256 
 
 3349 
 
 53 
 
 49 
 
 49 
 
 50 
 
 154 
 
 3443 
 
 3536 
 
 3630 
 
 3723 
 
 3817 
 
 3910 
 
 4004 
 
 4097 
 
 4190 
 
 4284 
 
 54 
 
 50 
 
 50 
 
 51 
 
 155 
 
 4377 
 
 4471 
 
 4564 
 
 4658 
 
 4751 
 
 4845 
 
 4938 
 
 5032 
 
 5125 
 
 5219 
 
 55 
 
 51 
 
 51 
 
 52 
 
 156 
 
 5312 
 
 5406 
 
 5499 
 
 5592 
 
 5686 
 
 5779 
 
 5873 
 
 5966 
 
 6060 
 
 6153 
 
 56 
 
 52 
 
 52 
 
 53 
 
 157 
 
 6247 
 
 6340 
 
 6434 
 
 6527 
 
 6621 
 
 6714 
 
 6808 
 
 6901 
 
 6995 
 
 7088 
 
 57 
 
 52 
 
 53 
 
 54 
 
 158 
 
 7182 
 
 7275 
 
 7369 
 
 7462 
 
 7556 
 
 7649 
 
 7743 
 
 7836 
 
 7930 
 
 8023 
 
 58 
 
 53 
 
 54 
 
 55 
 
 159 
 
 8117 
 
 8210 
 
 8304 
 
 8397 
 
 8491 
 
 8584 
 
 8678 
 
 8772 
 
 8865 
 
 8959 
 
 59 
 
 54 
 
 55 
 
 55 
 
 160 
 
 189052 
 
 9146 
 
 9239 
 
 9333 
 
 9426 
 
 9520 
 
 9613 
 
 9707 
 
 9800 
 
 9894 
 
 60 
 
 55 
 
 56 
 
 56 
 
 161 
 
 9987 
 
 0081 
 
 0175 
 
 0268 
 
 0362 
 
 0455 
 
 0549 
 
 0642 
 
 0736 
 
 0829 
 
 61 
 
 56 
 
 57 
 
 57 
 
 162 
 
 190923 
 
 1017 
 
 1110 
 
 1204 
 
 1297 
 
 1391 
 
 1484 
 
 1578 
 
 1671 
 
 1765 
 
 62 
 
 57 
 
 58 
 
 58 
 
 163 
 
 1859 
 
 1952 
 
 2046 
 
 2139 
 
 2233 
 
 2326 
 
 2420 
 
 2514 
 
 2607 
 
 2701 
 
 63 
 
 58 
 
 59 
 
 59 
 
 164- 
 
 2794 
 
 2888 
 
 2982 
 
 3075 
 
 3169 
 
 3262 
 
 3356 
 
 3449 
 
 3543 
 
 3637 
 
 64 
 
 59 
 
 60 
 
 60 
 
 165 
 
 3730 
 
 3824 
 
 3917 
 
 4011 
 
 4105 
 
 4198 
 
 4292 
 
 4386 
 
 4479 
 
 4573 
 
 65 
 
 60 
 
 60 
 
 61 
 
 166 
 
 4666 
 
 4760 
 
 4854 
 
 4947 
 
 5041 
 
 5134 
 
 5228 
 
 5322 
 
 5415 
 
 5509 
 
 66 
 
 61 
 
 61 
 
 62 
 
 167 
 
 5603 
 
 5696 
 
 5790 
 
 5883 
 
 5977 
 
 6071 
 
 6164 
 
 6258 
 
 6352 
 
 6445 
 
 67 
 
 62 
 
 62 
 
 63 
 
 168 
 
 6539 
 
 6633 
 
 6726 
 
 6820 
 
 6913 
 
 7007 
 
 7101 
 
 7194 
 
 7288 
 
 7382 
 
 68 
 
 63 
 
 63 
 
 64 
 
 169 
 
 7475 
 
 7569 
 
 7663 
 
 7756 
 
 7850 
 
 7944 
 
 8037 
 
 8131 
 
 8225 
 
 8318 
 
 69 
 
 63 
 
 64 
 
 65 
 
 170 
 
 198412 
 
 8506 
 
 8599 
 
 8693 
 
 8787 
 
 8880 
 
 8974 
 
 9068 
 
 9161 
 
 9255 
 
 70 
 
 64 
 
 65 
 
 66 
 
 171 
 
 9349 
 
 9442 
 
 9536 
 
 9630 
 
 9723 
 
 9817 
 
 9911 
 
 0004 
 
 0098 
 
 0192 
 
 71 
 
 65 
 
 66 
 
 67 
 
 172 
 
 20 0286 
 
 0379 
 
 0473 
 
 0567 
 
 0660 
 
 0754 
 
 0848 
 
 0941 
 
 1035 
 
 1129 
 
 72 
 
 66 
 
 67 
 
 68 
 
 173 
 
 1223 
 
 1316 
 
 1410 
 
 1504 
 
 1597 
 
 1691 
 
 1785 
 
 1879 
 
 1972 
 
 2066 
 
 73 
 
 67 
 
 68 
 
 69 
 
 174 
 
 2160 
 
 2253 
 
 2347 
 
 2441 
 
 2535 
 
 2628 
 
 2722 
 
 2816 
 
 2910 
 
 3003 
 
 74 
 
 68 
 
 69 
 
 70 
 
 175 
 
 3097 
 
 3191 
 
 3284 
 
 3378 
 
 3472 
 
 3566 
 
 3659 
 
 3753 
 
 3847 
 
 3941 
 
 75 
 
 69 
 
 70 
 
 70 
 
 176 
 
 4034 
 
 4128 
 
 4222 
 
 4316 
 
 4409 
 
 4503 
 
 4597 
 
 4691 
 
 4784 
 
 4878 
 
 76 
 
 70 
 
 71 
 
 71 
 
 177 
 
 4972 
 
 5066 
 
 5160 
 
 5253 
 
 5347 
 
 5441 
 
 5535 
 
 5628 
 
 5722 
 
 5816 
 
 77 
 
 71 
 
 72 
 
 72 
 
 178 
 
 5910 
 
 6003 
 
 6097 
 
 6191 
 
 6285 
 
 6379 
 
 6472 
 
 6566 
 
 6660 
 
 6754 
 
 78 
 
 72 
 
 73 
 
 73 
 
 179 
 
 6847 
 
 6941 
 
 7035 
 
 7129 
 
 7223 
 
 7316 
 
 7410 
 
 7504 
 
 7598 
 
 7692 
 
 79 
 
 73 
 
 73 
 
 74 
 
 180 
 
 20 7785 
 
 7879 
 
 7973 
 
 8067 
 
 8161 
 
 8254 
 
 8348 
 
 8442 
 
 8536 
 
 8630 
 
 80 
 
 74 
 
 74 
 
 75 
 
 181 
 
 8724 
 
 8817 
 
 8911 
 
 9005 
 
 9099 
 
 9193 
 
 9286 
 
 9380 
 
 9474 
 
 9568 
 
 81 
 
 75 
 
 75 
 
 76 
 
 182 
 
 9662 
 
 9756 
 
 9849 
 
 9943 
 
 0037 
 
 0131 
 
 0225 
 
 0319 
 
 0412 
 
 0506 
 
 82 
 
 75 
 
 76 
 
 77 
 
 183 
 
 21 0600 
 
 0694 
 
 0788 
 
 0882 
 
 0976 
 
 1069 
 
 1163 
 
 1257 
 
 1351 
 
 1445 
 
 83 
 
 76 
 
 77 
 
 78 
 
 184 
 
 1539 
 
 1632 
 
 1726 
 
 1820 
 
 1914 
 
 2008 
 
 2102 
 
 2196 
 
 2290 
 
 2383 
 
 84 
 
 77 
 
 78 
 
 79 
 
 185 
 
 2477 
 
 2571 
 
 2665 
 
 2759 
 
 2853 
 
 2947 
 
 3040 
 
 3134 
 
 3228 
 
 3322 
 
 85 
 
 78 
 
 79 
 
 80 
 
 186 
 
 3416 
 
 3510 
 
 3604 
 
 3698 
 
 3792 
 
 3885 
 
 3979 
 
 4073 
 
 4167 
 
 4261 
 
 86 
 
 79 
 
 80 
 
 81 
 
 187 
 
 4355 
 
 4449 
 
 4543 
 
 4637 
 
 4730 
 
 4824 
 
 4918 
 
 5012 
 
 5106 
 
 5200 
 
 87 
 
 80 
 
 81 
 
 82 
 
 188 
 
 5294 
 
 5388 
 
 5482 
 
 5576 
 
 5670 
 
 5763 
 
 5857 
 
 5951 
 
 6045 
 
 6139 
 
 88 
 
 81 
 
 82 
 
 83 
 
 189 
 
 6233 
 
 6327 
 
 6421 
 
 6515 
 
 6609 
 
 6703 
 
 6797 
 
 6891 
 
 6984 
 
 7078 
 
 89 
 
 82 
 
 83 
 
 84 
 
 190 
 
 21 7172 
 
 7266 
 
 7360 
 
 7454 
 
 7548 
 
 7642 
 
 7736 
 
 7830 
 
 7924 
 
 8018 
 
 90 
 
 83 
 
 84 
 
 85 
 
 191 
 
 8112 
 
 8206 
 
 8300 
 
 8394 
 
 8488 
 
 8582 
 
 8675 
 
 8769 
 
 8863 
 
 8957 
 
 91 
 
 84 
 
 85 
 
 86 
 
 192 
 
 9051 
 
 9145 
 
 9239 
 
 9333 
 
 9427 
 
 9521 
 
 9615 
 
 9709 
 
 9803 
 
 9897 
 
 92 
 
 85 
 
 86 
 
 86 
 
 193 
 
 9991 
 
 0085 
 
 0179 
 
 0273 
 
 o367 
 
 0461 
 
 0555 
 
 0649 
 
 0743 
 
 0837 
 
 93 
 
 86 
 
 86 
 
 87 
 
 194 
 
 22 0931 
 
 1025 
 
 1119 
 
 1213 
 
 1307 
 
 1401 
 
 1495 
 
 1589 
 
 1683 
 
 1777 
 
 94 
 
 86 
 
 87 
 
 88 
 
 195 
 
 1871 
 
 1965 
 
 2059 
 
 2153 
 
 2247 
 
 2341 
 
 2435 
 
 2529 
 
 2623 
 
 2717 
 
 95 
 
 87 
 
 88 
 
 89 
 
 196 
 
 2811 
 
 2905 
 
 2999 
 
 3093 
 
 3187 
 
 3281 
 
 3375 
 
 3469 
 
 3563 
 
 3657 
 
 96 
 
 88 
 
 89 
 
 90 
 
 197 
 
 3751 
 
 3845 
 
 3939 
 
 4033 
 
 4127 
 
 4221 
 
 4315 
 
 4409 
 
 4503 
 
 4597 
 
 97 
 
 89 
 
 90 
 
 91 
 
 198 
 
 4691 
 
 4785 
 
 4879 
 
 4973 
 
 5068 
 
 5162 
 
 5256 
 
 5350 
 
 5444 
 
 5538 
 
 98 
 
 90 
 
 91 
 
 92 
 
 199 
 
 5632 
 
 5726 
 
 5820 
 
 5914 
 
 6008 
 
 6102 
 
 6196 
 
 6290 
 
 6384 
 
 6478 
 
 99 
 
 91 
 
 92 
 
 93 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 92 
 
 93 
 
 94 
 
 25 
 
TABLE 1. 
 
 Log 
 
 Log (1+ar) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 94 
 
 95 
 
 96 
 
 1-200 
 
 1-226572 
 
 6666 
 
 6761 
 
 6855 
 
 6949 
 
 7043 
 
 7137 
 
 7231 
 
 7325 
 
 7419 
 
 00 
 
 
 
 
 
 
 
 201 
 
 7513 
 
 7607 
 
 7701 
 
 7795 
 
 7889 
 
 7983 
 
 8078 
 
 8172 
 
 8266 
 
 8360 
 
 01 
 
 1 
 
 1 
 
 1 
 
 202 
 
 8454 
 
 8548 
 
 8642 
 
 8736 
 
 8830 
 
 8924 
 
 9018 
 
 9113 
 
 9207 
 
 9301 
 
 02 
 
 2 
 
 2 
 
 2 
 
 203 
 
 9395 
 
 9489 
 
 9583 
 
 9677 
 
 9771 
 
 9865 
 
 9960 
 
 0054 
 
 0148 
 
 0242 
 
 03 
 
 g 
 
 3 
 
 3 
 
 204 
 
 23 0336 
 
 0430 
 
 0524 
 
 0618 
 
 0712 
 
 0807 
 
 0901 
 
 0995 
 
 1089 
 
 1183 
 
 04 
 
 4 
 
 4 
 
 4 
 
 205 
 
 1277 
 
 1371 
 
 1465 
 
 1560 
 
 1654 
 
 1748 
 
 1842 
 
 1936 
 
 2030 
 
 2124 
 
 GJ 
 
 t 
 
 5 
 
 5 
 
 206 
 
 2219 
 
 2313 
 
 2407 
 
 2501 
 
 2595 
 
 2689 
 
 2783 
 
 2878 
 
 2972 
 
 3066 
 
 06 
 
 6 
 
 6 
 
 6 
 
 207 
 
 3160 
 
 3254 
 
 3348 
 
 3443 
 
 3537 
 
 3631 
 
 3725 
 
 3819 
 
 3913 
 
 4007 
 
 07 
 
 7 
 
 7 
 
 7 
 
 208 
 
 4102 
 
 4196 
 
 4290 
 
 4384 
 
 4478 
 
 4573 
 
 4667 
 
 4761 
 
 4855 
 
 4949 
 
 08 
 
 8 
 
 8 
 
 8 
 
 209 
 
 5043 
 
 5138 
 
 5232 
 
 5326 
 
 5420 
 
 5514 
 
 5608 
 
 5703 
 
 5797 
 
 5891 
 
 09 
 
 e 
 
 9 
 
 9 
 
 210 
 
 23 5985 
 
 6079 
 
 6174 
 
 6268 
 
 6362 
 
 6456 
 
 6550 
 
 6645 
 
 6739 
 
 6833 
 
 10 
 
 9 
 
 10 
 
 10 
 
 211 
 
 6927 
 
 7021 
 
 7116 
 
 7210 
 
 7304 
 
 7398 
 
 7492 
 
 7587 
 
 7681 
 
 7775 
 
 11 
 
 10 
 
 10 
 
 11 
 
 212 
 
 7869 
 
 7964 
 
 8058 
 
 8152 
 
 8246 
 
 8340 
 
 8435 
 
 8529 
 
 8623 
 
 8717 
 
 12 
 
 11 
 
 11 
 
 12 
 
 213 
 
 8812 
 
 8906 
 
 9000 
 
 9094 
 
 9189 
 
 9283 
 
 9377 
 
 9471 
 
 9565 
 
 9660 
 
 13 
 
 12 
 
 12 
 
 12 
 
 214 
 
 9754 
 
 9848 
 
 9942 
 
 0037 
 
 0131 
 
 0225 
 
 0319 
 
 0414 
 
 0508 
 
 0602 
 
 14 
 
 13 
 
 13 
 
 13 
 
 215 
 
 24 0696 
 
 0791 
 
 0885 
 
 0979 
 
 1073 
 
 1168 
 
 1262 
 
 1356 
 
 1451 
 
 1545 
 
 1 f 
 
 14 
 
 14 
 
 14 
 
 216 
 
 1639 
 
 1733 
 
 1828 
 
 1922 
 
 2016 
 
 2110 
 
 2205 
 
 2299 
 
 2393 
 
 2487 
 
 16 
 
 15 
 
 15 
 
 15 
 
 217 
 
 2582 
 
 2676 
 
 2770 
 
 2865 
 
 2959 
 
 3053 
 
 3147 
 
 3242 
 
 3336 
 
 3430 
 
 17 
 
 16 
 
 16 
 
 16 
 
 218 
 
 3525 
 
 3619 
 
 3713 
 
 3808 
 
 3902 
 
 3996 
 
 4090 
 
 4185 
 
 4279 
 
 4373 
 
 18 
 
 17 
 
 1 1 
 
 17 
 
 219 
 
 4468 
 
 4562 
 
 4656 
 
 4751 
 
 4845 
 
 4939 
 
 5033 
 
 5128 
 
 5222 
 
 5316 
 
 If 
 
 18 
 
 18 
 
 18 
 
 220 
 
 245411 
 
 5505 
 
 5599 
 
 5694 
 
 5788 
 
 5882 
 
 5977 
 
 6071 
 
 6165 
 
 6260 
 
 20 
 
 19 
 
 19 
 
 19 
 
 221 
 
 6354 
 
 6448 
 
 6543 
 
 6637 
 
 6731 
 
 6826 
 
 6920 
 
 7014 
 
 7109 
 
 7203 
 
 21 
 
 20 
 
 20 
 
 20 
 
 222 
 
 7297 
 
 7392 
 
 7486 
 
 7580 
 
 7675 
 
 7769 
 
 7863 
 
 7958 
 
 8052 
 
 8146 
 
 22 
 
 21 
 
 21 
 
 21 
 
 223 
 
 8241 
 
 8335 
 
 8429 
 
 8524 
 
 8618 
 
 8713 
 
 8807 
 
 8901 
 
 8996 
 
 9090 
 
 23 
 
 22 
 
 22 
 
 22 
 
 224 
 
 9184 
 
 9279 
 
 9373 
 
 9467 
 
 9562 
 
 9656 
 
 9751 
 
 9845 
 
 9939 
 
 0034 
 
 24 
 
 23 
 
 23 
 
 23 
 
 225 
 
 250128 
 
 0222 
 
 0317 
 
 0411 
 
 0506 
 
 0600 
 
 0694 
 
 0789 
 
 0883 
 
 0978 
 
 25 
 
 24 
 
 24 
 
 24 
 
 226 
 
 1072 
 
 1166 
 
 1261 
 
 1355 
 
 1450 
 
 1544 
 
 1638 
 
 1733 
 
 1827 
 
 1922 
 
 26 
 
 24 
 
 25 
 
 25 
 
 227 
 
 2016 
 
 2110 
 
 2205 
 
 2299 
 
 2394 
 
 2488 
 
 2582 
 
 2677 
 
 2771 
 
 2866 
 
 27 
 
 25 
 
 '26 
 
 26 
 
 228 
 
 2960 
 
 3054 
 
 3149 
 
 3243 
 
 3338 
 
 3432 
 
 3527 
 
 3621 
 
 3715 
 
 3810 
 
 28 
 
 26 
 
 27 
 
 27 
 
 229 
 
 3904 
 
 3999 
 
 4093 
 
 4187 
 
 4282 
 
 4376 
 
 4471 
 
 4565 
 
 4660 
 
 4754 
 
 29 
 
 27 
 
 28 
 
 28 
 
 230 
 
 25 4849 
 
 4943 
 
 5037 
 
 5132 
 
 5226 
 
 5321 
 
 5415 
 
 5510 
 
 5604 
 
 5699 
 
 30 
 
 28 
 
 28 
 
 29 
 
 231 
 
 5793 
 
 5887 
 
 5982 
 
 6076 
 
 6171 
 
 6265 
 
 6360 
 
 6454 
 
 6549 
 
 6643 
 
 31 
 
 29 
 
 29 
 
 30 
 
 232 
 
 6738 
 
 6832 
 
 6926 
 
 7021 
 
 7115 
 
 7210 
 
 7304 
 
 7399 
 
 7493 
 
 7588 
 
 32 
 
 3( 
 
 30 
 
 31 
 
 233 
 
 7682 
 
 7777 
 
 7871 
 
 7966 
 
 8060 
 
 8155 
 
 8249 
 
 8344 
 
 8438 
 
 8533 
 
 33 
 
 31 
 
 31 
 
 32 
 
 234 
 
 8627 
 
 8722 
 
 8816 
 
 8911 
 
 9005 
 
 9100 
 
 9194 
 
 9289 
 
 9383 
 
 9477 
 
 34 
 
 32 
 
 32 
 
 33 
 
 235 
 
 9572 
 
 9666 
 
 9761 
 
 9855 
 
 9950 
 
 0045 
 
 o!39 
 
 0234 
 
 0328 
 
 o423 
 
 35 
 
 33 
 
 33 
 
 34 
 
 236 
 
 260517 
 
 0612 
 
 0706 
 
 0801 
 
 0895 
 
 0990 
 
 1084 
 
 1179 
 
 1273 
 
 1368 
 
 36 
 
 34 
 
 34 
 
 35 
 
 237 
 
 1462 
 
 1557 
 
 1651 
 
 1746 
 
 1840 
 
 1935 
 
 2029 
 
 2124 
 
 2218 
 
 2313 
 
 37 
 
 35 
 
 35 
 
 36 
 
 238 
 
 2408 
 
 2502 
 
 2597 
 
 2691 
 
 2786 
 
 2880 
 
 2975 
 
 3069 
 
 3164 
 
 3258 
 
 38 
 
 36 
 
 36 
 
 36 
 
 239 
 
 3353 
 
 3447 
 
 3542 
 
 3637 
 
 3731 
 
 3826 
 
 3920 
 
 4015 
 
 4109 
 
 4204 
 
 39 
 
 37 
 
 37 
 
 37 
 
 240 
 
 26 4298 
 
 4393 
 
 4488 
 
 4582 
 
 4677 
 
 4771 
 
 4866 
 
 4960 
 
 5055 
 
 5150 
 
 40 
 
 38 
 
 38 
 
 38 
 
 241 
 
 5244 
 
 5339 
 
 5433 
 
 5528 
 
 5622 
 
 5717 
 
 5812 
 
 5906 
 
 6001 
 
 6095 
 
 41 
 
 39 
 
 39 
 
 39 
 
 242 
 
 6190 
 
 6284 
 
 6379 
 
 6474 
 
 6568 
 
 6663 
 
 6757 
 
 6852 
 
 6947 
 
 7041 
 
 42 
 
 39 
 
 40 
 
 40 
 
 243 
 
 7136 
 
 7230 
 
 7325 
 
 7420 
 
 7514 
 
 7609 
 
 7703 
 
 7798 
 
 7893 
 
 7987 
 
 43 
 
 40 
 
 41 
 
 41 
 
 244 
 
 8082 
 
 8176 
 
 8271 
 
 8366 
 
 8460 
 
 8555 
 
 8649 
 
 8744 
 
 8839 
 
 8933 
 
 44 
 
 41 
 
 42 
 
 42 
 
 245 
 
 9028 
 
 9122 
 
 9217 
 
 9312 
 
 9406 
 
 9501 
 
 9596 
 
 9690 
 
 9785 
 
 9879 
 
 45 
 
 42 
 
 43 
 
 43 
 
 246 
 
 9974 
 
 0069 
 
 0163 
 
 0258 
 
 0353 
 
 0447 
 
 0542 
 
 0637 
 
 0731 
 
 0826 
 
 46 
 
 43 
 
 44 
 
 44 
 
 247 
 
 27 0920 
 
 1015 
 
 1110 
 
 1204 
 
 1299 
 
 1394 
 
 1488 
 
 1583 
 
 1678 
 
 1772 
 
 47 
 
 44 
 
 45 
 
 45 
 
 248 
 
 1867 
 
 1962 
 
 2056 
 
 2151 
 
 2246 
 
 2340 
 
 2435 
 
 2530 
 
 2624 
 
 2719 
 
 48 
 
 45 
 
 46 
 
 46 
 
 249 
 
 2813 
 
 2908 
 
 3003 
 
 3097 
 
 3192 
 
 3287 
 
 3382 
 
 3476 
 
 3571 
 
 3666 
 
 49 
 
 46 
 
 47 
 
 47 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 94 
 
 95 
 
 96 
 
 26 
 
TABLE I 
 
 
 Lo (}+x}~ 
 
 Pro. Parts. 
 
 Log 
 
 & \ ~T 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 94 
 
 95 
 
 96 
 
 1-250 
 
 1-27 3760 
 
 3855 
 
 3950 
 
 4044 
 
 4139 
 
 4234 
 
 4328 
 
 4423 
 
 4518 
 
 4612 
 
 50 
 
 47 
 
 48 
 
 48 
 
 251 
 
 4707 
 
 4802 
 
 4896 
 
 4991 
 
 5086 
 
 5180 
 
 5275 
 
 5370 
 
 5465 
 
 5559 
 
 51 
 
 48 
 
 48 
 
 49 
 
 252 
 
 5654 
 
 5749 
 
 5843 
 
 5938 
 
 6033 
 
 6127 
 
 6222 
 
 6317 
 
 6412 
 
 6506 
 
 52 
 
 49 
 
 49 
 
 50 
 
 253 
 
 6601 
 
 6696 
 
 6790 
 
 6885 
 
 6980 
 
 7075 
 
 7169 
 
 7264 
 
 7359 
 
 7453 
 
 53 
 
 50 
 
 50 
 
 51 
 
 254 
 
 7548 
 
 7643 
 
 7738 
 
 7832 
 
 7927 
 
 8022 
 
 8117 
 
 8211 
 
 8306 
 
 8401 
 
 54 
 
 51 
 
 51 
 
 52 
 
 255 
 
 8495 
 
 8590 
 
 8685 
 
 8780 
 
 8874 
 
 8969 
 
 9064 
 
 9159 
 
 9253 
 
 9348 
 
 55 
 
 52 
 
 52 
 
 53 
 
 256 
 
 9443 
 
 9538 
 
 9632 
 
 9727 
 
 9822 
 
 9917 
 
 0011 
 
 0106 
 
 0201 
 
 0296 
 
 56 
 
 53 
 
 53 
 
 54 
 
 257 
 
 28 0390 
 
 0485 
 
 0580 
 
 0675 
 
 0769 
 
 0864 
 
 0959 
 
 1054 
 
 1148 
 
 1243 
 
 57 
 
 54 
 
 54 
 
 55 
 
 258 
 
 1338 
 
 1433 
 
 1528 
 
 1622 
 
 1717 
 
 1812 
 
 1907 
 
 2001 
 
 2096 
 
 2191 
 
 58 
 
 55 
 
 55 
 
 56 
 
 259 
 
 2286 
 
 2380 
 
 2475 
 
 2570 
 
 2665 
 
 2760 
 
 2854 
 
 2949 
 
 3044 
 
 3139 
 
 59 
 
 55 
 
 56 
 
 57 
 
 260 
 
 28 3234 
 
 3328 
 
 3423 
 
 3518 
 
 3613 
 
 3708 
 
 3802 
 
 3897 
 
 3992 
 
 4087 
 
 60 
 
 56 
 
 57 
 
 58 
 
 261 
 
 4182 
 
 4276 
 
 4371 
 
 4466 
 
 4561 
 
 4656 
 
 4750 
 
 4845 
 
 4940 
 
 5035 
 
 61 
 
 57 
 
 58 
 
 59 
 
 262 
 
 5130 
 
 5224 
 
 5319 
 
 5414 
 
 5509 
 
 5604 
 
 5699 
 
 5793 
 
 5888 
 
 5983 
 
 62 
 
 58 
 
 59 
 
 60 
 
 263 
 
 6078 
 
 6173 
 
 6267 
 
 6362 
 
 6457 
 
 6552 
 
 6647 
 
 6742 
 
 6836 
 
 6931 
 
 63 
 
 59 
 
 60 
 
 60 
 
 264 
 
 7026 
 
 7121 
 
 7216 
 
 7311 
 
 7405 
 
 7500 
 
 7595 
 
 7690 
 
 7785 
 
 7880 
 
 64 
 
 60 
 
 61 
 
 61 
 
 265 
 
 7975 
 
 8069 
 
 8164 
 
 8259 
 
 8354 
 
 8449 
 
 8544 
 
 8638 
 
 8733 
 
 8828 
 
 65 
 
 61 
 
 62 
 
 62 
 
 266 
 
 8923 
 
 9018 
 
 9113 
 
 9208 
 
 9302 
 
 9397 
 
 9492 
 
 9587 
 
 9682 
 
 9777 
 
 66 
 
 62 
 
 63 
 
 63 
 
 267 
 
 9872 
 
 9967 
 
 0061 
 
 0156 
 
 0251 
 
 0346 
 
 0441 
 
 0536 
 
 0631 
 
 0726 
 
 67 
 
 63 
 
 64 
 
 64 
 
 268 
 
 29 0820 
 
 0915 
 
 1010 
 
 1105 
 
 1200 
 
 1295 
 
 1390 
 
 1485 
 
 1580 
 
 1674 
 
 68 
 
 64 
 
 65 
 
 65 
 
 269 
 
 1769 
 
 1864 
 
 1959 
 
 2054 
 
 2149 
 
 2244 
 
 2339 
 
 2434 
 
 2528 
 
 2623 
 
 69 
 
 65 
 
 66 
 
 66 
 
 270 
 
 292718 
 
 2813 
 
 2908 
 
 3003 
 
 3098 
 
 3193 
 
 3288 
 
 3383 
 
 3478 
 
 3572 
 
 70 
 
 66 
 
 66 
 
 67 
 
 271 
 
 3667 
 
 3762 
 
 3857 
 
 3952 
 
 4047 
 
 4142 
 
 4237 
 
 4332 
 
 4427 
 
 4522 
 
 71 
 
 67 
 
 67 
 
 68 
 
 272 
 
 4617 
 
 4712 
 
 4806 
 
 4901 
 
 4996 
 
 5091 
 
 5186 
 
 5281 
 
 5376 
 
 5471 
 
 72 
 
 08 
 
 68 
 
 69 
 
 273 
 
 5566 
 
 5661 
 
 5756 
 
 5851 
 
 5946 
 
 6041 
 
 6136 
 
 6230 
 
 6325 
 
 6420 
 
 73 
 
 69 
 
 69 
 
 70 
 
 274 
 
 6515 
 
 6610 
 
 6705 
 
 6800 
 
 6895 
 
 6990 
 
 7085 
 
 7180 
 
 7275 
 
 7370 
 
 74 
 
 70 
 
 70 
 
 71 
 
 275 
 
 7465 
 
 7560 
 
 7655 
 
 7750 
 
 7845 
 
 7940 
 
 8035 
 
 8130 
 
 8225 
 
 8320 
 
 75 
 
 70 
 
 71 
 
 72 
 
 276 
 
 8414 
 
 8509 
 
 8604 
 
 8699 
 
 8794 
 
 8889 
 
 8984 
 
 9079 
 
 9174 
 
 9269 
 
 76 
 
 71 
 
 72 
 
 73 
 
 277 
 
 9364 
 
 9459 
 
 9554 
 
 9649 
 
 9744 
 
 9839 
 
 9934 
 
 0029 
 
 0124 
 
 0219 
 
 77 
 
 72 
 
 73 
 
 74 
 
 278 
 
 30 0314 
 
 0409 
 
 0504 
 
 0599 
 
 0694 
 
 0789 
 
 0884 
 
 0979 
 
 1074 
 
 1169 
 
 78 
 
 73 
 
 74 
 
 75 
 
 279 
 
 1264 
 
 1359 
 
 1454 
 
 1549 
 
 1644 
 
 1739 
 
 1834 
 
 1929 
 
 2024 
 
 2119 
 
 79 
 
 74 
 
 75 
 
 76 
 
 280 
 
 302214 
 
 2309 
 
 2404 
 
 2499 
 
 2594 
 
 2689 
 
 2784 
 
 2879 
 
 2974 
 
 3069 
 
 80 
 
 75 
 
 76 
 
 77 
 
 281 
 
 3164 
 
 3259 
 
 3354 
 
 3449 
 
 3544 
 
 3639 
 
 3735 
 
 3830 
 
 3925 
 
 4020 
 
 81 
 
 76 
 
 77 
 
 78 
 
 282 
 
 4115 
 
 4210 
 
 4305 
 
 4400 
 
 4495 
 
 4590 
 
 4685 
 
 4780 
 
 4875 
 
 4970 
 
 82 
 
 77 
 
 78 
 
 79 
 
 283 
 
 5065 
 
 5160 
 
 5255 
 
 5350 
 
 5445 
 
 5540 
 
 5635 
 
 5730 
 
 5825 
 
 5921 
 
 83 
 
 78 
 
 79 
 
 80 
 
 284 
 
 6016 
 
 6111 
 
 6206 
 
 6301 
 
 6396 
 
 6491 
 
 6586 
 
 6681 
 
 6776 
 
 6871 
 
 84 
 
 79 
 
 80 
 
 81 
 
 285 
 
 6966 
 
 7061 
 
 7156 
 
 7251 
 
 7346 
 
 7442 
 
 7537 
 
 7632 
 
 7727 
 
 7822 
 
 85 
 
 80 
 
 81 
 
 82 
 
 286 
 
 7917 
 
 8012 
 
 8107 
 
 8202 
 
 8297 
 
 8392 
 
 8487 
 
 8583 
 
 8678 
 
 8773 
 
 86 
 
 81 
 
 82 
 
 83 
 
 287 
 
 8868 
 
 8963 
 
 9058 
 
 9153 
 
 9248 
 
 9343 
 
 9438 
 
 9533 
 
 9629 
 
 9724 
 
 87 
 
 82 
 
 83 
 
 84 
 
 288 
 
 9819 
 
 9914 
 
 0009 
 
 0104 
 
 0199 
 
 0294 
 
 0389 
 
 0484 
 
 0580 
 
 0675 
 
 88 
 
 83 
 
 84 
 
 84 
 
 289 
 
 31 0770 
 
 0865 
 
 0960 
 
 1055 
 
 1150 
 
 1245 
 
 1340 
 
 1436 
 
 1531 
 
 1626 
 
 89 
 
 84 
 
 85 
 
 85 
 
 290 
 
 31 1721 
 
 1816 
 
 1911 
 
 2006 
 
 2101 
 
 2197 
 
 2292 
 
 2387 
 
 2482 
 
 2577 
 
 90 
 
 85 
 
 86 
 
 86 
 
 291 
 
 2672 
 
 2767 
 
 2862 
 
 2958 
 
 3053 
 
 3148 
 
 3243 
 
 3338 
 
 3433 
 
 3528 
 
 91 
 
 86 
 
 86 
 
 87 
 
 292 
 
 3624 
 
 3719 
 
 3814 
 
 3909 
 
 4004 
 
 4099 
 
 4194 
 
 4290 
 
 4385 
 
 4480 
 
 92 
 
 86 
 
 87 
 
 88 
 
 293 
 
 4575 
 
 4670 
 
 4765 
 
 4861 
 
 4956 
 
 5051 
 
 5146 
 
 5241 
 
 5336 
 
 5431 
 
 93 
 
 87 
 
 88 
 
 89 
 
 294 
 
 5527 
 
 5622 
 
 5717 
 
 5812 
 
 5907 
 
 6002 
 
 6098 
 
 6193 
 
 6288 
 
 6383 
 
 94 
 
 88 
 
 89 
 
 90 
 
 295 
 
 6478 
 
 6574 
 
 6669 
 
 6764 
 
 6859 
 
 6954 
 
 7049 
 
 7145 
 
 7240 
 
 7335 
 
 95 
 
 89 
 
 90 
 
 91 
 
 296 
 
 7430 
 
 7525 
 
 7621 
 
 7716 
 
 7811 
 
 7906 
 
 8001 
 
 8096 
 
 8192 
 
 8287 
 
 96 
 
 90 
 
 91 
 
 92 
 
 297 
 
 8382 
 
 8477 
 
 8572 
 
 8668 
 
 8763 
 
 8858 
 
 8953 
 
 9048 
 
 9144 
 
 9239 
 
 97 
 
 91 
 
 92 
 
 93 
 
 298 
 
 9334 
 
 9429 
 
 9524 
 
 9620 
 
 9715 
 
 9810 
 
 9905 
 
 oOOl 
 
 0096 
 
 0191 
 
 98 
 
 92 
 
 93 
 
 94 
 
 299 
 
 32 0286 
 
 0381 
 
 0477 
 
 0572 
 
 0667 
 
 0762 
 
 0858 
 
 0953 
 
 1048 
 
 1143 
 
 99 
 
 93 
 
 94 
 
 95 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 94 
 
 95 
 
 96 
 
 27 
 
TABLE I. 
 
 Log 
 
 Log(l+*) 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9596 
 
 97 
 
 1-300 
 
 1-32 1238 
 
 1334 
 
 1429 
 
 1524 
 
 1619 
 
 1715 
 
 1810 
 
 1905 
 
 2000 
 
 2095 
 
 00 
 
 
 
 
 
 
 
 301 
 
 2191 
 
 2286 
 
 2381 
 
 2476 
 
 2572 
 
 2667 
 
 2762 
 
 2857 
 
 2953 
 
 3048 
 
 01 
 
 1 
 
 1 
 
 1 
 
 302 
 
 3143 
 
 3238 
 
 3334 
 
 3429 
 
 3524 
 
 3619 
 
 3715 
 
 3810 
 
 3905 
 
 4000 
 
 02 
 
 2 2: 2 
 
 303 
 
 4096 
 
 4191 
 
 4286 
 
 4381 
 
 4477 
 
 4572 
 
 4667 
 
 4763 
 
 4858 
 
 4953 
 
 03 
 
 3 3 3 
 
 304 
 
 5048 
 
 5144 
 
 5239 
 
 5334 
 
 5429 
 
 5525 
 
 5620 
 
 5715 
 
 5811 
 
 5906 
 
 04- 
 
 4 
 
 4 
 
 4 
 
 305 
 
 6001 
 
 6096 
 
 6192 
 
 6287 
 
 6382 
 
 6477 
 
 6573 
 
 6668 
 
 6763 
 
 6859 
 
 05 
 
 5 
 
 5 
 
 5 
 
 306 
 
 6954 
 
 7049 
 
 7144 
 
 7240 
 
 7335 
 
 7430 
 
 7526 
 
 7621 
 
 7716 
 
 7812 
 
 00 
 
 6 6 
 
 6 
 
 307 
 
 7907 
 
 8002 
 
 8097 
 
 8193 
 
 8288 
 
 8383 
 
 8479 
 
 8574 
 
 8669 
 
 8765 
 
 07 
 
 7| 7 
 
 7 
 
 308 
 
 8860 
 
 8955 
 
 9051 
 
 9146 
 
 9241 
 
 9336 
 
 9432 
 
 9527 
 
 9622 
 
 9718 
 
 08 
 
 888 
 
 309 
 
 9813 
 
 9908 
 
 0004 
 
 0099 
 
 0194 
 
 0290 
 
 0385 
 
 o480 
 
 0576 
 
 0671 
 
 09 
 
 9 
 
 9i 9 
 
 310 
 
 33 0766 
 
 0862 
 
 0957 
 
 1052 
 
 1148 
 
 1243 
 
 1338 
 
 1434 
 
 1529 
 
 1624 
 
 10 
 
 10 
 
 10 
 
 10 
 
 311 
 
 1720 
 
 1815 
 
 1910 
 
 2006 
 
 2101 
 
 2196 
 
 2292 
 
 2387 
 
 2482 
 
 2578 
 
 11 
 
 10 
 
 11 
 
 11 
 
 312 
 
 2673 
 
 2769 
 
 2864 
 
 2959 
 
 3055 
 
 3150 
 
 3245 
 
 3341 
 
 3436 
 
 3531 
 
 12 
 
 11 
 
 12 
 
 12 
 
 313 
 
 3627 
 
 3722 
 
 3817 
 
 3913 
 
 4008 
 
 4104 
 
 4199 
 
 4294 
 
 4390 
 
 4485 
 
 13 
 
 12 
 
 12 
 
 13 
 
 314 
 
 4580 
 
 4676 
 
 4771 
 
 4867 
 
 4962 
 
 5057 
 
 5153 
 
 5248 
 
 5343 
 
 5439 
 
 14 
 
 13 
 
 13 
 
 14 
 
 315 
 
 5534 
 
 5630 
 
 5725 
 
 5820 
 
 5916 
 
 6011 
 
 6106 
 
 6202 
 
 6297 
 
 6393 
 
 15 
 
 14 
 
 14 
 
 15 
 
 316 
 
 6488 
 
 6583 
 
 6679 
 
 6774 
 
 6870 
 
 6965 
 
 7060 
 
 7156 
 
 7251 
 
 7347 
 
 16 
 
 15 
 
 15 
 
 16 
 
 317 
 
 7442 
 
 7537 
 
 7633 
 
 7728 
 
 7824 
 
 7919 
 
 8014 
 
 8110 
 
 8205 
 
 8301 
 
 17 
 
 16 
 
 16 
 
 16 
 
 318 
 
 8396 
 
 8491 
 
 8587 
 
 8682 
 
 8778 
 
 8873 
 
 8969 
 
 9064 
 
 9159 
 
 9255 
 
 18 
 
 17 
 
 17 
 
 17 
 
 319 
 
 9350 
 
 9446 
 
 9541 
 
 9637 
 
 9732 
 
 9827 
 
 9923 
 
 0018 
 
 0114 
 
 0209 
 
 19 
 
 18 
 
 18 
 
 18 
 
 320 
 
 34 0305 
 
 0400 
 
 0495 
 
 0591 
 
 0686 
 
 0782 
 
 0877 
 
 0973 
 
 1068 
 
 1163 
 
 20 
 
 19 
 
 1919 
 
 321 
 
 1259 
 
 1354 
 
 1450 
 
 1545 
 
 1641 
 
 1736 
 
 1832 
 
 1927 
 
 2022 
 
 2118 
 
 21 
 
 20 
 
 2020 
 
 322 
 
 2213 
 
 2309 
 
 2404 
 
 2500 
 
 2595 
 
 2691 
 
 2786 
 
 2882 
 
 2977 
 
 3072 
 
 22 
 
 21 
 
 2121 
 
 323 
 
 3168 
 
 3263 
 
 3359 
 
 3454 
 
 3550 
 
 3645 
 
 3741 
 
 3836 
 
 3932 
 
 4027 
 
 23 
 
 22 
 
 2222 
 
 324 
 
 4123 
 
 4218 
 
 4314 
 
 4409 
 
 4505 
 
 4600 
 
 4695 
 
 4791 
 
 4886 
 
 4982 
 
 24 
 
 23 
 
 2323 
 
 325 
 
 5077 
 
 5173 
 
 5268 
 
 5364 
 
 5459 
 
 5555 
 
 5650 
 
 5746 
 
 5841 
 
 5937 
 
 25 
 
 24 
 
 24 24 
 
 326 
 
 6032 
 
 6128 
 
 6223 
 
 6319 
 
 6414 
 
 6510 
 
 6605 
 
 6701 
 
 6796 
 
 6892 
 
 2(i 
 
 25 
 
 25| 25 
 
 327 
 
 6987 
 
 7083 
 
 7178 
 
 7274 
 
 7369 
 
 7465 
 
 7560 
 
 7656 
 
 7751 
 
 7847 
 
 27 
 
 26 
 
 26 
 
 26 
 
 328 
 
 7942 
 
 8038 
 
 8133 
 
 8229 
 
 8324 
 
 8420 
 
 8515 
 
 86J1 
 
 8706 
 
 8802 
 
 28 
 
 27 
 
 27 
 
 27 
 
 329 
 
 8897 
 
 8993 
 
 9089 
 
 9184 
 
 9280 
 
 9375 
 
 9471 
 
 9566 
 
 9662 
 
 9757 
 
 29 
 
 28 
 
 28 
 
 28 
 
 330 
 
 34 9853 
 
 9948 
 
 0044 
 
 0139 
 
 0235 
 
 0330 
 
 0426 
 
 0521 
 
 0617 
 
 0713 
 
 30 
 
 28 
 
 2929 
 
 331 
 
 35 0808 
 
 0904 
 
 0999 
 
 1095 
 
 1190 
 
 1286 
 
 1381 
 
 1477 
 
 1572 
 
 1668 
 
 31 
 
 29 
 
 30 30 
 
 332 
 
 1764 
 
 1859 
 
 1955 
 
 2050 
 
 2146 
 
 2241 
 
 2337 
 
 2432 
 
 2528 
 
 2624 
 
 32 
 
 30 
 
 3131 
 
 333 
 
 2719 
 
 2815 
 
 2910 
 
 3006 
 
 3101 
 
 3197 
 
 3293 
 
 3388 
 
 3484 
 
 3579 
 
 33 
 
 31 
 
 32 
 
 32 
 
 334 
 
 3675 
 
 3770 
 
 3866 
 
 3961 
 
 4057 
 
 4153 
 
 4248 
 
 4344 
 
 4439 
 
 4535 
 
 34 
 
 32 
 
 33 
 
 33 
 
 335 
 
 4631 
 
 4726 
 
 4822 
 
 4917 
 
 5013 
 
 5108 
 
 5204 
 
 5300 
 
 5395 
 
 5491 
 
 35 
 
 33 
 
 34 
 
 34 
 
 336 
 
 5586 
 
 5682 
 
 5778 
 
 5873 
 
 5969 
 
 6064 
 
 6160 
 
 6256 
 
 6351 
 
 6447 
 
 36 
 
 34 
 
 35 
 
 35 
 
 337 
 
 6542 
 
 6638 
 
 6734 
 
 6829 
 
 6925 
 
 7020 
 
 7116 
 
 7212 
 
 7307 
 
 7403 
 
 37 
 
 35 
 
 36 
 
 36 
 
 338 
 
 7498 
 
 7594 
 
 7690 
 
 7785 
 
 7881 
 
 7976 
 
 8072 
 
 8168 
 
 8263 
 
 8359 
 
 38 
 
 36 
 
 36 
 
 37 
 
 339 
 
 8455 
 
 8550 
 
 8646 
 
 8741 
 
 8837 
 
 8933 
 
 9028 
 
 9124 
 
 9220 
 
 9315 
 
 39 
 
 37 
 
 37 
 
 38 
 
 340 
 
 359411 
 
 9506 
 
 9602 
 
 9698 
 
 9793 
 
 9889 
 
 9985 
 
 0080 
 
 0176 
 
 0271 
 
 40 
 
 38 
 
 38 
 
 39 
 
 341 
 
 36 0367 
 
 0463 
 
 0558 
 
 0654 
 
 0750 
 
 0845 
 
 0941 
 
 1037 
 
 1132 
 
 1228 
 
 41 
 
 39 
 
 3940 
 
 342 
 
 1324 
 
 1419 
 
 1515 
 
 1610 
 
 1706 
 
 1802 
 
 1897 
 
 1993 
 
 2089 
 
 2184 
 
 42 
 
 40 
 
 40 
 
 41 
 
 343 
 
 2280 
 
 2376 
 
 2471 
 
 2567 
 
 2663 
 
 2758 
 
 2854 
 
 2950 3045 
 
 3141 
 
 43 
 
 41 
 
 41 
 
 42 
 
 344 
 
 3237 
 
 3332 
 
 3428 
 
 3524 
 
 3619 
 
 3715 
 
 3811 
 
 3906 
 
 4002 
 
 4098 
 
 44 
 
 42 
 
 42 
 
 43 
 
 345 
 
 4193 
 
 4289 
 
 4385 
 
 44ftO 
 
 4576 
 
 4672 
 
 4767 
 
 4863 
 
 4959 
 
 5055 
 
 45 
 
 43 
 
 4344 
 
 T'TOU 
 
 346 
 
 5150 
 
 5246 
 
 5342 
 
 5437 
 
 5533 
 
 5629 
 
 5724 
 
 5820 
 
 5916 
 
 6011 
 
 46 
 
 44 
 
 44 45 
 
 347 
 
 6107 
 
 6203 
 
 6299 
 
 6394 
 
 6490 
 
 6586 
 
 6681 
 
 6777 
 
 6873 
 
 6968 
 
 47 
 
 45 
 
 4546 
 
 348 
 
 7064 
 
 7160 
 
 7256 
 
 7351 
 
 7447 
 
 7543 
 
 7638 
 
 7734 
 
 7830 
 
 7926 
 
 48 
 
 46 
 
 4647 
 
 349 
 
 8021 
 
 8117 
 
 8213 
 
 8308 
 
 8404 
 
 8500 
 
 8596 
 
 8691 
 
 8787 
 
 8883 
 
 49 
 
 47 
 
 47J48 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 95 
 
 9697 
 
 28 
 
TABLE I. 
 
 Log 
 
 Log(l+aO 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 95 
 
 96 
 
 97 
 
 1 350 
 
 1-368978 
 
 9074 
 
 9170 
 
 9266 
 
 9361 
 
 9457 
 
 9553 
 
 9649 
 
 9744 
 
 9840 
 
 50 
 
 48 
 
 48 
 
 49 
 
 351 
 
 9936 
 
 o031 
 
 0127 
 
 0223 
 
 0319 
 
 0414 
 
 0510 
 
 0606 
 
 0702 
 
 0797 
 
 51 
 
 48 
 
 49 
 
 49 
 
 352 
 
 37 0893 
 
 0989 
 
 1085 
 
 1180 
 
 1276 
 
 1372 
 
 1468 
 
 1563 
 
 1659 
 
 1755 
 
 52 
 
 49 
 
 50 
 
 50 
 
 353 
 
 1851 
 
 1946 
 
 2042 
 
 2138 
 
 2234 
 
 2329 
 
 2425 
 
 2521 
 
 2617 
 
 2712 
 
 53 
 
 50 
 
 51 
 
 51 
 
 354 
 
 2808 
 
 2904 
 
 3000 
 
 3095 
 
 3191 
 
 3287 
 
 3383 
 
 3479 
 
 3574 
 
 3670 
 
 54 
 
 51 
 
 52 
 
 52 
 
 355 
 
 3766 
 
 3862 
 
 3957 
 
 4053 
 
 4149 
 
 4245 
 
 4340 
 
 4436 
 
 4532 
 
 4628 
 
 55 
 
 52 
 
 53 
 
 53 
 
 356 
 
 4724 
 
 4819 
 
 4915 
 
 5011 
 
 5107 
 
 5202 
 
 5298 
 
 5394 
 
 5490 
 
 5586 
 
 56 
 
 53 
 
 54 
 
 54 
 
 357 
 
 5681 
 
 5777 
 
 5873 
 
 5969 
 
 6065 
 
 6160 
 
 6256 
 
 6352 
 
 6448 
 
 6544 
 
 57 
 
 54 
 
 55 
 
 55 
 
 358 
 
 6639 
 
 6735 
 
 6831 
 
 6927 
 
 7023 
 
 7118 
 
 7214 
 
 7310 
 
 7406 
 
 7502 
 
 58 
 
 55 
 
 56 
 
 56 
 
 359 
 
 7597 
 
 7693 
 
 7789 
 
 7885 
 
 7981 
 
 8076 
 
 8172 
 
 8268 
 
 8364 
 
 8460 
 
 59 
 
 56 
 
 57 
 
 57 
 
 360 
 
 37 8556 
 
 8651 
 
 8747 
 
 8843 
 
 8939 
 
 9035 
 
 9130 
 
 9226 
 
 9322 
 
 9418 
 
 60 
 
 57 
 
 58 
 
 58 
 
 361 
 
 9514 
 
 9610 
 
 9705 
 
 9801 
 
 9897 
 
 9993 
 
 0089 0185 
 
 0280 
 
 0376 
 
 61 
 
 58 
 
 59 
 
 59 
 
 362 
 
 38 0472 
 
 0568 
 
 0664 
 
 0760 
 
 0855 
 
 0951 
 
 1047 1143 
 
 1239 
 
 1335 
 
 62 
 
 59 
 
 60 
 
 60 
 
 363 
 
 1430 
 
 1526 
 
 1622 
 
 1718 
 
 1814 
 
 1910 
 
 2006 
 
 2101 
 
 2197 
 
 2293 
 
 63 
 
 60 
 
 60 
 
 61 
 
 364 
 
 2389 
 
 2485 
 
 2581 
 
 2677 
 
 2772 
 
 2868 
 
 2964 
 
 3060 
 
 3156 
 
 3252 
 
 64 
 
 61 
 
 61 
 
 62 
 
 365 
 
 3348 
 
 3443 
 
 3539 
 
 3635 
 
 3731 
 
 3827 
 
 3923 
 
 4019 
 
 4114 
 
 4210 
 
 65 
 
 62 
 
 62 
 
 63 
 
 366 
 
 4306 
 
 4402 
 
 4498 
 
 4594 
 
 4690 
 
 4786 
 
 4881 
 
 4977 
 
 5073 
 
 5169 
 
 66 
 
 63 
 
 63 
 
 64 
 
 367 
 
 5265 
 
 5361 
 
 5457 
 
 5553 
 
 5649 
 
 5744 
 
 5840 
 
 5936 
 
 6032 
 
 6128 
 
 67 
 
 64 
 
 64 
 
 65 
 
 368 
 
 6224 
 
 6320 
 
 6416 
 
 6512 
 
 6607 
 
 6703 
 
 6799 
 
 6895 
 
 6991 
 
 7087 
 
 68 
 
 65 
 
 65 
 
 66 
 
 369 
 
 7183 
 
 7279 
 
 7375 
 
 7471 
 
 7566 
 
 7662 
 
 7758 
 
 7854 
 
 7950 
 
 8046 
 
 69 
 
 66 
 
 66 
 
 67 
 
 370 
 
 388142 
 
 8238 
 
 8334 
 
 8430 
 
 8526 
 
 8621 
 
 8717 
 
 8813 
 
 8909 
 
 9005 
 
 70 
 
 66 
 
 67 
 
 68 
 
 371 
 
 9101 
 
 9197 
 
 9293 
 
 9389 
 
 9485 
 
 9581 
 
 9677 
 
 9772 
 
 9868 
 
 9964 
 
 71 
 
 67 
 
 68 
 
 69 
 
 372 
 
 39 0060 
 
 0156 
 
 0252 
 
 0348 
 
 0444 
 
 0540 
 
 0636 
 
 0732 
 
 0828 
 
 0924 
 
 72 
 
 68 
 
 69 
 
 70 
 
 373 
 
 1020 
 
 1115 
 
 1211 
 
 1307 
 
 1403 
 
 1499 
 
 1595 
 
 1691 
 
 1787 
 
 1883 
 
 73 
 
 69 
 
 70 
 
 71 
 
 374 
 
 1979 
 
 2075 
 
 2171 
 
 2267 
 
 2363 
 
 2459 
 
 2555 
 
 2651 
 
 2747 
 
 2842 
 
 74 
 
 70 
 
 71 
 
 72 
 
 375 
 
 2938 
 
 3034 
 
 3130 
 
 3226 
 
 3322 
 
 3418 
 
 3514 
 
 3610 
 
 3706 
 
 3802 
 
 75 
 
 71 
 
 72 
 
 73 
 
 376 
 
 3898 
 
 3994 
 
 4090 
 
 4186 
 
 4282 
 
 4378 
 
 4474 
 
 4570 
 
 4666 
 
 4762 
 
 76 
 
 72 
 
 73 
 
 74 
 
 377 
 
 4858 
 
 4954 
 
 5050 
 
 5146 
 
 5242 
 
 5338 
 
 5434 
 
 5529 
 
 5625 
 
 5721 
 
 77 
 
 73 
 
 74 
 
 75 
 
 378 
 
 5817 
 
 5913 
 
 6009 
 
 6105 
 
 6201 
 
 6297 
 
 6393 
 
 6489 
 
 6585 
 
 6681 
 
 78 
 
 74 
 
 75 
 
 76 
 
 379 
 
 6777 
 
 6873 
 
 6969 
 
 7065 
 
 7161 
 
 7257 
 
 7353 
 
 7449 
 
 7545 
 
 7641 
 
 79 
 
 75 
 
 76 
 
 77 
 
 380 
 
 39 7737 
 
 7833 
 
 7929 
 
 8025 
 
 8121 
 
 8217 
 
 8313 
 
 8409 
 
 8505 
 
 8601 
 
 80 
 
 76 
 
 77 
 
 78 
 
 381 
 
 8697 
 
 8793 
 
 8889 
 
 8985 
 
 9081 
 
 9177 
 
 9273 
 
 9369 
 
 9465 
 
 9561 
 
 81 
 
 77 
 
 78 
 
 79 
 
 382 
 
 9657 
 
 9753 
 
 9849 
 
 9945 
 
 0041 
 
 0137 
 
 0233 
 
 0329 
 
 0426 
 
 0522 
 
 82 
 
 78 
 
 79 
 
 80 
 
 -383 
 
 400618 
 
 0714 
 
 0810 
 
 0906 
 
 1002 
 
 1098 
 
 1194 
 
 1290 
 
 1386 
 
 1482 
 
 83 
 
 79 
 
 80 
 
 81 
 
 384 
 
 1578 
 
 1674 
 
 1770 
 
 1866 
 
 1962 
 
 2058 
 
 2154 
 
 2250 
 
 2346 
 
 2442 
 
 84 
 
 80 
 
 81 
 
 81 
 
 385 
 
 2538 
 
 2634 
 
 2730 
 
 2826 
 
 2922 
 
 3018 
 
 3114 
 
 3211 
 
 3307 
 
 3403 
 
 85 
 
 81 
 
 82 
 
 82 
 
 386 
 
 3499 
 
 3595 
 
 3691 
 
 3787 
 
 3883 
 
 3979 
 
 4075 
 
 4171 
 
 4267 
 
 4363 
 
 86 
 
 82 
 
 83 
 
 83 
 
 387 
 
 4459 
 
 4555 
 
 4651 
 
 4747 
 
 4843 
 
 4940 
 
 5036 
 
 5132 
 
 5228 
 
 5324 
 
 87 
 
 83 
 
 84 
 
 84 
 
 388 
 
 5420 
 
 5516 
 
 5612 
 
 5708 
 
 5804 
 
 5900 
 
 5996 
 
 6092 
 
 6188 
 
 6285 
 
 88 
 
 84 
 
 84 
 
 85 
 
 389 
 
 6381 
 
 6477 
 
 6573 
 
 6669 
 
 6765 
 
 6861 
 
 6957 
 
 7053 
 
 7149 
 
 7245 
 
 89 
 
 85 
 
 85 
 
 86 
 
 390 
 
 40 7341 
 
 7438 
 
 7534 
 
 7630 
 
 7726 
 
 7822 
 
 7918 
 
 8014 
 
 8110 
 
 8206 
 
 90 
 
 86 
 
 86 
 
 87 
 
 391 
 
 8302 
 
 8398 
 
 8495 
 
 8591 
 
 8687 
 
 8783 
 
 8879 
 
 8975 
 
 9071 
 
 9167 
 
 91 
 
 86 
 
 87 
 
 88 
 
 392 
 
 9263 
 
 9359 
 
 9456 
 
 9552 
 
 9648 
 
 9744 
 
 9840 
 
 9936 
 
 0032 
 
 0128 
 
 92 
 
 87 
 
 88 
 
 89 
 
 393 
 
 41 0224 
 
 0320 
 
 0417 
 
 0513 
 
 0609 
 
 0705 
 
 0801 
 
 0897 
 
 0993 
 
 1089 
 
 93 
 
 88 
 
 89 
 
 90 
 
 394 
 
 1186 
 
 1282 
 
 1378 
 
 1474 
 
 1570 
 
 1666 
 
 1762 
 
 1858 
 
 1955 
 
 2051 
 
 94 
 
 89 
 
 90 
 
 91 
 
 395 
 
 2147 
 
 2243 
 
 2339 
 
 2435 
 
 2531 
 
 2627 
 
 2724 
 
 2820 
 
 2916 
 
 3012 
 
 95 
 
 90 
 
 91 
 
 92 
 
 396 
 
 3108 
 
 3204 
 
 3300 
 
 3397 
 
 3493 
 
 3589 
 
 3685 
 
 3781 
 
 3877 
 
 3973 
 
 96 
 
 91 
 
 92 
 
 93 
 
 397 
 
 4070 
 
 4166 
 
 4262 
 
 4358 
 
 4454 
 
 4550 
 
 4646 
 
 4743 
 
 4839 
 
 4935 
 
 97 
 
 92 
 
 93 
 
 94 
 
 398 
 
 5031 
 
 5127 
 
 5223 
 
 5319 
 
 5416 
 
 5512 
 
 5608 
 
 5704 
 
 5800 
 
 5896 
 
 98 
 
 93 
 
 94 
 
 95 
 
 399 
 
 5993 
 
 6089 
 
 6185 
 
 6281 
 
 6377 
 
 6473 
 
 6570 
 
 6666 
 
 6762 
 
 6858 
 
 99 
 
 94 
 
 95 
 
 96 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 95 
 
 96 
 
 97 
 
 29 
 
TABLE I. 
 
 
 T /^. t\ i \ 
 
 Pro. 
 
 Log 
 
 
 
 .LJ"5 ^ 1 - 
 
 r*j 
 
 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 )6 
 
 97 
 
 1-400 
 
 1-41 6954 
 
 7050 
 
 7147 
 
 7243 
 
 7339 
 
 7435 
 
 7531 
 
 7628 
 
 7724 
 
 7820 
 
 00 
 
 
 
 
 
 401 
 
 7916 
 
 8012 
 
 8108 
 
 8205 
 
 8301 
 
 8397 
 
 8493 
 
 8589 
 
 8686 
 
 8782 
 
 01 
 
 1 
 
 1 
 
 402 
 
 8878 
 
 8974 
 
 9070 
 
 9166 
 
 9263 
 
 9359 
 
 9455 
 
 9551 
 
 9647 
 
 9744 
 
 02 
 
 2 
 
 2 
 
 403 
 
 9840 
 
 9936 
 
 0032 
 
 o!28 
 
 0225 
 
 0321 
 
 o417 
 
 0513 
 
 0609 
 
 0706 
 
 03 
 
 3 
 
 3 
 
 404 
 
 42 0802 
 
 0898 
 
 0994 
 
 1090 
 
 1187 
 
 1283 
 
 1379 
 
 1475 
 
 1571 
 
 1668 
 
 04 
 
 4 
 
 4 
 
 405 
 
 1764 
 
 1860 
 
 1956 
 
 2053 
 
 2149 
 
 2245 
 
 2341 
 
 2437 
 
 2534 
 
 2630 
 
 05 
 
 5 
 
 5 
 
 406 
 
 2726 
 
 2822 
 
 2919 
 
 3015 
 
 3111 
 
 3207 
 
 3303 
 
 3400 
 
 3496 
 
 3592 
 
 06 
 
 6 
 
 6 
 
 407 
 
 3688 
 
 3785 
 
 3881 
 
 3977 
 
 4073 
 
 4170 
 
 4266 
 
 4362 
 
 4458 
 
 4554 
 
 07 
 
 7 
 
 7 
 
 408 
 
 4651 
 
 4747 
 
 4843 
 
 4939 
 
 5036 
 
 5132 
 
 5228 
 
 5324 
 
 5421 
 
 5517 
 
 08 
 
 8 
 
 8 
 
 409 
 
 5613 
 
 5709 
 
 5806 
 
 5902 
 
 5998 
 
 6094 
 
 6191 
 
 6287 
 
 6383 
 
 6479 
 
 09 
 
 9 
 
 9 
 
 410 
 
 42 6576 
 
 6672 
 
 6768 
 
 6864 
 
 6961 
 
 7057 
 
 7153 
 
 7249 
 
 7346 
 
 7442 
 
 10 
 
 10 
 
 10 
 
 411 
 
 7538 
 
 7634 
 
 7731 
 
 7827 
 
 7923 
 
 8020 
 
 8116 
 
 8212 
 
 8308 
 
 8405 
 
 11 
 
 11 
 
 11 
 
 412 
 
 8501 
 
 8597 
 
 8693 
 
 8790 
 
 8886 
 
 8982 
 
 9079 
 
 9175 
 
 9271 
 
 9367 
 
 12 
 
 12 
 
 12 
 
 413 
 
 9464 
 
 9560 
 
 9656 
 
 9753 
 
 9849 
 
 9945 
 
 0041 
 
 0138 
 
 0234 
 
 0330 
 
 13 
 
 12 
 
 13 
 
 414 
 
 43 0427 
 
 0523 
 
 0619 
 
 0715 
 
 0812 
 
 0908 
 
 1004 
 
 1101 
 
 1197 
 
 1293 
 
 14 
 
 13 
 
 14 
 
 415 
 
 1389 
 
 1486 
 
 1582 
 
 1678 
 
 1775 
 
 1871 
 
 1967 
 
 2064 
 
 2160 
 
 2256 
 
 15 
 
 14 
 
 15 
 
 416 
 
 2352 
 
 2449 
 
 2545 
 
 2641 
 
 2738 
 
 2834 
 
 2930 
 
 3027 
 
 3123 
 
 3219 
 
 16 
 
 15 
 
 16 
 
 417 
 
 3316 
 
 3412 
 
 3508 
 
 3604 
 
 3701 
 
 3797 
 
 3893 
 
 3990 
 
 4086 
 
 4182 
 
 17 
 
 16 
 
 16 
 
 418 
 
 4279 
 
 4375 
 
 4471 
 
 4568 
 
 4664 
 
 4760 
 
 4857 
 
 4953 
 
 5049 
 
 5146 
 
 18 
 
 17 
 
 17 
 
 419 
 
 5242 
 
 5338 
 
 5435 
 
 5531 
 
 5627 
 
 5724 
 
 5820 
 
 5916 
 
 6013 
 
 6109 
 
 19 
 
 18 
 
 18 
 
 420 
 
 43 6205 
 
 6302 
 
 6398 
 
 6494 
 
 6591 
 
 6687 
 
 6783 
 
 6880 
 
 6976 
 
 7072 
 
 20 
 
 19 
 
 19 
 
 421 
 
 7169 
 
 7265 
 
 7361 
 
 7458 
 
 7554 
 
 7650 
 
 7747 
 
 7843 
 
 7939 
 
 8036 
 
 21 
 
 20 
 
 20 
 
 422 
 
 8132 
 
 8229 
 
 8325 
 
 8421 
 
 8518 
 
 8614 
 
 8710 
 
 8807 
 
 8903 
 
 8999 
 
 22 
 
 21 
 
 21 
 
 423 
 
 9096 
 
 9192 
 
 9288 
 
 9385 
 
 9481 
 
 9578 
 
 9674 
 
 9770 
 
 9867 
 
 9963 
 
 23 
 
 22 
 
 22 
 
 424 
 
 44 0059 
 
 0156 
 
 0252 
 
 0349 
 
 0445 
 
 0541 
 
 0638 
 
 0734 
 
 0830 
 
 0927 
 
 -24 
 
 23 
 
 23 
 
 425 
 
 1023 
 
 1120 
 
 1216 
 
 1312 
 
 1409 
 
 1505 
 
 1601 
 
 1698 
 
 1794 
 
 1891 
 
 25 
 
 -24 
 
 24 
 
 426 
 
 1987 
 
 2083 
 
 2180 
 
 2276 
 
 2373 
 
 2469 
 
 2565 
 
 2662 
 
 2758 
 
 2854 
 
 -26 
 
 25 
 
 25 
 
 427 
 
 2951 
 
 3047 
 
 3144 
 
 3240 
 
 3336 
 
 3433 
 
 3529 
 
 3626 
 
 3722 
 
 3818 
 
 -27 
 
 26 
 
 26 
 
 428 
 
 3915 
 
 4011 
 
 4108 
 
 4204 
 
 4300 
 
 4397 
 
 4493 
 
 4590 
 
 4686 
 
 4782 
 
 28 
 
 27 
 
 27 
 
 429 
 
 4879 
 
 4975 
 
 5072 
 
 5168 
 
 5265 
 
 5361 
 
 5457 
 
 5554 
 
 5650 
 
 5747 
 
 29 
 
 28 
 
 28 
 
 430 
 
 44 5843 
 
 5939 
 
 6036 
 
 6132 
 
 6229 
 
 6325 
 
 6422 
 
 6518 
 
 6614 
 
 6711 
 
 30 
 
 29 
 
 29 
 
 431 
 
 6807 
 
 6904 
 
 7000 
 
 7097 
 
 7193 
 
 7289 
 
 7386 
 
 7482 
 
 7579 
 
 7675 
 
 31 
 
 30 
 
 30 
 
 432 
 
 7772 
 
 7868 
 
 7964 
 
 8061 
 
 8157 
 
 8254 
 
 8350 
 
 8447 
 
 8543 
 
 8639 
 
 32 
 
 31 
 
 31 
 
 433 
 
 8736 
 
 8832 
 
 8929 
 
 9025 
 
 9122 
 
 9218 
 
 9315 
 
 9411 
 
 9507 
 
 9604 
 
 33 
 
 32 
 
 32 
 
 434 
 
 9700 
 
 9797 
 
 9893 
 
 9990 
 
 0086 
 
 0183 
 
 0279 
 
 0376 
 
 0472 
 
 0568 
 
 3-t 
 
 33 
 
 33 
 
 435 
 
 45 0665 
 
 0761 
 
 0858 
 
 0954 
 
 1051 
 
 1147 
 
 1244 
 
 1340 
 
 1437 
 
 1533 
 
 35 
 
 34 
 
 34 
 
 436 
 
 1630 
 
 1726 
 
 1822 
 
 1919 
 
 2015 
 
 2112 
 
 2208 
 
 2305 
 
 2401 
 
 2498 
 
 36 
 
 35 
 
 35 
 
 437 
 
 2594 
 
 2691 
 
 2787 
 
 2884 
 
 2980 
 
 3077 
 
 3173 
 
 3270 
 
 3366 
 
 3463 
 
 37 
 
 36 
 
 36 
 
 438 
 
 3559 
 
 3655 
 
 3752 
 
 3848 
 
 3945 
 
 4041 
 
 4138 
 
 4234 
 
 4-331 
 
 4427 
 
 38 
 
 36 
 
 37 
 
 439 
 
 4524 
 
 4620 
 
 4717 
 
 4813 
 
 4910 
 
 5006 
 
 5103 
 
 5199 
 
 5296 
 
 5392 
 
 39 
 
 37 
 
 38 
 
 440 
 
 45 5489 
 
 5585 
 
 5682 
 
 5778 
 
 5875 
 
 5971 
 
 6068 
 
 6164 
 
 6261 
 
 6357 
 
 4-0 
 
 38 
 
 39 
 
 441 
 
 6454 
 
 6550 
 
 6647 
 
 6743 
 
 6840 
 
 6936 
 
 7033 
 
 7129 
 
 7226 
 
 7322 
 
 41 
 
 39 
 
 40 
 
 442 
 
 7419 
 
 7515 
 
 7612 
 
 7708 
 
 7805 
 
 7901 
 
 7998 
 
 8094 
 
 8191 
 
 8287 
 
 42 
 
 40 
 
 41 
 
 443 
 
 8384 
 
 8481 
 
 8577 
 
 8674 
 
 8770 
 
 8867 
 
 8963 
 
 9060 
 
 9156 
 
 9253 
 
 43 
 
 4-1 
 
 42 
 
 444 
 
 9349 
 
 9446 
 
 9542 
 
 9639 
 
 9735 
 
 9832 
 
 9928 
 
 0025 
 
 0121 
 
 0218 
 
 44 
 
 4-2 
 
 4-3 
 
 445 
 
 460315 
 
 0411 
 
 0508 
 
 0604 
 
 0701 
 
 0797 
 
 0894 
 
 0990 
 
 1087 
 
 1183 
 
 45 
 
 43 
 
 44 
 
 446 
 
 1280 
 
 1376 
 
 1473 
 
 1570 
 
 1666 
 
 1763 
 
 1859 
 
 1956 
 
 2052 
 
 2149 
 
 46 
 
 44 
 
 45 
 
 447 
 
 2245 
 
 2342 
 
 2439 
 
 2535 
 
 2632 
 
 2728 
 
 2825 
 
 2921 
 
 3018 
 
 3114 
 
 47 
 
 45 
 
 46 
 
 448 
 
 3211 
 
 3308 
 
 3404 
 
 3501 
 
 3597 
 
 3694 
 
 3790 
 
 3887 
 
 3983 
 
 4080 
 
 48 
 
 46 
 
 47 
 
 449 
 
 4177 
 
 4273 
 
 4370 
 
 4466 
 
 4563 
 
 4659 
 
 4756 
 
 4853 
 
 4949 
 
 5046 
 
 49147 
 
 48 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 96 
 
 97 
 
 30 
 
TABLE I. 
 
 Log 
 
 Log(l+;r) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 6 
 
 7 
 
 1-450 
 
 465142 
 
 5239 
 
 5335 
 
 5432 
 
 5529 
 
 5625 
 
 5722 
 
 5818 
 
 5915 
 
 6011 
 
 50 
 
 8 
 
 9 
 
 451 
 
 6108 
 
 6205 
 
 6301 
 
 6398 
 
 6494 
 
 6591 
 
 6688 
 
 6784 
 
 6881 
 
 6977 
 
 51 
 
 9 
 
 9 
 
 452 
 
 7074 
 
 7170 
 
 7267 
 
 7364 
 
 7460 
 
 7557 
 
 7653 
 
 7750 
 
 7847 
 
 7943 
 
 52 
 
 
 
 
 
 453 
 
 8040 
 
 8136 
 
 8233 
 
 8330 
 
 8426 
 
 8523 
 
 8619 
 
 8716 
 
 8813 
 
 8909 
 
 53 
 
 1 
 
 1 
 
 454 
 
 9006 
 
 9102 
 
 9199 
 
 9296 
 
 9392 
 
 9489 
 
 9585 
 
 9682 
 
 9779 
 
 9875 
 
 54 
 
 2 
 
 2 
 
 455 
 
 9972 
 
 0069 
 
 0165 
 
 0262 
 
 0358 
 
 0455 
 
 0552 
 
 0648 
 
 0745 
 
 0841 
 
 55 
 
 3 
 
 3 
 
 456 
 
 47 0938 
 
 1035 
 
 1131 
 
 1228 
 
 1325 
 
 1421 
 
 1518 
 
 1614 
 
 1711 
 
 1808 
 
 56 
 
 4 
 
 4 
 
 457 
 
 1904 
 
 2001 
 
 2098 
 
 2194 
 
 2291 
 
 2387 
 
 2484 
 
 2581 
 
 2677 
 
 2774 
 
 57 
 
 5 
 
 5 
 
 458 
 
 2871 
 
 2967 
 
 3064 
 
 3160 
 
 3257 
 
 3354 
 
 3450 
 
 3547 
 
 3644 
 
 3740 
 
 58 
 
 6 
 
 6 
 
 459 
 
 3837 
 
 3934 
 
 4030 
 
 4127 
 
 4224 
 
 4320 
 
 4417 
 
 4513 
 
 4610 
 
 4707 
 
 59 
 
 7 
 
 7 
 
 460 
 
 47 4803 
 
 4900 
 
 4997 
 
 5093 
 
 5190 
 
 5287 
 
 5383 
 
 5480 
 
 5577 
 
 5673 
 
 60 
 
 8 
 
 58 
 
 461 
 
 5770 
 
 5867 
 
 5963 
 
 6060 
 
 6157 
 
 6253 
 
 6350 
 
 6447 
 
 6543 
 
 6640 
 
 61 
 
 9 
 
 59 
 
 462 
 
 6737 
 
 6833 
 
 6930 
 
 7027 
 
 7123 
 
 7220 
 
 7317 
 
 7413 
 
 7510 
 
 7607 
 
 62 
 
 
 
 60 
 
 463 
 
 7703 
 
 7800 
 
 7897 
 
 7993 
 
 8090 
 
 8187 
 
 8283 
 
 8380 
 
 8477 
 
 8573 
 
 63 
 
 
 
 61 
 
 464 
 
 8670 
 
 8767 
 
 8863 
 
 8960 
 
 9057 
 
 9153 
 
 9250 
 
 9347 
 
 9443 
 
 9540 
 
 64 
 
 61 
 
 62 
 
 465 
 
 9637 
 
 9733 
 
 9830 
 
 9927 
 
 0024 
 
 0120 
 
 0217 
 
 0314 
 
 0410 
 
 0507 
 
 65 
 
 62 
 
 63 
 
 466 
 
 48 0604 
 
 0700 
 
 0797 
 
 0894 
 
 0990 
 
 1087 
 
 1184 
 
 1281 
 
 1377 
 
 1474 
 
 66 
 
 'C 
 
 64 
 
 467 
 
 1571 
 
 1667 
 
 1764 
 
 1861 
 
 1957 
 
 2054 
 
 2151 
 
 2248 
 
 2344 
 
 2441 
 
 57 
 
 64 
 
 65 
 
 468 
 
 2538 
 
 2634 
 
 2731 
 
 ^828 
 
 2925 
 
 3021 
 
 3118 
 
 3215 
 
 3311 
 
 3408 
 
 68 
 
 65 
 
 66 
 
 469 
 
 3505 
 
 3602 
 
 3698 
 
 3795 
 
 3892 
 
 3988 
 
 4085 
 
 4182 
 
 4279 
 
 4375 
 
 69 
 
 66 
 
 67 
 
 470 
 
 48 4472 
 
 4569 
 
 4665 
 
 4762 
 
 4859 
 
 4956 
 
 5052 
 
 5149 
 
 5246 
 
 5343 
 
 70 
 
 17 
 
 68 
 
 471 
 
 5439 
 
 5536 
 
 5633 
 
 5729 
 
 5826 
 
 5923 
 
 6020 
 
 6116 
 
 6213 
 
 6310 
 
 71 
 
 68 
 
 69 
 
 472 
 
 6407 
 
 6503 
 
 6600 
 
 6697 
 
 6794 
 
 6890 
 
 6987 
 
 7084 
 
 7181 
 
 7277 
 
 72 
 
 69 
 
 70 
 
 473 
 
 7374 
 
 7471 
 
 7567 
 
 7664 
 
 7761 
 
 7858 
 
 7954 
 
 8051 
 
 8148 
 
 8245 
 
 73 
 
 70 
 
 71 
 
 474 
 
 8341 
 
 8438 
 
 8535 
 
 8632 
 
 8728 
 
 8825 
 
 8922 
 
 9019 
 
 9116 
 
 9212 
 
 74 
 
 71 
 
 72 
 
 475 
 
 9309 
 
 9406 
 
 9503 
 
 9599 
 
 9696 
 
 9793 
 
 9890 
 
 9986 
 
 0083 
 
 0180 
 
 75 
 
 72 
 
 73 
 
 476 
 
 49 0277 
 
 0373 
 
 0470 
 
 0567 
 
 0664 
 
 0760 
 
 0857 
 
 0954 
 
 1051 
 
 1148 
 
 76 
 
 ryf 
 
 74 
 
 477 
 
 1244 
 
 1341 
 
 1438 
 
 1535 
 
 1631 
 
 1728 
 
 1825 
 
 1922 
 
 2019 
 
 2115 
 
 77 
 
 74 
 
 75 
 
 478 
 
 2212 
 
 2309 
 
 2406 
 
 2502 
 
 2599 
 
 2696 
 
 2793 
 
 2890 
 
 2986 
 
 3083 
 
 78 
 
 75 
 
 76 
 
 479 
 
 3180 
 
 3277 
 
 3374 
 
 3470 
 
 3567 
 
 3664 
 
 3761 
 
 3857 
 
 3954 
 
 4051 
 
 79 
 
 76 
 
 77 
 
 480 
 
 494148 
 
 4245 
 
 4341 
 
 4438 
 
 4535 
 
 4632 
 
 4729 
 
 4825 
 
 4922 
 
 5019 
 
 80 
 
 i 
 
 78 
 
 481 
 
 5116 
 
 5213 
 
 5309 
 
 5406 
 
 5503 
 
 5600 
 
 5697 
 
 5793 
 
 5890 
 
 5987 
 
 81 
 
 78 
 
 79 
 
 482 
 
 6084 
 
 6181 
 
 6278 
 
 6374 
 
 6471 
 
 6568 
 
 6665 
 
 6762 
 
 6858 
 
 6955 
 
 82 
 
 '9 
 
 80 
 
 483 
 
 7052 
 
 7149 
 
 7246 
 
 7342 
 
 7439 
 
 7536 
 
 7633 
 
 7730 
 
 7827 
 
 7923 
 
 83 
 
 80 
 
 81 
 
 484 
 
 8020 
 
 8117 
 
 8214 
 
 8311 
 
 8408 
 
 8504 
 
 8601 
 
 8698 
 
 8795 
 
 8892 
 
 84 
 
 81 
 
 81 
 
 485 
 
 8989 
 
 9085 
 
 9182 
 
 9279 
 
 9376 
 
 9473 
 
 9569 
 
 9666 
 
 9763 
 
 9860 
 
 85 
 
 82 
 
 82 
 
 486 
 
 9957 
 
 0054 
 
 o!51 
 
 0247 
 
 0344 
 
 0441 
 
 0538 
 
 0635 
 
 0732 
 
 0828 
 
 86 
 
 83 
 
 83 
 
 487 
 
 50 0925 
 
 1022 
 
 1119 
 
 1216 
 
 1313 
 
 1409 
 
 1506 
 
 1603 
 
 1700 
 
 1797 
 
 87 
 
 84 
 
 84 
 
 488 
 
 1894 
 
 1991 
 
 2087 
 
 2184 
 
 2281 
 
 2378 
 
 2475 
 
 2572 
 
 2669 
 
 2765 
 
 88 
 
 84 
 
 8*5 
 
 489 
 
 2862 
 
 2959 
 
 3056 
 
 3153 
 
 3250 
 
 3347 
 
 3443 
 
 3540 
 
 3637 
 
 3734 
 
 89 
 
 85 
 
 86 
 
 490 
 
 503831 
 
 3928 
 
 4025 
 
 4122 
 
 4218 
 
 4315 
 
 4412 
 
 4509 
 
 4606 
 
 4703 
 
 90 
 
 86 
 
 87 
 
 491 
 
 4800 
 
 4896 
 
 4993 
 
 5090 
 
 5187 
 
 5284 
 
 5381 
 
 5478 
 
 5575 
 
 567 
 
 91 
 
 87 
 
 88 
 
 492 
 
 5768 
 
 5865 
 
 5962 
 
 6059 
 
 6156 
 
 6253 
 
 6350 
 
 6447 
 
 6543 
 
 6640 
 
 92 
 
 88 
 
 89 
 
 493 
 
 6737 
 
 6834 
 
 6931 
 
 7028 
 
 7125 
 
 7222 
 
 7319 
 
 7415 
 
 7512 
 
 7609 
 
 93 
 
 89 
 
 90 
 
 494 
 
 7706 
 
 7803 
 
 7900 
 
 7997 
 
 8094 
 
 8191 
 
 8287 
 
 8384 
 
 848 
 
 8578 
 
 94 
 
 90 
 
 91 
 
 495 
 
 8673 
 
 8772 
 
 8869 
 
 8966 
 
 9063 
 
 9160 
 
 9256 
 
 9353 
 
 9450 
 
 954 
 
 95 
 
 9 
 
 92 
 
 496 
 
 9644 
 
 974 
 
 9838 
 
 993o 
 
 0032 
 
 o!29 
 
 0226 
 
 0322 
 
 o41 
 
 051 
 
 96 
 
 92 
 
 93 
 
 497 
 
 51 0612 
 
 0710 
 
 0807 
 
 0904 
 
 1001 
 
 109S 
 
 1195 
 
 1292 
 
 138 
 
 148 
 
 9 
 
 93 
 
 94 
 
 498 
 
 1585 
 
 167 
 
 1776 
 
 1873 
 
 1970 
 
 206 r 
 
 2164 
 
 226 
 
 235 
 
 245 
 
 9 
 
 94 
 
 95 
 
 499 
 
 255$ 
 
 264 
 
 2745 
 
 2842 
 
 2939 
 
 3036 
 
 3132 
 
 3230 
 
 332 
 
 342 
 
 9 
 
 9 
 
 96 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9 
 
 97 
 
TABLE I. 
 
 Log 
 
 Log (1+a) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 1-500 
 
 1-513521 
 
 3618 
 
 3715 
 
 3812 
 
 3909 
 
 4006 
 
 4103 
 
 4199 
 
 4296 
 
 4393 
 
 00 
 
 
 
 
 
 501 
 
 4490 
 
 4587 
 
 4684 
 
 4781 
 
 4878 
 
 4975 
 
 5072 
 
 5169 
 
 5266 
 
 5363 
 
 01 
 
 1 
 
 1 
 
 502 
 
 5460 
 
 5557 
 
 5654 
 
 5751 
 
 5848 
 
 5945 
 
 6041 
 
 6138 
 
 6235 
 
 6332 
 
 02 
 
 2 
 
 2 
 
 503 
 
 6429 
 
 6526 
 
 6623 
 
 6720 
 
 6817 
 
 6914 
 
 7011 
 
 7108 
 
 7205 
 
 7302 
 
 03 
 
 
 2 
 
 504- 
 
 7399 
 
 7496 
 
 7593 
 
 7690 
 
 7787 
 
 7884 
 
 7981 
 
 8078 
 
 8175 
 
 8272 
 
 04 
 
 4 
 
 4 
 
 505 
 
 8369 
 
 8465 
 
 8562 
 
 8659 
 
 8756 
 
 8853 
 
 8950 
 
 9047 
 
 9144 
 
 9241 
 
 05 
 
 I 
 
 K 
 
 506 
 
 9338 
 
 9435 
 
 9532 
 
 9629 
 
 9726 
 
 9823 
 
 9920 
 
 0017 
 
 0114 
 
 0211 
 
 06 
 
 6 
 
 6 
 
 507 
 
 52 0308 
 
 0405 
 
 0502 
 
 0599 
 
 0696 
 
 0793 
 
 0890 
 
 0987 
 
 1084 
 
 1181 
 
 07 
 
 r i 
 
 7 
 
 508 
 
 1278 
 
 1375 
 
 1472 
 
 1569 
 
 1666 
 
 1763 
 
 1860 
 
 1957 
 
 2054 
 
 2151 
 
 08 
 
 S 
 
 8 
 
 509 
 
 2248 
 
 2345 
 
 2442 
 
 2539 
 
 2636 
 
 2733 
 
 2830 
 
 2927 
 
 3024 
 
 3121 
 
 09 
 
 c 
 
 
 
 510 
 
 523218 
 
 3315 
 
 3412 
 
 3509 
 
 3606 
 
 3703 
 
 3800 
 
 3897 
 
 3994 
 
 4091 
 
 10 
 
 10 
 
 10 
 
 511 
 
 4188 
 
 4285 
 
 4382 
 
 4479 
 
 4576 
 
 4673 
 
 4770 
 
 4867 
 
 4964 
 
 5061 
 
 11 
 
 11 
 
 11 
 
 512 
 
 5158 
 
 5255 
 
 5352 
 
 5449 
 
 5546 
 
 5643 
 
 5740 
 
 5837 
 
 5934 
 
 6031 
 
 12 
 
 12 
 
 12 
 
 513 
 
 6128 
 
 6225 
 
 6322 
 
 6419 
 
 6516 
 
 6613 
 
 6710 
 
 6807 
 
 6904 
 
 7001 
 
 13 
 
 13 
 
 I ^ 
 
 514 
 
 7098 
 
 7195 
 
 7292 
 
 7390 
 
 7487 
 
 7584 
 
 7681 
 
 7778 
 
 7875 
 
 7972 
 
 14 
 
 14 
 
 14 
 
 515 
 
 8069 
 
 8166 
 
 8263 
 
 8360 
 
 8457 
 
 8554 
 
 8651 
 
 8748 
 
 8845 
 
 8942 
 
 15 
 
 15 
 
 15 
 
 516 
 
 9039 
 
 9136 
 
 9233 
 
 9330 
 
 9427 
 
 9524 
 
 9621 
 
 9718 
 
 9815 
 
 9913 
 
 16 
 
 16 
 
 16 
 
 517 
 
 530010 
 
 0107 
 
 0204 
 
 0301 
 
 0398 
 
 0495 
 
 0592 
 
 0689 
 
 0786 
 
 0883 
 
 17 
 
 16 
 
 17 
 
 518 
 
 0980 
 
 1077 
 
 1174 
 
 1271 
 
 1368 
 
 1465 
 
 1562 
 
 1660 
 
 1757 
 
 1854 
 
 18 
 
 17 
 
 18 
 
 519 
 
 1951 
 
 2048 
 
 2145 
 
 2242 
 
 2339 
 
 2436 
 
 2533 
 
 2630 
 
 2727 
 
 2824 
 
 19 
 
 18 
 
 19 
 
 520 
 
 53 2921 
 
 3018 
 
 3115 
 
 3213 
 
 3310 
 
 3407 
 
 3504 
 
 3601 
 
 3698 
 
 3795 
 
 20 
 
 19 
 
 20 
 
 521 
 
 3892 
 
 3989 
 
 4086 
 
 4183 
 
 4280 
 
 4377 
 
 4475 
 
 4572 
 
 4669 
 
 4766 
 
 21 
 
 20 
 
 21 
 
 522 
 
 4863 
 
 4960 
 
 5057 
 
 5154 
 
 5251 
 
 5348 
 
 KJ,1C 
 (LlTTTtJ 
 
 5542 
 
 5640 
 
 5737 
 
 22 
 
 21 
 
 22 
 
 523 
 
 5834 
 
 5931 
 
 6028 
 
 6125 
 
 6222 
 
 6319 
 
 6416 
 
 6513 
 
 6610 
 
 6708 
 
 23 
 
 22 
 
 23 
 
 524 
 
 6805 
 
 6902 
 
 6999 
 
 7096 
 
 7193 
 
 7290 
 
 7387 
 
 7484 
 
 7581 
 
 7678 
 
 24 
 
 23 
 
 24 
 
 525 
 
 7776 
 
 7873 
 
 7970 
 
 8067 
 
 8164 
 
 8261 
 
 8358 
 
 8455 
 
 8552 
 
 8650 
 
 25 
 
 24 
 
 25 
 
 526 
 
 8747 
 
 8844 
 
 8941 
 
 9038 
 
 9135 
 
 9232 
 
 9329 
 
 9426 
 
 9524 
 
 9621 
 
 26 
 
 25 
 
 25 
 
 527 
 
 9718 
 
 9815 
 
 9912 
 
 0009 
 
 0106 
 
 0203 
 
 0300 
 
 0398 
 
 0495 
 
 0592 
 
 27 
 
 26 
 
 26 
 
 528 
 
 54 0689 
 
 0786 
 
 0883 
 
 0980 
 
 1077 
 
 1175 
 
 1272 
 
 1369 
 
 1466 
 
 1563 
 
 28 
 
 27 
 
 27 
 
 529 
 
 1660 
 
 1757 
 
 1854 
 
 1952 
 
 2049 
 
 2146 
 
 2243 
 
 2340 
 
 2437 
 
 2534 
 
 29 
 
 28 
 
 28 
 
 530 
 
 54 2631 
 
 2729 
 
 2826 
 
 2923 
 
 3020 
 
 3117 
 
 3214 
 
 3311 
 
 3409 
 
 3506 
 
 30 
 
 29 
 
 29 
 
 531 
 
 3603 
 
 3700 
 
 3797 
 
 3894 
 
 3991 
 
 4089 
 
 4186 
 
 4283 
 
 4380 
 
 4477 
 
 31 
 
 30 
 
 30 
 
 532 
 
 4574 
 
 4671 
 
 4769 
 
 4866 
 
 4963 
 
 5060 
 
 5157 
 
 5254 
 
 5351 
 
 5449 
 
 32 
 
 31 
 
 31 
 
 533 
 
 5546 
 
 5643 
 
 5740 
 
 5837 
 
 5934 
 
 6032 
 
 6129 
 
 6226 
 
 6323 
 
 6420 
 
 33 
 
 32 
 
 32 
 
 534- 
 
 6517 
 
 6614 
 
 6712 
 
 6809 
 
 6906 
 
 7003 
 
 7100 
 
 7197 
 
 7295 
 
 7392 
 
 34 
 
 33 
 
 33 
 
 535 
 
 7489 
 
 7586 
 
 7683 
 
 7780 
 
 7878 
 
 7975 
 
 8072 
 
 8169 
 
 8266 
 
 8363 
 
 35 
 
 34 
 
 34 
 
 536 
 
 8461 
 
 8558 
 
 8655 
 
 8752 
 
 8849 
 
 8946 
 
 9044 
 
 9141 
 
 9238 
 
 9335 
 
 36 
 
 35 
 
 35 
 
 537 
 
 9432 
 
 9530 
 
 9627 
 
 9724 
 
 9821 
 
 9918 
 
 0015 
 
 0113 
 
 0210 
 
 0307 
 
 37 
 
 36 
 
 36 
 
 538 
 
 55 0404 
 
 0501 
 
 0599 
 
 0696 
 
 0793 
 
 0890 
 
 0987 
 
 1084 
 
 1182 
 
 1279 
 
 38 
 
 37 
 
 37 
 
 539 
 
 1376 
 
 1473 
 
 1570 
 
 1668 
 
 1765 
 
 1862 
 
 1959 
 
 2056 
 
 2154 
 
 2251 
 
 39 
 
 38 
 
 38 
 
 54-0 
 
 55 2348 
 
 2445 
 
 2542 
 
 2640 
 
 2737 
 
 2834 
 
 2931 
 
 3028 
 
 3126 
 
 3223 
 
 40 
 
 39 
 
 39 
 
 541 
 
 3320 
 
 3417 
 
 3514 
 
 3612 
 
 3709 
 
 3806 
 
 3903 
 
 4000 
 
 4098 
 
 4195 
 
 41 
 
 40 
 
 40 
 
 542 
 
 4292 
 
 4389 
 
 4486 
 
 4584 
 
 4681 
 
 4778 
 
 4875 
 
 4973 
 
 5070 
 
 5167 
 
 42 
 
 41 
 
 41 
 
 543 
 
 5264 
 
 5361 
 
 5459 
 
 5556 
 
 5653 
 
 5750 
 
 5847 
 
 5945 
 
 6042 
 
 6139 
 
 43 
 
 42 
 
 42 
 
 544 
 
 6236 
 
 6334 
 
 6431 
 
 6528 
 
 6625 
 
 6722 
 
 6820 
 
 6917 
 
 7014 
 
 7111 
 
 44 
 
 43 
 
 43 
 
 545 
 
 7209 
 
 7306 
 
 7403 
 
 7500 
 
 7598 
 
 7695 
 
 7792 
 
 7889 
 
 7986 
 
 8084 
 
 45 
 
 44 
 
 44 
 
 546 
 
 8181 
 
 8278 
 
 8375 
 
 8473 
 
 8570 
 
 8667 
 
 8764 
 
 8862 
 
 8959 
 
 9056 
 
 46 
 
 45 
 
 45 
 
 547 
 
 9153 
 
 9251 
 
 9348 
 
 9445 
 
 9542 
 
 9639 
 
 9737 
 
 9834 
 
 9931 
 
 0028 
 
 47 
 
 46 
 
 46 
 
 548 
 
 560126 
 
 0223 
 
 0320 
 
 0417 
 
 0515 
 
 0612 
 
 0709 
 
 0806 
 
 0904 
 
 1001 
 
 48 
 
 47 
 
 47 
 
 549 
 
 1098 
 
 1195 
 
 1293 
 
 1390 
 
 1487 
 
 1584 
 
 1682 
 
 1779 
 
 1876 
 
 1974. 
 
 49 
 
 48 
 
 48 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 32 
 
TABLE I. 
 
 Log 
 
 
 Log (1+9) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 1-550 
 
 1-562071 
 
 2168 
 
 2265 
 
 2363 
 
 2460 
 
 2557 
 
 2654 
 
 2752 
 
 2849 
 
 2946 
 
 50 
 
 49 
 
 49 
 
 551 
 
 3043 
 
 3141 
 
 3238 
 
 3335 
 
 3432 
 
 3530 
 
 3627 
 
 3724 
 
 3822 
 
 3919 
 
 51 
 
 49 
 
 50 
 
 552 
 
 4016 
 
 4113 
 
 4211 
 
 4308 
 
 4405 
 
 4502 
 
 4600 
 
 4697 
 
 4794 
 
 4892 
 
 52 
 
 50 
 
 51 
 
 553 
 
 4989 
 
 5086 
 
 5183 
 
 5281 
 
 5378 
 
 5475 
 
 5572 
 
 567C 
 
 5767 
 
 5864 
 
 53 
 
 51 
 
 52 
 
 554 
 
 5962 
 
 6059 
 
 6156 
 
 6253 
 
 6351 
 
 6448 
 
 6545 
 
 6643 
 
 6740 
 
 6837 
 
 54 
 
 52 
 
 53 
 
 555 
 
 6934 
 
 7032 
 
 7129 
 
 7226 
 
 7324 
 
 7421 
 
 7518 
 
 7616 
 
 7713 
 
 7810 
 
 55 
 
 53 
 
 5* 
 
 556 
 
 7907 
 
 8005 
 
 8102 
 
 8199 
 
 8297 
 
 8394 
 
 849 
 
 858S 
 
 8686 
 
 8783 
 
 56 
 
 54 
 
 55 
 
 557 
 
 8880 
 
 8978 
 
 9075 
 
 9172 
 
 9270 
 
 9367 
 
 9464 
 
 9562 
 
 9659 
 
 9756 
 
 57 
 
 5o 
 
 56 
 
 558 
 
 9853 
 
 9951 
 
 0048 
 
 0145 
 
 0243 
 
 0340 
 
 0437 
 
 0535 
 
 0632 
 
 0729 
 
 58 
 
 56 
 
 57 
 
 559 
 
 57 0827 
 
 0924 
 
 1021 
 
 1118 
 
 1216 
 
 1313 
 
 1410 
 
 150S 
 
 1605 
 
 1702 
 
 59 
 
 57 
 
 58 
 
 560 
 
 57 1800 
 
 1897 
 
 1994 
 
 2092 
 
 2189 
 
 2286 
 
 2384 
 
 248 
 
 2578 
 
 2676 
 
 ao 
 
 58 
 
 59 
 
 561 
 
 2773 
 
 2870 
 
 2968 
 
 3065 
 
 3162 
 
 3260 
 
 3357 
 
 3454 
 
 3552 
 
 3649 
 
 61 
 
 59 
 
 60 
 
 562 
 
 3746 
 
 3844 
 
 3941 
 
 4038 
 
 4136 
 
 4233 
 
 4330 
 
 4428 
 
 4525 
 
 4622 
 
 6260 
 
 61 
 
 563 
 
 4720 
 
 4817 
 
 4914 
 
 5012 
 
 5109 
 
 5206 
 
 5304 
 
 540 
 
 5498 
 
 5596 
 
 63 
 
 6 
 
 62 
 
 564 
 
 5693 
 
 5790 
 
 5888 
 
 5985 
 
 6082 
 
 6180 
 
 6277 
 
 6374 
 
 6472 
 
 6569 
 
 64 
 
 62 
 
 63 
 
 565 
 
 6666 
 
 6764 
 
 6861 
 
 6958 
 
 7056 
 
 7153 
 
 7251 
 
 7348 
 
 7445 
 
 7543 
 
 65 
 
 63 
 
 64 
 
 566 
 
 7640 
 
 7737 
 
 7835 
 
 7932 
 
 8029 
 
 8127 
 
 8224 
 
 832 
 
 8418 
 
 8516 
 
 6664 
 
 65 
 
 567 
 
 8614 
 
 8711 
 
 8808 
 
 8906 
 
 9003 
 
 9100 
 
 9198 
 
 9295 
 
 9392 
 
 9490 
 
 67 
 
 65 
 
 66 
 
 568 
 
 9587 
 
 9685 
 
 9782 
 
 9879 
 
 9977 
 
 0074 
 
 0171 
 
 026S 
 
 0366 
 
 0464 
 
 68 
 
 66 
 
 67 
 
 569 
 
 580561 
 
 0658 
 
 0756 
 
 0853 
 
 0950 
 
 1048 
 
 1145 
 
 1243 
 
 1340 
 
 1437 
 
 69 
 
 67 
 
 68 
 
 570 
 
 58 1535 
 
 1632 
 
 1729 
 
 1827 
 
 192"4 
 
 2022 
 
 2119 
 
 2216 
 
 2314 
 
 2411 
 
 70 
 
 68 
 
 69 
 
 571 
 
 2508 
 
 2606 
 
 2703 
 
 2801 
 
 2898 
 
 2995 
 
 3093 
 
 3190 
 
 3288 
 
 3385 
 
 71 
 
 69 
 
 70 
 
 572 
 
 3482 
 
 3580 
 
 3677 
 
 3775 
 
 3872 
 
 3969 
 
 4067 
 
 4164 
 
 4261 
 
 4359 
 
 72 
 
 70 
 
 71 
 
 573 
 
 4456 
 
 4554 
 
 4651 
 
 4748 
 
 4846 
 
 4943 
 
 5041 
 
 5138 
 
 5235 
 
 5333 
 
 73 
 
 71 
 
 72 
 
 574 
 
 5430 
 
 5528 
 
 5625 
 
 5722 
 
 5820 
 
 5917 
 
 6015 
 
 6112 
 
 6210 
 
 6307 
 
 74 
 
 72 
 
 73 
 
 575 
 
 6404 
 
 6502 
 
 6599 
 
 6697 
 
 6794 
 
 6891 
 
 6989 
 
 7086 
 
 7184 
 
 7281 
 
 75 
 
 73 
 
 73 
 
 576 
 
 7378 
 
 7476 
 
 7573 
 
 7671 
 
 7768 
 
 7866 
 
 7963 
 
 8060 
 
 8158 
 
 8255 
 
 76 
 
 74 
 
 74 
 
 577 
 
 8353 
 
 8450 
 
 8547 
 
 8645 
 
 8742 
 
 8840 
 
 8937 
 
 9035 
 
 9132 
 
 9229 
 
 77 
 
 75 
 
 75 
 
 578 
 
 9327 
 
 9424 
 
 9522 
 
 9619 
 
 9717 
 
 9814 
 
 9911 
 
 0009 
 
 0106 
 
 0204 
 
 78 
 
 76 
 
 76 
 
 579 
 
 590301 
 
 0399 
 
 0496 
 
 0593 
 
 0691 
 
 0788 
 
 0886 
 
 0983 
 
 1081 
 
 1178 
 
 79 
 
 77 
 
 77 
 
 580 
 
 59 1275 
 
 1373 
 
 1470 
 
 1568 
 
 1665 
 
 1763 
 
 1860 
 
 1958 
 
 2055 
 
 2152 
 
 80 
 
 78 
 
 78 
 
 58 i 
 
 2250 
 
 2347 
 
 2445 
 
 2542 
 
 2640 
 
 2737 
 
 2835 
 
 2932 
 
 3029 
 
 3127 
 
 81 
 
 79 
 
 79 
 
 582 
 
 3224 
 
 3322 
 
 3419 
 
 3517 
 
 3614 
 
 3712 
 
 3809 
 
 3906 
 
 4004 
 
 4101 
 
 82 
 
 80 
 
 80 
 
 583 
 
 4199 
 
 4296 
 
 4394 
 
 4491 
 
 4589 
 
 4686 
 
 4784 
 
 4881 
 
 4978 
 
 5076 
 
 83 
 
 81 
 
 81 
 
 584 
 
 5173 
 
 5271 
 
 5368 
 
 5466 
 
 5563 
 
 5661 
 
 5758 
 
 5856 
 
 5953 
 
 6051 
 
 84 
 
 81 
 
 82 
 
 585 
 
 6148 
 
 6246 
 
 6343 
 
 6440 
 
 6538 
 
 6635 
 
 6733 
 
 C830 
 
 6928 
 
 7025 
 
 85 
 
 82 
 
 83 
 
 586 
 
 7123 
 
 7220 
 
 7318 
 
 7415 
 
 7513 
 
 7610 
 
 7708 
 
 7805 
 
 7903 
 
 8000 
 
 86 
 
 83 
 
 84 
 
 587 
 
 8097 
 
 8195 
 
 8292 
 
 8390 
 
 8487 
 
 8585 
 
 8682 
 
 8780 
 
 8877 
 
 8975 
 
 87 
 
 84 
 
 85 
 
 588 
 
 9072 
 
 9170 
 
 9267 
 
 9365 
 
 9462 
 
 9560 
 
 9657 
 
 9755 
 
 9852 
 
 9950 
 
 88 
 
 85 
 
 86 
 
 589 
 
 60 0047 
 
 0145 
 
 0242 
 
 0340 
 
 0437 
 
 0535 
 
 0632 
 
 0730 
 
 0827 
 
 0925 
 
 89 
 
 86 
 
 87 
 
 590 
 
 60 1022 
 
 1120 
 
 1217 
 
 1315 
 
 1412 
 
 1510 
 
 1607 
 
 1705 
 
 1802 
 
 1900 
 
 90 
 
 87 
 
 88 
 
 591 
 
 1997 
 
 2094 
 
 2192 
 
 2290 
 
 2387 
 
 2485 
 
 2582 
 
 2680 
 
 2777 
 
 2875 
 
 91 
 
 88 
 
 89 
 
 592 
 
 2972 
 
 3070 
 
 3167 
 
 3265 
 
 3362 
 
 3460 
 
 3557 
 
 3655 
 
 3752 
 
 3850 
 
 92 
 
 B9 
 
 90 
 
 593 
 
 3947 
 
 4045 
 
 4142 
 
 4240 
 
 4337 
 
 4435 
 
 4532 
 
 4630 
 
 4727 
 
 4825 
 
 93 
 
 90 
 
 91 
 
 594 
 
 4922 
 
 5020 
 
 5117 
 
 5215 
 
 5312 
 
 5410 
 
 5507 
 
 5605 
 
 5702 
 
 5800 
 
 94 
 
 91 
 
 92 
 
 595 
 
 5897 
 
 5995 
 
 6092 
 
 6190 
 
 6288 
 
 6385 
 
 6483 
 
 6580 
 
 6678 
 
 6775 
 
 95 
 
 92 
 
 93 
 
 596 
 
 6873 
 
 6970 
 
 7068 
 
 7165 
 
 7263 
 
 7360 
 
 7458 
 
 7555 
 
 7653 
 
 7750 
 
 96 
 
 93 
 
 94 
 
 597 
 
 7848 
 
 7946 
 
 8043 
 
 8141 
 
 8238 
 
 8336 
 
 8433 
 
 8531 
 
 8628 
 
 8726 
 
 97 
 
 34 
 
 95 
 
 598 
 
 8823 
 
 8921 
 
 9018 
 
 9116 
 
 9214 
 
 9311 
 
 9409 
 
 9506 
 
 9604 
 
 9701 
 
 98 
 
 )5 
 
 96 
 
 599 
 
 9799 
 
 9896 
 
 9994 
 
 0091 
 
 0189 
 
 0286 
 
 0384 
 
 0482 
 
 0579 
 
 0677 
 
 9 
 
 )6 
 
 97 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 i 
 
 ?7 
 
 8 
 
 33 
 
TABLE I. 
 
 Log 
 
 Log (1+ar) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 1-600 
 
 1-61 0774 
 
 0872 
 
 0969 
 
 1067 
 
 1164 
 
 1262 
 
 1360 
 
 1457 
 
 1555 
 
 1652 
 
 00 
 
 
 
 
 
 601 
 
 1750 
 
 1847 
 
 1945 
 
 2042 
 
 2140 
 
 2238 
 
 2335 
 
 2433 
 
 2530 
 
 2628 
 
 01 
 
 1 
 
 1 
 
 602 
 
 2725 
 
 2823 
 
 2920 
 
 3018 
 
 3116 
 
 3213 
 
 3311 
 
 3408 
 
 3506 
 
 3603 
 
 02 
 
 2 
 
 2 
 
 603 
 
 3701 
 
 3799 
 
 3896 
 
 3994 
 
 4091 
 
 4189 
 
 4286 
 
 4384 
 
 4482 
 
 4579 
 
 03 
 
 2 
 
 3 
 
 604 
 
 4677 
 
 4774 
 
 4872 
 
 4969 
 
 5067 
 
 5165 
 
 5262 
 
 5360 
 
 5457 
 
 5555 
 
 04 
 
 4 
 
 4 
 
 605 
 
 5652 
 
 5750 
 
 5848 
 
 5945 
 
 6043 
 
 6140 
 
 6238 
 
 6335 
 
 6433 
 
 6531 
 
 05 
 
 K 
 
 5 
 
 606 
 
 6628 
 
 6726 
 
 6823 
 
 6921 
 
 7019 
 
 7116 
 
 7214 
 
 7311 
 
 7409 
 
 7506 
 
 06 
 
 6 
 
 6 
 
 607 
 
 7604 
 
 7702 
 
 7799 
 
 7897 
 
 7994 
 
 8092 
 
 8190 
 
 8287 
 
 8385 
 
 8482 
 
 n^7 
 
 7 
 
 7 
 
 608 
 
 8580 
 
 8678 
 
 8775 
 
 8873 
 
 8970 
 
 9068 
 
 9166 
 
 9263 
 
 9361 
 
 9458 
 
 08 8 
 
 8 
 
 609 
 
 9556 
 
 9654 
 
 9751 
 
 9849 
 
 9946 
 
 0044 
 
 o!42 
 
 0239 
 
 0337 
 
 0434 
 
 09 9 
 
 9 
 
 610 
 
 62 0532 
 
 0630 
 
 0727 
 
 0825 
 
 0922 
 
 1020 
 
 1118 
 
 1215 
 
 1313 
 
 1410 
 
 10 
 
 10 
 
 10 
 
 611 
 
 1508 
 
 1606 
 
 1703 
 
 1801 
 
 1898 
 
 1996 
 
 2094 
 
 2191 
 
 2289 
 
 2386 
 
 11 
 
 11 
 
 11 
 
 612 
 
 2484 
 
 2582 
 
 2679 
 
 2777 
 
 2875 
 
 2972 
 
 3070 
 
 3167 
 
 3265 
 
 3363 
 
 12 
 
 12 
 
 12 
 
 613 
 
 3460 
 
 3558 
 
 3656 
 
 3753 
 
 3851 
 
 3948 
 
 4046 
 
 4144 
 
 4241 
 
 4339 
 
 13 
 
 13 
 
 13 
 
 614 
 
 4437 
 
 4534 
 
 4632 
 
 4729 
 
 4827 
 
 4925 
 
 5022 
 
 5120 
 
 5218 
 
 5315 
 
 14 
 
 14 
 
 14 
 
 615 
 
 5413 
 
 5510 
 
 5608 
 
 5706 
 
 5803 
 
 5901 
 
 5999 
 
 6096 
 
 6194 
 
 6291 
 
 15 
 
 15 
 
 15 
 
 616 
 
 6389 
 
 6487 
 
 6584 
 
 6682 
 
 6780 
 
 6877 
 
 6975 
 
 7073 
 
 7170 
 
 7268 
 
 16 
 
 16 
 
 16 
 
 617 
 
 7366 
 
 7463 
 
 7561 
 
 7658 
 
 7756 
 
 7854 
 
 7951 
 
 8049 
 
 8147 
 
 8244 
 
 17 
 
 16 
 
 17 
 
 618 
 
 8342 
 
 8440 
 
 8537 
 
 8635 
 
 8733 
 
 8830 
 
 8928 
 
 9026 
 
 9123 
 
 9221 
 
 18 
 
 17 
 
 18 
 
 619 
 
 9318 
 
 9416 
 
 9514 
 
 9611 
 
 9709 
 
 9807 
 
 9904 
 
 0002 
 
 olOO 
 
 0197 
 
 19 
 
 1! 
 
 19 
 
 620 
 
 63 0295 
 
 0393 
 
 0490 
 
 0588 
 
 0686 
 
 0783 
 
 0881 
 
 0979 
 
 1076 
 
 1174 
 
 2019 
 
 20 
 
 621 
 
 1272 
 
 1369 
 
 1467 
 
 1565 
 
 1662 
 
 1760 
 
 1858 
 
 1955 
 
 2053 
 
 2151 
 
 2120 
 
 21 
 
 622 
 
 2248 
 
 2346 
 
 2444 
 
 2541 
 
 2639 
 
 2737 
 
 2834 
 
 2932 
 
 3030 
 
 3127 
 
 2221 
 
 22 
 
 623 
 
 3225 
 
 3323 
 
 3420 
 
 3518 
 
 3616 
 
 3713 
 
 3811 
 
 3909 
 
 4006 
 
 4104 
 
 2322 
 
 23 
 
 624 
 
 4202 
 
 4299 
 
 4397 
 
 4495 
 
 4592 
 
 4690 
 
 4788 
 
 4885 
 
 4983 
 
 5081 
 
 24|23 
 
 24 
 
 625 
 
 5179 
 
 5276 
 
 5374 
 
 5472 
 
 5569 
 
 5667 
 
 5765 
 
 5862 
 
 5960 
 
 6058 
 
 25 
 
 24 
 
 25 
 
 626 
 
 6155 
 
 6253 
 
 6351 
 
 6448 
 
 6546 
 
 6644 
 
 6742 
 
 6839 
 
 6937 
 
 7035 
 
 26 
 
 25 
 
 25 
 
 627 
 
 7132 
 
 7230 
 
 7328 
 
 7425 
 
 7523 
 
 7621 
 
 7718 
 
 7816 
 
 7914 
 
 8012 
 
 27 
 
 26 
 
 26 
 
 628 
 
 8109 
 
 8207 
 
 8305 
 
 8402 
 
 8500 
 
 8598 
 
 8695 
 
 8793 
 
 8891 
 
 8989 
 
 28 
 
 27 
 
 27 
 
 629 
 
 9086 
 
 9184 
 
 9282 
 
 9379 
 
 9477 
 
 9575 
 
 9673 
 
 9770 
 
 9868 
 
 9966 
 
 29 
 
 28 
 
 28 
 
 630 
 
 64 0063 
 
 0161 
 
 0259 
 
 0356 
 
 0454 
 
 0552 
 
 0650 
 
 0747 
 
 0845 
 
 0943 
 
 30 
 
 29 
 
 29 
 
 631 
 
 1040 
 
 1138 
 
 1236 
 
 1334 
 
 1431 
 
 1529 
 
 1627 
 
 1724 
 
 1822 
 
 1920 
 
 31 
 
 30 
 
 30 
 
 632 
 
 2018 
 
 2115 
 
 2213 
 
 2311 
 
 2409 
 
 2506 
 
 2604 
 
 2702 
 
 2799 
 
 2897 
 
 32 
 
 31 
 
 31 
 
 633 
 
 2995 
 
 3093 
 
 3190 
 
 3288 
 
 3386 
 
 3484 
 
 3581 
 
 3679 
 
 3777 
 
 3874 
 
 33 
 
 32 
 
 32 
 
 634 
 
 3972 
 
 4070 
 
 4168 
 
 4265 
 
 4363 
 
 4461 
 
 4559 
 
 4656 
 
 4754 
 
 4852 
 
 34 
 
 33 
 
 33 
 
 635 
 
 4949 
 
 5047 
 
 5145 
 
 5243 
 
 5340 
 
 5438 
 
 5536 
 
 5634 
 
 5731 
 
 5829 
 
 35|34 
 
 34 
 
 636 
 
 5927 
 
 6025 
 
 6122 
 
 6220 
 
 6318 
 
 6416 
 
 6513 
 
 6611 
 
 6709 
 
 6807 
 
 3635 
 
 35 
 
 637 
 
 6904 
 
 7002 
 
 7100 
 
 7198 
 
 7295 
 
 7393 
 
 7491 
 
 7589 
 
 7686 
 
 7784 
 
 37|36 
 
 36 
 
 638 
 
 7882 
 
 7980 
 
 8077 
 
 8175 
 
 8273 
 
 8371 
 
 8468 
 
 8566 
 
 8664 
 
 8762 
 
 38 
 
 37 
 
 37 
 
 639 
 
 8859 
 
 8957 
 
 9055 
 
 9153 
 
 9250 
 
 9348 
 
 9446 
 
 9544 
 
 9641 
 
 9739 
 
 39 
 
 38 
 
 38 
 
 640 
 
 64 9837 
 
 9935 
 
 0032 
 
 0130 
 
 0228 
 
 0326 
 
 0423 
 
 0521 
 
 0619 
 
 0717 
 
 40 
 
 39 
 
 39 
 
 641 
 
 650814 
 
 0912 
 
 1010 
 
 1108 
 
 1206 
 
 1303 
 
 1401 
 
 1499 
 
 1597 
 
 1694 
 
 41 
 
 40 
 
 40 
 
 642 
 
 1792 
 
 1890 
 
 1988 
 
 2085 
 
 2183 
 
 2281 
 
 2379 
 
 2477 
 
 2574 
 
 2672 
 
 42 
 
 41 
 
 41 
 
 643 
 
 2770 
 
 2868 
 
 2965 
 
 3063 
 
 3161 
 
 3259 
 
 3357 
 
 3454 
 
 3552 
 
 3650 
 
 43 
 
 42 
 
 42 
 
 644 
 
 3748 
 
 3845 
 
 3943 
 
 4041 
 
 4139 
 
 4237 
 
 4334 
 
 4432 
 
 4530 
 
 4628 
 
 44 
 
 43 
 
 43 
 
 645 
 
 4726 
 
 4823 
 
 4921 
 
 5019 
 
 5117 
 
 5214 
 
 5312 
 
 5410 
 
 5508 
 
 5606 
 
 45 
 
 44 
 
 44 
 
 646 
 
 5703 
 
 5801 
 
 5899 
 
 5997 
 
 6095 
 
 6192 
 
 6290 
 
 6388 
 
 6486 
 
 6584 
 
 46 
 
 45 
 
 45 
 
 647 
 
 6681 
 
 6779 
 
 6877 
 
 6975 
 
 7073 
 
 7170 
 
 7268 
 
 7366 
 
 7464 
 
 7562 
 
 47 
 
 46 
 
 46 
 
 648 
 
 7659 
 
 7757 
 
 7855 
 
 7953 
 
 8051 
 
 8148 
 
 8246 
 
 8344 
 
 8442 
 
 8540 
 
 48 
 
 47 
 
 47 
 
 649 
 
 8637 
 
 8735 
 
 8833 
 
 8931 
 
 9029 
 
 9126 
 
 9224 
 
 9322 
 
 9420 
 
 9518 
 
 49 
 
 48 
 
 48 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ~7~ 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 34 
 
TABLE I. 
 
 Log 
 
 
 Log(H-) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6' 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 1-650 
 
 1-659615 
 
 9713 
 
 9811 
 
 9909 
 
 0007 
 
 o!04 
 
 0202 
 
 0300 
 
 0398 
 
 0496 
 
 50 
 
 49 
 
 49 
 
 651 
 
 66 0594 
 
 0691 
 
 0789 
 
 0887 
 
 0985 
 
 1083 
 
 1180 
 
 1278 
 
 1376 
 
 1474 
 
 51 
 
 49 
 
 50 
 
 652 
 
 1572 
 
 1670 
 
 1767 
 
 1865 
 
 1963 
 
 2061 
 
 2159 
 
 2256 
 
 2354 
 
 2452 
 
 52 
 
 50 
 
 51 
 
 653 
 
 2550 
 
 2648 
 
 2746 
 
 2843 
 
 2941 
 
 3039 
 
 3137 
 
 3235 
 
 3333 
 
 3430 
 
 53 
 
 51 
 
 52 
 
 654 
 
 3528 
 
 3626 
 
 3724 
 
 3822 
 
 3920 
 
 4017 
 
 4115 
 
 4213 
 
 4311 
 
 4409 
 
 54 
 
 52 
 
 53 
 
 655 
 
 4507 
 
 4604 
 
 4702 
 
 4800 
 
 4898 
 
 4996 
 
 5094 
 
 5191 
 
 5289 
 
 5387 
 
 55 
 
 53 
 
 54 
 
 656 
 
 5485 
 
 5583 
 
 5681 
 
 5778 
 
 5876 
 
 5974 
 
 6072 
 
 6170 
 
 6268 
 
 6365 
 
 56 
 
 54 
 
 55 
 
 657 
 
 6463 
 
 6561 
 
 6659 
 
 6757 
 
 6855 
 
 6953 
 
 7050 
 
 7148 
 
 72<f6 
 
 7344 
 
 57 
 
 55 
 
 56 
 
 658 
 
 7442 
 
 7540 
 
 7638 
 
 7735 
 
 7833 
 
 7931 
 
 8029 
 
 8127 
 
 8225 
 
 8322 
 
 58 
 
 56 
 
 57 
 
 659 
 
 8420 
 
 8518 
 
 8616 
 
 8714 
 
 8812 
 
 8910 
 
 9007 
 
 9105 
 
 9203 
 
 9301 
 
 59 
 
 57 
 
 58 
 
 660 
 
 66 9399 
 
 9497 
 
 9595 
 
 9692 
 
 9790 
 
 9888 
 
 9986 
 
 0084 
 
 0182 
 
 0280 
 
 60 
 
 58 
 
 59 
 
 661 
 
 67 0378 
 
 0475 
 
 0573 
 
 0671 
 
 0769 
 
 0867 
 
 0965 
 
 1063 
 
 1160 
 
 1258 
 
 61 
 
 59 
 
 60 
 
 662 
 
 1356 
 
 1454 
 
 1552 
 
 1650 
 
 1748 
 
 1846 
 
 1943 
 
 2041 
 
 2139 
 
 2237 
 
 62 
 
 60 
 
 61 
 
 663 
 
 2335 
 
 2433 
 
 2531 
 
 2628 
 
 2726 
 
 2824 
 
 2922 
 
 3020 
 
 3118 
 
 3216 
 
 63 
 
 61 
 
 62 
 
 664 
 
 3314 
 
 3412 
 
 3509 
 
 3607 
 
 3705 
 
 3803 
 
 3901 
 
 3999 
 
 4097 
 
 4195 
 
 64 
 
 62 
 
 63 
 
 665 
 
 4292 
 
 4390 
 
 4488 
 
 4586 
 
 4684 
 
 4782 
 
 4880 
 
 4978 
 
 5076 
 
 5173 
 
 65 
 
 63 
 
 64 
 
 666 
 
 5271 
 
 5369 
 
 5467 
 
 5565 
 
 5663 
 
 5761 
 
 5859 
 
 5957 
 
 6054 
 
 6152 
 
 66 
 
 64 
 
 65 
 
 667 
 
 6250 
 
 6348 
 
 6446 
 
 6544 
 
 6642 
 
 6740 
 
 6838 
 
 6935 
 
 7033 
 
 7131 
 
 67 
 
 65 
 
 66 
 
 668 
 
 7229 
 
 7327 
 
 7425 
 
 7523 
 
 7621 
 
 7719 
 
 7817 
 
 7914 
 
 8012 
 
 8110 
 
 68 
 
 66 
 
 67 
 
 669 
 
 8208 
 
 8306 
 
 8404 
 
 8502 
 
 8600 
 
 8698 
 
 8796 
 
 8893 
 
 8991 
 
 9089 
 
 69 
 
 67 
 
 68 
 
 670 
 
 679187 
 
 9285 
 
 9383 
 
 9481 
 
 9579 
 
 9677 
 
 9775 
 
 9873 
 
 9970 
 
 0068 
 
 70 
 
 68 
 
 69 
 
 671 
 
 680166 
 
 0264 
 
 0362 
 
 0460 
 
 0558 
 
 0656 
 
 0754 
 
 0852 
 
 0950 
 
 1048 
 
 71 
 
 69 
 
 70 
 
 672 
 
 1145 
 
 1243 
 
 1341 
 
 1439 
 
 1537 
 
 1635 
 
 1733 
 
 1831 
 
 1929 
 
 2027 
 
 72 
 
 70 
 
 71 
 
 673 
 
 2125 
 
 2223 
 
 2320 
 
 2418 
 
 2516 
 
 2614 
 
 2712 
 
 2810 
 
 2908 
 
 3006 
 
 73 
 
 71 
 
 72 
 
 674 
 
 3104 
 
 3202 
 
 3300 
 
 3398 
 
 3496 
 
 3593 
 
 3691 
 
 3789 
 
 3887 
 
 3985 
 
 74 
 
 72 
 
 73 
 
 6Z5 
 
 4083 
 
 4181 
 
 4279 
 
 4377 
 
 4475 
 
 4573 
 
 4671 
 
 4769 
 
 4867 
 
 4965 
 
 75 
 
 73 
 
 73 
 
 676 
 
 5062 
 
 5160 
 
 5258 
 
 5356 
 
 5454 
 
 5552 
 
 5650 
 
 5748 
 
 5846 
 
 5944 
 
 76 
 
 74 
 
 74 
 
 677 
 
 6042 
 
 6140 
 
 6238 
 
 6336 
 
 6434 
 
 6532 
 
 6629 
 
 6727 
 
 6825 
 
 6923 
 
 77 
 
 75 
 
 75 
 
 678 
 
 7021 
 
 7119 
 
 7217 
 
 7315 
 
 7413 
 
 7511 
 
 7609 
 
 7707 
 
 7805 
 
 7903 
 
 78 
 
 76 
 
 76 
 
 679 
 
 8001 
 
 8099 
 
 8197 
 
 8295 
 
 8392 
 
 8490 
 
 8588 
 
 8686 
 
 8784 
 
 8882 
 
 79 
 
 77 
 
 77 
 
 680 
 
 68 8980 
 
 9078 
 
 9176 
 
 9274 
 
 9372 
 
 9470 
 
 9568 
 
 9666 
 
 9764 
 
 9862 
 
 80 
 
 78 
 
 78 
 
 681 
 
 9960 
 
 0058 
 
 0156 
 
 0254 
 
 0352 
 
 0450 
 
 0548 
 
 0645 
 
 0743 
 
 0841 
 
 81 
 
 79 
 
 79 
 
 682 
 
 69 0939 
 
 1037 
 
 1135 
 
 1233 
 
 1331 
 
 1429 
 
 1527 
 
 1625 
 
 1723 
 
 1821 
 
 82 
 
 80 
 
 BO 
 
 683 
 
 1919 
 
 2017 
 
 2115 
 
 2213 
 
 2311 
 
 2409 
 
 2507 
 
 2605 
 
 2703 
 
 2801 
 
 83 
 
 81 
 
 31 
 
 684 
 
 2899 
 
 2997 
 
 3095 
 
 3193 
 
 3291 
 
 3389 
 
 3487 
 
 3585 
 
 3683 
 
 3780 
 
 84 
 
 81 
 
 2 
 
 685 
 
 3878 
 
 3976 
 
 4074 
 
 4172 
 
 4270 
 
 4368 
 
 4466 
 
 4564 
 
 4662 
 
 4760 
 
 85 
 
 82 
 
 S3 
 
 686 
 
 4858 
 
 4956 
 
 5054 
 
 5152 
 
 5250 
 
 5348 
 
 5446 
 
 5544 
 
 5642 
 
 5740 
 
 86 
 
 83 
 
 B4 
 
 687 
 
 5838 
 
 5936 
 
 6034 
 
 6132 
 
 6230 
 
 6328 
 
 6426 
 
 6524 
 
 6622 
 
 6720 
 
 87 
 
 84 
 
 85 
 
 688 
 
 6818 
 
 6916 
 
 7014 
 
 7112 
 
 7210 
 
 7308 
 
 7406 
 
 7504 
 
 7602 
 
 7700 
 
 88 
 
 85 
 
 S6 
 
 689 
 
 7798 
 
 7896 
 
 7994 
 
 8092 
 
 8190 
 
 8288 
 
 8386 
 
 8484 
 
 8582 
 
 8680 
 
 89 
 
 86 
 
 B7 
 
 690 
 
 69 8778 
 
 8876 
 
 8974 
 
 9072 
 
 9170 
 
 9268 
 
 9366 
 
 9464 
 
 9562 
 
 9660 
 
 90 
 
 87 
 
 B8 
 
 691 
 
 9758 
 
 9856 
 
 9954 
 
 0052 
 
 0150 
 
 0248 
 
 0346 
 
 0444 
 
 0542 
 
 0640 
 
 91 
 
 88 
 
 B9 
 
 692 
 
 70 0738 
 
 0836 
 
 0934 
 
 1032 
 
 1130 
 
 1228 
 
 1326 
 
 1424 
 
 1522 
 
 1620 
 
 92 
 
 69 
 
 90 
 
 693 
 
 1718 
 
 1816 
 
 1914 
 
 2012 
 
 2110 
 
 2208 
 
 2306 
 
 2404 
 
 2502 
 
 2600 
 
 93 
 
 90 
 
 91 
 
 694 
 
 2698 
 
 2796 
 
 2894 
 
 2992 
 
 3090 
 
 3188 
 
 3286 
 
 3384 
 
 3482 
 
 3580 
 
 94 
 
 91 
 
 92 
 
 695 
 
 3678 
 
 3776 
 
 3874 
 
 3972 
 
 4070 
 
 4168 
 
 4267 
 
 4365 
 
 4463 
 
 4561 
 
 95 
 
 92 
 
 93 
 
 696 
 
 4659 
 
 4757 
 
 4855 
 
 4953 
 
 5051 
 
 5149 
 
 5247 
 
 5345 
 
 5443 
 
 5541 
 
 96 
 
 93 
 
 94 
 
 697 
 
 5639 
 
 5737 
 
 5835 
 
 5933 
 
 6031 
 
 6129 
 
 6227 
 
 6325 
 
 6423 
 
 6521 
 
 97 
 
 94 
 
 95 
 
 698 
 
 6619 
 
 6717 
 
 6815 
 
 6913 
 
 7011 
 
 7109 
 
 7207 
 
 7305 
 
 7404 
 
 7502 
 
 98 
 
 95 
 
 96 
 
 699 
 
 7600 
 
 7698 
 
 7796 
 
 7894 
 
 7992 
 
 8090 
 
 8188 
 
 8286 
 
 8384 
 
 8482 
 
 99 
 
 96 
 
 97 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 97 
 
 98 
 
 35 
 
TABLE I. 
 
 Log 
 
 
 Log(l+*) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 1-700 
 
 1-708580 
 
 8678 
 
 8776 
 
 8874 
 
 8972 
 
 9070 
 
 9168 
 
 9266 
 
 9364 
 
 9462 
 
 00 
 
 
 
 
 
 701 
 
 9560 
 
 9659 
 
 9757 
 
 9855 
 
 9953 
 
 0051 
 
 0149 
 
 0247 
 
 0345 
 
 0443 
 
 01 
 
 1 
 
 1 
 
 702 
 
 .71 0541 
 
 0639 
 
 0737 
 
 0835 
 
 0933 
 
 1031 
 
 1129 
 
 1227 
 
 1325 
 
 1423 
 
 02 
 
 2 
 
 2 
 
 703 
 
 1522 
 
 1620 
 
 1718 
 
 1816 
 
 1914 
 
 2012 
 
 2110 
 
 2208 
 
 2306 
 
 2404 
 
 03 
 
 3 
 
 3 
 
 704 
 
 2502 
 
 2600 
 
 2698 
 
 2796 
 
 2894 
 
 2992 
 
 3090 
 
 3189 
 
 3287 
 
 3385 
 
 04 
 
 4 
 
 4 
 
 705 
 
 3483 
 
 3581 
 
 3679 
 
 3777 
 
 3875 
 
 3973 
 
 4071 
 
 4169 
 
 4267 
 
 4365 
 
 05 
 
 5 
 
 K 
 
 706 
 
 4463 
 
 4561 
 
 4660 
 
 4758 
 
 4856 
 
 4954 
 
 5052 
 
 5150 
 
 5248 
 
 5346 
 
 06 
 
 6 
 
 6 
 
 707 
 
 5444 
 
 5542 
 
 5640 
 
 5738 
 
 5836 
 
 5935 
 
 6033 
 
 6131 
 
 6229 
 
 6327 
 
 07 
 
 7 
 
 7 
 
 708 
 
 6425 
 
 6523 
 
 6621 
 
 6719 
 
 6817 
 
 6915 
 
 7013 
 
 7111 
 
 7210 
 
 7308 
 
 08 
 
 8 
 
 8 
 
 709 
 
 7406 
 
 7504 
 
 7602 
 
 7700 
 
 7798 
 
 7896 
 
 7994 
 
 8092 
 
 8190 
 
 8288 
 
 09 
 
 9 
 
 9 
 
 710 
 
 718387 
 
 8485 
 
 8583 
 
 8681 
 
 8779 
 
 8877 
 
 8975 
 
 9073 
 
 9171 
 
 9269 
 
 10 
 
 10 
 
 10 
 
 711 
 
 9367 
 
 9466 
 
 9564 
 
 9662 
 
 9760 
 
 9858 
 
 9956 
 
 0054 
 
 0152 
 
 0250 
 
 11 
 
 11 
 
 11 
 
 712 
 
 72 0348 
 
 0446 
 
 0545 
 
 0643 
 
 0741 
 
 0839 
 
 0937 
 
 1035 
 
 1133 
 
 1231 
 
 12 
 
 12 
 
 12 
 
 713 
 
 1329 
 
 1427 
 
 1526 
 
 1624 
 
 1722 
 
 1820 
 
 1918 
 
 2016 
 
 2114 
 
 2212 
 
 13 
 
 13J13 
 
 714 
 
 2310 
 
 2409 
 
 2507 
 
 2605 
 
 2703 
 
 2801 
 
 2899 
 
 2997 
 
 3095 
 
 3193 
 
 14 
 
 14 
 
 14 
 
 715 
 
 3291 
 
 3390 
 
 3488 
 
 3586 
 
 3684 
 
 3782 
 
 3880 
 
 3978 
 
 4076 
 
 4174 
 
 15 
 
 15 
 
 15 
 
 716 
 
 4273 
 
 4371 
 
 4469 
 
 4567 
 
 4665 
 
 4763 
 
 4861 
 
 4959 
 
 5058 
 
 5156 
 
 1C 
 
 16 
 
 16 
 
 717 
 
 5254 
 
 5352 
 
 5450 
 
 5548 
 
 5646 
 
 5744 
 
 5842 
 
 5941 
 
 6039 
 
 6137 
 
 17 
 
 17 
 
 17 
 
 718 
 
 6235 
 
 6333 
 
 6431 
 
 6529 
 
 6627 
 
 6726 
 
 6824 
 
 6922 
 
 7020 
 
 7118 
 
 IS 
 
 18 
 
 18 
 
 719 
 
 7216 
 
 7314 
 
 7412 
 
 7511 
 
 7609 
 
 7707 
 
 7805 
 
 7903 
 
 8001 
 
 8099 
 
 19 
 
 19 
 
 19 
 
 720 
 
 728197 
 
 8296 
 
 8394 
 
 8492 
 
 8590 
 
 8688 
 
 8786 
 
 8884 
 
 8983 
 
 9081 
 
 20 
 
 20 
 
 20 
 
 721 
 
 9179 
 
 9277 
 
 9375 
 
 9473 
 
 9571 
 
 9669 
 
 9768 
 
 9866 
 
 9964 
 
 0062 
 
 21 
 
 21 
 
 21 
 
 722 
 
 730160 
 
 0258 
 
 0356 
 
 0455 
 
 0553 
 
 0651 
 
 0749 
 
 0847 
 
 0945 
 
 1043 
 
 22 
 
 22 
 
 22 
 
 723 
 
 1142 
 
 1240 
 
 1338 
 
 1436 
 
 1534 
 
 1632 
 
 1730 
 
 1829 
 
 1927 
 
 2025 
 
 23 
 
 23 
 
 23 
 
 724 
 
 2123 
 
 2221 
 
 2319 
 
 2417 
 
 2516 
 
 2614 
 
 2712 
 
 2810 
 
 2908 
 
 3006 
 
 24 
 
 24 
 
 24 
 
 725 
 
 3104 
 
 3203 
 
 3301 
 
 3399 
 
 3497 
 
 3595 
 
 3693 
 
 3792 
 
 3890 
 
 3988 
 
 25 
 
 -36 
 
 25 
 
 726 
 
 4086 
 
 4184 
 
 4282 
 
 4380 
 
 4479 
 
 4577 
 
 4675 
 
 4773 
 
 4871 
 
 4969 
 
 26 
 
 25 
 
 26 
 
 727 
 
 5068 
 
 5166 
 
 5264 
 
 5362 
 
 5460 
 
 5558 
 
 5657 
 
 5755 
 
 5853 
 
 5951 
 
 27 
 
 26 
 
 27 
 
 728 
 
 6049 
 
 6147 
 
 6246 
 
 6344 
 
 6442 
 
 6540 
 
 6638 
 
 6736 
 
 6835 
 
 6933 
 
 28 
 
 27 
 
 28 
 
 729 
 
 7031 
 
 7129 
 
 7227 
 
 7325 
 
 7424 
 
 7522 
 
 7620 
 
 7718 
 
 7816 
 
 7914 
 
 29 
 
 28 
 
 29 
 
 730 
 
 73 8013 
 
 8111 
 
 8209 
 
 8307 
 
 8405 
 
 8503 
 
 8602 
 
 8700 
 
 8798 
 
 8896 
 
 30 
 
 29 
 
 30 
 
 731 
 
 8994 
 
 9092 
 
 9191 
 
 9289 
 
 9387 
 
 9485 
 
 9583 
 
 9682 
 
 9780 
 
 9878 
 
 31 
 
 30 
 
 31 
 
 732 
 
 9976 
 
 0074 
 
 o!72 
 
 0271 
 
 0369 
 
 0467 
 
 0565 
 
 0663 
 
 0762 
 
 0860 
 
 32 
 
 31 
 
 32 
 
 733 
 
 74 0958 
 
 1056 
 
 1154 
 
 1252 
 
 1351 
 
 1449 
 
 1547 
 
 1645 
 
 1743 
 
 1842 
 
 33 
 
 32 
 
 33 
 
 734 
 
 1940 
 
 2038 
 
 2136 
 
 2234 
 
 2333 
 
 2431 
 
 2529 
 
 2627 
 
 2725 
 
 2823 
 
 34 
 
 33 
 
 34 
 
 735 
 
 2922 
 
 3020 
 
 3118 
 
 3216 
 
 3314 
 
 3413 
 
 3511 
 
 3609 
 
 3707 
 
 3805 
 
 35 
 
 34 
 
 35 
 
 736 
 
 3904 
 
 4002 
 
 4100 
 
 4198 
 
 4296 
 
 4395 
 
 4493 
 
 4591 
 
 4689 
 
 4787 
 
 36|35 
 
 36 
 
 737 
 
 4886 
 
 4984 
 
 5082 
 
 5180 
 
 5278 
 
 5377 
 
 5475 
 
 5573 
 
 5671 
 
 5769 
 
 37 
 
 36 
 
 37 
 
 738 
 
 5868 
 
 5966 
 
 6064 
 
 6162 
 
 6260 
 
 6359 
 
 6457 
 
 6555 
 
 6653 
 
 6752 
 
 38 
 
 37 
 
 38 
 
 739 
 
 6850 
 
 6948 
 
 7046 
 
 7144 
 
 7243 
 
 7341 
 
 7439 
 
 7537 
 
 7635 
 
 7734 
 
 39 
 
 38 
 
 39 
 
 740 
 
 74 7832 
 
 7930 
 
 8028 
 
 8126 
 
 8225 
 
 8323 
 
 8421 
 
 8519 
 
 8618 
 
 8716 
 
 40 
 
 39 
 
 40 
 
 741 
 
 8814 
 
 8912 
 
 9010 
 
 9109 
 
 9207 
 
 9305 
 
 9403 
 
 9501 
 
 9600 
 
 9698 
 
 41 
 
 40 
 
 41 
 
 742 
 
 9796 
 
 9894 
 
 9993 
 
 0091 
 
 0189 
 
 0287 
 
 0385 
 
 0484 
 
 0582 
 
 0680 
 
 42 
 
 41 
 
 42 
 
 743 
 
 75 0778 
 
 0877 
 
 0975 
 
 1073 
 
 1171 
 
 1270 
 
 1368 
 
 1466 
 
 1564 
 
 1662 
 
 43 
 
 42 
 
 43 
 
 744 
 
 1761 
 
 1859 
 
 1957 
 
 2055 
 
 2154 
 
 2252 
 
 2350 
 
 2448 
 
 2546 
 
 2645 
 
 4443 
 
 44 
 
 745 
 
 2743 
 
 2841 
 
 2939 
 
 3038 
 
 3136 
 
 3234 
 
 3332 
 
 3431 
 
 3529 
 
 3627 
 
 45 
 
 44 
 
 45 
 
 746 
 
 3725 
 
 3824 
 
 3922 
 
 4020 
 
 4118 
 
 4216 
 
 4315 
 
 4413 
 
 4511 
 
 4609 
 
 46 
 
 45 
 
 46 
 
 747 
 
 4708 
 
 4806 
 
 4904 
 
 5002 
 
 5101 
 
 5199 
 
 5297 
 
 5395 
 
 5494 
 
 5592 
 
 47 
 
 46 
 
 47 
 
 748 
 
 5690 
 
 5788 
 
 5887 
 
 5985 
 
 6083 
 
 6181 
 
 6280 
 
 6378 
 
 6476 
 
 6574 
 
 48 
 
 47 
 
 48 
 
 749 
 
 6673 
 
 6771 
 
 6869 
 
 6967 
 
 7066 
 
 7164 
 
 7262 
 
 7360 
 
 7459 
 
 7557 
 
 49 
 
 48 
 
 49 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
TABLE I. 
 
 Log 
 
 LOJJ (1+,) 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 1-750 
 
 1-75 7655 
 
 7753 
 
 7852 
 
 7950 
 
 8048 
 
 8146 
 
 8245 
 
 8343 
 
 8441 
 
 8539 
 
 50 
 
 49 
 
 50 
 
 751 
 
 8638 
 
 8736 
 
 8834 
 
 8932 
 
 9031 
 
 9129 
 
 9227 
 
 9325 
 
 9424 
 
 9522 
 
 51 
 
 50 
 
 50 
 
 752 
 
 9620 
 
 9719 
 
 9817 
 
 9915 
 
 0013 
 
 0112 
 
 0210 
 
 0308 
 
 0406 
 
 0505 
 
 52 
 
 51 
 
 51 
 
 753 
 
 76 0603 
 
 0701 
 
 0799 
 
 0898 
 
 0996 
 
 1094 
 
 1192 
 
 1291 
 
 1389 
 
 1487 
 
 53 
 
 52 
 
 52 
 
 754 
 
 1586 
 
 1684 
 
 1782 
 
 1880 
 
 1979 
 
 2077 
 
 2175 
 
 2273 
 
 2372 
 
 2470 
 
 54 
 
 53 
 
 53 
 
 755 
 
 2568 
 
 2667 
 
 2765 
 
 2863 2961 
 
 3060 
 
 3158 
 
 3256 
 
 3354 
 
 3453 
 
 55 
 
 54 
 
 54 
 
 756 
 
 3551 
 
 3649 
 
 3748 
 
 3846! 3944 
 
 4042 
 
 4141 
 
 4239 
 
 4337 
 
 4435 
 
 56 
 
 55 
 
 55 
 
 757 
 
 4534 
 
 4632 
 
 4730 
 
 4829 
 
 4927 
 
 5025 
 
 5123 
 
 5222 
 
 5320 
 
 5418 
 
 57 
 
 56 
 
 56 
 
 758 
 
 5517 
 
 5615 
 
 5713 
 
 58UJ5910 
 
 6008 
 
 6106 
 
 6205 
 
 6303 
 
 6401 
 
 58 
 
 57 
 
 57 
 
 759 
 
 6499 
 
 6558 
 
 6696 
 
 6794 
 
 6893 
 
 6991 
 
 7089 
 
 7187 
 
 7286 
 
 7384 
 
 59 
 
 58 
 
 58 
 
 760 
 
 76 7482 
 
 7581 
 
 7679 
 
 7777 
 
 7876 
 
 7974 
 
 8072 
 
 8170 
 
 8269 
 
 8367 
 
 60 
 
 59 
 
 59 
 
 761 
 
 8465 
 
 8564 
 
 8662 
 
 8760 
 
 8858 
 
 8957 
 
 9055 
 
 9153 
 
 9252 
 
 9350 
 
 61 
 
 60 
 
 60 
 
 762 
 
 9448 
 
 9547 
 
 9645 
 
 9743 
 
 9841 
 
 9940 
 
 0038 
 
 0136 
 
 0235 
 
 0333 
 
 62 
 
 61 
 
 61 
 
 763 
 
 770431 
 
 0530 
 
 0628 
 
 0726 
 
 0824 
 
 0923 
 
 1021 
 
 1119 
 
 1218 
 
 1316 
 
 63 
 
 62 
 
 62 
 
 764 
 
 1414 
 
 1513 
 
 1611 
 
 1709 
 
 1808 
 
 1906 
 
 2004 
 
 2102 
 
 2201 
 
 2299 
 
 64 
 
 63 
 
 63 
 
 765 
 
 . 2397 
 
 2496 
 
 2594 
 
 2692 
 
 2791 
 
 2889 
 
 2987 
 
 3086 
 
 3184 
 
 3282 
 
 65 
 
 64 
 
 64 
 
 766 
 
 3381 
 
 3479 
 
 3577 
 
 3675 
 
 3774 
 
 3872 
 
 3970 
 
 4069 
 
 4167 
 
 4265 
 
 66 
 
 65 
 
 65 
 
 767 
 
 4364 
 
 4462 
 
 4560 
 
 4659 
 
 4757 
 
 4855 
 
 4954 
 
 5052 
 
 5150 
 
 5249 
 
 67 
 
 66 
 
 66 
 
 768 
 
 5347 
 
 5445 
 
 5544 
 
 5642 
 
 5740 
 
 5839 
 
 5937 
 
 6035 
 
 6134 
 
 6232 
 
 68 
 
 67 
 
 67 
 
 769 
 
 6330 
 
 6428 
 
 6527 
 
 6625 
 
 6723 
 
 6822 
 
 6920 
 
 7018 
 
 7117 
 
 7215 
 
 69 
 
 68 
 
 68 
 
 770 
 
 777313 
 
 7412 
 
 7510 
 
 7608 
 
 7707 
 
 7805 
 
 7903 
 
 8002 
 
 8100 
 
 8198 
 
 70 
 
 69 
 
 69 
 
 771 
 
 8297 
 
 8395 
 
 8493 
 
 8592 
 
 8690 
 
 8788 
 
 8887 
 
 8985 
 
 9083 
 
 9182 
 
 71 
 
 70 
 
 70 
 
 772 
 
 9280 
 
 9378 
 
 9477 
 
 9575 
 
 9673 
 
 9772 
 
 9870 
 
 9969 
 
 0067 
 
 0165 
 
 72 
 
 71 
 
 71 
 
 773 
 
 78 0264 
 
 0362 
 
 0460 
 
 0559 
 
 0657 
 
 0755 
 
 0854 
 
 0952 
 
 1050 
 
 1149 
 
 73 
 
 72 
 
 72 
 
 774 
 
 1247 
 
 1345 
 
 1444 
 
 1542 
 
 1640 
 
 1739 
 
 1837 
 
 1935 
 
 2034 
 
 2132 
 
 74 
 
 73 
 
 73 
 
 775 
 
 2230 
 
 2329 
 
 2427 
 
 2525 
 
 2624 
 
 2722 
 
 2821 
 
 2919 
 
 3017 
 
 3116 
 
 75 
 
 73 
 
 74 
 
 776 
 
 3214 
 
 3312 
 
 3411 
 
 3509 
 
 3607 
 
 3706 
 
 3604 
 
 3902 
 
 4001 
 
 4099 
 
 76 
 
 74 
 
 75 
 
 777 
 
 4197 
 
 4296 
 
 4394 
 
 4493 
 
 4591 
 
 4689 
 
 4788 
 
 4886 
 
 4984 
 
 5083 
 
 77 
 
 75 
 
 76 
 
 778 
 
 5181 
 
 5279 
 
 5378 
 
 5476 
 
 5575 
 
 5673 
 
 5771 
 
 5870 
 
 5968 
 
 6066 
 
 78 
 
 76 
 
 77 
 
 779 
 
 6165 
 
 6263 
 
 6361 
 
 6460 
 
 6558 
 
 6657 
 
 6755 
 
 6853 
 
 6952 
 
 7050 
 
 79 
 
 77 
 
 78 
 
 780 
 
 787148 
 
 7247 
 
 7345 
 
 7443 
 
 7542 
 
 7640 
 
 7739 
 
 7837 
 
 7935 
 
 8034 
 
 80 
 
 78 
 
 79 
 
 781 
 
 8132 
 
 8230 
 
 8329 
 
 8427 
 
 8526 
 
 8624 
 
 8722 
 
 8821 
 
 8919 
 
 9017 
 
 81 
 
 79 
 
 80 
 
 782 
 
 9116 
 
 9214 
 
 9313 
 
 9411 
 
 9509 
 
 9608 
 
 9706 
 
 9804 
 
 9903 
 
 oOOl 
 
 S2 
 
 80 
 
 81 
 
 783 
 
 790100 
 
 0198 
 
 0296 
 
 0395 
 
 0493 
 
 0591 
 
 0690 
 
 0788 
 
 0887 
 
 0985 
 
 83 
 
 81 
 
 82 
 
 784 
 
 1083 
 
 1182 
 
 1280 
 
 1378 
 
 1477 
 
 1575 
 
 1674 
 
 1772 
 
 1870 
 
 1969 
 
 84 
 
 82 
 
 83 
 
 785 
 
 2067 
 
 2166 
 
 2264 
 
 2362 
 
 2461 
 
 2559 
 
 2657 
 
 2756 
 
 2854 
 
 2953 
 
 85 
 
 83 
 
 84 
 
 786 
 
 3051 
 
 3149 
 
 3248 
 
 3346 
 
 3445 
 
 3543 
 
 3641 
 
 3740 
 
 3838 
 
 3937 
 
 86 
 
 84 
 
 85 
 
 787 
 
 4035 
 
 4133 
 
 4232 
 
 4330 
 
 4429 
 
 4527 
 
 4625 
 
 4724 
 
 4822 
 
 4921 
 
 87 
 
 85 
 
 86 
 
 788 
 
 5019 
 
 5117 
 
 5216 
 
 5314 
 
 5413 
 
 5511 
 
 5609 
 
 5708 
 
 5806 
 
 5905 
 
 88 
 
 86 
 
 8? 
 
 789 
 
 6003 
 
 6101 
 
 6200 
 
 6298 
 
 6397 
 
 6495 
 
 6593 
 
 6692 
 
 6790 
 
 6889 
 
 89 
 
 87 
 
 88 
 
 790 
 
 79 6987 
 
 7085 
 
 7184 
 
 7282 
 
 7381 
 
 7479 
 
 7577 
 
 7676 
 
 7774 
 
 7873 
 
 90 
 
 88 
 
 89 
 
 791 
 
 7971 
 
 8069 
 
 8168 
 
 8266 
 
 8365 
 
 8463 
 
 8561 
 
 8660 
 
 8758 
 
 8857 
 
 91 
 
 89 
 
 90 
 
 792 
 
 8955 
 
 9053 
 
 9152 
 
 9250 
 
 9349 
 
 944? 
 
 9546 
 
 964 t 
 
 9742 
 
 9841 
 
 92 
 
 90 
 
 91 
 
 793 
 
 9939 
 
 0038 
 
 0136 
 
 0234 
 
 0333 
 
 0431 
 
 0530 
 
 0628 
 
 0727 
 
 0825 
 
 93 
 
 91 
 
 92 
 
 794 
 
 80 0923 
 
 1022 
 
 1120 
 
 1219 
 
 1317 
 
 1415 
 
 1514 
 
 1612 
 
 1711 
 
 1809 
 
 94 
 
 92 
 
 93 
 
 795 
 
 1908 
 
 2006 
 
 2104 
 
 2203 
 
 2301 
 
 2400 
 
 2498 
 
 2597 
 
 2695 
 
 2793 
 
 95 
 
 93 
 
 94 
 
 796 
 
 2892 
 
 2990 
 
 3089 
 
 3187 
 
 3-286 
 
 3384 
 
 3482 
 
 3581 
 
 3679 
 
 3778 
 
 96 
 
 94 
 
 95 
 
 797 
 
 3876 
 
 3975 
 
 4073 
 
 4171 
 
 4270 
 
 4368 
 
 4467 
 
 4565 
 
 4664 
 
 4762 
 
 97 
 
 95 
 
 96 
 
 798 
 
 4860 
 
 4959 
 
 5057 
 
 5156 
 
 5254 
 
 5353 
 
 5451 
 
 5549 
 
 5648 
 
 5746 
 
 98 
 
 96 
 
 97 
 
 799 
 
 5845 
 
 5943 
 
 6042 
 
 6140 
 
 6238 
 
 6337 
 
 6435 
 
 6534 
 
 6632 
 
 6731 
 
 99 
 
 97 
 
 98 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 37 
 
TABLE I. 
 
 
 T _ /I I \ 
 
 Pro. 
 
 Log 
 
 -L<O 
 
 8 V 1 "! 
 
 -*) 
 
 
 
 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 1-800 
 
 1-80 6829 
 
 6928 
 
 7026 
 
 7124 
 
 7223 
 
 7321 
 
 7420 
 
 7518 
 
 7617 
 
 7715 
 
 00 
 
 
 
 
 
 801 
 
 7814 
 
 7912 
 
 8010 
 
 8109 
 
 8207 
 
 8306 
 
 8404 
 
 8503 
 
 8601 
 
 8700 
 
 01 
 
 1 
 
 1 
 
 802 
 
 8798 
 
 8896 
 
 8995 
 
 9093 
 
 9192 
 
 9290 
 
 9389 
 
 9487 
 
 9586 
 
 9684 
 
 02 
 
 2 
 
 2 
 
 803 
 
 9782 
 
 9881 
 
 9979 
 
 0078 
 
 o!76 
 
 0275 
 
 0373 
 
 0472 
 
 0570 
 
 0669 
 
 03 
 
 3 
 
 3 
 
 804 
 
 81 0767 
 
 0865 
 
 0964 
 
 1062 
 
 1161 
 
 1259 
 
 1358 
 
 1456 
 
 1555 
 
 1653 
 
 04 
 
 4 
 
 4 
 
 805 
 
 1752 
 
 1850 
 
 1948 
 
 2047 
 
 2145 
 
 2244 
 
 2342 
 
 2441 
 
 2539 
 
 2638 
 
 05 
 
 5 
 
 5 
 
 806 
 
 2736 
 
 2835 
 
 2933 
 
 3032 
 
 3130 
 
 3228 
 
 3327 
 
 3425 
 
 3524 
 
 3622 
 
 06 
 
 6 
 
 6 
 
 807 
 
 3721 
 
 3819 
 
 3918 
 
 4016 
 
 4115 
 
 4213 
 
 4312 
 
 4410 
 
 4509 
 
 4607 
 
 07 
 
 7 
 
 7 
 
 808 
 
 4705 
 
 4804 
 
 4902 
 
 5001 
 
 5099 
 
 5198 
 
 5296 
 
 5395 
 
 5493 
 
 5592 
 
 08 
 
 8 
 
 8 
 
 809 
 
 5690 
 
 5789 
 
 5887 
 
 5986 
 
 6084 
 
 6182 
 
 6281 
 
 6379 
 
 6478 
 
 6576 
 
 09 
 
 9 
 
 9 
 
 810 
 
 81 6675 
 
 6773 
 
 6872 
 
 6970 
 
 7069 
 
 7167 
 
 7266 
 
 7364 
 
 7463 
 
 7561 
 
 10 
 
 10 
 
 10 
 
 811 
 
 7660 
 
 7758 
 
 7857 
 
 7955 
 
 8054 
 
 8152 
 
 8251 
 
 8349 
 
 8447 
 
 8546 
 
 11 
 
 11 
 
 11 
 
 812 
 
 8644 
 
 8743 
 
 8841 
 
 8940 
 
 9038 
 
 9137 
 
 9235 
 
 9334 
 
 9432 
 
 9531 
 
 12 
 
 12 
 
 12 
 
 813 
 
 9629 
 
 9728 
 
 9826 
 
 9925 
 
 0023 
 
 0122 
 
 0220 
 
 0319 
 
 0417 
 
 0516 
 
 13 
 
 13 
 
 13 
 
 814 
 
 820614 
 
 0713 
 
 0811 
 
 0910 
 
 1008 
 
 1107 
 
 1205 
 
 1304 
 
 1402 
 
 1501 
 
 14 
 
 14 
 
 14 
 
 815 
 
 1599 
 
 1698 
 
 1796 
 
 1895 
 
 1993 
 
 2092 
 
 2190 
 
 2288 
 
 2387 
 
 2485 
 
 15 
 
 15 
 
 15 
 
 816 
 
 2584 
 
 2682 
 
 2781 
 
 2879 
 
 2978 
 
 3076 
 
 3175 
 
 3273 
 
 3372 
 
 3470 
 
 16 
 
 16 
 
 16 
 
 817 
 
 3569 
 
 3667 
 
 3766 
 
 3864 
 
 3963 
 
 4061 
 
 4160 
 
 4258 
 
 4357 
 
 4455 
 
 17 
 
 17 
 
 17 
 
 818 
 
 4554 
 
 4652 
 
 4751 
 
 4849 
 
 4948 
 
 5046 
 
 5145 
 
 5243 
 
 5342 
 
 5440 
 
 18 
 
 18 
 
 18 
 
 819 
 
 5539 
 
 5637 
 
 5736 
 
 5835 
 
 5933 
 
 6032 
 
 6130 
 
 6229 
 
 6327 
 
 6426 
 
 19 
 
 19 
 
 19 
 
 820 
 
 82 6524 
 
 6623 
 
 6721 
 
 6820 
 
 6918 
 
 7017 
 
 7115 
 
 7214 
 
 7312 
 
 7411 
 
 20 
 
 20 
 
 20 
 
 821 
 
 7509 
 
 7608 
 
 7706 
 
 7805 
 
 7903 
 
 8002 
 
 8100 
 
 8199 
 
 8297 
 
 8396 
 
 21 
 
 21 
 
 21 
 
 822 
 
 8494 
 
 8593 
 
 8691 
 
 8790 
 
 8888 
 
 8987 
 
 9085 
 
 9184 
 
 9282 
 
 9381 
 
 22 
 
 22 
 
 22 
 
 823 
 
 9479 
 
 9578 
 
 9677 
 
 9775 
 
 9874 
 
 9972 
 
 0071 
 
 0169 
 
 0268 
 
 0366 
 
 23 
 
 23 
 
 23 
 
 824 
 
 83 0465 
 
 0563 
 
 0662 
 
 0760 
 
 0859 
 
 0957 
 
 1056 
 
 1154 
 
 1253 
 
 1351 
 
 24 
 
 24 
 
 24 
 
 825 
 
 1450 
 
 1548 
 
 1647 
 
 1746 
 
 1844 
 
 1943 
 
 2041 
 
 2140 
 
 2238 
 
 2337 
 
 25 
 
 25 
 
 25 
 
 826 
 
 2435 
 
 2534 
 
 2632 
 
 2731 
 
 2829 
 
 2928 
 
 3026 
 
 3125 
 
 3223 
 
 3322 
 
 26 
 
 25 
 
 26 
 
 827 
 
 3421 
 
 3519 
 
 3618 
 
 3716 
 
 3815 
 
 3913 
 
 4012 
 
 4110 
 
 4209 
 
 4307 
 
 27 
 
 26 
 
 27 
 
 828 
 
 4406 
 
 4504 
 
 4603 
 
 4701 
 
 4800 
 
 4899 
 
 4997 
 
 5096 
 
 5194 
 
 5293 
 
 28 
 
 27 
 
 28 
 
 829 
 
 5391 
 
 5490 
 
 5588 
 
 5687 
 
 5785 
 
 5884 
 
 5982 
 
 6081 
 
 6180 
 
 6278 
 
 29 
 
 28 
 
 29 
 
 830 
 
 83 6377 
 
 6475 
 
 6574 
 
 6672 
 
 6771 
 
 6869 
 
 6968 
 
 7066 
 
 7165 
 
 7264 
 
 30 
 
 29 
 
 30 
 
 831 
 
 7362 
 
 7461 
 
 7559 
 
 7658 
 
 7756 
 
 7855 
 
 7953 
 
 8052 
 
 8150 
 
 8249 
 
 31 
 
 30 
 
 31 
 
 832 
 
 8348 
 
 8446 
 
 8545 
 
 8643 
 
 8742 
 
 8840 
 
 8939 
 
 9037 
 
 9136 
 
 9235 
 
 32 
 
 31 
 
 32 
 
 833 
 
 9333 
 
 9432 
 
 9530 
 
 9629 
 
 9727 
 
 9826 
 
 9924 
 
 0023 
 
 0121 
 
 0220 
 
 33 
 
 32 
 
 33 
 
 834 
 
 840319 
 
 0417 
 
 0516 
 
 0614 
 
 0713 
 
 0811 
 
 0910 
 
 1009 
 
 1107 
 
 1206 
 
 34 
 
 33 
 
 34 
 
 835 
 
 1304 
 
 1403 
 
 1501 
 
 1600 
 
 1698 
 
 1797 
 
 1896 
 
 1994 
 
 2093 
 
 2191 
 
 35 
 
 34 
 
 35 
 
 836 
 
 2290 
 
 2388 
 
 2487 
 
 2585 
 
 2684 
 
 2783 
 
 2881 
 
 2980 
 
 3078 
 
 3177 
 
 36 
 
 35 
 
 36 
 
 837 
 
 3275 
 
 3374 
 
 3473 
 
 3571 
 
 3670 
 
 3768 
 
 3867 
 
 3965 
 
 4064 
 
 4163 
 
 37 
 
 36 
 
 37 
 
 838 
 
 4261 
 
 4360 
 
 4458 
 
 4557 
 
 4655 
 
 4754 
 
 4853 
 
 4951 
 
 5050 
 
 5148 
 
 38 
 
 37 
 
 38 
 
 839 
 
 5247 
 
 5345 
 
 ^4.44 
 
 t/TTT 
 
 5543 
 
 5641 
 
 5740 
 
 5838 
 
 5937 
 
 6035 
 
 6134 
 
 39 
 
 38 
 
 39 
 
 840 
 
 84 6233 
 
 6331 
 
 6430 
 
 6528 
 
 6627 
 
 6725 
 
 6824 
 
 6923 
 
 7021 
 
 7120 
 
 40 
 
 39 
 
 40 
 
 841 
 
 7218 
 
 7317 
 
 7415 
 
 7514 
 
 7613 
 
 7711 
 
 7810 
 
 7908 
 
 8007 
 
 8106 
 
 41 
 
 40 
 
 41 
 
 842 
 
 8204 
 
 8303 
 
 8401 
 
 8500 
 
 8598 
 
 8697 
 
 8796 
 
 8894 
 
 8993 
 
 9091 
 
 42 
 
 41 
 
 42 
 
 843 
 
 9190 
 
 9289 
 
 9387 
 
 9486 
 
 9584 
 
 9683 
 
 9781 
 
 9880 
 
 9979 
 
 0077 
 
 43 
 
 42 
 
 43 
 
 844 
 
 850176 
 
 0274 
 
 0373 
 
 0472 
 
 0570 
 
 0669 
 
 0767 
 
 0866 
 
 0965 
 
 1063 
 
 44 
 
 43 
 
 44 
 
 845 
 
 1162 
 
 1260 
 
 1359 
 
 1457 
 
 1556 
 
 1655 
 
 1753 
 
 1852 
 
 1950 
 
 2049 
 
 45 
 
 44 
 
 45 
 
 846 
 
 2148 
 
 2246 
 
 2345 
 
 2443 
 
 2542 
 
 2641 
 
 2739 
 
 2838 
 
 2936 
 
 3035 
 
 46 
 
 45 
 
 46 
 
 847 
 
 3134 
 
 3232 
 
 3331 
 
 3429 
 
 3528 
 
 3627 
 
 3725 
 
 3824 
 
 3922 
 
 4021 
 
 47 
 
 46 
 
 47 
 
 848 
 
 4120 
 
 4218 
 
 4317 
 
 4415 
 
 4514 
 
 4613 
 
 4711 
 
 4810 
 
 4908 
 
 5007 
 
 48 
 
 47 
 
 48 
 
 849 
 
 5106 
 
 5204 
 
 5303 
 
 5401 
 
 5500 
 
 5599 
 
 5697 
 
 5796 
 
 5894 
 
 5993 
 
 49 
 
 48 
 
 49 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 38 
 
OF THE 
 
 TABLE I. 
 
 
 . .,.,,.. 
 
 Pro. 
 
 Log 
 
 g '( +#) 
 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 1-850 
 
 1-856092 
 
 6190 
 
 6289 
 
 6387 
 
 6486 
 
 6585 
 
 6683 
 
 6782 
 
 6881 
 
 6979 
 
 50 
 
 49 
 
 50 
 
 851 
 
 7078 
 
 7176 
 
 7275 
 
 7374 
 
 7472 
 
 7571 
 
 7669 
 
 7768 
 
 7867! 7965 
 
 51 
 
 50 
 
 50 
 
 852 
 
 8064 
 
 8162 
 
 8261 
 
 8360 
 
 8458 
 
 8557 
 
 8656 
 
 8754 
 
 8853 
 
 8951 
 
 5251 
 
 51 
 
 853 
 
 9050 
 
 9149 
 
 9247 
 
 9346 
 
 9444 
 
 9543 
 
 9642 
 
 9740 
 
 9839 
 
 9938 
 
 53 152 
 
 52 
 
 854 
 
 86 0036 
 
 0135 
 
 0233 
 
 0332 
 
 0431 
 
 0529 
 
 0628 
 
 0727 
 
 0825 
 
 0924 
 
 54 
 
 53 
 
 53 
 
 855 
 
 1022 
 
 1121 
 
 1220 
 
 1318 
 
 1417 
 
 1516 
 
 1614 
 
 1713 
 
 1811 
 
 1910 
 
 55 
 
 54 
 
 54 
 
 856 
 
 2009 
 
 2107 
 
 2206 
 
 2305 
 
 2403 
 
 2502 
 
 2600 
 
 2699 
 
 2798 
 
 2896 
 
 56 
 
 55 
 
 55 
 
 857 
 
 2995 
 
 3094 
 
 3192 
 
 3291 
 
 3389 
 
 3488 
 
 3587 
 
 3685 
 
 3784 
 
 3883 
 
 57 
 
 56 
 
 56 
 
 858 
 
 3981 
 
 4080 
 
 4178 
 
 4277 
 
 4376 
 
 4474 
 
 4573 
 
 4672 
 
 4770 
 
 4869 
 
 58 
 
 57 
 
 57 
 
 859 
 
 4968 
 
 5066 
 
 5165 
 
 5263 
 
 5362 
 
 5461 
 
 5559 
 
 5658 
 
 5757 
 
 5855 
 
 59 
 
 58 
 
 58 
 
 860 
 
 86 5954 
 
 6053 
 
 6151 
 
 6250 
 
 6348 
 
 6447 
 
 6546 
 
 6644 
 
 6743 
 
 6842 
 
 60 
 
 59 
 
 59 
 
 861 
 
 6940 
 
 7039 
 
 7138 
 
 7236 
 
 7335 
 
 7434 
 
 7532 
 
 7631 
 
 7729 
 
 7828 
 
 61 
 
 60 
 
 60 
 
 862 
 
 7927 
 
 8025 
 
 8124 
 
 8223 
 
 8321 
 
 8420 
 
 8519 
 
 8617 
 
 8716 
 
 8815 
 
 62'61 
 
 61 
 
 863 
 
 8913 
 
 9012 
 
 9111 
 
 9209 
 
 9308 
 
 9406 
 
 9505 
 
 9604 
 
 9702 
 
 9801 
 
 6362 
 
 62 
 
 864 
 
 9900 
 
 9998 
 
 0097 
 
 0196 
 
 0294 
 
 0393 
 
 0492 
 
 0590 
 
 0689 
 
 0788 
 
 6463 
 
 63 
 
 865 
 
 87 0886 
 
 0985 
 
 1084 
 
 1182 
 
 1281 
 
 1380 
 
 1478 
 
 1577 
 
 1675 
 
 1774 
 
 6564 
 
 64 
 
 866 
 
 1873 
 
 1971 
 
 2070 
 
 2169 
 
 2267 
 
 2366 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 6665 
 
 65 
 
 867 
 
 2859 
 
 2958 
 
 3057 
 
 3155 
 
 3254 
 
 3353 
 
 3451 
 
 3550 
 
 3649 
 
 3747 
 
 67 
 
 66 
 
 66 
 
 868 
 
 3846 3945 
 
 4043 
 
 4142 
 
 4241 
 
 4339 
 
 4438 
 
 4537 
 
 4635 
 
 4734 
 
 68 
 
 67 
 
 67 
 
 869 
 
 4833 
 
 4931 
 
 5030 
 
 5129 
 
 5227 
 
 5326 
 
 5425 
 
 5523 
 
 5622 
 
 5721 
 
 69 
 
 68 
 
 68 
 
 870 
 
 875819 
 
 5918 
 
 6017 
 
 6115 
 
 6214 
 
 6313 
 
 6411 
 
 6510 
 
 6609 
 
 6707 
 
 70 
 
 69 
 
 69 
 
 871 
 
 6806 
 
 6905 
 
 7003 
 
 7102 
 
 7201 
 
 7299 
 
 7398 
 
 7497 
 
 7595 
 
 7694 
 
 71 
 
 70 
 
 70 
 
 872 
 
 7793 
 
 7891 
 
 7990 
 
 8089 
 
 8187 
 
 8286 
 
 8385 
 
 8483 
 
 8582 
 
 8681 
 
 72 
 
 71 
 
 71 
 
 873 
 
 8780 
 
 8878 
 
 8977 
 
 9076 
 
 9174 
 
 9273 
 
 9372 
 
 9470 
 
 9569 
 
 9668 
 
 73 
 
 72 
 
 72 
 
 874 
 
 9766 
 
 9865 
 
 9964 
 
 0062 
 
 0161 
 
 0260 
 
 0358 
 
 0457 
 
 0556 
 
 0654 
 
 74 
 
 73 
 
 73 
 
 875 
 
 88 0753 
 
 0852 
 
 0951 
 
 1049 
 
 1148 
 
 1247 
 
 1345 
 
 1444 
 
 1543 
 
 1641 
 
 75 
 
 73 
 
 74 
 
 876 
 
 1740 
 
 1839 
 
 1937 
 
 2036 
 
 2135 
 
 2233 
 
 2332 
 
 2431 
 
 2529 
 
 2628 
 
 76 
 
 74 
 
 75 
 
 877 
 
 2727 
 
 2826 
 
 2924 
 
 3023 
 
 3122 
 
 3220 
 
 3319 
 
 3418 
 
 3516 
 
 3615 
 
 77 
 
 75 
 
 76 
 
 878 
 
 3714 
 
 3812 
 
 3911 
 
 4010 
 
 4109 
 
 4207 
 
 4306 
 
 4405 
 
 4503 
 
 4602 
 
 78 
 
 76 
 
 77 
 
 879 
 
 4701 
 
 4799 
 
 4898 
 
 4997 
 
 5096 
 
 5194 
 
 5293 
 
 5392 
 
 5490 
 
 5589 
 
 79 
 
 77 
 
 78 
 
 880 
 
 88 5688 
 
 5786 
 
 5885 
 
 5984 
 
 6083 
 
 6181 
 
 6280 
 
 6379 
 
 6477 
 
 6576 
 
 80 
 
 78 
 
 79 
 
 881 
 
 6675 
 
 6773 
 
 6872 
 
 6971 
 
 7070 
 
 7168 
 
 7267 
 
 7366 
 
 7464 
 
 7563 
 
 81 
 
 79 
 
 80 
 
 882 
 
 7662 
 
 7760 
 
 7859 
 
 7958 
 
 8057 
 
 8155 
 
 8254 
 
 8353 
 
 8451 
 
 8550 
 
 82 
 
 80 
 
 81 
 
 883 
 
 8649 
 
 8748 
 
 8846 
 
 8945 
 
 9044 
 
 9142 
 
 9241 
 
 9340 
 
 9438 
 
 9537 
 
 8381 
 
 82 
 
 884 
 
 9636 
 
 9735 
 
 9833 
 
 9932 
 
 0031 
 
 0129 
 
 0228 
 
 0327 
 
 0426 
 
 0524 
 
 8482 
 
 83 
 
 885 
 
 89 0623 
 
 0722 
 
 0820 
 
 0919 
 
 1018 
 
 1117 
 
 1215 
 
 1314 
 
 1413 
 
 1511 
 
 8583 
 
 84 
 
 886 
 
 1610 
 
 1709 
 
 1808 
 
 1906! 2005 
 
 2104 
 
 2202 
 
 230! 
 
 2400 
 
 2499 
 
 8684 
 
 85 
 
 887 
 
 2597 
 
 2696 
 
 2795 
 
 2894 
 
 2992 
 
 3091 
 
 3190 
 
 3288 
 
 3387 
 
 3486 
 
 87 
 
 85 
 
 86 
 
 888 
 
 3585 
 
 3683 
 
 3782 
 
 3881 
 
 3979 
 
 4078 
 
 4177 
 
 4276 
 
 4374 
 
 4473 
 
 88 
 
 86 
 
 87 
 
 889 
 
 4572 
 
 4671 
 
 4769 
 
 4868 
 
 4967 
 
 5065 
 
 5164 
 
 5263 
 
 5362 
 
 5460 
 
 8987 
 
 88 
 
 890 
 
 89 5559 
 
 5658 
 
 5757 
 
 5855 
 
 5954 
 
 6053 
 
 6151 
 
 6250 
 
 6349 
 
 6448 
 
 90 
 
 88 
 
 89 
 
 891 
 
 6546 
 
 6645 
 
 6744 
 
 6843 
 
 6941 
 
 7040 
 
 7139 
 
 7237 
 
 7336 
 
 7435 
 
 91 
 
 89 
 
 90 
 
 892 
 
 7534 
 
 7632 
 
 7731 
 
 7830 
 
 7929 
 
 8027 
 
 8126 
 
 8225 
 
 8324 
 
 8422 
 
 92 
 
 90 
 
 91 
 
 893 
 
 8521 
 
 8620 
 
 8719 
 
 8817 
 
 8916 
 
 9015 
 
 9113 
 
 9212 
 
 9311 
 
 9410 
 
 93 
 
 91 
 
 92 
 
 894 
 
 9508 
 
 9607 
 
 9706 
 
 9805 
 
 9903 
 
 0002 
 
 0101 
 
 0200 
 
 0298 
 
 0397 
 
 94- 
 
 92 
 
 93 
 
 895 
 
 90 0496 
 
 0595 
 
 0693 
 
 0792 
 
 0891 
 
 0990 
 
 1088 
 
 1187 
 
 1286 
 
 1385 
 
 95 
 
 93 
 
 94 
 
 896 
 
 1483 
 
 1582 
 
 1681 
 
 1780 1878 
 
 1977 
 
 2076 
 
 2174 
 
 2273 
 
 2372 
 
 9694 
 
 95 
 
 897 
 
 2471 
 
 2569 
 
 2668 
 
 2767 
 
 2866 
 
 2964 
 
 3063 
 
 3162 
 
 3261 
 
 3359 
 
 97 
 
 95 
 
 96 
 
 898 
 
 3458 
 
 3557 
 
 3656 
 
 3754 
 
 3853 
 
 3952 
 
 4051 
 
 4150 
 
 4248 
 
 4347 
 
 98 
 
 96 
 
 97 
 
 899 
 
 4446 
 
 4545 
 
 4643 
 
 4742 
 
 4841 
 
 4940 
 
 5038 
 
 5137 
 
 5236 
 
 5335 
 
 99 
 
 97 
 
 98 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 ~9~ 
 
 
 98 
 
 99 
 
TABLE I. 
 
 Log 
 
 Log(l+aO 
 
 Pro. 
 Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 1-900 
 
 1-905433 
 
 5532 
 
 5631 
 
 5730 
 
 5828 
 
 5927 
 
 6026 
 
 6125 
 
 6223 
 
 6322 
 
 00 
 
 
 
 
 
 901 
 
 6421 
 
 6520 
 
 6618 
 
 6717 
 
 6816 
 
 6915 
 
 7013 
 
 7112 
 
 7211 
 
 7310 
 
 01 
 
 j 
 
 ] 
 
 902 
 
 7409 
 
 7507 
 
 7606 
 
 7705 
 
 7804 
 
 7902 
 
 8001 
 
 8100 
 
 8199 
 
 8297 
 
 02 
 
 2 
 
 2 
 
 903 
 
 8396 
 
 8495 
 
 8594 
 
 8692 
 
 8791 
 
 8890 
 
 8989 
 
 9088 
 
 9186 
 
 9285 
 
 03 
 
 J 
 
 <! 
 
 904 
 
 9384 
 
 9483 
 
 9581 
 
 9680 
 
 9779 
 
 9878 
 
 9976 
 
 0075 
 
 0174 
 
 0273 
 
 04 
 
 A 
 
 4 
 
 905 
 
 91 0372 
 
 0470 
 
 0569 
 
 0668 
 
 0767 
 
 0865 
 
 0964 
 
 1063 
 
 1162 
 
 1260 
 
 05 
 
 I 
 
 e 
 
 906 
 
 1359 
 
 1458 
 
 1557 
 
 1656 
 
 1754 
 
 1853 
 
 1952 
 
 205 
 
 2149 
 
 2248 
 
 or 
 
 6 
 
 6 
 
 907 
 
 2347 
 
 2446 
 
 2545 
 
 2643 
 
 2742 
 
 2841 
 
 2940 
 
 3038 
 
 3137 
 
 3236 
 
 07 
 
 7 
 
 7 
 
 908 
 
 3335 
 
 3434 
 
 3532 
 
 3631 
 
 3730 
 
 3829 
 
 3927 
 
 4026 
 
 4125 
 
 4224 
 
 08 
 
 8 
 
 8 
 
 909 
 
 4323 
 
 4421 
 
 4520 
 
 4619 
 
 4718 
 
 4816 
 
 4915 
 
 5014 
 
 5113 
 
 5212 
 
 09 
 
 ( 
 
 9 
 
 910 
 
 915310 
 
 5409 
 
 5508 
 
 5607 
 
 5706 
 
 5804 
 
 5903 
 
 6002 
 
 6101 
 
 6199 
 
 10 
 
 10 
 
 10 
 
 911 
 
 6298 
 
 6397 
 
 6496 
 
 6595 
 
 6693 
 
 6792 
 
 6891 
 
 6990 
 
 7089 
 
 7187 
 
 11 
 
 11 
 
 11 
 
 912 
 
 7286 
 
 7385 
 
 7484 
 
 7583 
 
 7681 
 
 7780 
 
 7879 
 
 7978 
 
 8076 
 
 8175 
 
 12 
 
 1 9 
 
 12 
 
 913 
 
 8274 
 
 8373 
 
 8472 
 
 8570 
 
 8669 
 
 8768 
 
 8867 
 
 8966 
 
 9064 
 
 9163 
 
 13 
 
 13 
 
 13 
 
 914 
 
 9262 
 
 9361 
 
 9460 
 
 9558 
 
 9657 
 
 9756 
 
 9855 
 
 9954 
 
 o052 
 
 0151 
 
 14 
 
 14 
 
 14 
 
 915 
 
 92 0250 
 
 0349 
 
 0448 
 
 0546 
 
 0645 
 
 0744 
 
 0843 
 
 0942 
 
 1040 
 
 1139 
 
 15 
 
 1 ^ 
 
 15 
 
 916 
 
 1238 
 
 1337 
 
 1436 
 
 1534 
 
 1633 
 
 1732 
 
 1831 
 
 1930 
 
 2028 
 
 2127 
 
 16 
 
 16 
 
 16 
 
 917 
 
 2226 
 
 2325 
 
 2424 
 
 2522 
 
 2621 
 
 2720 
 
 2819 
 
 2918 
 
 3016 
 
 3115 
 
 17 
 
 17 
 
 17 
 
 918 
 
 3214 
 
 3313 
 
 3412 
 
 3510 
 
 3609 
 
 3708 
 
 3807 
 
 3906 
 
 4005 
 
 4103 
 
 18 
 
 18 
 
 18 
 
 919 
 
 4202 
 
 4301 
 
 4400 
 
 4499 
 
 4597 
 
 4696 
 
 4795 
 
 4894 
 
 4993 
 
 5091 
 
 19 
 
 19 
 
 19 
 
 920 
 
 92 5190 
 
 5289 
 
 5388 
 
 5487 
 
 5585 
 
 5684 
 
 5783 
 
 5882 
 
 5981 
 
 6080 
 
 20 
 
 20 
 
 20 
 
 921 
 
 6178 
 
 6277 
 
 6376 
 
 6475 
 
 6574 
 
 6672 
 
 6771 
 
 6870 
 
 6969 
 
 7068 
 
 21 
 
 21 
 
 21 
 
 922 
 
 7167 
 
 7265 
 
 7364 
 
 7463 
 
 7562 
 
 7661 
 
 7759 
 
 7858 
 
 7957 
 
 8056 
 
 22 
 
 22 
 
 22 
 
 923 
 
 8155 
 
 8254 
 
 8352 
 
 8451 
 
 8550 
 
 8649 
 
 8748 
 
 8846 
 
 8945 
 
 9044 
 
 23 
 
 23 
 
 23 
 
 924 
 
 9143 
 
 9242 
 
 9341 
 
 9439 
 
 9538 
 
 9637 
 
 9736 
 
 9835 
 
 9934 
 
 0032 
 
 24 
 
 24 
 
 24 
 
 925 
 
 930131 
 
 0230 
 
 0329 
 
 0428 
 
 0526 
 
 0625 
 
 0724 
 
 0823 
 
 0922 
 
 1021 
 
 25 
 
 25 
 
 25 
 
 926 
 
 1119 
 
 1218 
 
 1317 
 
 1416 
 
 1515 
 
 1614 
 
 1712 
 
 1811 
 
 1910 
 
 2009 
 
 26 
 
 25 
 
 26 
 
 927 
 
 2108 
 
 2207 
 
 2305 
 
 2404 
 
 2503 
 
 2602 
 
 2701 
 
 2800 
 
 2898 
 
 2997 
 
 27 
 
 26 
 
 27 
 
 928 
 
 3096 
 
 3195 
 
 3294 
 
 3393 
 
 3491 
 
 3590 
 
 3689 
 
 3788 
 
 3887 
 
 3986 
 
 28 
 
 27 
 
 28 
 
 929 
 
 4084 
 
 4183 
 
 4282 
 
 4381 
 
 4480 
 
 4579 
 
 4677 
 
 4776 
 
 4875 
 
 4974 
 
 29 
 
 28 
 
 29 
 
 930 
 
 93 5073 
 
 5172 
 
 5270 
 
 5369 
 
 5468 
 
 5567 
 
 5666 
 
 5765 
 
 5863 
 
 5962 
 
 30 
 
 29 
 
 30 
 
 931 
 
 6061 
 
 6160 
 
 6259 
 
 6358 
 
 645? 
 
 6555 
 
 6654 
 
 6753 
 
 6852 
 
 6951 
 
 81 
 
 30 
 
 31 
 
 932 
 
 7050 
 
 7148 
 
 7247 
 
 7346 
 
 7445 
 
 7544 
 
 7643 
 
 7742 
 
 7840 
 
 7939 
 
 32 
 
 31 
 
 32 
 
 933 
 
 8038 
 
 8137 
 
 8236 
 
 8335 
 
 8433 
 
 8532 
 
 8631 
 
 8730 
 
 8829 
 
 8928 
 
 33 
 
 32 
 
 33 
 
 934 
 
 9027 
 
 9125 
 
 9224 
 
 9323 
 
 9422 
 
 9521 
 
 9620 
 
 9718 
 
 9817 
 
 9916 
 
 34 
 
 33 
 
 34 
 
 935 
 
 940015 
 
 0114 
 
 0213 
 
 0312 
 
 0410 
 
 0509 
 
 0608 
 
 0707 
 
 0806 
 
 0905 
 
 35 
 
 34 
 
 35 
 
 936 
 
 1004 
 
 1102 
 
 1201 
 
 1300 
 
 1399 
 
 1498 
 
 1597 
 
 1696 
 
 1794 
 
 1893 
 
 36 
 
 35 
 
 36 
 
 937 
 
 1992 
 
 2091 
 
 2190 
 
 2289 
 
 2388 
 
 2486 
 
 2585 
 
 2684 
 
 2783 
 
 2882 
 
 37 
 
 36 
 
 37 
 
 938 
 
 2981 
 
 3080 
 
 3178 
 
 3277 
 
 3376 
 
 3475 
 
 3574 
 
 3673 
 
 3772 
 
 3870 
 
 38 
 
 37 
 
 38 
 
 939 
 
 3969 
 
 4068 
 
 4167 
 
 4266 
 
 4365 
 
 4464 
 
 4563 
 
 4661 
 
 4760 
 
 4859 
 
 39 
 
 38 
 
 39 
 
 940 
 
 944958 
 
 5057 
 
 5156 
 
 5255 
 
 5353 
 
 5452 
 
 5551 
 
 5650 
 
 5749 
 
 5848 
 
 40 
 
 3 
 
 40 
 
 941 
 
 5947 
 
 6045 
 
 6144 
 
 6243 
 
 6342 
 
 6441 
 
 6540 
 
 6639 
 
 6738 
 
 6836 
 
 41 
 
 40 
 
 41 
 
 942 
 
 6935 
 
 7034 
 
 7133 
 
 7232 
 
 7331 
 
 7430 
 
 7529 
 
 7627 
 
 7726 
 
 7825 
 
 42 
 
 41 
 
 42 
 
 943 
 
 7924 
 
 8023 
 
 8122 
 
 8221 
 
 8320 
 
 8418 
 
 8517 
 
 8616 
 
 8715 
 
 88J4 
 
 43 
 
 42 
 
 43 
 
 944 
 
 8913 
 
 9012 
 
 9111 
 
 9209 
 
 9308 
 
 9407 
 
 9506 
 
 9605 
 
 9704 
 
 9803 
 
 44 
 
 43 
 
 44 
 
 945 
 
 9902 
 
 oOOO 
 
 0099 
 
 0198 
 
 0297 
 
 0396 
 
 0495 
 
 0594 
 
 0693 
 
 0791 
 
 45 
 
 44 
 
 45 
 
 946 
 
 95 0890 
 
 0989 
 
 1088 
 
 1187 
 
 1286 
 
 1385 
 
 1484 
 
 1582 
 
 1681 
 
 1780 
 
 46 
 
 45 
 
 46 
 
 947 
 
 1879 
 
 1978 
 
 2077 
 
 2176 
 
 2275 
 
 2374 
 
 2472 
 
 2571 
 
 2670 
 
 2769 
 
 47 
 
 46 
 
 47 
 
 948 
 
 2868 
 
 2967 
 
 3066 
 
 3165 
 
 3264 
 
 3362 
 
 3461 
 
 3560 
 
 2659 
 
 3758 
 
 48 
 
 47 
 
 48 
 
 949 
 
 3857 
 
 3956 
 
 4055 
 
 4154 
 
 4252 
 
 4351 
 
 4450 
 
 4549 
 
 4648 
 
 4747 
 
 49 
 
 48 
 
 49 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 40 
 
TABLE I. 
 
 
 Log 
 
 
 Log(l+) 
 
 Pro. 
 Parts. 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 9899 
 
 
 1-950 
 
 1-954846 
 
 4945 
 
 5044 
 
 5142 
 
 5241 
 
 5340 
 
 5439 
 
 5538 
 
 5637 
 
 5736 
 
 50 
 
 49 
 
 50 
 
 
 951 
 
 5835 
 
 5934 
 
 6032 
 
 6131 
 
 6230 
 
 6329 
 
 6428 
 
 6527 
 
 6626 6725 
 
 51 
 
 5050 
 
 
 952 
 
 6824 
 
 6922 
 
 7021 
 
 7120 
 
 7219 
 
 7318 
 
 7417 
 
 7516 
 
 7615 
 
 7714 
 
 525151 
 
 
 953 
 
 7813 
 
 7911 
 
 8010 
 
 8109 
 
 8208 
 
 8307 
 
 8406 
 
 8505 
 
 8604 
 
 8703 
 
 53|52 52 
 
 
 954 
 
 8802 
 
 8900 
 
 8999 
 
 9098 
 
 9197 
 
 9296 
 
 9395 
 
 9494 
 
 9593 
 
 9692 
 
 54 53 53 
 
 
 955 
 
 9791 
 
 9889 
 
 9988 
 
 0087 
 
 0186 
 
 0285 
 
 0384 
 
 0483 
 
 0582 
 
 0681 
 
 55154154 
 
 
 956 
 
 96 0780 
 
 0879 
 
 0977 
 
 1076 
 
 1175 
 
 1274 
 
 1373 
 
 1472 
 
 1571 
 
 1670 
 
 5655i55 
 
 
 957 
 
 1769 
 
 1868 
 
 1966 
 
 2065 
 
 2164 
 
 2263 
 
 2362 
 
 2461 
 
 2560 
 
 2659 
 
 57156 56 
 
 
 958 
 
 2758 
 
 2857 
 
 2956 
 
 3055 
 
 3153 
 
 3252 
 
 3351 
 
 3450 
 
 3549 
 
 3648 
 
 58|57J57 
 
 
 959 
 
 3747 
 
 3846 
 
 3945 
 
 4044 
 
 4143 
 
 4241 
 
 4340 
 
 4439 
 
 4538 
 
 4637 
 
 59 
 
 
 
 960 
 
 96 4736 
 
 4835 
 
 4934 
 
 5033 
 
 5132 
 
 5231 
 
 5330 
 
 5428 
 
 5527 
 
 5626 
 
 60 
 
 5959 
 
 
 961 
 
 5725 
 
 5824 
 
 5923 
 
 6022 
 
 6121 
 
 6220 
 
 6319 
 
 6418 
 
 6517 
 
 6615 
 
 61 
 
 60160 
 
 
 962 
 
 6714 
 
 6813 
 
 6912 
 
 7011 
 
 7110 
 
 7209 
 
 7308 
 
 7407 
 
 7506 
 
 7605 
 
 62 
 
 61 
 
 61 
 
 
 963 
 
 7704 
 
 7803 
 
 7901 
 
 8000 
 
 8099 
 
 8198 
 
 8297 
 
 8396 
 
 8495 
 
 8594 
 
 63 
 
 62 
 
 62 
 
 
 964 
 
 8693 
 
 8792 
 
 8891 
 
 8990 
 
 9089 
 
 9187 
 
 9286 
 
 9385 
 
 9484 
 
 9583 
 
 64 
 
 63 
 
 63 
 
 
 965 
 
 9682 
 
 9781 
 
 9880 
 
 9979 
 
 0078 
 
 0177 
 
 0276 
 
 0375 
 
 o474 
 
 0572 
 
 65 
 
 64 
 
 64 
 
 
 966 
 
 970671 
 
 0770 
 
 0869 
 
 0968 
 
 1067 
 
 1166 
 
 1265 
 
 1364 
 
 1463 
 
 1562 
 
 66 
 
 65 
 
 65 
 
 
 967 
 
 1661 
 
 1760 
 
 1859 
 
 1958 
 
 2056 
 
 2155 
 
 2254 
 
 2353 
 
 2452 
 
 2551 
 
 67 
 
 66 
 
 66 
 
 
 968 
 
 2650 
 
 2749 
 
 2848 
 
 2947 
 
 3046 
 
 3145 
 
 3244 
 
 3343 
 
 3442 
 
 3540 
 
 68J67 
 
 67 
 
 
 969 
 
 3639 
 
 3738 
 
 3837 
 
 3936 
 
 4035 
 
 4134 
 
 4233 
 
 4332 
 
 4431 
 
 4530 
 
 69 
 
 68 
 
 68 
 
 
 970 
 
 97 4629 
 
 4728 
 
 4827 
 
 4926 
 
 5025 
 
 5123 
 
 5222 
 
 5321 
 
 5420 
 
 5519 
 
 70 
 
 69 
 
 69 
 
 
 971 
 
 5618 
 
 5717 
 
 5816 
 
 5915 
 
 6014 
 
 6113 
 
 6212 
 
 6311 
 
 6410 
 
 6509 
 
 71 
 
 70 
 
 70 
 
 
 972 
 
 6608 
 
 6707 
 
 6806 
 
 6904 
 
 7003 
 
 7102 
 
 7201 
 
 7300 
 
 7399 
 
 7498 
 
 72 
 
 71 
 
 71 
 
 
 973 
 
 7597 
 
 7696 
 
 7795 
 
 7894 
 
 7993 
 
 8092 
 
 8191 
 
 8290 
 
 8389 
 
 8488 
 
 73 
 
 72 
 
 72 
 
 
 974 
 
 8587 
 
 8686 
 
 8784 
 
 8883 
 
 8982 
 
 9081 
 
 9180 
 
 9279 
 
 9378 
 
 9477 
 
 74 
 
 73 
 
 73 
 
 
 975 
 
 9576 
 
 9675 
 
 9774 
 
 9873 
 
 9972 
 
 0071 
 
 0170 
 
 0269 
 
 0368 
 
 0467 
 
 75 
 
 73 
 
 74 
 
 
 976 
 
 98 0566 
 
 0665 
 
 0764 
 
 0862 
 
 0961 
 
 1060 
 
 1159 
 
 1258 
 
 1357 
 
 1456 
 
 76 
 
 74 
 
 75 
 
 
 977 
 
 1555 
 
 1654 
 
 1753 
 
 1852 
 
 1951 
 
 2050 
 
 2149 
 
 2248 
 
 2347 
 
 2446 
 
 77 
 
 75 
 
 76 
 
 
 978 
 
 2545 
 
 2644 
 
 2743 
 
 2842 
 
 2941 
 
 3040 
 
 3139 
 
 3237 
 
 3336 
 
 3435 
 
 78 
 
 76 
 
 77 
 
 
 979 
 
 3534 
 
 3633 
 
 3732 
 
 3831 
 
 3930 
 
 4029 
 
 4128 
 
 4227 
 
 4326 
 
 4425 
 
 79 
 
 77 
 
 78 
 
 
 980 
 
 98 4524 
 
 4623 
 
 4722 
 
 4821 
 
 4920 
 
 5019 
 
 5118 
 
 5217 
 
 5316 
 
 5415 
 
 80 
 
 78 
 
 79 
 
 
 981 
 
 5514 
 
 5613 
 
 5712 
 
 5811 
 
 5909 
 
 6008 
 
 6107 
 
 6206 
 
 6305 
 
 6404 
 
 8l|79 
 
 80 
 
 
 982 
 
 6503 
 
 6602 
 
 6701 
 
 6800 
 
 6899 
 
 6998 
 
 7097 
 
 7196 
 
 7295 
 
 7394 
 
 82)80 
 
 81 
 
 
 983 
 
 7493 
 
 7592 
 
 7691 
 
 7790 
 
 7889 
 
 7988 
 
 8087 
 
 8186 
 
 8285 
 
 8384 
 
 8381 
 
 82 
 
 
 984 
 
 8483 
 
 8582 
 
 8681 
 
 8780 
 
 8879 
 
 8978 
 
 9077 
 
 9176 
 
 9275 
 
 9373 
 
 84|82 
 
 83 
 
 
 985 
 
 9472 
 
 9571 
 
 9670 
 
 9769 
 
 9868 
 
 9967 
 
 0066 
 
 0165 
 
 0264 
 
 0363 
 
 85J83 
 
 84 
 
 
 986 
 
 99 0462 
 
 0561 
 
 0660 
 
 0759 
 
 0858 
 
 0957 
 
 1056 
 
 1155 
 
 1254 
 
 1353 
 
 86184 
 
 85 
 
 
 987 
 
 1452 
 
 1551 
 
 1650 
 
 1749 
 
 1848 
 
 1947 
 
 2046 
 
 2145 
 
 2244 
 
 2343 
 
 87 S.> 
 
 86 
 
 
 988 
 
 2442 
 
 2541 
 
 2640 
 
 2739 
 
 2838 
 
 2937 
 
 3036 
 
 3135 
 
 3234 
 
 3333 
 
 6886187 
 
 
 989 
 
 3432 
 
 3531 
 
 3630 
 
 3729 
 
 3828 
 
 3927 
 
 4026 
 
 4125 
 
 4224 
 
 4323 
 
 89 
 
 87:88 
 
 
 990 
 
 99 4422 
 
 4521 
 
 4619 
 
 4718 
 
 4817 
 
 4916 
 
 5015 
 
 5114 
 
 5213 
 
 5312 
 
 90 
 
 88189 
 
 
 991 
 
 5411 
 
 5510 
 
 5609 
 
 5708 
 
 5807 
 
 5906 
 
 6005 
 
 6104 
 
 6203 
 
 6302 
 
 91 
 
 8990 
 
 
 992 
 
 6401 
 
 6500 
 
 6599 
 
 6698 
 
 6797 
 
 6896 
 
 6995 
 
 7094 
 
 7193 
 
 7292 
 
 92 
 
 9091 
 
 
 993 
 
 7391 
 
 7490 
 
 7589 
 
 7688 
 
 7787 
 
 7886 
 
 7985 
 
 8084 
 
 8183 
 
 8282 
 
 93 
 
 91192 
 
 
 994 
 
 8381 
 
 8480 
 
 8579 
 
 8678 
 
 8777 
 
 8876 
 
 8975 
 
 9074 
 
 9173 
 
 9272 
 
 94 
 
 92 
 
 93 
 
 
 995 
 
 9371 
 
 9470 
 
 9569 
 
 9668 
 
 9767 
 
 9866 
 
 9965 
 
 0064 
 
 0163 
 
 0262 
 
 95 
 
 93 
 
 94 
 
 
 996 
 
 2-000361 
 
 0460 
 
 0559 
 
 0658 
 
 0757 
 
 0856 
 
 0955 
 
 1054 
 
 1153 
 
 1252 
 
 96 
 
 94 
 
 95 
 
 
 997 
 
 1351 
 
 1450 
 
 1549 
 
 1648 
 
 1747 
 
 1846 
 
 1945 
 
 2044 
 
 2143 
 
 2242 
 
 97 
 
 95 
 
 96 
 
 
 998 
 
 2341 
 
 2440 
 
 2539 
 
 2638 
 
 2737 
 
 2836 
 
 2935 
 
 3034 
 
 3133 
 
 3232 
 
 98 
 
 96 
 
 97 
 
 'QC 
 
 999 
 
 3331 
 
 3430 
 
 3529 
 
 3628 
 
 3727 
 
 3826 
 
 3925 
 
 4024 
 
 4123 
 
 4222 
 
 99 
 
 97 
 
 98 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 98 
 
 99 
 
 41 
 
TABLE II. 
 
TABLE II. 
 
 Log 
 
 Log (1*) 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3-00 
 
 I 99 9565 
 
 9564 
 
 9563 
 
 9562 
 
 9561 
 
 9560 
 
 9559 
 
 9558 
 
 9557 
 
 9556 
 
 i 
 
 01 
 
 9555 
 
 9554 
 
 9553 
 
 9552 
 
 9551 
 
 9550 
 
 9549 
 
 9548 
 
 9547 
 
 9546 
 
 
 02 
 
 9545 
 
 9544 
 
 9543 
 
 9542 
 
 9541 
 
 9540 
 
 9539 
 
 9538 
 
 9537 
 
 9535 
 
 
 03 
 
 9534 
 
 9533 9532 
 
 9531 
 
 9530 
 
 9529 
 
 9528 
 
 9527J 9526 
 
 9525 
 
 
 04 
 
 9524 
 
 9522 
 
 9521 
 
 9520 
 
 9519 
 
 9518 
 
 9517 
 
 9516 
 
 9515 
 
 9514 
 
 
 05 
 
 9512 
 
 9511 
 
 9510 
 
 9509 
 
 9508 
 
 9507 
 
 9506 
 
 9505 
 
 9503 
 
 9502 
 
 
 06 
 
 9501 
 
 9500 
 
 9499 
 
 9498 
 
 9496 
 
 9495 
 
 9494 
 
 9493 9492 
 
 9491 
 
 
 07 
 
 9489 
 
 9488 
 
 9487 
 
 9486 
 
 9485 
 
 9484 
 
 9482 
 
 9481! 9480 
 
 9479 
 
 
 08 
 
 9478 
 
 9476 
 
 9475 
 
 9474 
 
 9473 
 
 9471 
 
 9470 
 
 9469 
 
 9468 
 
 9467 
 
 
 09 
 
 9465 
 
 9464 
 
 9463 
 
 9462 
 
 9460 
 
 9459 
 
 9458 
 
 9457 
 
 9455 
 
 9454 
 
 
 10 
 
 99 9453 
 
 9452 
 
 9450 
 
 9449 
 
 9448 
 
 9447 
 
 9445 
 
 9444 
 
 9443 
 
 9441 
 
 
 11 
 
 9440 
 
 9439 
 
 9438 
 
 9436 
 
 9435 
 
 9434 
 
 9432 
 
 9431 
 
 9430 
 
 9428 
 
 
 12 
 
 9427 
 
 9426 
 
 9424 
 
 9423 
 
 9422 
 
 9420 
 
 9419 
 
 9418 
 
 9416 
 
 9415 
 
 
 13 
 
 9414 
 
 9412 
 
 9411 
 
 9410 
 
 9408 
 
 9407 
 
 9406 
 
 9404 
 
 9403 
 
 9401 
 
 
 14 
 
 9400 
 
 9399 
 
 9397 
 
 9396 
 
 9395 
 
 9393 
 
 9392 
 
 9390 
 
 9389 
 
 9388 
 
 
 15 
 
 9386 
 
 9385 
 
 9383 
 
 9382 
 
 9380 
 
 9379 
 
 9378 
 
 9376 
 
 9375 
 
 9373 
 
 
 16 
 
 9372 
 
 9370 
 
 9369 
 
 9367 
 
 9366 
 
 9365 
 
 9363 
 
 9362 
 
 9360 
 
 9359 
 
 
 17 
 
 9357 
 
 9356 
 
 9354 
 
 9353 
 
 9351 
 
 9350 
 
 9348 
 
 9347 
 
 9345 
 
 9344 
 
 2 
 
 18 
 
 9342 
 
 9341 
 
 9339 
 
 9338 
 
 9336 
 
 9335 
 
 9333 
 
 9331 
 
 9330 
 
 9328 
 
 
 19 
 
 9327 
 
 9325 
 
 9324 
 
 9322 
 
 9321 
 
 9319 
 
 9317 
 
 9316 
 
 9314 
 
 9313 
 
 
 20 
 
 999311 
 
 9310 
 
 9308 
 
 9306 
 
 9305 
 
 9303 
 
 9302 
 
 9300 
 
 9298 
 
 9297 
 
 
 21 
 
 9295 
 
 9293 
 
 9292 
 
 9290 
 
 9289 
 
 9287 
 
 9285 
 
 9284 
 
 9282 
 
 9280 
 
 
 22 
 
 9279 
 
 9277 
 
 9275 
 
 9274 
 
 9272 
 
 9270 
 
 9269 
 
 9267 
 
 9265 
 
 9264 
 
 
 23 
 
 9262 
 
 9260 
 
 9258 
 
 9257 
 
 9255 
 
 9253 
 
 9252 
 
 9250 
 
 9248 
 
 9246 
 
 
 24 
 
 9245 
 
 9243 
 
 9241 
 
 9239 
 
 9238 
 
 9236 
 
 9234 
 
 9232 
 
 9231 
 
 9229 
 
 
 25 
 
 9227 
 
 9225 
 
 9223 
 
 9222 
 
 9220 
 
 9218 
 
 9216 
 
 9214 
 
 9213 
 
 9211 
 
 
 26 
 
 9209 
 
 9207 
 
 9205 
 
 9204 
 
 9202 
 
 9200 
 
 9198 
 
 9196 
 
 9194 
 
 9192 
 
 
 27 
 
 9191 
 
 9189 
 
 9187 
 
 9185 
 
 9183 
 
 9181 
 
 9179 
 
 9177 
 
 9175 
 
 9174 
 
 
 28 
 
 9172 
 
 9170 
 
 9168 
 
 9166 
 
 9164 
 
 9162 
 
 9160 
 
 9158 
 
 9156 
 
 9154 
 
 
 29 
 
 9152 
 
 9150 
 
 9148 
 
 9146 
 
 9145 
 
 9143 
 
 9141 
 
 9139 
 
 9137 
 
 9135 
 
 
 30 
 
 99 9133 
 
 9131 
 
 9129 
 
 9127 
 
 9125 
 
 9123 
 
 9121 
 
 9118 
 
 9116 
 
 9114 
 
 
 31 
 
 9112 
 
 9110 
 
 9108 
 
 9106 
 
 9104 
 
 9102 
 
 9100 
 
 9098 
 
 9096 
 
 9094 
 
 
 32 
 
 9092 
 
 9090 
 
 9087 
 
 9085 
 
 9083 
 
 9081 
 
 9079 
 
 9077 
 
 9075 
 
 9073 
 
 
 33 
 
 9071 
 
 9068 
 
 9066 
 
 9064 
 
 9062 
 
 9060 
 
 9058 
 
 9055 
 
 9053 
 
 9051 
 
 
 34 
 
 9049 
 
 9047 
 
 9044 
 
 9042 
 
 9040 
 
 9038 
 
 9036 
 
 9033 
 
 9031 
 
 9029 
 
 
 35 
 
 9027 
 
 9024 
 
 9022 
 
 9020 
 
 9018 
 
 9015 
 
 9013 
 
 9011 
 
 9009 
 
 9006 
 
 
 36 
 
 9004 
 
 9002 
 
 8999 
 
 8997 
 
 8995 
 
 8992 
 
 8990 
 
 8988 
 
 8985 
 
 8983 
 
 
 37 
 
 8981 
 
 8978 
 
 8976 
 
 8974 
 
 8971 
 
 8969 
 
 8967 
 
 8964 
 
 8962 
 
 8959 
 
 
 38 
 
 8957 
 
 8955 
 
 8952 
 
 8950 
 
 8947 
 
 8945 
 
 8942 
 
 8940 
 
 8938 
 
 8935 
 
 
 39 
 
 8933 
 
 8930 
 
 8928 
 
 8925 
 
 8923 
 
 8920 
 
 8918 
 
 8915 
 
 8913 
 
 8910 
 
 3 
 
 40 
 
 99 8908 
 
 8905 
 
 8903 
 
 8900 
 
 8898 
 
 8895 
 
 8893 
 
 8890 
 
 8887 
 
 8885 
 
 
 41 
 
 8882 
 
 8880 
 
 8877 
 
 8874 
 
 8872 
 
 8869 
 
 8867 
 
 8864 
 
 8861 
 
 8859 
 
 
 42 
 
 8856 
 
 8854 
 
 8851 
 
 8848 
 
 8846 
 
 8843 
 
 8840 
 
 8838 
 
 8835 
 
 8832 
 
 
 43 
 
 8830 
 
 8827 
 
 8824 
 
 8821 
 
 8819 
 
 8816 
 
 8813 
 
 8810 
 
 8808 
 
 8805 
 
 
 44 
 
 8802 
 
 8799 
 
 8797 
 
 8794 
 
 8791 
 
 8788 
 
 8786 
 
 8783 
 
 8780 
 
 8777 
 
 
 45 
 
 8774 
 
 8771 
 
 8769 
 
 8766 
 
 8763 
 
 8760 
 
 8757 
 
 8754 
 
 8751 
 
 8749 
 
 
 46 
 
 8746 
 
 8743 
 
 8740 
 
 8737 
 
 8734 
 
 8731 
 
 8728 
 
 8725 
 
 8722 
 
 8719 
 
 
 47 
 
 8716 
 
 8713 
 
 8710 
 
 8707 
 
 8705 
 
 8702 
 
 8699 
 
 8696 
 
 8693 
 
 8689 
 
 
 48 
 
 8686 
 
 8683 
 
 8680 
 
 8677 
 
 8674 
 
 8671 
 
 8668 
 
 8665 
 
 8662 
 
 8659 
 
 
 49 
 
 8656 
 
 8653 
 
 8650 
 
 8646 
 
 8643 
 
 8640 
 
 8637 
 
 8634 
 
 8631 
 
 8628 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 43 
 
TABLE II. 
 
 Log 
 
 
 Log (1 a-) 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3-50 
 
 1-99 8624 
 
 8621 
 
 8618 
 
 8615 
 
 8612 
 
 8609 
 
 8605 
 
 8602 
 
 8599 
 
 8596 
 
 
 51 
 
 8592 
 
 8589 
 
 8586 
 
 8583 
 
 8579 
 
 8576 
 
 8573 
 
 8569 
 
 8566 
 
 8563 
 
 
 52 
 
 8560 
 
 8556 
 
 8553 
 
 8550 
 
 8546 
 
 8543 
 
 8539 
 
 8536 
 
 8533 
 
 8529 
 
 
 53 
 
 8526 
 
 8523 
 
 8519 
 
 8516 
 
 8512 
 
 8509 
 
 8505 
 
 8502 
 
 8498 
 
 8495 
 
 
 54 
 
 8492 
 
 8488 
 
 8485 
 
 8481 
 
 8478 
 
 8474 
 
 8471 
 
 8467 
 
 8463 
 
 8460 
 
 4 
 
 55 
 
 8456 
 
 8453 
 
 8449 
 
 8446 
 
 8442 
 
 8438 
 
 8435 
 
 8431 
 
 8428 
 
 8424 
 
 
 56 
 
 8420 
 
 8417 
 
 8413 
 
 8409 
 
 8406 
 
 8402 
 
 8398 
 
 8395 
 
 8391 
 
 8387 
 
 
 57 
 
 8383 
 
 8380 
 
 8376 
 
 8372 
 
 8368 
 
 8365 
 
 8361 
 
 8357 
 
 8353 
 
 8350 
 
 
 58 
 
 8346 
 
 8342 
 
 8338 
 
 8334 
 
 8330 
 
 8327 
 
 8323 
 
 8319 
 
 8315 
 
 8311 
 
 
 59 
 
 8307 
 
 8303 
 
 8299 
 
 8295 
 
 8291 
 
 8287 
 
 8284 
 
 8280 
 
 8276 
 
 8272 
 
 
 60 
 
 99 8268 
 
 8264 
 
 8260 
 
 8256 
 
 8252 
 
 8247 
 
 8243 
 
 8239 
 
 8235 
 
 8231 
 
 
 61 
 
 8227 
 
 8223 
 
 8219 
 
 8215 
 
 8211 
 
 8207 
 
 8202 
 
 8198 
 
 8194 
 
 8190 
 
 
 62 
 
 8186 
 
 8182 
 
 8177 
 
 8173 
 
 8169 
 
 8165 
 
 8160 
 
 8156 
 
 8152 
 
 8148 
 
 
 63 
 
 8143 
 
 8139 
 
 8135 
 
 8131 
 
 8126 
 
 8122 
 
 8118 
 
 8113 
 
 8109 
 
 8104 
 
 
 64 
 
 8100 
 
 8096 
 
 8091 
 
 8087 
 
 8082 
 
 8078 
 
 8074 
 
 8069 
 
 8065 
 
 8060 
 
 
 65 
 
 8056 
 
 8051 
 
 8047 
 
 8042 
 
 8038 
 
 8033 
 
 8029 
 
 8024 
 
 8020 
 
 8015 
 
 5 
 
 66 
 
 8010 
 
 8006 
 
 8001 
 
 7997 
 
 7992 
 
 7987 
 
 7983 
 
 7978 
 
 7973 
 
 7969 
 
 
 67 
 
 7964 
 
 7959 
 
 7954 
 
 7950 
 
 7945 
 
 7940 
 
 7935 
 
 7931 
 
 7926 
 
 7921 
 
 
 68 
 
 7916 
 
 7912 
 
 7907 
 
 7902 
 
 7897 
 
 7892 
 
 7887 
 
 7882 
 
 7878 
 
 7873 
 
 
 69 
 
 7868 
 
 7863 
 
 7858 
 
 7853 
 
 7848 
 
 7843 
 
 7838 
 
 7833 
 
 7828 
 
 7823 
 
 
 70 
 
 99 7818 
 
 7813 
 
 7808 
 
 7803 
 
 7798 
 
 7793 
 
 7787 
 
 7782 
 
 7777 
 
 7772 
 
 
 71 
 
 7767 
 
 7762 
 
 7757 
 
 7751 
 
 7746 
 
 7741 
 
 7736 
 
 7731 
 
 7725 
 
 7720 
 
 
 72 
 
 7715 
 
 7710 
 
 7704 
 
 7699 
 
 7694 
 
 7688 
 
 7683 
 
 7678 
 
 7672 
 
 7667 
 
 
 73 
 
 7661 
 
 7656 
 
 7651 
 
 7645 
 
 7640 
 
 7634 
 
 7629 
 
 7623 
 
 7618 
 
 7612 
 
 
 74 
 
 7607 
 
 7601 
 
 7596 
 
 7590 
 
 7585 
 
 7579 
 
 7573 
 
 7568 
 
 7562 
 
 7557 
 
 3 
 
 75 
 
 7551 
 
 7545 
 
 7540 
 
 7534 
 
 7528 
 
 7522 
 
 7517 
 
 7511 
 
 7505 
 
 7499 
 
 
 76 
 
 7494 
 
 7488 
 
 7482 
 
 7476 
 
 7470 
 
 7465 
 
 7459 
 
 7453 
 
 7447 
 
 7441 
 
 
 77 
 
 7435 
 
 7429 
 
 7423 
 
 7417 
 
 7411 
 
 7405 
 
 7399 
 
 7393 
 
 7387 
 
 7381 
 
 
 78 
 
 7375 
 
 7369 
 
 7363 
 
 7357 
 
 7351 
 
 7345 
 
 7339 
 
 7332 
 
 7326 
 
 7320 
 
 
 79 
 
 7314 
 
 7308 
 
 7301 
 
 7295 
 
 7289 
 
 7283 
 
 7276 
 
 7270 
 
 7264 
 
 7257 
 
 
 .80 
 
 99 7251 
 
 7245 
 
 7238 
 
 7232 
 
 7226 
 
 7219 
 
 7213 
 
 7206 
 
 7200 
 
 7193 
 
 
 81 
 
 7187 
 
 7180 
 
 7174 
 
 7167 
 
 7161 
 
 7154 
 
 7148 
 
 7141 
 
 7134 
 
 7128 
 
 7 
 
 82 
 
 7121 
 
 7114 
 
 7108 
 
 7101 
 
 7094 
 
 7088 
 
 7081 
 
 7074 
 
 7067 
 
 7061 
 
 
 83 
 
 7054 
 
 7047 
 
 7040 
 
 7033 
 
 7026 
 
 7020 
 
 7013 
 
 7006 
 
 6999 
 
 6992 
 
 
 84 
 
 6985 
 
 6978 
 
 6971 
 
 6964 
 
 6957 
 
 6950 
 
 6943 
 
 6936 
 
 6929 
 
 6922 
 
 
 85 
 
 6914 
 
 6907 
 
 6900 
 
 6893 
 
 6886 
 
 6879 
 
 6871 
 
 6864 
 
 6857 
 
 6850 
 
 
 86 
 
 6842 
 
 6835 
 
 6828 
 
 6820 
 
 6813 
 
 6806 
 
 6798 
 
 6791 
 
 6783 
 
 6776 
 
 
 87 
 
 6769 
 
 6761 
 
 6754 
 
 6746 
 
 6739 
 
 6731 
 
 6723 
 
 6716 
 
 6708 
 
 6701 
 
 8 
 
 88 
 
 6693 
 
 6685 
 
 6678 
 
 6670 
 
 6662 
 
 6655 
 
 6647 
 
 6639 
 
 6631 
 
 6623 
 
 
 89 
 
 6616 
 
 6608 
 
 6600 
 
 6592 
 
 6584 
 
 6576 
 
 6568 
 
 6560 
 
 6552 
 
 6544 
 
 
 90 
 
 99 6537 
 
 6528 
 
 6520 
 
 6512 
 
 6504 
 
 6496 
 
 6488 
 
 6480 
 
 6472 
 
 6464 
 
 
 91 
 
 6455 
 
 6447 
 
 6439 
 
 6431 
 
 6423 
 
 6414 
 
 6406 
 
 6398 
 
 6389 
 
 6381 
 
 
 92 
 
 6373 
 
 6364 
 
 6356 
 
 6347 
 
 6339 
 
 6330 
 
 6322 
 
 6313 
 
 6305 
 
 6296 
 
 1 
 
 93 
 
 6288 
 
 6279 
 
 6271 
 
 6262 
 
 6253 
 
 6245 
 
 6236 
 
 6227 
 
 6218 
 
 6210 
 
 
 94 
 
 6201 
 
 6192 
 
 6183 
 
 6174 
 
 6166 
 
 6157 
 
 6148 
 
 6139 
 
 6130 
 
 6121 
 
 
 95 
 
 6112 
 
 6103 
 
 6094 
 
 6085 
 
 6076 
 
 6067 
 
 6058 
 
 6049 
 
 6040 
 
 6030 
 
 
 96 
 
 6021 
 
 6012 
 
 6003 
 
 5993 
 
 5984 
 
 5975 
 
 5965 
 
 5956 
 
 5947 
 
 5937 
 
 
 97 
 
 5928 
 
 5918 
 
 5909 
 
 5900 
 
 5890 
 
 5881 
 
 5871 
 
 5861 
 
 5852 
 
 5842 
 
 
 
 98 
 
 5833 
 
 5823 
 
 5813 
 
 5804 
 
 5794 
 
 5784 
 
 5774 
 
 5765 
 
 5755 
 
 5745 
 
 
 99 
 
 5735 
 
 5725 
 
 5715 
 
 5705 
 
 5695 
 
 5685 
 
 5675 
 
 5665 
 
 5655 
 
 5645 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 6 
 
 7 
 
 8 
 
 9 
 
 ). 
 
 44 
 
TABLE II. 
 
 Log 
 
 Lpg(l-) 
 
 Biff. 
 
 Pro. Parts. 
 
 i" 
 
 1 1 
 2 2 
 3 3 
 4 4 
 5 6 
 6 7 
 7 8 
 8 9 
 910 
 
 12 
 
 1 
 2 
 4 
 5 
 6 
 7 
 8 
 10 
 11 
 
 13 
 
 I 
 3 
 4 
 5 
 
 7 
 8 
 9 
 10 
 12 
 
 X 
 
 
 
 1 
 
 2 3 
 
 5615 5605 
 5512 5502 
 5407! 5397 
 5300; 5289 
 5190 5178 
 
 5077 5065 
 4961 4950 
 4843 4831 
 4723 4710 
 4599| 4586 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2-00 
 01 
 02 
 03 
 04 
 
 05 
 06 
 07 
 08 
 09 
 
 1-99 5635 
 5533 
 5428 
 5321 
 5212 
 
 5100 
 4985 
 4867 
 4747 
 4624 
 
 5625 
 5523 
 5418 
 5310 
 5201 
 
 5088 
 4973 
 4855 
 4735 
 4611 
 
 5595 
 5491 
 5386 
 5278 
 5167 
 
 5054 
 4938 
 4820 
 4698 
 4574 
 
 5584 
 5481 
 5375 
 5267 
 5156 
 
 5043 
 4926 
 4807 
 4686 
 4561 
 
 5574 
 5471 
 5364 
 5256 
 5145 
 
 5031 
 4915 
 4795 
 4673 
 4549 
 
 5564 
 5460 
 5354 
 5245 
 5134 
 
 5019 
 4903 
 4783 
 4661 
 4536 
 
 5554 
 5450 
 5343 
 5234 
 5122 
 
 5008 
 4891 
 4771 
 4649 
 4523 
 
 5543 
 5439 
 5332 
 5223 
 5111 
 
 4996 
 4879 
 4759 
 4636 
 4511 
 
 11 
 
 12 
 13 
 
 14 
 
 15 
 16 
 
 17 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 23 
 
 24 
 
 25 
 26 
 
 27 
 28 
 
 29 
 30 
 
 31 
 
 32 
 33 
 
 3 
 
 i i 
 
 2 3 
 3 4 
 4 6 
 5 7 
 6 8 
 7 10 
 811 
 913 
 
 15,16 
 
 "11 
 5 5 
 
 6 6 
 8 8 
 910 
 
 nil 
 
 1213 
 1414 
 
 10 
 11 
 12 
 13 
 14 
 
 15 
 16 
 1? 
 18 
 19 
 
 99 4498 
 4369 
 4237 
 4102 
 3963 
 
 3822 
 3677 
 3528 
 3376 
 3221 
 
 4485 
 4356 
 4223 
 
 4088 
 3949 
 
 3807 
 3662 
 3513 
 3361 
 3205 
 
 4472 
 4343 
 4210 
 4074 
 3935 
 
 3793 
 3647 
 3498 
 3346 
 3189 
 
 4459 
 4330 
 4197 
 4060 
 3921 
 
 3779 
 3633 
 3483 
 3330 
 3174 
 
 4447 
 4316 
 4183 
 4047 
 3907 
 
 3764 
 3618 
 3468 
 3315 
 3158 
 
 4434 
 4303 
 4170 
 4033 
 3893 
 
 3750 
 3603 
 3453 
 3299 
 3142 
 
 4421 
 4290 
 4156 
 4019 
 3879 
 
 3735 
 3588 
 3438 
 3284 
 3126 
 
 4408 
 4277 
 4143 
 4005 
 3865 
 
 3721 
 3573 
 3422 
 3268 
 3110 
 
 4395 
 4263 
 4129 
 3991 
 3850 
 
 3706 
 3558 
 3407 
 3252 
 3094 
 
 4382 
 4250 
 4115 
 3977 
 3836 
 
 3691 
 3543 
 3392 
 3237 
 3078 
 
 17J1819 
 
 1 222 
 2 3' 4 ! 4 
 3 5! 5 6 
 47 78 
 5 9 910 
 6 1011 11 
 7 121313 
 8141415 
 9 15116 17 
 
 20 
 21 
 
 22 
 23 
 24 
 
 25 
 26 
 
 27 
 28 
 29 
 
 99 3062 
 2899 
 2732 
 2561 
 2386 
 
 2208 
 2024 
 1837 
 1645 
 1448 
 
 3046 
 
 2882 
 2715 
 2544 
 2369 
 
 2189 
 2006 
 1818 
 1625 
 1428 
 
 3029 
 2866 
 2698 
 2527 
 2351 
 
 2171 
 1987 
 1799 
 1606 
 1408 
 
 3013 
 2849 
 2681 
 2509 
 2333 
 
 2153 
 1969 
 
 1780 
 1586 
 1388 
 
 2997 
 2833 
 2664 
 2492 
 2315 
 
 2135 
 1950 
 1761 
 1567 
 1368 
 
 2981 
 2816 
 2647 
 2474 
 2298 
 
 2116 
 1931 
 1741 
 1547 
 1348 
 
 2964 
 2799 
 2630 
 2457 
 2280 
 
 2098 
 1912 
 1722 
 1527 
 1328 
 
 2948 
 
 2782 
 2613 
 2439 
 
 2262 
 
 2080 
 1894 
 1703 
 1508 
 1308 
 
 2932 
 2766 
 2596 
 2422 
 2244 
 
 2061 
 
 1875 
 1684 
 1488 
 1288 
 
 2915 
 
 2749 
 2579 
 2404 
 2226 
 
 2043 
 1856 
 1664 
 1468 
 1267 
 
 20 
 
 1 2 
 2 4 
 3 6 
 4 8 
 5 10 
 612 
 7 14 
 8 16 
 9 is 
 
 2122 
 
 2~2 
 4 4 
 6 7 
 8 9 
 11 11 
 1313 
 1515 
 1718 
 1920 
 
 30 
 31 
 32 
 33 
 34 
 
 35 
 36 
 37 
 
 38 
 39 
 
 99 1247 
 1041 
 0830 
 0614 
 0393 
 
 0167 
 98 9935 
 9698 
 9455 
 9206 
 
 1227 
 1020 
 0809 
 0592 
 0371 
 
 0144 
 9912 
 9674 
 9430 
 9181 
 
 1206 
 0999 
 0787 
 0570 
 0348 
 
 0121 
 
 9888 
 9650 
 9406 
 9156 
 
 1186 
 0978 
 0766 
 0549 
 0326 
 
 0098 
 9865 
 9626 
 9381 
 9131 
 
 1165 
 0957 
 0744 
 0526 
 0303 
 
 0075 
 9841 
 9601 
 9356 
 9105 
 
 1145 
 
 0936 
 0723 
 0504 
 0281 
 
 0052 
 9817 
 9577 
 9331 
 9080 
 
 1124 1103 
 0915 0894 
 070 1| 0680 
 0482 0460 
 0258 0235 
 
 0029 0005 
 9793 9770 
 9553 9528 
 9307 9282 
 9054 9029 
 
 1083 
 0873 
 0658 
 0438 
 0213 
 
 9982 
 9746 
 9504 
 9257 
 9003 
 
 1062 
 0851 
 0636 
 0416 
 0190 
 
 9959 
 9722 
 9480 
 9231 
 8977 
 
 23 
 
 1|2 
 2 5 
 3 7 
 4 9 
 
 512 
 6 14 
 716 
 
 818 
 921 
 
 26 
 
 3 
 2 1 5 
 3 8 
 410 
 5 13 
 616 
 7'18 
 821 
 923 
 
 24 
 
 12 
 5 
 7 
 10 
 12 
 14 
 7 
 19 
 22 
 
 mmm 
 
 27 
 
 3 
 5 
 8 
 11 
 
 14 
 16 
 19 
 22 
 24 
 
 25 
 
 1 
 5 
 
 8 
 10 
 13 
 15 
 
 18 
 20 
 
 23 
 
 28 
 
 3 
 6 
 8 
 11 
 14 
 17 
 20 
 22 
 25 
 
 40 
 41 
 42 
 43 
 44 
 
 45 
 46 
 47 
 
 48 
 49 
 
 98 8952 
 8691 
 8424 
 8151 
 
 7871 
 
 7584 
 7291 
 6990 
 6682 
 6367 
 
 8926 
 8665 
 8397 
 8123 
 7842 
 
 7555 
 7261 
 6960 
 6651 
 6335 
 
 8900 
 8638 
 8370 
 8095 
 
 7814 
 
 7526 
 7231 
 6929 
 6620 
 6303 
 
 8874 
 8612 
 8343 
 8067 
 
 7785 
 
 7497 
 7201 
 6899 
 6589 
 6271 
 
 8848 
 8585 
 8315 
 8039 
 
 7757 
 
 7468 
 7171 
 6868 
 6557 
 6239 
 
 8822 
 8558 
 8288 
 8012 
 7728 
 
 7438 
 7141 
 6837 
 6526 
 6207 
 
 8796 
 8532 
 8261 
 7983 
 7700 
 
 7409 
 7111 
 6806 
 6494 
 6175 
 
 8770 
 8505 
 8233 
 7955 
 7671 
 
 7379 
 7081 
 6776 
 6463 
 6142 
 
 8744 
 8478 
 8206 
 7927 
 7642 
 
 7350 
 7051 
 6745 
 6431 
 6110 
 
 8717 
 8451 
 8178 
 7899 
 7613 
 
 7320 
 7020 
 6714 
 6399 
 
 6077 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. Parts. 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(l~*) 
 
 Diff 
 
 Pro. 
 
 Parts. 
 
 
 
 1 
 
 6012 
 5681 
 5341 
 4994 
 4638 
 
 4274 
 3901 
 3518 
 3127 
 2726 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 | 
 
 29 
 
 3 
 6 
 9 
 12 
 15 
 17 
 20 
 
 i 
 
 3C 
 
 3 
 
 e 
 
 
 12 
 15 
 16 
 21 
 24 
 27 
 
 31 
 
 f 
 12 
 
 16 
 19 
 22 
 
 M 
 
 2-50 
 51 
 52 
 53 
 54 
 
 55 
 56 
 57 
 
 58 
 59 
 
 T-98 6045 
 5714 
 5376 
 5029 
 4674 
 
 4311 
 3938 
 3557 
 3167 
 2767 
 
 5979 
 5647 
 5307 
 4959 
 4602 
 
 4237 
 3863 
 3480 
 3087 
 2685 
 
 5946 
 5613 
 5273 
 4923 
 4566 
 
 4200 
 3825 
 3441 
 3048 
 2645 
 
 5913 
 5580 
 5238 
 4888 
 4530 
 
 4163 
 
 3787 
 3402 
 3008 
 2604 
 
 5880 
 5546 
 5203 
 4853 
 4493 
 
 4126 
 3749 
 3363 
 2968 
 2563 
 
 5847 
 5512 
 5169 
 4817 
 4457 
 
 4088 
 3711 
 3324 
 
 2928 
 2522 
 
 5814 
 5478 
 5134 
 
 478 
 442 
 
 405 
 367 f 
 
 328c 
 2886 
 248 
 
 5781 
 5444 
 5099 
 4746 
 4384 
 
 4014 
 3634 
 3245 
 
 2847 
 2440 
 
 5747 
 5410 
 5064 
 4710 
 4347 
 
 3976 
 3596 
 3206 
 
 2807 
 2398 
 
 34 
 35 
 36 
 37 
 
 38 
 
 39 
 40 
 41 
 
 42 
 43 
 44 
 45 
 46 
 47 
 49 
 50 
 51 
 52 
 
 53 
 55 
 56 
 
 58 
 59 
 
 60 
 62 
 63 
 65 
 67 
 
 68 
 70 
 72 
 73 
 75 
 
 77 
 79 
 81 
 
 83 
 85 
 
 87 
 90 
 92 
 94 
 97 
 
 99 
 102 
 104 
 107 
 109 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 
 1 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 9 
 
 32 
 
 c 
 
 6 
 10 
 13 
 16 
 IP 
 22 
 26 
 29 
 
 33 
 
 3 
 7 
 10 
 13 
 
 17 
 20 
 23 
 26 
 30 
 
 34 
 
 1 
 14 
 
 5 
 
 27 
 31 
 
 60 
 61 
 62 
 63 
 64 
 
 65 
 66 
 67 
 68 
 69 
 
 98 2357 
 1937 
 1507 
 1067 
 0616 
 
 0154 
 97 9681 
 9196 
 8699 
 8191 
 
 2315 
 
 1895 
 1464 
 1023 
 0570 
 
 0107 
 9633 
 9147 
 8649 
 8139 
 
 2274 
 1852 
 1420 
 0978 
 0525 
 
 0060 
 9585 
 9098 
 8599 
 8087 
 
 2232 
 1809 
 1376 
 0933 
 0479 
 
 0013 
 9537 
 9048 
 
 8548 
 8036 
 
 2190 
 1767 
 1333 
 
 0888 
 0433 
 
 9966 
 9488 
 8999 
 8497 
 7984 
 
 2148 
 1724 
 
 1289 
 0843 
 0387 
 
 9919 
 9440 
 8949 
 8447 
 7932 
 
 2106 
 1681 
 1245 
 0798 
 0340 
 
 9872 
 9391 
 8900 
 8396 
 7879 
 
 2064 
 163 r 
 120C 
 0753 
 0294 
 
 9824 
 9343 
 8850 
 834o 
 
 7827 
 
 2022 
 1594 
 1156 
 0707 
 0247 
 
 9776 
 9294 
 8800 
 8293 
 
 7775 
 
 1980 
 155 
 1112 
 0662 
 0201 
 
 9729 
 9245 
 
 8750 
 8242 
 
 7722 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 35 
 
 4 
 7 
 11 
 14 
 
 18 
 21 
 25 
 
 28 
 32 
 
 36 
 
 4 
 
 7 
 11 
 14 
 
 18 
 22 
 25 
 29 
 32 
 
 37 
 
 1 
 
 19 
 22 
 26 
 30 
 33 
 
 70 
 71 
 
 72 
 73 
 74 
 
 75 
 76 
 
 77 
 78 
 79 
 
 97 7669 
 7135 
 6588 
 6027 
 5453 
 
 4864 
 4261 
 3643 
 3010 
 2361 
 
 7616 
 7081 
 6533 
 5971 
 5395 
 
 4805 
 4200 
 3580 
 2945 
 2295 
 
 7564 
 7027 
 6477 
 5914 
 5336 
 
 4745 
 4139 
 3518 
 
 2881 
 2229 
 
 7510 
 6972 
 6421 
 5856 
 
 5278 
 
 4685 
 4077 
 3455 
 2816 
 2163 
 
 7457 
 6918 
 6365 
 5799 
 5219 
 
 4625 
 4016 
 3391 
 
 2752 
 2096 
 
 7404 
 6863 
 6309 
 5742 
 5160 
 
 4564 
 3454 
 3328 
 2687 
 2030 
 
 7350 
 6809 
 6253 
 5684 
 5101 
 
 4504 
 3892 
 3265 
 2622 
 1963 
 
 7297 
 6754 
 6197 
 5627 
 5042 
 
 4,J./Lt 
 
 7243 
 6699 
 6141 
 5569 
 4983 
 
 4383 
 3768 
 3137 
 2492 
 1830 
 
 7189 
 6643 
 6084 
 5511 
 4924 
 
 4322 
 3706 
 3074 
 2426 
 1763 
 
 
 38 
 
 39 
 
 40 
 
 3830 
 3201 
 2557 
 1896 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 4 
 
 8 
 11 
 15 
 19 
 
 23 
 27 
 30 
 34 
 
 4 
 
 8 
 12 
 16 
 
 20 
 23 
 27 
 31 
 35 
 
 8 
 12 
 16 
 
 20 
 24 
 28 
 32 
 36 
 
 80 
 81 
 
 82 
 83 
 .84 
 
 85 
 86 
 87 
 88 
 89 
 
 97 1695 
 1013 
 0315 
 96 9598 
 8864 
 
 8112 
 7340 
 6550 
 5739 
 4908 
 
 1628 
 0944 
 0244 
 9526 
 8790 
 
 8035 
 7262 
 6469 
 5657 
 4823 
 
 1560 
 0875 
 0173 
 9453 
 8715 
 
 7959 
 7184 
 6389 
 5574 
 4739 
 
 1492 
 0806 
 0102 
 9380 
 8640 
 
 7882 
 7105 
 6308 
 5492 
 4654 
 
 1424 
 0736 
 0030 
 9307 
 8565 
 
 7805 
 7026 
 6228 
 5409 
 4569 
 
 1356 
 0666 
 9959 
 9234 
 8490 
 
 7728 
 6947 
 6147 
 5326 
 4484 
 
 1288 
 0596 
 9887 
 9160 
 8415 
 
 7651 
 6868 
 6065 
 5243 
 4399 
 
 1220 
 0526 
 
 9815 
 9086 
 8339 
 
 7574 
 6789 
 5984 
 5159 
 4313 
 
 1151 
 0456 
 9743 
 9013 
 8264 
 
 7496 
 6709 
 5903 
 5076 
 
 4228 
 
 1082 
 0385 
 9671 
 8939 
 
 8188 
 
 7418 
 6629 
 5821 
 4992 
 4142 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 41 
 
 4 
 
 8 
 12 
 16 
 21 
 25 
 29 
 33 
 37 
 
 42 
 
 4 
 8 
 13 
 17 
 21 
 25 
 29 
 34 
 38 
 
 43 
 
 4 
 c 
 
 13 
 17 
 
 22 
 26 
 30 
 34 
 39 
 
 90 
 91 
 92 
 93 
 94 
 
 95 
 96 
 97 
 98 
 99 
 
 96 4055 
 3182 
 2286 
 1367 
 0425 
 
 95 9459 
 8468 
 7451 
 6409 
 5340 
 
 3969 
 3093 
 2195 
 1274 
 0329 
 
 9361 
 8367 
 7348 
 6303 
 5231 
 
 3882 
 3004 
 2104 
 1181 
 0234 
 
 9263 
 8267 
 7245 
 6197 
 5122 
 
 3796 
 2915 
 2013 
 1087 
 0138 
 
 9164 
 8166 
 7141 
 6091 
 5013 
 
 3709 
 2826 
 1921 
 0993 
 0041 
 
 9065 
 8064 
 7038 
 5985 
 4904 
 
 3621 
 
 2737 
 1829 
 0899 
 9945 
 8966 
 7963 
 6933 
 5878 
 4794 
 
 3534 
 
 2647 
 1737 
 
 0805 
 9848 
 
 8867 
 7861 
 6829 
 5771 
 4685 
 
 3446 
 2557 
 1645 
 0710 
 9751 
 
 8768 
 7759 
 6725 
 5663 
 4574 
 
 3358 
 2467 
 1553 
 0615 
 9654 
 
 8668 
 7657 
 6620 
 5556 
 4464 
 
 3270 
 2376 
 1460 
 0520 
 9556 
 
 8568 
 7554 
 6514 
 5448 
 4353 
 
 1 
 2 
 3 
 4 
 5 
 6' 
 7; 
 SL 
 94 
 
 44 
 
 4 
 9 
 13 
 18 
 22 
 16 
 $1 
 55; 
 
 to^ 
 
 45 
 
 5 
 9 
 14 
 18 
 23 
 >7 
 J2 
 56 
 H 
 
 46 
 
 5 
 9 
 14 
 
 18 
 23 
 28 
 32 
 J7 
 I! 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pr. Parts. 
 
 46 
 
PROPORTIONAL PARTS. 
 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 H 
 
 1 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 5 
 
 6 
 
 6 
 
 6 
 
 6 
 
 2 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 3 
 
 14 
 
 14 
 
 14 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 17 
 
 17 
 
 4 
 
 18 
 
 19 
 
 19 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 22 
 
 22 
 
 22 
 
 23 
 
 23 
 
 5 
 
 23 
 
 24 
 
 24 
 
 25 
 
 25 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 29 
 
 29 
 
 6 
 
 28 
 
 28 
 
 29 
 
 29 
 
 30 
 
 31 
 
 31 
 
 32 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 7 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 36 
 
 36 
 
 37 
 
 38 
 
 39 
 
 39 
 
 40 
 
 41 
 
 8 
 
 37 
 
 38 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 46 
 
 9 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 50 
 
 51 
 
 52 
 
 
 
 59 
 ~T 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 71 
 
 2 
 3 
 
 o 
 12 
 18 
 
 12 
 
 18 
 
 12 
 18 
 
 12 
 19 
 
 13 
 19 
 
 13 
 19 
 
 13 
 
 20 
 
 13 
 
 20 
 
 13 
 
 20 
 
 14 
 
 20 
 
 14 
 21 
 
 14 
 21 
 
 14 
 21 
 
 4 
 
 24 
 
 24 
 
 24 
 
 25 
 
 25 
 
 26 
 
 26 
 
 26 
 
 27 
 
 27 
 
 28 
 
 28 
 
 28 
 
 5 
 
 30 
 
 30 
 
 31 
 
 31 
 
 32 
 
 32 
 
 33 
 
 33 
 
 34 
 
 34 
 
 35 
 
 35 
 
 36 
 
 6 
 
 35 
 
 36 
 
 37 
 
 37 
 
 38 
 
 38 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 43 
 
 7 
 
 41 
 
 42 
 
 43 
 
 43 
 
 44 
 
 45 
 
 46 
 
 46 
 
 47 
 
 48 
 
 48 
 
 49 
 
 50 
 
 8 
 
 47 
 
 48 
 
 49 
 
 50 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 54 
 
 55 
 
 56 
 
 57 
 
 9 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 59 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 1 
 
 7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 2 
 
 14 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 17 
 
 17 
 
 3 
 
 22 
 
 22 
 
 22 
 
 23 
 
 23 
 
 23 
 
 23 
 
 24 
 
 24 
 
 24 
 
 25 
 
 25 
 
 25 
 
 4 
 
 29 
 
 29 
 
 30 
 
 30 
 
 30 
 
 31 
 
 31 
 
 32 
 
 32 
 
 32 
 
 33 
 
 33 
 
 34 
 
 5 
 
 36 
 
 37 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 6 
 
 43 
 
 44 
 
 44 
 
 45 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 49 
 
 49 
 
 50 
 
 50 
 
 7 
 
 50 
 
 51 
 
 52 
 
 53 
 
 53 
 
 54 
 
 55 
 
 55 
 
 56 
 
 57 
 
 57 
 
 58 
 
 59 
 
 8 
 
 58 
 
 58 
 
 59 
 
 60 
 
 61 
 
 62 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 
 
 66 
 
 67 
 
 9 
 
 65 
 
 66 
 
 67 
 
 68 
 
 68 
 
 69 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 
 85 
 
 86 
 
 87 
 
 88 
 
 89 
 
 90 
 
 91 
 
 92 
 
 93 
 
 94 
 
 95 
 
 96 
 
 97 
 
 1 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 ~~9~ 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 10 
 
 2 
 
 17 
 
 17 
 
 17 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 19 
 
 19 
 
 19 
 
 3 
 
 26 
 
 26 
 
 26 
 
 26 
 
 27 
 
 27 
 
 27 
 
 28 
 
 28 
 
 28 
 
 29 
 
 29 
 
 29 
 
 4 
 
 34 
 
 34 
 
 35 
 
 35 
 
 36 
 
 36 
 
 36 
 
 37 
 
 37 
 
 38 
 
 38 
 
 38 
 
 39 
 
 5 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 48 
 
 49 
 
 6 
 
 51 
 
 52 
 
 52 
 
 53 
 
 53 
 
 54 
 
 55 
 
 55 
 
 56 
 
 56 
 
 57 
 
 58 
 
 58 
 
 7 
 
 60 
 
 60 
 
 61 
 
 62 
 
 62 
 
 63 
 
 64 
 
 64 
 
 65 
 
 66 
 
 67 
 
 67 
 
 68 
 
 8 
 
 68 
 
 69 
 
 70 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 9 
 
 77 
 
 77 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 85 
 
 86 
 
 86 
 
 87 
 
 
 98 
 
 99 
 
 100 
 
 101 
 
 102 
 
 103 
 
 104 
 
 105 
 
 106 
 
 107 
 
 108 
 
 109 
 
 110 
 
 1 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 2 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 22 
 
 22 
 
 22 
 
 3 
 
 29 
 
 30 
 
 30 
 
 30 
 
 31 
 
 31 
 
 31 
 
 32 
 
 32 
 
 32 
 
 32 
 
 33 
 
 33 
 
 4 
 
 39 
 
 40 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 5 
 
 49 
 
 50 
 
 50 
 
 51 
 
 51 
 
 52 
 
 52 
 
 53 
 
 53 
 
 54 
 
 54 
 
 55 
 
 55 
 
 6 
 
 59 
 
 59 
 
 60 
 
 61 
 
 61 
 
 62 
 
 62 
 
 63 
 
 64 
 
 64 
 
 65 
 
 65 
 
 66 
 
 7 
 
 69 
 
 69 
 
 70 
 
 71 
 
 71 
 
 72 
 
 73 
 
 74 
 
 74 
 
 75 
 
 76 
 
 76 
 
 77 
 
 8 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 82 
 
 83 
 
 84 
 
 85 
 
 86 
 
 86 
 
 87 
 
 88 
 
 9 
 
 88 
 
 89 
 
 90 
 
 91 
 
 92 
 
 93 
 
 94 
 
 95 
 
 95 
 
 96 
 
 97 
 
 98 
 
 99 
 
 [The Table is continued on the following pages. The portion which 
 succeeds has the common difference in the argument column, one-tenth 
 of that in the preceding portion of the Table,] 
 
 47 
 
TABLE II. 
 
 Log 
 
 Log (I-*) 
 
 
 Pro. 
 
 & 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 9 
 
 Diff. 
 
 Parts. 
 
 i-ooo 
 
 1-95 4243 
 
 4231 
 
 4220 
 
 4209 
 
 4198 
 
 4187 
 
 4176 
 
 4165 
 
 4154 
 
 4142 
 
 11 
 
 
 
 001 
 
 4131 
 
 4120 
 
 4109 
 
 4098 
 
 4087 
 
 4076 
 
 4064 
 
 4053 
 
 4042 
 
 4031 
 
 
 
 
 002 
 003 
 
 4020 
 3908 
 
 4009 
 3897 
 
 3997 
 
 3885 
 
 3986 
 3874 
 
 3975 
 3863 
 
 3964 
 3852 
 
 3953 3941 
 3841 3829 
 
 3930 
 3818 
 
 3919 
 
 3807 
 
 
 
 
 004 
 
 3796 
 
 3785 
 
 3773 
 
 3762 
 
 3751 
 
 3740 
 
 3728 
 
 3717 
 
 3706 
 
 3695 
 
 
 
 
 005 
 
 3683 
 
 3672 
 
 3661 
 
 3650 
 
 3638 
 
 3627 
 
 3616 
 
 3605 
 
 3593 
 
 3582 
 
 
 
 
 006 
 
 3571 
 
 3559 
 
 3548 
 
 3537 
 
 3526 
 
 3514 
 
 3503 
 
 3492 
 
 3480 
 
 3469 
 
 
 U 
 
 007 
 
 3458 3446 
 
 3435 
 
 3424 
 
 3412 
 
 3401 
 
 3390 
 
 3378 
 
 3367 
 
 3356 
 
 
 1 
 
 ] 
 
 008 
 
 3344| 3333 
 
 3322 
 
 3310 
 
 3299 
 
 3288 
 
 3276 
 
 3265 
 
 3254 
 
 3242 
 
 
 2 
 
 2 
 
 009 
 
 3231 3220 
 
 3208 
 
 3197 
 
 3185 
 
 3174 
 
 3163 
 
 3151 
 
 3140 
 
 3128 
 
 
 3 
 
 ; 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 4 
 
 010 
 
 953117 3106 
 
 3094 
 
 3083 
 
 3071 
 
 3060 
 
 3049 
 
 3037 
 
 3026 
 
 3014 
 
 
 5 
 
 6 
 
 on 
 
 3003 2991 
 
 2980 
 
 2969 
 
 2957 
 
 2946 
 
 2934 
 
 2923 
 
 2911 
 
 2900 
 
 
 6 
 
 7 
 
 012 
 
 2888 
 
 2877 
 
 2866 
 
 2854 
 
 2843 
 
 2831 
 
 2820 
 
 2808 
 
 2797 
 
 2785 
 
 
 7 
 
 8 
 
 013 
 
 2774 2762 
 
 2751 
 
 2739 
 
 2728 
 
 2716 
 
 2705 
 
 2693 
 
 2682 
 
 2670 
 
 12 
 
 8 
 
 c 
 
 014 
 
 2659 2647 
 
 2636 
 
 2624 
 
 2613 
 
 2601 
 
 2590 
 
 2578 
 
 2566 
 
 2555 
 
 
 9 
 
 
 
 10 
 
 mmm 
 
 015 
 
 2543 
 
 2532 
 
 2520 
 
 2509 
 
 2497 
 
 2486 
 
 2474 
 
 2463 
 
 2451 
 
 2439 
 
 
 
 
 016 
 
 2428 
 
 2416 
 
 2405 
 
 2393 
 
 2381 
 
 2370 
 
 2358 
 
 2347 
 
 2335 
 
 2323 
 
 
 
 
 017 
 
 2312 
 
 2300 
 
 2289 
 
 2277 
 
 2265 
 
 2254 
 
 2242 
 
 2231 
 
 2219 
 
 2207 
 
 
 
 
 018 
 
 2196 
 
 2184 
 
 2172 
 
 2161 
 
 2149 
 
 2137 
 
 2126 
 
 2114 
 
 2102 
 
 2091 
 
 
 
 
 019 
 
 2079 
 
 2067 
 
 2056 
 
 2044 
 
 2032 
 
 2021 
 
 2009 
 
 1997 
 
 1986 
 
 1974 
 
 
 
 
 020 
 
 95 1962 
 
 i o/i r 
 
 1951 
 
 1 OO,I 
 
 1939 
 
 1 QOO 
 
 1927 
 
 1 O 1 /k 
 
 1916 
 
 1 C'AO 
 
 1904 
 
 1 '7Q r 7 
 
 1892 
 
 1 77?i 
 
 1880 
 
 1 7fJ*7 
 
 1869 
 
 1 1*1 K. 1 
 
 1857 
 
 1 ri A f\ 
 
 
 12 
 
 021 
 022 
 
 1845 
 1728 
 
 1834 
 1716 
 
 loss 
 
 1704 
 
 1810 
 1693 
 
 1 <yo 
 1681 
 
 I/O/ 
 
 1669 
 
 1 i tO 
 
 1657 
 
 1 <O.J 
 
 1645 
 
 1 131 
 
 1634 
 
 1740 
 1622 
 
 
 1 
 
 
 023 
 
 1610 
 
 1598 
 
 1587 
 
 1575 
 
 1563 
 
 1551 
 
 1539 
 
 1528 
 
 1516 
 
 1504 
 
 
 2 
 3 
 
 
 024 
 
 1492 
 
 1480 
 
 1468 
 
 1457 
 
 1445 
 
 1433 
 
 1421 
 
 1409 
 
 1397 
 
 1386 
 
 
 4 
 
 5 
 
 025 
 
 1374 
 
 1362 
 
 1350 
 
 1338 
 
 1326 
 
 1314 
 
 1303 
 
 1291 
 
 1279 
 
 1267 
 
 
 5 
 
 6 
 
 026 
 
 1255 
 
 1243 
 
 1231 
 
 1219 
 
 1208 
 
 1196 
 
 1184 
 
 1172 
 
 1160 
 
 1148 
 
 
 6 
 
 
 027 
 
 1136 
 
 1124 
 
 1112 
 
 1100 
 
 1089 
 
 1077 
 
 1065 
 
 1053 
 
 1041 
 
 1029 
 
 
 7 
 
 
 028 
 
 1017 
 
 1005 
 
 0993 
 
 0981 
 
 0969 
 
 0957 
 
 0945 
 
 0933 
 
 0921 
 
 0909 
 
 
 8 
 9 
 
 10 
 1 
 
 029 
 
 0897 
 
 0885 
 
 0873 
 
 0861 
 
 0850 
 
 0838 
 
 0826 
 
 0814 
 
 0802 
 
 0790 
 
 
 
 
 
 
 030 
 
 95 0778 
 
 0766 
 
 0754 
 
 0742 
 
 0730 
 
 0718 
 
 0706 
 
 0693 
 
 0681 
 
 0669 
 
 
 
 
 031 
 
 0657 
 
 0645 
 
 0633 
 
 0621 
 
 0609 
 
 0597 
 
 0585 
 
 0573 
 
 0561 
 
 0549 
 
 
 
 
 032 
 
 0537 
 
 0525 
 
 0513 
 
 0501 
 
 0489 
 
 0477 
 
 0464 
 
 0452 
 
 0440 
 
 0428 
 
 
 
 
 033 
 
 0416 
 
 0404 
 
 0392 
 
 0380 
 
 0368 
 
 0356 
 
 0344 
 
 0331 
 
 0319 
 
 0307 
 
 
 
 
 034 
 
 0295 
 
 0283 
 
 0271 
 
 0259 
 
 0247 
 
 0234 
 
 0222 
 
 0210 
 
 0198 
 
 0186 
 
 
 
 
 035 
 
 0174 
 
 A A r O 
 
 0161 
 
 AA/I A 
 
 0149 
 
 AAQQ 
 
 0137 
 
 AA 1 K. 
 
 0125 
 
 AAAQ 
 
 0113 
 
 oQQl 
 
 0101 
 
 nQ7Q 
 
 0088 
 
 nQflK 
 
 0076 
 
 nO Pi /I 
 
 0064 
 
 nft 1 O 
 
 
 13 
 
 036 
 037 
 
 (j(JOZ 
 
 94 9930 
 
 UU*U 
 
 9918 
 
 (J(Jco 
 9905 
 
 OOlO 
 
 9893 
 
 UUUo 
 
 9881 
 
 yyy i 
 9869 
 
 yy /y 
 9856 
 
 yyoo 
 9844 
 
 9yo4? 
 9832 
 
 9yvi 
 
 9820 
 
 
 1 
 
 
 038 
 
 9807 
 
 9795 
 
 9783 
 
 9771 
 
 9758 
 
 9746 
 
 9734 
 
 9722 
 
 9709 
 
 9697 
 
 
 2 
 3 
 
 * 
 
 039 
 
 9685 
 
 9673 
 
 9660 
 
 9648 
 
 9636 
 
 9623 
 
 9611 
 
 9599 
 
 9586 
 
 9574 
 
 
 4 
 
 5 
 
 040 
 
 94 9562 
 
 9550 
 
 9537 
 
 9525 
 
 9513 
 
 9500 
 
 9488 
 
 9476 
 
 9463 
 
 9451 
 
 
 5 
 
 6 
 
 8 
 
 041 
 
 9439 
 
 9426 
 
 9414 
 
 9401 
 
 9389 
 
 9377 
 
 9364 
 
 9352 
 
 9340 
 
 9327 
 
 
 7 
 
 
 0*2 
 
 9315 
 
 9303 
 
 9290 
 
 9278 
 
 9265 
 
 9253 
 
 9241 
 
 9228 
 
 9216 
 
 9203 
 
 
 8 
 
 1( 
 
 043 
 
 9191 
 
 9179 
 
 9166 
 
 9154 
 
 9141 
 
 9129 
 
 9116 
 
 9104 
 
 9092 
 
 9079 
 
 
 9 
 
 J2 
 
 044 
 
 9067 
 
 9054 
 
 9042 
 
 9029 
 
 9017 
 
 9004 
 
 8992 
 
 8979 
 
 8967 
 
 8955 
 
 
 
 
 045 
 
 8942 
 
 8930 
 
 8917 
 
 8905 
 
 8892 
 
 8880 
 
 8867 
 
 8855 
 
 8842 
 
 8830 
 
 13 
 
 
 
 046 
 
 8817 
 
 8805 
 
 8792 
 
 8780 
 
 8767 
 
 8755 
 
 8742 
 
 8730 
 
 8717 
 
 8704 
 
 
 
 
 047 
 
 8692 
 
 8679 
 
 8667 
 
 8654 
 
 8642 
 
 8629 
 
 8617 
 
 8604 
 
 8591 
 
 8579 
 
 
 
 
 048 
 
 856G 
 
 8554 
 
 8541 
 
 8529 
 
 8516 
 
 8503 
 
 8491 
 
 8478 
 
 8466 
 
 8453 
 
 
 
 
 049 
 
 8440 
 
 8428 
 
 8415 
 
 8403 8390 
 
 8377 
 
 8365 
 
 .8352 
 
 8340 
 
 8327 
 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 48 
 
TABLE II. 
 
 Log 
 
 Log .(*-*) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff 
 
 Parts. 
 
 1-050 
 
 1-948314 
 
 8302 
 
 8289 
 
 8276 
 
 8264 
 
 8251 
 
 8238 
 
 8226 
 
 8213 
 
 8200 
 
 13 
 
 
 
 051 
 
 8188 
 
 8175 
 
 8162 
 
 8150 
 
 8137 
 
 8124 
 
 8112 8099 
 
 8086 
 
 8074 
 
 
 
 
 052 
 
 8061 
 
 8048 
 
 8035 
 
 8023 
 
 8010 
 
 7997 
 
 7985 7972 
 
 7959 
 
 7946 
 
 
 
 
 053 
 
 7934 
 
 7921 7908 
 
 7895 
 
 7883 
 
 7870 
 
 7857 7844 
 
 7832 
 
 7819 
 
 
 
 
 054 
 
 7806 
 
 7793 7781 
 
 7768 
 
 7755 
 
 7742 
 
 7729 7717 
 
 7704 7691 
 
 
 
 
 055 
 
 7678 
 
 7665 ! 7653 
 
 7640 
 
 7627 
 
 7614 
 
 7601 7589 
 
 7576 7563 
 
 
 
 
 056 
 
 7550 
 
 7537) 7524 
 
 7511 
 
 7499 
 
 7486 
 
 7473 7460 
 
 7447 i 7434 
 
 
 13 
 
 057 
 
 7421 
 
 7409 7396 
 
 7383 
 
 7370 
 
 7357 
 
 7344: 7331 
 
 7318 7306 
 
 
 
 
 
 058 
 
 7293 
 
 7280 726? 
 
 7254 
 
 7241 
 
 7228 
 
 7215 
 
 7202 
 
 7189 7176 
 
 
 1 
 
 ; 
 
 059 
 
 7163 
 
 7150! 7138 
 
 7125 
 
 7112 
 
 7099 
 
 7086 
 
 7073 
 
 7060 
 
 7047 
 
 
 3 
 
 I 
 
 060 
 
 94 7034 
 
 7021 
 
 7008 
 
 6995 
 
 6982 
 
 6969 
 
 6956 
 
 6943 
 
 6930 6917 
 
 
 
 
 
 
 061 
 
 6904 
 
 6891 
 
 6878 6865 
 
 6852 
 
 6839 
 
 68261 6813 
 
 6800 6787 
 
 
 6 
 
 8 
 
 062 
 
 6774 6761 
 
 6748 6735 
 
 6722 
 
 6709 
 
 6695 6682 
 
 6669 6656 
 
 
 7 
 
 ( 
 
 063 
 
 6643 
 
 6630 
 
 6617 
 
 6604 
 
 6591 
 
 6578 
 
 6565 655-2 
 
 6539 6525 
 
 
 8 
 
 10 
 
 064 
 
 6512 
 
 6499 
 
 6486 
 
 6473 
 
 6460 
 
 6447 
 
 6434! 6420 
 
 6407 
 
 6394 
 
 
 9 
 
 mam 
 
 12 
 
 mm 
 
 065 
 
 6381 
 
 6368 
 
 6355 
 
 6342 
 
 6328 
 
 6315 
 
 6302 6289 
 
 6276 6263 
 
 
 
 
 066 
 
 6250 
 
 6236 
 
 6223 
 
 6210 
 
 6197 
 
 6184 
 
 6170 6157 
 
 6144 6131 
 
 
 
 
 067 
 
 6118 
 
 6104 
 
 6091 6078 
 
 6065 
 
 6052 
 
 6038 6025 
 
 6012 
 
 5999 
 
 
 
 
 068 
 
 5985 
 
 5972 
 
 5959! 5946 
 
 5932 
 
 5919 
 
 5906 5893 
 
 5879 
 
 5866 
 
 
 
 
 069 
 
 5853 
 
 5839 
 
 5826 5813 
 
 5800 
 
 5786 
 
 5773 
 
 5760 
 
 5746 
 
 5733 
 
 
 
 
 070 
 
 94 5720 
 
 5706 
 
 5693 5680 
 
 5666 
 
 5653 
 
 5640 
 
 5626 
 
 5613 
 
 5600 
 
 
 14 
 
 071 
 
 5586 
 
 5573 5560 5546 
 
 5533 
 
 5520 
 
 5506 5493 
 
 5480 
 
 5466 
 
 
 
 
 072 
 
 5453 
 
 5439 5426 5413 
 
 5399 
 
 5386 
 
 5372 5359 
 
 5346 
 
 5332 
 
 
 1 
 
 } 
 
 073 
 
 5319 
 
 5305 5292! 5279 
 
 5265 
 
 5252 
 
 5238 5225 
 
 5211 
 
 5198 
 
 
 * * 
 
 074 
 
 5184 
 
 5171 
 
 5158 5144 
 
 5131 
 
 5117 
 
 5104 5090 
 
 5077 
 
 5063 
 
 
 3 
 
 4 
 
 4 
 
 6 
 
 075 
 
 5050 
 
 5036 
 
 5023 
 
 5009 
 
 4996 
 
 4982 
 
 4969 
 
 4955 
 
 4942 
 
 4928 
 
 14 
 
 5 
 
 7 
 
 076 
 
 4915 
 
 4901 
 
 4888 
 
 4874 
 
 4861 
 
 4847 
 
 4833 
 
 4820 
 
 4806 
 
 4793 
 
 
 6 
 
 8 
 
 077 
 
 4779 
 
 4766 
 
 4752 
 
 4739 
 
 4725 
 
 4711 
 
 4698 
 
 4684 
 
 4671 
 
 4657 
 
 
 7 10 
 
 078 
 
 4644 
 
 4630 
 
 4616 
 
 4603 
 
 4589 
 
 4575 
 
 4562 
 
 4548 
 
 4535 
 
 4521 
 
 
 8 
 
 11 
 
 079 
 
 4507 
 
 4494 
 
 4480 
 
 4466 
 
 4453 
 
 4439 
 
 4426 
 
 4412 
 
 4398 4385 
 
 
 9 
 
 JHHBI 
 
 13 
 
 M 
 
 080 
 
 944371 
 
 4357 
 
 4344 
 
 4330 
 
 4316 
 
 4303 
 
 4289 
 
 4275 
 
 4261 
 
 4248 
 
 
 
 
 081 
 
 4234 
 
 4220 
 
 4207 
 
 4193 
 
 4179 
 
 4166 4152 4138 
 
 4124J4111 
 
 
 
 
 082 
 
 4097 
 
 4083 
 
 4069 
 
 4056 
 
 4042 
 
 4028 
 
 4014 
 
 4001 
 
 3 087 3973 
 
 
 
 
 083 
 
 3959 
 
 3946 
 
 3932 
 
 3918 
 
 3904 
 
 3890 
 
 3877 
 
 38G3 
 
 3849 3835 
 
 
 
 
 084 
 
 3821 
 
 3808 
 
 3794 
 
 3780 
 
 3766 
 
 3752 
 
 3738 
 
 3725 
 
 3711 3697 
 
 
 
 
 085 
 
 3683 
 
 3669 
 
 3655 
 
 3642 
 
 3628 
 
 3614 3600 
 
 3586 
 
 3572 3558 
 
 
 15 
 
 086 
 
 3544 
 
 3531 
 
 3517 
 
 3503 
 
 3489 
 
 3475 
 
 3461 
 
 3447 
 
 3433 3419 
 
 
 
 
 
 
 087 
 
 3405 
 
 3392 
 
 3378 
 
 3364 
 
 3350 
 
 3336 
 
 3322 
 
 3308 
 
 3294 3280 
 
 
 1 
 
 2 
 
 088 
 
 3266 
 
 3252 
 
 3238 
 
 3224 
 
 3210 
 
 3196 
 
 3182 
 
 3168 
 
 3154 3140 
 
 
 2 
 
 3 
 
 089 
 
 3126 
 
 3112 
 
 3098 
 
 3084 
 
 3070 
 
 3056 
 
 3042 
 
 3028 
 
 3014 3000 
 
 
 4 
 
 o 
 
 6 
 
 090 
 
 94 2986 
 
 2972 
 
 2958 
 
 2944 
 
 2930 
 
 291612902 
 
 2888 
 
 2874 2860 
 
 
 5 
 6 
 
 8 
 9 
 
 091 
 
 2846 2832 
 
 2818 
 
 2804 
 
 2790 
 
 2775 2761 
 
 2747 
 
 2733 2719 
 
 
 7 
 
 11 
 
 092 
 
 2705 2691 
 
 2677 
 
 2663 
 
 2649 
 
 2634 2620 2606 
 
 2592 2578 
 
 
 8 
 
 12 
 
 093 
 
 2564 
 
 2550 
 
 2535 
 
 2521 
 
 2507 
 
 2493 2479 2465 
 
 2451 
 
 2436 
 
 
 9 
 
 14 
 
 094 
 
 2422 
 
 2408 
 
 2394 2380 
 
 2365 
 
 235 1 12337 
 
 2323 
 
 2309 
 
 2294 
 
 
 
 am 
 
 
 
 
 
 
 
 
 
 
 
 
 095 
 
 2280 
 
 2266 
 
 2252 2238 
 
 2223 
 
 2209 2195 2181 
 
 2166 2152 
 
 
 
 
 096 
 
 2138 
 
 2124 
 
 2109 2095 
 
 2081 
 
 2067 2052 
 
 2038 
 
 2024 
 
 2009 
 
 
 
 
 097 
 
 1995 
 
 1981 
 
 1967 
 
 1952 
 
 1938 
 
 1924 1909 
 
 1895 
 
 1881 
 
 1866 
 
 
 
 
 098 
 
 1852 
 
 1838 
 
 1823 
 
 1809 
 
 1795 
 
 1780 
 
 1766 
 
 1752 
 
 1737 
 
 1723 
 
 
 
 
 099 
 
 1709 
 
 1694 
 
 1680 
 
 1666 
 
 1651 
 
 1637 
 
 1622 
 
 1608 
 
 1594 
 
 1579 
 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 IT 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 
 Parts. 
 
 49 
 
TABLE II. 
 
 Log 
 
 Log (l-x) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Parts. 
 
 1-100 
 
 1-94 1565 
 
 1550 
 
 1536 
 
 1522 
 
 1507 
 
 1493 
 
 1478 
 
 1464 
 
 1449 
 
 1435 
 
 
 
 101 
 
 1421 
 
 1406 
 
 1392 
 
 1377 
 
 1363 
 
 1348 
 
 1334 
 
 1319 
 
 1305 
 
 1290 
 
 
 
 102 
 
 1276 
 
 1262 
 
 1247 
 
 1233 
 
 1218 
 
 1204 
 
 1189 
 
 1175 
 
 1160 
 
 1146 
 
 15 
 
 
 103 
 
 1131 
 
 1117 
 
 1102 
 
 1087 
 
 1073 
 
 1058 
 
 1044 
 
 1029 
 
 1015 
 
 1000 
 
 
 14 
 
 104 
 
 0986 
 
 0971 
 
 0957 
 
 0942 
 
 0927 
 
 0913 
 
 0898 
 
 0884 
 
 0869 
 
 0855 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 105 
 
 0840 
 
 0825 
 
 0811 
 
 0796 
 
 0782 
 
 0767 
 
 0752 
 
 0738 
 
 0723 
 
 0708 
 
 
 2 
 
 3 
 
 106 
 
 0694 
 
 0679 
 
 0665 
 
 0650 
 
 0635 
 
 0621 
 
 0606 
 
 0591 
 
 0577 
 
 0562 
 
 
 a 
 
 4 
 
 107 
 
 0547 
 
 0533 
 
 0518 
 
 0503 
 
 0489 
 
 0474 
 
 0459 
 
 0445 
 
 0430 
 
 0415 
 
 
 4 
 
 6 
 
 108 
 
 0400 
 
 0386 
 
 0371 
 
 0356 
 
 0342 
 
 0327 
 
 0312 
 
 0297 
 
 0283 
 
 0268 
 
 
 5 
 
 7 
 
 109 
 
 0253 
 
 0238 
 
 0224 
 
 0209 
 
 0194 
 
 0179 
 
 0165 
 
 0150 
 
 0135 
 
 0120 
 
 
 6 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 10 
 
 110 
 
 940105 
 
 0091 
 
 0076 
 
 0061 
 
 0046 
 
 0031 
 
 0017 
 
 0002 
 
 9987 
 
 9972 
 
 
 8 
 
 11 
 
 111 
 
 93 9957 
 
 9943 
 
 9928 
 
 9913 
 
 9898 
 
 9883 
 
 9868 
 
 9853 
 
 9839 
 
 9824 
 
 
 9 
 
 ^i 
 
 13 
 
 MB 
 
 112 
 
 9809 
 
 9794 
 
 9779 
 
 9764 
 
 9749 
 
 9735 
 
 9720 
 
 9705 
 
 9690 
 
 9675 
 
 
 
 113 
 
 9660 
 
 9645 
 
 9630 
 
 9615 
 
 9600 
 
 9585 
 
 9571 
 
 9556 
 
 9541 
 
 9526 
 
 
 
 114 
 
 9511 
 
 9496 
 
 9481 
 
 9466 
 
 9451 
 
 9436 
 
 9421 
 
 9406 
 
 9391 
 
 9376 
 
 
 
 115 
 
 9361 
 
 9346 
 
 9331 
 
 9316 
 
 9301 
 
 9286 
 
 9271 
 
 9256 
 
 9241 
 
 9226 
 
 
 15 
 
 116 
 
 9211 
 
 9196 
 
 9181 
 
 9166 
 
 9151 
 
 9136 
 
 9121 
 
 9106 
 
 9091 
 
 9076 
 
 
 j 
 
 2 
 
 117 
 
 9061 
 
 9046 
 
 9031 
 
 9015 
 
 9000 
 
 8985 
 
 8970 
 
 8955 
 
 8940 
 
 8925 
 
 
 2 
 
 3 
 
 118 
 
 8910 
 
 8895 
 
 8880 
 
 8864 
 
 8849 
 
 8834 
 
 8819 
 
 8804 
 
 8789 
 
 8774 
 
 
 3 
 
 
 119 
 
 8759 
 
 8743 
 
 8728 
 
 8713 
 
 8698 
 
 8683 
 
 8668 
 
 8652 
 
 8637 
 
 8622 
 
 
 4 
 
 6 
 
 120 
 
 93 8607 
 
 8592 
 
 8577 
 
 8561 
 
 8546 
 
 8531 
 
 8516 
 
 8501 
 
 8485 
 
 8470 
 
 
 5 
 
 6 
 
 8 
 9 
 
 121 
 
 8455 
 
 8440 
 
 8424 
 
 8409 
 
 8394 
 
 8379 
 
 8363 
 
 8348 
 
 8333 
 
 8318 
 
 
 7 
 
 11 
 
 122 
 
 8302 
 
 8287 
 
 8272 
 
 8257 
 
 8241 
 
 8226 
 
 8211 
 
 8195 
 
 8180 
 
 8165 
 
 
 8 
 
 12 
 
 123 
 
 8150 
 
 8134 
 
 8119 
 
 8104 
 
 8088 
 
 8073 
 
 8058 
 
 8042 
 
 8027 
 
 8012 
 
 
 9 
 
 
 
 14 
 
 mmm 
 
 124 
 
 7996 
 
 7981 
 
 7966 
 
 7950 
 
 7935 
 
 7920 
 
 7904 
 
 7889 
 
 7873 
 
 7858 
 
 
 
 125 
 
 7843 
 
 7827 
 
 7812 
 
 7796 
 
 7781 
 
 7766 
 
 7750 
 
 7735 
 
 7719 
 
 7704 
 
 
 
 126 
 
 7689 
 
 7673 
 
 7658 
 
 7642 
 
 7627 
 
 7611 
 
 7596 
 
 7580 
 
 7565 
 
 7550 
 
 
 
 127 
 
 7534 
 
 7519 
 
 7503 
 
 7488 
 
 7472 
 
 7457 
 
 7441 
 
 7426 
 
 7410 
 
 7395 
 
 16 
 
 16 
 
 128 
 
 7379 
 
 7364 
 
 7348 
 
 7333 
 
 7317 
 
 7302 
 
 7286 
 
 7271 
 
 7255 
 
 7239 
 
 
 . 
 
 - 
 
 129 
 
 7224 
 
 7208 
 
 7193 
 
 7177 
 
 7162 
 
 7146 
 
 7131 
 
 7115 
 
 7099 
 
 7084 
 
 
 2 
 
 <q 
 
 130 
 
 93 7068 
 
 7053 
 
 7037 
 
 7021 
 
 7006 
 
 6990 
 
 6975 
 
 6959 
 
 6943 
 
 6928 
 
 
 4 
 
 6 
 
 131 
 
 6912 
 
 6896 
 
 6881 
 
 6865 
 
 6849 
 
 6834 
 
 6818 
 
 6802 
 
 6787 
 
 6771 
 
 
 5 
 
 8 
 
 132 
 
 6755 
 
 6740 
 
 6724 
 
 6708 
 
 6693 
 
 6677 
 
 6661 
 
 6646 
 
 6630 
 
 6614 
 
 
 6 
 
 10 
 
 133 
 
 6599 
 
 6583 
 
 6567 
 
 6551 
 
 6536 
 
 6520 
 
 6504 
 
 6488 
 
 6473 
 
 6457 
 
 
 7 
 
 11 
 
 134 
 
 6441 
 
 6425 
 
 6410 
 
 6394 
 
 6378 
 
 6362 
 
 6346 
 
 6331 
 
 6315 
 
 6299 
 
 
 9 
 
 13 
 14 
 
 135 
 
 6283 
 
 6268 
 
 6252 
 
 6236 
 
 6220 
 
 6204- 
 
 6188 
 
 6173 
 
 6157 
 
 6141 
 
 
 
 136 
 
 6125 
 
 6109 
 
 6093 
 
 6078 
 
 6062 
 
 6046 
 
 6030 
 
 6014 
 
 5998 
 
 5982 
 
 
 
 137 
 
 5966 
 
 5951 
 
 5935 
 
 5919 
 
 5903 
 
 5887 
 
 5871 
 
 5855 
 
 5839 
 
 5823 
 
 
 
 138 
 
 5807 
 
 5791 
 
 5775 
 
 5760 
 
 5744 
 
 5728 
 
 5712 
 
 5696 
 
 5680 
 
 5664 
 
 
 
 139 
 
 5648 
 
 5632 
 
 5616 
 
 5600 
 
 5584 
 
 5568 
 
 5552 
 
 5536 
 
 5520 
 
 5504 
 
 
 17 
 
 140 
 
 93 5488 
 
 5472 
 
 5456 
 
 5440 
 
 5424 
 
 5408 
 
 5392 
 
 5376 
 
 5360 
 
 5344 
 
 
 1 
 
 2 
 
 2 
 
 
 
 141 
 
 5328 
 
 5311 
 
 5295 
 
 5279 
 
 5263 
 
 5247 
 
 5231 
 
 5215 
 
 5199 
 
 5183 
 
 
 3 
 
 5 
 
 142 
 
 5167 
 
 5151 
 
 5135 
 
 5118 
 
 5103 
 
 5086 
 
 5070 
 
 5054 
 
 5038 
 
 5022 
 
 
 4 
 
 7 
 
 143 
 
 5006 
 
 4989 
 
 4973 
 
 4957 
 
 4941 
 
 4925 
 
 4909 
 
 4892 
 
 4876 
 
 4860 
 
 
 5 
 
 9 
 
 144 
 
 4844 
 
 4828 
 
 4812 
 
 4795 
 
 4779 
 
 4763 
 
 4747 
 
 4730 
 
 4714 
 
 4698 
 
 
 6 
 
 10 
 
 145 
 
 4682 
 
 4666 
 
 4649 
 
 4633 
 
 4617 
 
 4601 
 
 4584 
 
 4568 
 
 4552 
 
 4536 
 
 
 7 
 
 8 
 
 12 
 14 
 
 146 
 
 4519 
 
 4503 
 
 4487 
 
 4470 
 
 4454 
 
 4438 
 
 4422 
 
 4405 
 
 4389 
 
 4373 
 
 
 9 
 
 15 
 
 147 
 
 4356 
 
 4340 
 
 4324 
 
 4307 
 
 4291 
 
 4275 
 
 4258 
 
 4242 
 
 4226 
 
 4209 
 
 
 "" 
 
 148 
 
 4193 
 
 4177 
 
 4160 
 
 4144 
 
 4127 
 
 4111 
 
 4095 
 
 4078 
 
 4062 
 
 4046 
 
 
 
 149 
 
 4029 
 
 4013 
 
 3996 
 
 3980 
 
 3963 
 
 3947 
 
 3931 
 
 3914 
 
 3898 
 
 3881 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 
 'arts. 
 
 50 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(l-;r) 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 3750 
 3585 
 3419 
 3253 
 
 3087 
 
 2920 
 2753 
 
 2585 
 2417 
 2248 
 
 8 
 
 9 
 
 T-150 
 151 
 152 
 153 
 154 
 
 155 
 156 
 157 
 158 
 159 
 
 1-93 3865 
 3700 
 3535 
 3369 
 3203 
 
 3037 
 
 2870 
 2703 
 2535 
 2366 
 
 3848 
 3684 
 3518 
 3353 
 3187 
 
 3020 
 2853 
 2686 
 2518 
 2350 
 
 3832 
 3667 
 3502 
 3336 
 3170 
 3004 
 2836 
 2669 
 2501 
 2333 
 
 3815 
 3651 
 3485 
 3320 
 3153 
 
 2987 
 2820 
 2652 
 2484 
 2316 
 
 3799 
 3634 
 3469 
 3303 
 3137 
 
 2970 
 2803 
 2635 
 2467 
 2299 
 
 3783 
 3618 
 3452 
 3286 
 3120 
 
 2953 
 2786 
 2619 
 2451 
 
 2282 
 
 3766 
 3601 
 3436 
 3270 
 3104 
 
 2937 
 2770 
 2602 
 2434 
 2265 
 
 3733 
 
 3568 
 3403 
 3237 
 3070 
 
 2903 
 2736 
 2568 
 2400 
 2231 
 
 3717 
 3552 
 3386 
 3220 
 3054 
 
 2887 
 2719 
 2552 
 2383 
 2215 
 
 17 
 
 18 
 19 
 
 ] 
 
 ] 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 mmm 
 
 1 
 
 5 
 
 2 
 3 
 5 
 6 
 
 8 
 10 
 11 
 13 
 
 14 
 
 = 
 
 r 
 
 160 
 161 
 162 
 163 
 164- 
 
 165 
 166 
 167 
 168 
 169 
 
 932198 
 2028 
 1859 
 1689 
 1518 
 
 1347 
 1176 
 1004 
 0831 
 0658 
 
 2181 
 2012 
 1842 
 1672 
 1501 
 
 1330 
 1158 
 0986 
 0814 
 0641 
 
 2164 
 1995 
 1825 
 1655 
 1484 
 
 1313 
 1141 
 0969 
 0797 
 0624 
 
 2147 
 1978 
 1808 
 1638 
 1467 
 
 1296 
 1124 
 0952 
 0779 
 0606 
 
 2130 
 1961 
 1791 
 1621 
 1450 
 
 1279 
 1107 
 0935 
 0762 
 0589 
 
 2113 
 1944 
 1774 
 1604 
 1433 
 
 1261 
 1090 
 0918 
 0745 
 
 0572 
 
 2096 
 1927 
 1757 
 1586 
 1416 
 
 1244 
 1073 
 0900 
 0728 
 0554 
 
 2079 
 1910 
 1740 
 1569 
 1398 
 
 122? 
 1055 
 0883 
 0710 
 0537 
 
 2062 
 1893 
 1723 
 1552 
 1381 
 
 1210 
 1038 
 
 0866 
 0693 
 0520 
 
 2045 
 1876 
 1706 
 1535 
 1364 
 
 1193 
 1021 
 0849 
 0676 
 0502 
 
 1 
 2 
 3 
 4 
 
 o 
 6 
 7 
 8 
 9 
 
 2 
 3 
 
 5 
 
 ij 
 
 10 
 12 
 14 
 15 
 
 170 
 171 
 172 
 173 
 174 
 
 175 
 176 
 177 
 178 
 179 
 
 93 0485 
 0311 
 0137 
 92 9962 
 9787 
 
 9611 
 9435 
 9258 
 9081 
 8904 
 
 0468 
 0294 
 0119 
 9945 
 9769 
 
 9594 
 9417 
 9241 
 9063 
 
 8886 
 
 0450 
 0276 
 0102 
 9927 
 9752 
 
 9576 
 9400 
 9223 
 9046 
 
 8868 
 
 0433 
 0259 
 0085 
 9910 
 9734 
 
 9558 
 9382 
 9205 
 9028 
 8850 
 
 0416 
 0242 
 0067 
 9892 
 9717 
 
 9541 
 9364 
 9188 
 9010 
 
 8832 
 
 0398 
 0224 
 0050 
 9875 
 9699 
 
 9523 
 9347 
 9170 
 8992 
 8815 
 
 0381 
 0207 
 0032 
 9857 
 9682 
 9506 
 9329 
 9152 
 8975 
 8797 
 
 0363 
 0189 
 0015 
 9840 
 9664 
 
 9488 
 9311 
 9134 
 8957 
 8779 
 
 0346 
 0172 
 9997 
 
 9822 
 9646 
 
 9470 
 9294 
 9117 
 8939 
 8761 
 
 0329 
 0154 
 9980 
 9804 
 9629 
 
 9453 
 9276 
 9099 
 8921 
 8743 
 
 18 
 
 1 
 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 smf* 
 1 
 
 2 
 4 
 
 5 
 
 7 
 
 9 
 11 
 13 
 14 
 
 1C 
 
 KB 
 
 9 
 
 180 
 181 
 182 
 183 
 184 
 
 185 
 186 
 
 187 
 188 
 189 
 
 92 8725 
 
 8547 
 8368 
 8188 
 8008 
 
 7828 
 7647 
 7465 
 7283 
 7101 
 
 8708 
 8529 
 8350 
 8170 
 7990 
 
 7810 
 7628 
 
 7447 
 7265 
 7082 
 
 6899 
 6716 
 6532 
 6347 
 6162 
 
 5976 
 5790 
 5604 
 5417 
 5229 
 
 8690 
 8511 
 8332 
 8152 
 7972 
 
 7791 
 7610 
 7429 
 
 7247 
 7064 
 
 8672 
 8493 
 8314 
 8134 
 7954 
 
 7773 
 7592 
 7411 
 
 7228 
 7046 
 
 6863 
 6679 
 6495 
 6310 
 6125 
 
 5939 
 5753 
 5566 
 5379 
 5191 
 
 8654 
 8475 
 8296 
 8116 
 7936 
 
 7755 
 7574 
 7392 
 7210 
 7027 
 
 6844 
 6660 
 6476 
 6292 
 6106 
 
 5921 
 5734 
 5548 
 5360 
 5173 
 
 8636 
 8457 
 
 8278 
 8098 
 7918 
 
 7737 
 7556 
 7374 
 7192 
 7009 
 
 8618 
 8439 
 8260 
 8080 
 7900 
 
 7719 
 7538 
 7356 
 7174 
 6991 
 
 8601 
 8422 
 8242 
 8062 
 
 7882 
 
 7701 
 7520 
 7338 
 7155 
 6972 
 
 8583 
 8404 
 8224 
 8044 
 7864 
 
 7683 
 7501 
 7320 
 7137 
 6954 
 
 8565 
 8386 
 8206 
 8026 
 7846 
 
 7665 
 7483 
 7301 
 7119 
 6936 
 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 9 
 
 H 
 
 2 
 
 4 
 6 
 
 : 
 
 11 
 
 13 
 15 
 17 
 
 IMV 
 
 190 
 191 
 192 
 193 
 194 
 
 195 
 196 
 197 
 198 
 199 
 
 926918 
 6734 
 6550 
 6365 
 6180 
 
 5995 
 5809 
 5622 
 5435 
 5248 
 
 6881 
 6697 
 6513 
 6329 
 6143 
 
 5958 
 5772 
 5585 
 5398 
 5210 
 
 6826 
 6642 
 6458 
 6273 
 6088 
 
 5902 
 5716 
 5529 
 5342 
 5154 
 
 6807 
 6624 
 6439 
 6255 
 6069 
 
 5883 
 5697 
 5510 
 5323 
 5135 
 
 6789 
 6605 
 6421 
 6236 
 6051 
 
 5865 
 5678 
 5491 
 5304 
 5116 
 
 6771 
 6587 
 6402 
 6217 
 6032 
 
 5846 
 56GO 
 5473 
 5285 
 5097 
 
 6752 
 6568 
 6384 
 6199 
 6013 
 
 5828 
 5641 
 5454 
 5266 
 
 5078 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 51 
 
TABLE II. 
 
 Log 
 
 Log(l:-*) 
 
 -r\'S 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Din 
 
 Parts. 
 
 F-200 
 
 T-92 5060 
 
 5041 
 
 5022 
 
 5003 
 
 4984 
 
 4965 
 
 4947 
 
 4928 
 
 4909 
 
 4890 
 
 19 
 
 
 201 
 
 4871 
 
 4852 
 
 4833 
 
 4814 
 
 4795 
 
 4777 
 
 4758 
 
 4739 
 
 4720 
 
 4701 
 
 
 
 202 
 
 4682 
 
 4663 
 
 4644 
 
 4625 
 
 4606 
 
 4587 
 
 4568 
 
 4549 
 
 4530 
 
 4511 
 
 
 
 203 
 
 4492 
 
 4473 
 
 4454 
 
 4435 
 
 4416 
 
 4397 
 
 4378 
 
 4359 
 
 4340 
 
 4321 
 
 
 19 
 
 204 
 
 4302 
 
 4283 
 
 4264 
 
 4245 
 
 4226 
 
 4207 
 
 4188 
 
 4169 
 
 4150 
 
 4131 
 
 
 
 
 205 
 
 4111 
 
 4092 
 
 4073 
 
 4054 
 
 4035 
 
 4016 
 
 3997 
 
 3978 
 
 3959 
 
 3939 
 
 
 2 
 
 t 
 
 206 
 
 3920 
 
 3901 
 
 3882 
 
 3863 
 
 3844 
 
 3824 
 
 3805 
 
 3786 
 
 3767 
 
 3748 
 
 
 3 
 
 6 
 
 207 
 
 3729 
 
 3709 
 
 3690 3671 
 
 3652 
 
 3632 
 
 3613 
 
 3594 
 
 3575 
 
 3556 
 
 
 4 
 
 8 
 
 208 
 
 3536 
 
 3517 
 
 3498 
 
 3479 
 
 3459 
 
 3440 
 
 3421 
 
 3401 
 
 3382 
 
 3363 
 
 
 5 
 
 10 
 
 209 
 
 3344 
 
 3324 
 
 3305 
 
 3286 
 
 3266 
 
 3247 
 
 3228 
 
 3208 
 
 3189 
 
 3170 
 
 
 ti 
 
 1 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 j<: 
 
 210 
 
 923150 
 
 3131 
 
 3111 
 
 3092 
 
 3073 
 
 3053 
 
 3034 
 
 3015 
 
 2995 
 
 2976 
 
 
 8 
 
 15 
 
 211 
 
 2956 
 
 2937 
 
 2918 
 
 2898 
 
 2879 
 
 2859 
 
 2840 2820 2801 
 
 2781 
 
 
 9 i / 
 
 212 
 
 2762 
 
 2743 
 
 2723 
 
 2704 
 
 2684 
 
 2665 
 
 2645 
 
 2626 2606 
 
 2587 
 
 20 
 
 
 213 
 
 2567 
 
 2548 
 
 2528 
 
 2508 
 
 2489 
 
 2469 
 
 2450 
 
 2430 1 24 11 
 
 2391 
 
 
 
 214 
 
 2372 
 
 2352 
 
 2332 
 
 2313 
 
 2293 
 
 2274 
 
 2254 
 
 2234 2215 
 
 2195 
 
 
 
 215 
 
 2176 
 
 2156 
 
 2136 
 
 2117 
 
 2097 
 
 2077 
 
 2058 
 
 2038 2018 
 
 1999 
 
 
 20 
 
 216 
 
 1979 
 
 1959 
 
 1940 
 
 1920 
 
 1900 
 
 1881 
 
 1861 
 
 1841 
 
 1821 
 
 1802 
 
 
 1 1 '> 
 
 217 
 
 1782 
 
 1762 
 
 1743 
 
 1723 
 
 1703 
 
 1683 
 
 1664 
 
 1644 
 
 1624 
 
 1604 
 
 
 2 
 
 4 
 
 218 
 
 1584 
 
 1565 
 
 1545 
 
 1525 
 
 1505 
 
 1485 
 
 1466 
 
 1446 
 
 1426 
 
 1406 
 
 
 3 
 
 6 
 
 219 
 
 1386 
 
 1366 
 
 1347 
 
 1327 
 
 1307 
 
 1287 
 
 1267 
 
 1247 
 
 1227 
 
 1207 
 
 
 4 
 
 8 
 
 Ul 
 
 220 
 
 921188 
 
 1168 
 
 1148 
 
 1128 
 
 1108 
 
 1088 
 
 1068 
 
 1048 
 
 1028 
 
 1008 
 
 
 5 
 6 
 
 12 
 
 221 
 
 0988 
 
 0968 
 
 0948 
 
 0928 
 
 0908 
 
 0888 
 
 0868 
 
 0848 
 
 0829 
 
 0809 
 
 
 7 
 
 14 
 
 222 
 
 0788 
 
 0768 
 
 0748 
 
 0728 
 
 0708 
 
 0688 
 
 0668 
 
 0648 
 
 0628 
 
 0608 
 
 
 8 
 
 16 
 
 223 
 
 0588 
 
 0568 
 
 0548 
 
 0528 
 
 0508 
 
 0488 
 
 0468 
 
 0448 
 
 0427 
 
 0407 
 
 
 9 is 
 
 224 
 
 0387 
 
 0367 
 
 0347 
 
 0327 
 
 0307 
 
 0287 
 
 0266 
 
 0246 
 
 0226 
 
 0206 
 
 
 
 225 
 
 0186 
 
 0166 
 
 0145 
 
 0125 
 
 0105 
 
 0085 
 
 0065 
 
 0044 
 
 0024 
 
 0004 
 
 
 
 226 
 
 91 9984 
 
 9963 
 
 9943 
 
 9923 
 
 9903 
 
 9882 
 
 9862 
 
 9842 
 
 9822 
 
 9801 
 
 
 
 227 
 
 9781 
 
 9761 
 
 9741 
 
 9720 
 
 9700 
 
 9680 
 
 9659 
 
 9639 
 
 9619 
 
 9598 
 
 
 21 
 
 228 
 
 9578 
 
 9558 
 
 9537 
 
 9517 
 
 9497 
 
 9476 
 
 9456 
 
 9435 
 
 9415 
 
 9395 
 
 
 1 2 
 
 229 
 
 9374 
 
 9354 
 
 9333 
 
 9313 
 
 9293 
 
 9272 
 
 9252 
 
 9231 
 
 9211 
 
 9190 
 
 
 2 4 
 
 3C 
 
 230 
 
 91 9170 
 
 9150 
 
 9129 
 
 9109 
 
 9088 
 
 9068 
 
 9047 
 
 9027 
 
 9006 
 
 8986 
 
 21 
 
 4 
 
 u 
 
 8 
 
 231 
 
 8965 
 
 8945 
 
 8924 
 
 8904 
 
 8883 
 
 8862 
 
 8842 
 
 8821 
 
 8801 
 
 8780 
 
 
 5 11 
 
 232 
 
 8760 
 
 8739 
 
 8719 
 
 8698 
 
 8677 
 
 8657 
 
 8636 
 
 8616 
 
 8595 
 
 8574 
 
 
 6 13 
 
 233 
 
 8554 
 
 8533 
 
 8512 
 
 8492 
 
 8471 
 
 8451 
 
 8430 
 
 8409 
 
 8389 
 
 8368 
 
 
 7 15 
 
 234 
 
 8347 
 
 8326 
 
 8306 
 
 8285 
 
 8264 
 
 8244 
 
 8223 
 
 8202 
 
 8182 
 
 8161 
 
 
 8 17 
 9 19 
 
 235 
 
 8140 
 
 8119 
 
 8099 
 
 8078 
 
 8057 
 
 8036 
 
 8015 
 
 7995 
 
 7974 
 
 7953 
 
 
 
 236 
 
 7932 
 
 7912 
 
 7891 
 
 7870 
 
 7849 
 
 7828 
 
 7807 
 
 7787 
 
 7766 
 
 7745 
 
 
 
 237 
 
 7724 
 
 7703 
 
 7682 
 
 7661 
 
 7641 
 
 7620 
 
 7599 
 
 7578 
 
 7557 
 
 7536 
 
 
 
 238 
 
 7515 
 
 7494 
 
 7473 
 
 7452 
 
 7431 
 
 7410 
 
 7390 
 
 7369 
 
 7348 
 
 7327 
 
 
 
 239 
 
 7306 
 
 7285 
 
 7264 
 
 7243 
 
 7222 
 
 7201 
 
 7180 
 
 7159 
 
 7138 
 
 7117 
 
 
 22 
 
 240 
 
 91 7096 
 
 7075 
 
 7054 
 
 7033 
 
 7011 
 
 6990 
 
 6969 
 
 6948 
 
 6927 
 
 6906 
 
 
 2 
 
 c 
 4 
 
 241 
 
 6885 
 
 6864 
 
 6843 
 
 6822 
 
 6801 
 
 6780 
 
 6758 
 
 6737 
 
 6716 
 
 6695 
 
 
 3l 7 
 
 242 
 
 6674 
 
 6653 
 
 6632 
 
 6610 
 
 6589 
 
 6568 
 
 6547 
 
 6526 
 
 6504 
 
 6483 
 
 
 4 
 
 9 
 
 243 
 
 6462 
 
 6441 
 
 6420 
 
 6398 
 
 6377 
 
 6356 
 
 6335 
 
 6313 
 
 6292 
 
 6271 
 
 
 5 
 
 11 
 
 244 
 
 6250 
 
 6228 
 
 6207 
 
 6186 
 
 6164 
 
 6143 
 
 6122 
 
 6101 
 
 6079 
 
 6058 
 
 
 6 
 
 13 
 
 245 
 
 6037 
 
 6015 
 
 5994 
 
 5973 
 
 5951 
 
 5930 
 
 5909 
 
 5887 
 
 5866 
 
 5844 
 
 
 7 
 8 
 
 15 
 
 IS 
 
 246 
 
 5823 
 
 5802 
 
 5780 
 
 5759 
 
 5737 
 
 5716 
 
 5695 
 
 5673 
 
 5652 
 
 5630 
 
 
 9 
 
 20 
 
 247 
 
 5609 
 
 5587 
 
 5566 
 
 5544 
 
 5523 
 
 5502 
 
 5480 
 
 5459 
 
 5437 
 
 5416 
 
 22 
 
 *m 
 
 248 
 
 5394 
 
 5373 
 
 5351 
 
 5330 
 
 5308 
 
 5286 
 
 5265 
 
 5243 
 
 5222 
 
 5200 
 
 
 
 249 
 
 5179 
 
 5157 
 
 5136 
 
 5114 
 
 5092 
 
 5071 
 
 5049 
 
 5028 
 
 5006 
 
 4984 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 52 
 
TABLE II. 
 
 Log 
 
 Log(l-ar) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 'artSi 
 
 f-250 
 
 1-914963 
 
 4941 
 
 4919 
 
 4898 
 
 4876 
 
 4854 
 
 4833 
 
 4811 
 
 4789 
 
 4768 
 
 22 
 
 
 251 
 
 4746 
 
 4724 
 
 4703 
 
 4681 
 
 4659 
 
 4638 
 
 4616 4594 
 
 4572 
 
 4551 
 
 
 21 
 
 252 
 
 4529 
 
 4507 
 
 4485 
 
 4464 
 
 4442 
 
 4420 
 
 43981 4377 
 
 4355 
 
 4333 
 
 
 1 ^ 
 
 253 
 
 4311 
 
 4289 
 
 4267 
 
 4246 
 
 4224 
 
 4202 
 
 4180 4158 
 
 4136 
 
 4115 
 
 
 2 
 
 4 
 
 254 
 
 4093 
 
 4071 
 
 4049 
 
 4027 
 
 4005 
 
 3983 
 
 3961 
 
 3939 
 
 3918 
 
 3896 
 
 
 3 
 
 6 
 
 255 
 
 3874 
 
 3852 
 
 3830 
 
 3808 
 
 3786 
 
 3764 
 
 3742 
 
 3720 
 
 3698 
 
 3676 
 
 
 4 
 
 8 
 
 256 
 
 3654 
 
 3632 
 
 S610 
 
 3588 
 
 3566 
 
 3544 
 
 3522 3500 
 
 3478 
 
 3456 
 
 
 5 
 
 11 
 
 257 
 
 3434 
 
 3412 
 
 3390 
 
 3368 
 
 3345 
 
 3323 
 
 3301 3279 
 
 3257 
 
 3235 
 
 
 6 
 
 7 
 
 13 
 15 
 
 258 
 
 3213 
 
 3191 
 
 3169 
 
 3146 
 
 3124 
 
 3102 
 
 3080; 3058 
 
 3036 
 
 3013 
 
 
 8 
 
 17 
 
 259 
 
 2991 
 
 2969 
 
 2947 
 
 2925 
 
 2903 
 
 2880 
 
 2858 2836 
 
 2814 
 
 2791 
 
 
 9 
 
 MKB 
 
 19 
 
 
 
 260 
 
 91 2769 
 
 2747 
 
 2725 
 
 2702 
 
 2680 
 
 2658 
 
 2636 2613 
 
 2591 
 
 2569 
 
 
 
 261 
 
 2546 
 
 2524 
 
 2502 
 
 2479 
 
 2457 
 
 2435 
 
 2412 2390 
 
 2368 
 
 2345 
 
 
 22 
 
 262 
 
 2323 
 
 2301 
 
 2278 
 
 2256 
 
 2233 
 
 2211 
 
 2189 
 
 2166 
 
 2144 
 
 2121 
 
 
 1 
 
 2 
 
 263 
 
 2099 
 
 2077 
 
 2054 
 
 2032 
 
 2009 
 
 1987 
 
 1964 
 
 1942 
 
 1919 
 
 1897 
 
 
 2 
 
 4 
 
 264 
 
 1874 
 
 1852 
 
 1829 
 
 1807 
 
 1784 
 
 1762 
 
 1739 
 
 1717 
 
 1694 
 
 1672 
 
 23 
 
 3 
 
 4 
 
 7 
 9 
 
 265 
 
 1649 
 
 1627 
 
 1604 
 
 1581 
 
 1559 
 
 1536 
 
 1514 
 
 1491 
 
 1468 
 
 1446 
 
 
 5 
 
 H 
 
 266 
 
 1423 
 
 1401 
 
 1378 
 
 1355 
 
 1333 
 
 1310 
 
 1287 
 
 1265 
 
 1242 
 
 1219 
 
 
 6 
 
 13 
 
 267 
 
 1197 
 
 1174 
 
 1151 
 
 1128 
 
 1106 
 
 1083 
 
 1060 
 
 1038 
 
 1015 
 
 0992 
 
 
 7 
 
 15 
 
 268 
 
 0969 
 
 0947 
 
 0924 
 
 0901 
 
 0878 
 
 0856 
 
 0833 
 
 0810 
 
 0787 
 
 0764 
 
 
 8 
 
 18 
 
 269 
 
 0742 
 
 0719 
 
 0696 
 
 0673 
 
 0650 
 
 0627 
 
 0605 
 
 0582 
 
 0559 
 
 0536 
 
 
 9 
 
 mmn 
 
 20 
 
 
 
 270 
 
 910513 
 
 0490 
 
 0467 
 
 0444 
 
 0421 
 
 0399 
 
 0376 
 
 0353 
 
 0330 
 
 0307 
 
 
 23 
 
 271 
 
 0284 
 
 0261 
 
 0238 
 
 0215 
 
 0192 
 
 0169 
 
 0146 
 
 0123 
 
 0100 
 
 0077 
 
 
 
 272 
 
 0054 
 
 0031 
 
 0008 
 
 9985 
 
 9962 
 
 9939 
 
 9916 
 
 9893 
 
 9870 
 
 9847 
 
 
 1 2 
 2 5 
 
 273 
 
 90 9824 
 
 9801 
 
 9777 
 
 9754 
 
 9731 
 
 9708 
 
 9685 
 
 9662 
 
 9639 
 
 9616 
 
 
 3 7 
 
 274 
 
 9593 
 
 9569 
 
 9546 
 
 9523 
 
 9500 
 
 9477 
 
 9454 
 
 9430 
 
 9407 
 
 9384 
 
 
 4 9 
 
 275 
 
 9361 
 
 9338 
 
 9314 
 
 9291 
 
 9268 
 
 9245 
 
 9221 
 
 9198 
 
 9175 
 
 9152 
 
 
 5 12 
 
 276 
 
 9128 
 
 9105 
 
 9082 
 
 9059 
 
 9035 
 
 9012 
 
 8989 
 
 8965 
 
 8942 
 
 8919 
 
 
 6 14 
 
 277 
 
 8895 
 
 8872 
 
 8849 
 
 8825 
 
 8802 
 
 8779 
 
 8755 8732 
 
 8708 
 
 8685 
 
 
 7 16 
 
 81 C 
 
 278 
 
 8662 
 
 8638 
 
 8615 
 
 8591 
 
 8568 
 
 8544 
 
 8521 
 
 8498 
 
 8474 
 
 8451 
 
 
 lo 
 
 9 2] 
 
 279 
 
 8427 
 
 8404 
 
 8380 
 
 8357 
 
 8333 
 
 8310 
 
 8286 
 
 8263 
 
 8239 
 
 8216 
 
 24 
 
 
 280 
 
 908192 
 
 8169 
 
 8145 
 
 8121 
 
 8098 
 
 8074 
 
 8051 
 
 8027 
 
 8004 
 
 7980 
 
 
 24 
 
 281 
 
 7956 
 
 7933 
 
 7909 
 
 7886 
 
 7862 
 
 7838 
 
 7815 
 
 7791 
 
 7767 
 
 7744 
 
 
 1 
 
 2 
 
 282 
 
 7720 
 
 7696 
 
 7673 
 
 7649 
 
 7625 
 
 7602 
 
 7578 
 
 7554 
 
 7530 7507 
 
 
 2 
 
 5 
 
 283 
 
 7483 
 
 7459 
 
 7435 
 
 7412 
 
 7388 
 
 7364 
 
 7340 
 
 7317 
 
 7293 7269 
 
 
 3 
 
 7 
 
 284 
 
 7245 
 
 7221 
 
 7198 
 
 7174 
 
 7150 
 
 7126 
 
 7102 
 
 7078 
 
 7054 7031 
 
 
 4 
 
 10 
 
 285 
 
 7007 
 
 6983 6959 
 
 6935 
 
 6911 
 
 6887 
 
 6863 
 
 6839 
 
 6815 6792 
 
 
 5 
 
 12 
 
 1 A 
 
 286 
 
 6768 
 
 6744 6720 
 
 6696 
 
 6672 
 
 6648 
 
 6624 
 
 6600 
 
 6576 
 
 6552 
 
 
 7 
 
 14 
 
 17 
 
 287 
 
 6528 
 
 6504 6480 
 
 6456 
 
 6432 
 
 6408 
 
 6384 
 
 6360 
 
 6335 
 
 6311 
 
 
 8 
 
 L i 
 
 19 
 
 288 
 
 6287 
 
 6263 C239 
 
 6215 
 
 6191 
 
 6167 
 
 6143 
 
 6119 
 
 6094 
 
 6070 
 
 
 9|22 
 
 289 
 
 6046 
 
 6022 
 
 5998 
 
 5974 
 
 5949 
 
 5925 
 
 5901 
 
 5877 
 
 5853 
 
 5828 
 
 
 
 2QO 
 
 90 5804 
 
 5780 
 
 5756 
 
 5732 
 
 5707 
 
 56831 5659 
 
 5635 
 
 5610 
 
 5586 
 
 
 25 
 
 &U\r 
 
 291 
 
 5562 
 
 5537 
 
 5513 
 
 5489 
 
 5464 
 
 5440(54 16 
 
 5391 
 
 5367 
 
 5343 
 
 
 1 
 
 g 
 
 292 
 
 5318 
 
 5294 
 
 5270 
 
 5245 
 
 5221 
 
 5197 5172 
 
 5148 
 
 5123 
 
 5099 
 
 
 2 
 
 5 
 
 293 
 
 5075 
 
 5050 
 
 5026 
 
 5001 
 
 4977 
 
 4952 
 
 4928 
 
 4903 
 
 4879 
 
 4854 
 
 
 3 
 4 
 
 8 
 1C 
 
 294 
 
 4830 
 
 4805 
 
 4781 
 
 4756 
 
 4732 
 
 4707 
 
 4683 
 
 4658 
 
 4634 
 
 4609 
 
 25 
 
 5 
 
 13 
 
 295 
 
 4584 
 
 4560 
 
 4535 
 
 4511 
 
 4486 
 
 4462 
 
 4437 
 
 4412 
 
 4388 
 
 4363 
 
 
 6 
 
 15 
 
 296 
 
 4338 
 
 4314 
 
 4289 
 
 4264 
 
 4240 
 
 4215 
 
 4190 
 
 4166 
 
 4141 
 
 4116 
 
 
 7il8 
 
 297 
 
 4092 
 
 4067 
 
 4042 
 
 4018 
 
 3993 
 
 3968 
 
 3943 
 
 3919 
 
 3894 
 
 3869 
 
 
 8 20 
 
 298 
 
 3844 
 
 3819 
 
 3795 
 
 3770 
 
 3745 
 
 3720 
 
 3695 
 
 3671 
 
 3646 
 
 3621 
 
 
 9 23 
 
 299 
 
 3596 
 
 3571 
 
 3546 
 
 3521 
 
 3497 
 
 3472 
 
 3447 
 
 3422 
 
 3397 
 
 3372 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
TABLE II. 
 
 Log 
 
 Log (I-*) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 ! 9 
 
 Diff 
 
 Parts. 
 
 T-300 
 
 1-90 3347 
 
 3322 
 
 3297 
 
 3272 
 
 3247 
 
 3222 
 
 3197 
 
 3172 
 
 3147 
 
 3122 
 
 25 
 
 
 301 
 
 orfco 
 
 3097 
 
 3072 
 
 3047 
 
 3022 
 
 2997 
 
 2972 
 
 2947 
 
 2922 
 
 2897 
 
 2872 
 
 
 25 
 
 302 
 
 2847 
 
 2822 
 
 2797 
 
 2772 
 
 2747 
 
 2722 
 
 2697 
 
 2671 
 
 2646 
 
 2621 
 
 
 
 
 303 
 
 2596 
 
 2571 
 
 2546 
 
 2521 
 
 2495 
 
 2470 
 
 2445 
 
 2420 
 
 2395 
 
 2370 
 
 
 1 
 
 2 
 
 ' 
 
 304 
 
 2344 
 
 2319 
 
 2294 
 
 2269 
 
 2243 
 
 2218 
 
 2193 
 
 2168 
 
 2142 
 
 2117 
 
 
 3 
 
 8 
 
 305 
 
 2092 
 
 2066 
 
 2041 
 
 2016 
 
 1991 
 
 1965 
 
 1940 
 
 1915 
 
 1889 
 
 1864 
 
 
 4 
 
 10 
 
 306 
 
 1839 
 
 1813 
 
 1788 
 
 1762 
 
 1737 
 
 1712 
 
 1686 
 
 1661 
 
 1635 
 
 1610 
 
 
 5 
 
 13 
 
 307 
 
 1585 
 
 1559 
 
 1534 
 
 1508 
 
 1483 
 
 1457 
 
 1432 
 
 1406 
 
 1381 
 
 1355 
 
 26 
 
 6 
 
 15 
 
 308 
 
 1330 
 
 1304 
 
 1279 
 
 1253 
 
 1228 
 
 1202 
 
 1177 
 
 1151 
 
 1126 
 
 1100 
 
 
 7 
 8 
 
 20 
 
 309 
 
 1074 
 
 1049 
 
 1023 
 
 0998 
 
 0972 
 
 0946 
 
 0921 
 
 0895 
 
 0870 
 
 0844 
 
 
 9 
 
 ! 
 
 23 
 
 M 
 
 310 
 
 900818 
 
 0793 
 
 0767 
 
 0741 
 
 0716 
 
 0690 
 
 0664 
 
 0638 
 
 0613 
 
 0587 
 
 
 
 311 
 
 0561 
 
 0536 
 
 0510 
 
 0484 
 
 0458 
 
 0433 
 
 0407 
 
 0381 
 
 0355 0329 
 
 
 26 
 
 312 
 
 0304 
 
 0278 
 
 0252 
 
 0226 
 
 0200 
 
 0175 
 
 0149 
 
 0123 
 
 0097| 0071 
 
 
 I 
 
 
 
 313 
 
 0045 
 
 0019 
 
 9993 
 
 9968 
 
 9942 
 
 9916 
 
 9890 
 
 9864 
 
 9838 
 
 9812 
 
 
 2 
 
 t 
 
 314 
 
 89 9786 
 
 9760 
 
 9734 
 
 9708 
 
 9682 
 
 9656 
 
 9630 
 
 9604 
 
 9578 
 
 9552 
 
 
 3 
 
 8 
 
 315 
 
 9526 
 
 9500 
 
 9474 
 
 9448 
 
 9422 
 
 9396 
 
 9370 
 
 9344 
 
 9318 
 
 9292 
 
 
 4 10 
 
 316 
 
 9265 
 
 9239 
 
 9213 
 
 9187 
 
 9161 
 
 9135 
 
 9109 
 
 9083 
 
 9056 
 
 9030 
 
 
 5 13 
 
 6 16 
 
 317 
 
 9004 
 
 8978 
 
 8952 
 
 8925 
 
 8899 
 
 8873 
 
 8847 
 
 8821 
 
 8794 
 
 8768 
 
 
 7 18 
 
 318 
 
 8742 
 
 8716 
 
 8689 
 
 8663 
 
 8637 
 
 8610 
 
 8584 
 
 8558 
 
 8532 
 
 8505 
 
 
 a 21 
 
 319 
 
 8479 
 
 8453 
 
 8426 
 
 8400 
 
 8373 
 
 8347 
 
 8321 
 
 8294 
 
 8268 
 
 8242 
 
 
 9 23 
 
 320 
 
 89 8215 
 
 8189 
 
 8162 
 
 8136 
 
 8109 
 
 8083 
 
 8057 
 
 8030 
 
 8004 
 
 7977 
 
 
 27 
 
 321 
 
 7951 
 
 7924 
 
 7898 
 
 7871 
 
 7845 
 
 7818 
 
 7792 
 
 7765 
 
 7738 
 
 7712 
 
 27 
 
 
 322 
 
 7685 
 
 7659 
 
 7632 
 
 7606 
 
 7579 
 
 7552 
 
 7526 
 
 7499 
 
 7473 
 
 7446 
 
 
 1 
 
 j 
 
 323 
 
 7419 
 
 7393 
 
 7366 
 
 7339 
 
 7313 
 
 7286 
 
 7259 
 
 7233 
 
 7206 
 
 7179 
 
 
 2 
 
 5 
 
 324 
 
 7153 
 
 7126 
 
 7099 
 
 7072 
 
 7046 
 
 7019 
 
 6992 
 
 6965 
 
 6938 
 
 6912 
 
 
 3 
 
 4 
 
 8 
 
 325 
 
 6885 
 
 6858 
 
 6831 
 
 6804 
 
 6778 
 
 6751 
 
 6724 
 
 6697 
 
 6670 
 
 6643 
 
 
 5 
 
 14 
 
 326 
 
 6617 
 
 6590 
 
 6563 
 
 6536 
 
 6509 
 
 6482 
 
 6455 
 
 6428 
 
 6401 
 
 6374 
 
 
 6 
 
 l(i 
 
 327 
 
 6347 
 
 6320 
 
 6293 
 
 6266 
 
 6239 
 
 6212 
 
 6185 
 
 6158 
 
 6131 
 
 6104 
 
 
 7 
 
 19 
 
 328 
 
 6077 
 
 6050 
 
 6023 
 
 5996 
 
 5969 
 
 5942 
 
 5915 
 
 5888 
 
 5861 
 
 5834 
 
 
 8 
 
 22 
 
 329 
 
 5807 
 
 5780 
 
 5752 
 
 5725 
 
 5698 
 
 5671 
 
 5644 
 
 5617 
 
 5590 
 
 5562 
 
 
 9 
 
 ^mm 
 
 24 
 
 Mb- 
 
 330 
 
 89 5535 
 
 5508 
 
 5481 
 
 5454 
 
 5426 
 
 5399 
 
 5372 
 
 5345 
 
 5317 
 
 5290 
 
 
 28 
 
 331 
 
 5263 
 
 5236 
 
 5208 
 
 5181 
 
 5154 
 
 5126 
 
 5099 
 
 5072 
 
 5044 
 
 5017 
 
 
 
 
 
 332 
 
 4990 
 
 4962 
 
 4935 
 
 4908 
 
 4880 
 
 4853 
 
 4825 
 
 4798 
 
 4771 
 
 4743 
 
 
 1 
 
 3 
 
 333 
 
 4716 
 
 4688 
 
 4661 
 
 4633 
 
 4606 
 
 4578 
 
 4551 
 
 4523 
 
 4496 
 
 4468 
 
 28 
 
 2 
 
 
 
 6 
 
 8 
 
 334 
 
 4441 
 
 4413 
 
 4386 
 
 4358 
 
 4331 
 
 4303 
 
 4276 
 
 4248 
 
 4221 
 
 4193 
 
 
 4 
 
 11 
 
 335 
 
 4165 
 
 4138 
 
 4110 
 
 4083 
 
 4055 
 
 4027 
 
 4000 
 
 3972 
 
 3944 
 
 3917 
 
 
 5 
 
 14 
 
 336 
 
 3889 
 
 3861 
 
 3834 
 
 3806 
 
 3778 
 
 3751 
 
 3723 
 
 3695 
 
 3667 
 
 3640 
 
 
 6 
 
 17 
 
 337 
 
 3612 
 
 3584 
 
 3556 
 
 3529 
 
 3501 
 
 3473 
 
 04,4* 
 OTTtJ 
 
 3417 
 
 3390 
 
 3362 
 
 
 g 
 
 20 
 22 
 
 338 
 
 3334 
 
 3306 
 
 3278 
 
 3250 
 
 3222 
 
 3195 
 
 3167 
 
 3139 
 
 3111 
 
 3083 
 
 
 9 
 
 25 
 
 339 
 
 3055 
 
 3027 
 
 2999 
 
 2971 
 
 2943 
 
 2915 
 
 2887 
 
 2859 
 
 2831 
 
 2803 
 
 
 
 340 
 
 89 2775 
 
 2747 
 
 2719 
 
 2691 
 
 2663 
 
 2635 
 
 2607 
 
 2579 
 
 2551 
 
 2523 
 
 
 29 
 
 341 
 
 2495 
 
 2467 
 
 2439 
 
 2411 
 
 2383 
 
 2354 
 
 2326 
 
 2298 
 
 2270 
 
 2242 
 
 
 1 
 
 3 
 
 342 
 
 2214 
 
 2186 
 
 2157 
 
 2129 
 
 2101 
 
 2073 
 
 2045 
 
 2016 
 
 1988 
 
 1960 
 
 
 2 
 
 6 
 
 343 
 
 1932 
 
 1903 
 
 1875 
 
 1847 
 
 1819 
 
 1790 
 
 1762 
 
 1734 
 
 1705 
 
 1677 
 
 
 3 
 
 9 
 
 344 
 
 1649 
 
 1620 
 
 1592 
 
 1564 
 
 1535 
 
 1507 
 
 1479 
 
 1450 
 
 1422 
 
 1393 
 
 
 4 
 
 12 
 
 345 
 
 1365 
 
 1336 
 
 1308 
 
 1280 
 
 1251 
 
 1223 
 
 1194 
 
 1166 
 
 1137 
 
 1109 
 
 29 
 
 5 
 
 6 
 
 15 
 17 
 
 346 
 
 1080 
 
 1052 
 
 1023 
 
 0995 
 
 0966 
 
 0938 
 
 0909 
 
 0881 
 
 0852 
 
 0823 
 
 
 7 
 
 20 
 
 347 
 
 0795 
 
 0766 
 
 0738 
 
 0709 
 
 0680 
 
 0652 
 
 0623 
 
 0594 
 
 0566 
 
 0537 
 
 
 8 
 
 23 
 
 348 
 
 0508 
 
 0480 
 
 0451 
 
 0422 
 
 0394 
 
 0365 
 
 0336 
 
 0308 
 
 0279 
 
 0250 
 
 
 j!^ 
 
 26 
 
 349 
 
 0221 
 
 0193 
 
 0164 
 
 0135 
 
 0106 
 
 0077 
 
 0049 
 
 0020 
 
 9991 
 
 9962 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 'arts. 
 
 54 
 
TABLE II. 
 
 Log 
 
 Log(l-,) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 ! 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Parts. 
 
 1-350 
 
 T-88 993; 
 
 } 9904 
 
 1. 9876! 9847 
 
 9818 
 
 9789 
 
 976C 
 
 I 9731 
 
 9702 
 
 9673 
 
 29 
 
 
 
 351 
 
 9644 
 
 t 9615 9587 9558 
 
 9529 
 
 950C 
 
 9471 
 
 9442 9413 
 
 9384 
 
 
 29 
 
 352 
 
 935^ 
 
 > 9326! 9297 
 
 9268 9239 
 
 9209 
 
 9180 9151 9122 
 
 9093 
 
 
 
 
 353 
 
 9064 
 
 , 9035| 900C 
 
 8977 8948 
 
 8918 
 
 8889! 8860 883 
 
 8802 
 
 
 
 
 
 354 
 
 877.: 
 
 1 8743 
 
 8714 
 
 8685 
 
 8656 
 
 8627 
 
 8597 8568 
 
 8539 
 
 8510 
 
 
 3 
 
 
 355 
 
 848C 
 
 18451 
 
 842S 
 
 8392 
 
 8363 
 
 8334 
 
 8304 8275 
 
 8246 
 
 8216 
 
 
 4 
 
 12 
 
 356 
 
 8187 
 
 8158 
 
 8128 
 
 8099 8070 
 
 8040 
 
 8011 7981 
 
 7952 
 
 7922 
 
 
 5 
 
 15 
 
 357 
 
 789S 
 
 7864 
 
 7834 
 
 7805 
 
 7775 
 
 7746 
 
 7716 7687 
 
 7657 
 
 7628 
 
 30 
 
 6 
 
 17 
 
 358 
 
 7596 
 
 7569 
 
 7539 
 
 7509 
 
 7480 
 
 7450 
 
 7421 
 
 7391 
 
 7361 
 
 7332 
 
 
 7 
 
 8 
 
 20 
 23 
 
 359 
 
 7302 
 
 7273 
 
 7243 
 
 7213 
 
 7184 
 
 7154 
 
 7124 
 
 7095 
 
 7065 
 
 7035 
 
 
 9 
 
 M 
 
 26 
 
 mmtrn 
 
 360 
 
 88 7006 
 
 6976 
 
 6946 
 
 6916 
 
 6887 
 
 6857 
 
 6827 
 
 6797 
 
 6767 
 
 6738 
 
 
 
 
 361 
 
 6708' 6678 
 
 6648 
 
 6618 
 
 6589 
 
 6559 
 
 6529 
 
 6499 
 
 6469 
 
 6439 
 
 
 30 
 
 362 
 
 6409 6379 
 
 6350 
 
 6320 
 
 6290 
 
 6260 
 
 6230 
 
 6200 
 
 6170 
 
 6140 
 
 
 1 
 
 J 
 
 363 
 
 6110 
 
 6080 
 
 6050 
 
 6020 
 
 5990 
 
 5960 
 
 5930 
 
 5900 
 
 5870 
 
 5840 
 
 
 2 
 
 6 
 
 364 
 
 5810 
 
 5780 
 
 5750 
 
 5719 
 
 5689 
 
 5659 
 
 5629 
 
 5599 
 
 5569 
 
 5539 
 
 
 3 
 4 
 
 1 
 
 12 
 
 365 
 
 5509 
 
 5478 
 
 5448 
 
 5418 
 
 5388 
 
 5358 
 
 5327 
 
 5297 
 
 5267 
 
 5237 
 
 
 
 
 366 
 
 5206 
 
 5176 
 
 5146 
 
 5116 
 
 5085 
 
 5055 
 
 5025 
 
 4994 
 
 4964 
 
 4934 
 
 
 6 
 
 18 
 
 367 
 
 4903 
 
 4873 
 
 4843 
 
 4812 
 
 4782 
 
 4752 
 
 4721 
 
 4691 
 
 4660 
 
 4630 
 
 
 7 
 
 21 
 
 368 
 
 4600 
 
 4569 
 
 45391 4508 
 
 4478 
 
 4447 
 
 4417 
 
 4386 
 
 4356 
 
 4325 
 
 31 
 
 8 
 
 24 
 
 369 
 
 4295 
 
 4264 
 
 4234 4203 
 
 4172 
 
 4142 
 
 4111 
 
 4081 
 
 4050 
 
 4020 
 
 
 9 
 
 ** 
 
 27 
 
 mmm 
 
 370 
 
 88 3989 
 
 3958 
 
 3928 j 3897 
 
 3866 
 
 3836 
 
 3805 
 
 3774 
 
 3744 
 
 3713 
 
 
 31 
 
 371 
 
 3682 
 
 3652 
 
 3621 3590 
 
 3559 
 
 3529 
 
 3498 
 
 3467 
 
 3436 
 
 3405 
 
 
 
 
 372 
 
 3375 
 
 3344 
 
 3313 3282 
 
 3251 
 
 3221 
 
 3190 
 
 3159 
 
 3128 
 
 3097 
 
 
 1 
 
 
 
 
 373 
 
 3066 
 
 3035 
 
 3004 2973 
 
 2943 
 
 2912 
 
 2881 
 
 2850 
 
 2819 
 
 2788 
 
 
 3 
 
 9 
 
 374 
 
 2757 
 
 2726 
 
 2695 
 
 2664 
 
 2633 
 
 2602 
 
 2571 2540 
 
 2508 
 
 2477 
 
 
 4 
 
 12 
 
 375 
 
 2446 
 
 2415 
 
 2384 
 
 2353 
 
 2322 
 
 2291 
 
 2260 2229 
 
 2197 
 
 2166 
 
 
 I 
 5jl6 
 
 376 
 
 2135 
 
 2104 
 
 2073 2041 
 
 2010 
 
 1979 
 
 1948 1917 
 
 1885 
 
 1854 
 
 
 6 
 
 19 
 
 377 
 
 1823 
 
 1791 
 
 1760 1729 
 
 1698 
 
 1666 
 
 1635) 1604 
 
 1572 
 
 1541 
 
 
 7 
 a 
 
 22 
 
 378 
 
 1510 
 
 1478 
 
 1447 
 
 1415 
 
 1384 
 
 1353 
 
 1321 1290 1258 
 
 1227 
 
 
 o 
 9 
 
 28 
 
 379 
 
 1195 
 
 1164 
 
 1132 
 
 1101 
 
 1069 
 
 1038 
 
 1006 
 
 0975 
 
 0943 
 
 0912 
 
 32 
 
 **mm 
 
 MB 
 
 380 
 
 88 0880 
 
 0849 
 
 0817 
 
 0786 
 
 0754 
 
 0722 
 
 0691 
 
 0659 
 
 0628 
 
 0596 
 
 
 32 
 
 381 
 
 0564 
 
 0533 
 
 0501 
 
 0469 
 
 0438 
 
 0406 
 
 0374 
 
 0342 
 
 0311 
 
 0279 
 
 
 j 
 
 3 
 
 382 
 
 0247 
 
 0215 
 
 0184 
 
 0152 
 
 0120 
 
 0088 
 
 0057 
 
 0025 9993 
 
 9961 
 
 
 2 
 
 6 
 
 383 
 
 87 9929 
 
 9897 
 
 9865 
 
 9834 
 
 9802 
 
 9770 
 
 9738 
 
 9706 
 
 9674 
 
 9642 
 
 
 3 
 
 10 
 
 384 
 
 9610 
 
 9578 
 
 9546 
 
 9514 
 
 9482 
 
 9450 
 
 9418 
 
 9386 
 
 9354 
 
 9322 
 
 
 4 
 
 13 
 
 385 
 
 9290 
 
 9258 
 
 9226 
 
 9194 
 
 9162 
 
 9130 
 
 9098 
 
 9066 
 
 9034 
 
 9002 
 
 
 5 
 
 16 
 m 
 
 386 
 
 8969 8937 
 
 8905 
 
 8873 
 
 8841 
 
 8809 
 
 8776J 8744 
 
 8712 
 
 8680 
 
 
 3 , 1 C7 
 
 7 22 
 
 387 
 
 8648 8615 
 
 8583 8551 
 
 8519 
 
 8486 
 
 8454J 8422 
 
 8389 
 
 8357 
 
 
 8 26 
 
 388 
 
 8325 8292 8260 8228 
 
 8195 
 
 8163 
 
 8130 8098 
 
 8066 
 
 8033 
 
 
 9 . 
 
 >9 
 
 389 
 
 8001 7968! 7936 
 
 7903 
 
 7871 
 
 7839 
 
 7806 
 
 7774 
 
 7741 
 
 7709 
 
 33 
 
 
 > 
 
 
 67 7676 7643 7611 7578 
 
 jt-if* 
 
 7ci q 
 
 74ft 1 
 
 7443 741 ?i 
 
 7quq 
 
 
 33 
 
 
 391 
 
 7350 7318 7285 7252 
 
 7220 
 
 i O id 
 
 7187 
 
 TO JL 
 
 7154 
 
 1 T?TO 1 T 1 ij 
 
 7122 7089 
 
 ooo 
 7056 
 
 
 I \ 
 
 -. 
 
 3 
 
 392 
 
 7023 6991 6958 
 
 6925 
 
 6892 
 
 6860 
 
 6827 
 
 6794 6761 
 
 6728 
 
 
 2 7 
 
 3! i /\ 
 
 393 
 
 6696 6663 6630 
 
 6597 
 
 6564 
 
 6531 
 
 6498 
 
 6465 
 
 6433 
 
 6400 
 
 
 1U 
 
 4 13 
 
 394 
 
 6367 
 
 6334 6301 
 
 6268 
 
 6235 
 
 6202 
 
 6169 
 
 6136 6103 
 
 6070 
 
 
 I 
 5 17 
 
 395 
 
 6037 6004 5971 
 
 5938 
 
 5905 
 
 5872 
 
 5838 
 
 5805 5772 
 
 5739 
 
 
 6 20 
 
 396 
 397 
 
 5706 
 5374 
 
 5673 5640J 5607 
 5341 5308 5274 
 
 5573 
 5241 
 
 5540 
 5208 
 
 5507 
 5175 
 
 5474 5441 
 5141 5108 
 
 5407 
 5075 
 
 
 7 23 
 826 
 q on 
 
 398 
 
 5041 
 
 5008 4975 
 
 4941 
 
 4908 
 
 4874 
 
 4841 
 
 4808 4774 
 
 4741 
 
 
 mn 
 
 f\J 
 
 
 
 399 
 
 4707 
 
 4674 
 
 4640 
 
 4607 
 
 4574 
 
 4540 
 
 4507 
 
 4473 
 
 4439 
 
 4406 
 
 34 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff'. 
 
 ro. 
 arts. 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(l-ff) 
 
 Diff. 
 
 Pro. 
 
 Parts. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 f-400 
 401 
 402 
 403 
 404 
 
 405 
 406 
 407 
 408 
 409 
 
 T-87 4372 
 4036 
 3699 
 3361 
 3022 
 
 2682 
 2341 
 1999 
 1655 
 1311 
 
 4339 
 4003 
 3666 
 3328 
 
 2988 
 
 2648 
 2307 
 1964 
 1621 
 1277 
 
 4305 
 3969 
 3632 
 3294 
 2954 
 
 2614 
 2273 
 1930 
 1587 
 1242 
 
 4272 
 3935 
 3598 
 3260 
 2920 
 2580 
 2238 
 1896 
 1552 
 1208 
 
 4238 
 3902 
 3564 
 3226 
 
 2886 
 
 2546 
 2204 
 1862 
 1518 
 1173 
 
 4205 
 3868 
 3531 
 3192 
 
 2852 
 
 2512 
 2170 
 1827 
 1483 
 1139 
 
 4171 
 3834 
 3497 
 3158 
 
 2818 
 
 2478 
 2136 
 1793 
 1449 
 1104 
 
 4137 
 3801 
 3463 
 3124 
 
 2784 
 
 2443 
 2102 
 1759 
 1415 
 1069 
 
 4104 
 3767 
 3429 
 3090 
 2750 
 
 2409 
 2007 
 1724 
 1380 
 1035 
 
 4070 
 3733 
 3395 
 3056 
 2716 
 
 2375 
 2033 
 1690 
 1346 
 1000 
 
 34 
 35 
 
 36 
 37 
 
 
 33 
 
 11 
 2 7 
 3 10 
 4 13 
 
 517 
 
 620 
 723 
 
 826 
 930 
 
 34 
 
 3 
 
 7 
 10 
 14 
 
 17 
 
 20 
 24 
 27 
 31 
 
 410 
 411 
 412 
 413 
 414 
 
 415 
 416 
 417 
 418 
 419 
 
 87 0966 
 0619 
 0272 
 86 9923 
 9573 
 
 9222 
 8870 
 8517 
 8163 
 
 7808 
 
 0931 
 0584 
 0237 
 9888 
 9538 
 
 9187 
 
 8835 
 8482 
 8128 
 
 7772 
 
 0896 
 0550 
 0202 
 9853 
 9503 
 
 9152 
 
 8800 
 8447 
 8092 
 
 7737 
 
 0862 
 0515 
 0167 
 9818 
 9468 
 
 9117 
 8765 
 8411 
 8057 
 7701 
 
 0827 
 0480 
 0132 
 9783 
 9433 
 
 9082 
 8729 
 8376 
 8021 
 7666 
 
 0793 
 0446 
 0097 
 9748 
 9398 
 
 9047 
 8694 
 8340 
 7986 
 7630 
 
 0758 
 0411 
 0063 
 9713 
 9363 
 
 9011 
 8659 
 8305 
 7950 
 7594 
 
 0723 
 0376 
 0028 
 9678 
 9328 
 
 8976 
 8623 
 8270 
 7915 
 7559 
 
 0689 
 0341 
 9993 
 9643 
 9293 
 
 8941 
 
 8588 
 8234 
 7879 
 7523 
 
 7166 
 6807 
 6448 
 6087 
 5726 
 
 5363 
 4999 
 4634 
 4267 
 3900 
 
 0654 
 0306 
 9958 
 9608 
 9257 
 
 8906 
 8553 
 8199 
 
 7844 
 7487 
 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 35 
 
 4 
 7 
 11 
 14 
 
 18 
 21 
 25 
 28 
 32 
 
 36 
 
 11 
 14 
 18 
 22 
 25 
 29 
 32 
 
 420 
 421 
 422 
 423 
 424 
 
 425 
 426 
 427 
 
 428 
 429 
 
 86 7452 
 7094 
 6736 
 6376 
 6015 
 
 5653 
 5290 
 4926 
 4560 
 4194 
 
 7416 
 7058 
 6700 
 6340 
 5979 
 
 5617 
 5254 
 4889 
 4524 
 4157 
 
 7380 
 7023 
 6664 
 6304 
 5943 
 
 5581 
 5217 
 4853 
 
 4487 
 4120 
 
 7345 
 6987 
 6628 
 6268 
 5907 
 
 5544 
 5181 
 4816 
 4451 
 
 4084 
 
 7309 
 6951 
 6592 
 6232 
 5870 
 
 5508 
 5144 
 4780 
 4414 
 4047 
 
 7273 
 6915 
 6556 
 6196 
 5834 
 
 5472 
 5108 
 4743 
 4377 
 4010 
 
 7237 
 6879 
 6520 
 6160 
 5798 
 
 5435 
 5072 
 4707 
 4341 
 3973 
 
 7202 
 6843 
 6484 
 6123 
 5762 
 
 5399 
 5035 
 4670 
 4304 
 3937 
 
 7130 
 6772 
 6412 
 6051 
 5689 
 
 5326 
 4962 
 4597 
 4231 
 3863 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 \ 
 
 37 
 
 4 
 7 
 11 
 15 
 19 
 22 
 26 
 30 
 33 
 
 38 
 
 4 
 8 
 11 
 15 
 19 
 23 
 27 
 30 
 34 
 
 430 
 431 
 432 
 433 
 434 
 
 435 
 436 
 437 
 438 
 439 
 
 86 3826 
 3457 
 3087 
 2716 
 2344 
 
 1970 
 1595 
 1220 
 
 0842 
 0464 
 
 3789 
 3420 
 3050 
 2679 
 2306 
 
 1933 
 1558 
 1182 
 0805 
 0426 
 
 3753 
 3383 
 3013 
 2642 
 2269 
 
 1895 
 1520 
 1144 
 0767 
 
 0388 
 
 3716 
 3346 
 2976 
 2605 
 2232 
 
 1858 
 1483 
 1107 
 0729 
 0350 
 
 3679 
 3309 
 2939 
 2567 
 2194 
 
 1820 
 1445 
 1069 
 0691 
 0312 
 
 3642 
 3272 
 2902 
 2530 
 2157 
 
 1783 
 1408 
 1031 
 0653 
 0275 
 
 3605 
 3235 
 2865 
 2493 
 2120 
 
 1745 
 1370 
 0993 
 0616 
 0237 
 
 3568 
 3198 
 
 2828 
 2456 
 
 2082 
 
 1708 
 1332 
 0956 
 0578 
 0199 
 
 3531 
 3161 
 2790 
 2418 
 2045 
 
 1671 
 1295 
 0918 
 0540 
 0161 
 
 3494 
 3124 
 2753 
 2381 
 
 2008 
 
 1633 
 1257 
 0880 
 0502 
 0123 
 
 38 
 39 
 
 
 1 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 
 I 
 
 39 
 
 4 
 8 
 12 
 16 
 20 
 23 
 27 
 31 
 35 
 
 40 
 
 4 
 8 
 12 
 16 
 20 
 24 
 28 
 32 
 36 
 
 440 
 441 
 442 
 443 
 444 
 
 445 
 446 
 447 
 
 448 
 449 
 
 86 0085 
 85 9704 
 9322 
 8939 
 8554 
 
 8169 
 7782 
 7391 
 7005 
 6614 
 
 0047 
 9666 
 9284 
 8900 
 8516 
 
 8130 
 7743 
 7355 
 6966 
 6575 
 
 0009 
 9628 
 9245 
 
 8862 
 8477 
 
 8092 
 7704 
 7316 
 6927 
 6536 
 
 9971 
 9589 
 9207 
 8824 
 8439 
 
 8053 
 7666 
 
 7277 
 6888 
 6497 
 
 9932 
 9551 
 9169 
 
 8785 
 8400 
 
 8014 
 7627 
 7238 
 6849 
 6457 
 
 9894 
 9513 
 9131 
 
 8747 
 8362 
 
 7976 
 7588 
 7199 
 6809 
 6418 
 
 9856 
 9475 
 9092 
 8708 
 8323 
 
 7937 
 7549 
 7160 
 6770 
 6379 
 
 9818 
 9437 
 9054 
 8670 
 
 8285 
 
 7898 
 7510 
 7122 
 6731 
 6340 
 
 9780 
 9398 
 9016 
 8631 
 
 8246 
 
 7859 
 7472 
 7083 
 6692 
 6301 
 
 9742 
 9360 
 8977 
 8593 
 8207 
 
 7821 
 7433 
 7044 
 6653 
 6261 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(l-;r) 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6065 
 5672 
 5277 
 4881 
 4483 
 
 4085 
 3685 
 3283 
 2881 
 2477 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1-450 
 451 
 452 
 453 
 454 
 
 455 
 456 
 457 
 
 458 
 459 
 
 1-85 6222 
 5829 
 5435 
 5039 
 4642 
 
 4244 
 3845 
 3444 
 3042 
 2639 
 
 6183 
 5790 
 5395 
 5000 
 4603 
 
 4204 
 3805 
 3404 
 3002 
 2598 
 
 6144 
 
 5750 
 5356 
 4960 
 4563 
 
 4161 
 3765 
 3364 
 2961 
 2558 
 
 6104 
 5711 
 5316 
 4920 
 4523 
 
 4124 
 3725 
 3324 
 2921 
 2517 
 
 6026 
 5632 
 5237 
 4841 
 4443 
 
 4045 
 3645 
 3243 
 2840 
 2436 
 
 2031 
 1625 
 1217 
 0807 
 0397 
 
 9985 
 9572 
 9157 
 8741 
 8324 
 
 5987 
 5593 
 5198 
 4801 
 4404 
 
 4005 
 3604 
 3203 
 2800 
 2396 
 
 5947 
 5553 
 5158 
 4762 
 4364 
 
 3965 
 3564 
 3163 
 2760 
 2356 
 
 5908 
 5514 
 5118 
 
 4722 
 4324 
 
 3925 
 3524 
 3122 
 2719 
 2315 
 
 5869 
 5474 
 5079 
 4682 
 4284 
 
 3885 
 3484 
 3082 
 2679 
 2275 
 
 40 
 41 
 
 
 1 
 
 2 
 3 
 
 6 
 
 6 
 
 7 
 8 
 9 
 
 40 
 
 4 
 8 
 12 
 16 
 20 
 24 
 28 
 32 
 36 
 
 41 
 
 4 
 
 8 
 12 
 16 
 21 
 25 
 29 
 33 
 37 
 
 460 
 461 
 462 
 463 
 464 
 
 465 
 466 
 467 
 468 
 469 
 
 85 2234 
 1828 
 1421 
 1012 
 0602 
 
 0191 
 
 84 9778 
 9365 
 8949 
 8533 
 
 2193 
 
 1787 
 1380 
 0971 
 0561 
 
 0150 
 9737 
 9323 
 
 8908 
 8491 
 
 2153 
 1747 
 1339 
 0930 
 0520 
 
 0109 
 9696 
 9282 
 8866 
 8449 
 
 2112 
 1706 
 1298 
 0889 
 0479 
 
 0067 
 9654 
 9240 
 
 8824 
 8407 
 
 2072 
 1665 
 1258 
 0848 
 0438 
 
 0026 
 9613 
 9199 
 8783 
 8366 
 
 1991 
 1584 
 1176 
 0766 
 0356 
 
 9944 
 9530 
 9116 
 8699 
 
 8282 
 
 1950 
 1543 
 1135 
 0725 
 0315 
 
 9902 
 9489 
 9074 
 8658 
 8240 
 
 1909 
 1502 
 1094 
 0684 
 0273 
 
 9861 
 9447 
 9032 
 8616 
 
 8198 
 
 1869 
 1462 
 1053 
 0643 
 0232 
 
 9820 
 9406 
 8991 
 8574 
 8156 
 
 42 
 43 
 
 44 
 45 
 
 46 
 
 Diff. 
 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 1? 
 4 
 8 
 13 
 17 
 21 
 25 
 29 
 34 
 38 
 
 43 
 
 4 
 9 
 13 
 17 
 22 
 26 
 30 
 34 
 39 
 
 470 
 471 
 472 
 473 
 474 
 
 475 
 476 
 477 
 
 478 
 479 
 
 848115 
 7695 
 
 7275 
 6852 
 6429 
 
 6004 
 5578 
 5150 
 4721 
 4290 
 
 8073 
 7653 
 7232 
 6810 
 6386 
 
 5961 
 5535 
 5107 
 4678 
 4247 
 
 8031 
 7611 
 7190 
 6768 
 6344 
 
 5919 
 5492 
 5064 
 4635 
 4204 
 
 7989 
 7569 
 7148 
 6725 
 6302 
 
 5876 
 5450 
 5021 
 4592 
 4161 
 
 7947 
 
 7527 
 7106 
 6683 
 6259 
 
 5834 
 5407 
 4979 
 4549 
 4118 
 
 7905 
 7485 
 7064 
 6641 
 6217 
 
 5791 
 5364 
 4936 
 4506 
 4075 
 
 7863 
 7443 
 7021 
 6598 
 6174 
 
 5748 
 5321 
 4893 
 4463 
 4031 
 
 7821 
 7401 
 6979 
 6556 
 6132 
 
 5706 
 5278 
 4850 
 4420 
 3988 
 
 7779 
 7359 
 6937 
 6514 
 6089 
 
 5663 
 5236 
 4807 
 4377 
 3945 
 
 7737 
 7317 
 6895 
 6471 
 6047 
 
 5620 
 5193 
 4764 
 4333 
 3902 
 
 3468 
 3034 
 2598 
 2160 
 1721 
 
 1281 
 0839 
 0395 
 9951 
 9504 
 
 
 2 
 3 
 4 
 5 
 6 
 
 : 
 
 4445 
 
 4 5 
 9 9 
 1314 
 1818 
 2223 
 2627 
 31J32 
 3536 
 4041 
 
 480 
 481 
 482 
 483 
 
 484 
 
 485 
 486 
 487 
 488 
 489 
 
 84 3858 
 3425 
 2990 
 2554 
 2116 
 
 1677 
 1237 
 0795 
 035J 
 83 9906 
 
 3815 
 3382 
 2947 
 2510 
 2072 
 
 1633 
 1192 
 0750 
 0307 
 9861 
 
 3772 
 3338 
 2903 
 2467 
 2029 
 
 1589 
 1148 
 0706 
 0262 
 9817 
 
 3729 
 3295 
 2860 
 2423 
 1985 
 
 1545 
 1104 
 0662 
 0218 
 9772 
 
 3685 
 3251 
 2816 
 2379 
 1941 
 
 1501 
 1060 
 0617 
 0173 
 9728 
 
 3642 
 
 3208 
 2772 
 2335 
 1897 
 
 1457 
 1016 
 0573 
 0129 
 9683 
 
 3599 
 3164 
 2729 
 2292 
 1853 
 
 1413 
 0972 
 0529 
 0084 
 9638 
 
 3555 
 3121 
 
 2685 
 2248 
 1809 
 
 1369 
 0927 
 0484 
 0040 
 9594 
 
 3512 
 3077 
 2641 
 2204 
 1765 
 
 1325 
 0883 
 0440 
 9995 
 9549 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 46,47 
 
 5 5 
 9 9 
 1414 
 18,19 
 2324 
 28,28 
 3233 
 3738 
 41 42 
 
 490 
 491 
 492 
 493 
 494 
 
 495 
 496 
 497 
 498 
 499 
 
 83 9459 
 9011 
 8562 
 8111 
 7659 
 
 7205 
 6749 
 6292 
 5833 
 5373 
 
 9415 
 8967 
 8517 
 8066 
 7613 
 
 7159 
 6703 
 6246 
 
 5787 
 5327 
 
 9370 
 8922 
 8472 
 8021 
 7568 
 
 7114 
 6658 
 6200 
 5741 
 5281 
 
 9325 
 
 8877 
 8427 
 7975 
 7522 
 
 7068 
 6612 
 6154 
 5695 
 5235 
 
 9280 
 
 8832 
 8382 
 7930 
 
 7477 
 
 7022 
 6566 
 6109 
 5649 
 5189 
 
 9236 
 8787 
 8337 
 7885 
 7432 
 
 6977 
 6521 
 6063 
 5603 
 5142 
 
 9191 
 
 8742 
 8292 
 7840 
 7386 
 
 6931 
 6475 
 6017 
 5557 
 5096 
 
 9146 
 8697 
 8246 
 7794 
 7341 
 
 6886 
 6429 
 5971 
 5511 
 5050 
 
 9101 
 8652 
 8201 
 7749 
 7295 
 
 6840 
 6383 
 5925 
 5465 
 5004 
 
 9056 
 8607 
 8156 
 7704 
 7250 
 
 6795 
 6338 
 5879 
 5419 
 
 4958 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Pro. 
 
 Parts. 
 
 57 
 
TABLE II. 
 
 Log 
 
 Log(l-ar) 
 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 Parts. 
 
 r-500 
 
 834911 
 
 4865 
 
 4819 
 
 4773 
 
 4726 
 
 4680 
 
 4634 
 
 4587 
 
 4541 
 
 4495 
 
 
 
 501 
 
 4448 
 
 4402 
 
 4355 
 
 4309 
 
 4262 
 
 4216 
 
 4170 
 
 4123 
 
 4076 
 
 4030 
 
 47 
 
 4 
 
 17 
 
 18 
 
 502 
 
 3983 
 
 3937 
 
 3890 
 
 3844 
 
 3797 
 
 3750 
 
 3704 
 
 3657 
 
 3610 
 
 3564 
 
 
 
 
 
 
 503 
 
 3517 
 
 3470 
 
 3424 
 
 3377 
 
 3330 
 
 3283 
 
 3236 
 
 3190 
 
 3143 
 
 3096 
 
 
 o 
 
 b 
 n 
 
 b 
 10 
 
 504 
 
 3049 
 
 3002 
 
 2955 
 
 2908 
 
 2861 
 
 2814 
 
 2767 
 
 2721 
 
 2674 
 
 2626 
 
 
 314 
 
 14 
 
 505 
 
 2579 
 
 2532 
 
 2485 
 
 2438 
 
 2391 
 
 2344 
 
 2297 
 
 2250 
 
 2203 
 
 2156 
 
 
 419 
 
 
 506 
 
 2108 
 
 2061 
 
 2014 
 
 1967 
 
 1919 
 
 1872 
 
 1825 
 
 1778 
 
 1730 
 
 1683 
 
 
 524 
 
 24 
 
 507 
 
 1636 
 
 1588 
 
 1541 
 
 1493 
 
 1446 
 
 1399 
 
 1351 
 
 1304 
 
 1256 
 
 1209 
 
 
 628 
 
 29 
 
 508 
 509 
 
 1161 
 
 0685 
 
 1114 
 0638 
 
 1066 
 0590 
 
 1019 
 0542 
 
 0971 
 0494 
 
 0923 
 0447 
 
 0876 
 0399 
 
 0828 
 0351 
 
 0781 
 0303 
 
 0733 
 0256 
 
 48 
 
 733 
 838 
 9 ( 42 
 
 34 
 
 38 
 43 
 
 510 
 
 83 0208 
 
 0160 
 
 0112 
 
 0064 
 
 0016 
 
 9968 
 
 9920 
 
 9872 
 
 9824 
 
 9776 
 
 
 
 
 511 
 
 82 9728 
 
 9680 
 
 9632 
 
 9584 
 
 9536 
 
 9488 
 
 9440 
 
 9392 
 
 9344 
 
 9296 
 
 
 49J50 
 
 512 
 
 9248 
 
 9199 
 
 9151 
 
 9103 
 
 9055 
 
 9007 
 
 8958 
 
 8910 
 
 8862 
 
 8813 
 
 
 1 
 
 5 
 
 c 
 
 513 
 
 8765 
 
 8717 
 
 8668 
 
 8620 
 
 8572 
 
 8523 
 
 8475 
 
 8426 
 
 8378 
 
 8329 
 
 
 2 
 
 10 
 
 10 
 
 514 
 
 8281 
 
 8232 
 
 8184 
 
 8135 
 
 8087 
 
 8038 
 
 7990 
 
 7941 
 
 7892 
 
 7844 
 
 49 
 
 3 
 
 lb 
 
 15 
 
 515 
 
 7795 
 
 7747 
 
 7698 
 
 7649 
 
 7600 
 
 7552 
 
 7503 
 
 7454 
 
 7405 
 
 7357 
 
 
 4 
 
 20 
 
 20 
 
 516 
 
 7308 
 
 7259 
 
 7210 
 
 7161 
 
 7112 
 
 7063 
 
 7014 
 
 6965 
 
 6917 6868 
 
 
 b 
 
 6 
 
 2b 
 
 9 
 
 2b 
 30 
 
 517 
 
 6819 
 
 6770 
 
 6721 
 
 667J 
 
 6622 
 
 6573 
 
 6524 
 
 6475 
 
 6426 
 
 6377 
 
 
 7 
 
 34 
 
 35 
 
 518 
 
 6328 
 
 6279 
 
 6229 
 
 6180 
 
 6131 
 
 6082 
 
 6032 
 
 5983 
 
 5934 
 
 5885 
 
 
 8 
 
 39 
 
 40 
 
 519 
 
 5835 
 
 5786 
 
 5736 
 
 5687 
 
 5638 
 
 5588 
 
 5539 
 
 5489 
 
 5440 
 
 5390 
 
 
 9 
 
 44 
 
 45 
 
 520 
 
 82 5341 
 
 5292 
 
 5242 
 
 5192 
 
 5143 
 
 5093 
 
 5044 
 
 4994 
 
 4944 
 
 4895 
 
 50 
 
 
 51 52 
 
 521 
 
 4845 
 
 4795 
 
 4746 
 
 4696 
 
 4646 
 
 4596 
 
 4547 
 
 4497 
 
 4447 
 
 439 r 
 
 
 
 D1 r* 
 
 522 
 
 4347 
 
 4298 
 
 4248 
 
 4198 
 
 4148 
 
 4098 
 
 4048 
 
 3998 
 
 3948 
 
 3898 
 
 
 1 
 
 5 
 
 
 523 
 
 3848 
 
 379b 
 
 3748 
 
 3698 
 
 3648 
 
 3598 
 
 3548 
 
 3498 
 
 3447 
 
 3397 
 
 
 2 
 
 1010 
 
 524 
 
 3347 
 
 3297 
 
 3247 
 
 3196 
 
 3146 
 
 3096 
 
 3046 
 
 299o 
 
 2945 
 
 2895 
 
 
 3 
 
 4 
 
 15 16 
 2021 
 
 525 
 
 2844 
 
 2794 
 
 2743 
 
 2693 
 
 2643 
 
 2592 
 
 2542 
 
 249 
 
 2441 
 
 2390 
 
 51 
 
 5 
 
 2626 
 
 526 
 
 2340 
 
 2289 
 
 2239 
 
 2188 
 
 2137 
 
 2087 
 
 2036 
 
 1985 
 
 1935 
 
 1884 
 
 
 (i 
 
 3l!31 
 
 527 
 
 1833 
 
 1783 
 
 1732 
 
 1681 
 
 1630 
 
 1580 
 
 1529 
 
 1478 
 
 1427 
 
 1376 
 
 
 7 
 
 3636 
 
 528 
 
 1325 
 
 1274 
 
 1223 
 
 1173 
 
 1122 
 
 1071 
 
 1020 
 
 0969 
 
 0918 
 
 0867 
 
 
 8 
 
 41 42 
 
 529 
 
 0815 
 
 0764 
 
 0713 
 
 0662 
 
 0611 
 
 0560 
 
 0509 
 
 0458 
 
 0106 
 
 0355 
 
 
 9 
 
 4647 
 
 
 
 
 
 
 0099 
 
 
 
 
 
 9842 
 
 
 
 530 
 
 82 0304 
 
 0253 
 
 0201 
 
 0150 
 
 0047 
 
 9996 
 
 9945 
 
 9893 
 
 
 5354 
 
 531 
 
 81 9790 
 
 9739 
 
 9688 
 
 9636 
 
 9,585 
 
 9533 
 
 948? 
 
 943C 
 
 9378 
 
 9327 
 
 52 
 
 
 
 
 
 532 
 
 9275 
 
 9224 
 
 9172 
 
 9120 
 
 9069 
 
 9017 
 
 8965 
 
 8914 
 
 8862 
 
 8810 
 
 
 I 
 
 5 
 
 
 533 
 
 8758 
 
 8706 
 
 8655 
 
 8603 
 
 8551 
 
 8499 
 
 8447 
 
 8395 
 
 8343 
 
 8291 
 
 
 2 
 
 3 
 
 HIM 
 
 16 }K 
 
 534 
 
 8239 
 
 8187 
 
 8135 
 
 8083 
 
 8031 
 
 7979 
 
 7927 
 
 787j 
 
 7823 
 
 7771 
 
 
 4 
 
 21 
 
 22 
 
 535 
 
 7719 
 
 7667 
 
 7614 
 
 7562 
 
 7510 
 
 7458 
 
 7406 
 
 7353 
 
 7301 
 
 7249 
 
 
 5 
 
 27 
 
 27 
 
 536 
 
 7196 
 
 7144 
 
 7092 
 
 7039 
 
 6987 
 
 6934 
 
 6882 
 
 6830 
 
 6777 
 
 6725 
 
 53 
 
 6 
 
 32 
 
 32 
 
 537 
 
 6672 
 
 6620 
 
 6567 
 
 6514 
 
 6462 
 
 6409 
 
 6357 
 
 6304 
 
 6251 
 
 6199 
 
 
 V 
 
 37 
 
 38 
 
 538 
 
 6146 
 
 6093 
 
 6041 
 
 c c i o 
 
 5988 
 
 f 4 : iQ 
 
 5935 
 t/i r\f> 
 
 5882 
 
 5829 
 
 577^ 
 
 IKO/1O 
 
 5724 
 
 C I QA 
 
 5671 
 
 K I A.] 
 
 
 8 
 !> 
 
 42 
 
 48 
 
 43 
 49 
 
 ooy 
 
 OOio 
 
 oooo 
 
 
 
 0*1)0 
 
 4876 
 
 dOOJ 
 
 OoUU 
 
 
 
 
 
 
 540 
 
 81 5088 
 
 5035 
 
 4982 
 
 4929 
 
 4823 
 
 4769 
 
 4716 
 
 4663 
 
 4610 
 
 
 5556 
 
 541 
 
 4556 
 
 4503 
 
 4450 
 
 4397 
 
 4343 
 
 4290 
 
 4237 
 
 418, 
 
 4130 
 
 4076 
 
 
 
 
 
 
 
 542 
 
 4023 
 
 3969 
 
 3916 
 
 3862 
 
 3809 
 
 3755 
 
 3702 
 
 364S 
 
 3595 
 
 3541 
 
 54 
 
 1 
 
 
 
 (j 
 
 n 
 
 |] 
 
 543 
 
 3487 
 
 3434 
 
 3380 
 
 3326 
 
 3273 
 
 3219 
 
 3165 
 
 311 
 
 3058 
 
 3004 
 
 
 3 
 
 17 
 
 17 
 
 544 
 
 2950 
 
 2896 
 
 2842 
 
 2788 
 
 2734 
 
 2681 
 
 2627 
 
 2573 
 
 2519 
 
 2465 
 
 
 4 
 
 2222 
 
 545 
 
 2411 
 
 2357 
 
 2303 
 
 2249 
 
 2194 
 
 2140 
 
 2086 
 
 2032 
 
 1978 
 
 1924 
 
 
 5 
 
 2828 
 
 546 
 
 1870 
 
 1815 
 
 1761 
 
 1707 
 
 1652 
 
 1598 
 
 1544 
 
 1490 
 
 1435 
 
 1381 
 
 
 b 
 
 7 
 
 3334 
 
 }Q QQ 
 
 547 
 
 1326 
 
 1272 
 
 1218 
 
 1163 
 
 1109 
 
 1054 
 
 1000 
 
 0945 
 
 0890 
 
 0836 
 
 55 
 
 8 
 
 44 45 
 
 548 
 
 0781 
 
 0727 
 
 0672 
 
 0617 
 
 0563 
 
 0508 
 
 0453 
 
 0399 
 
 0344 
 
 0289 
 
 
 9 50 50 
 
 549 
 
 0234 
 
 01V9 
 
 0125 
 
 0070 
 
 0015 
 
 9960 
 
 9905 
 
 9850 
 
 9795 
 
 9740 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff 
 
 Pro. 
 
 Parts. 
 
 58 
 
TABLE II. 
 
 
 Log(l-*) 
 
 
 
 Log 
 
 
 Diff 
 
 Pro. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 Parts. 
 
 550 
 
 ~ 80 9685 
 
 9630 
 
 9575 
 
 9520 
 
 9465 
 
 9410 
 
 9355 
 
 9300 
 
 9245 
 
 9190 
 
 
 
 551 
 
 9134 
 
 9079 
 
 9024 
 
 8969 
 
 8914 
 
 8858 
 
 8803 
 
 8748 
 
 8692 
 
 8637 
 
 
 
 5758 
 
 552 
 
 8582 
 
 8526 
 
 8471 
 
 8415 
 
 8360 
 
 8304 
 
 8249 
 
 8193 
 
 8138 
 
 8082 
 
 56 
 
 
 
 
 
 
 
 553 
 
 8027 
 
 7971 
 
 7915 
 
 7860 
 
 7804 
 
 7748 
 
 7693 
 
 7637 
 
 7581 
 
 7526 
 
 
 1 
 
 6 
 
 -6 
 
 554 
 
 7470 
 
 7414 
 
 7358 
 
 7302 
 
 7246 
 
 7191 
 
 7135 
 
 7079 
 
 7023 
 
 6967 
 
 
 2 
 3 
 
 1 1 
 17 
 
 12 
 17 
 
 555 
 
 6911 
 
 6855 
 
 6799 
 
 6743 
 
 6687 
 
 6631 
 
 6575 
 
 6519 
 
 6462 
 
 6406 
 
 
 42323 
 
 556 
 
 6350 
 
 6294 
 
 6238 
 
 6181 
 
 6125 
 
 6069 
 
 6013 
 
 5956 
 
 5900 
 
 5844 
 
 
 52929 
 
 557 
 
 5787 
 
 5731 
 
 5674 
 
 5618 
 
 5561 
 
 5505 
 
 5448 
 
 5392 
 
 5335 
 
 5279 
 
 57 
 
 6 34 35 
 
 558 
 
 5222 
 
 5166 
 
 5109 
 
 5052 
 
 4996 
 
 4939 
 
 4882 
 
 4826 
 
 4769 
 
 4712 
 
 
 7 40 41 
 8 4fi 4fi 
 
 559 
 
 4655 
 
 4598 
 
 4542 
 
 4485 
 
 4428 
 
 4371 
 
 4314 
 
 4257 
 
 4200 
 
 4143 
 
 
 O rtO TtO 
 
 95l|52 
 
 560 
 
 80 4086 
 
 4029 
 
 3972 
 
 3915 
 
 3858 
 
 3801 
 
 3744 
 
 3687 
 
 3630 
 
 3572 
 
 
 
 
 561 
 
 3515 
 
 3458 
 
 3401 
 
 3343 
 
 3286 
 
 3229 
 
 3171 
 
 3114 
 
 3057 
 
 2999 
 
 
 _|59 
 
 60 
 
 562 
 
 2942 
 
 2885 
 
 2827 
 
 2770 
 
 2712 
 
 2655 
 
 2597 
 
 2540 
 
 2482 
 
 2424 
 
 58 
 
 1 
 
 Q 
 
 5 
 
 563 
 
 2367 
 
 2309 
 
 2251 
 
 2194 
 
 2136 
 
 2078 
 
 2021 
 
 1963 
 
 1905 
 
 1847 
 
 
 2 
 
 12 
 
 12 
 
 564 
 
 1789 
 
 1732 
 
 1674 
 
 1616 
 
 1558 
 
 1500 
 
 1442 
 
 1384 
 
 1326 
 
 1268 
 
 
 3 
 
 18 
 
 18 
 
 565 
 
 1210 
 
 1152 
 
 1094 
 
 1036 
 
 0978 
 
 0919 
 
 0861 
 
 0803 
 
 0745 
 
 0687 
 
 
 4 
 
 24 
 
 24 
 
 566 
 
 0628 
 
 0570 
 
 0512 
 
 0454 
 
 0395 
 
 0337 
 
 0279 
 
 0220 
 
 0162 
 
 0103 
 
 
 5 
 
 30 
 
 30 
 
 567 
 
 0045 
 
 9986 
 
 9928 
 
 9869 
 
 9811 
 
 9752 
 
 9694 
 
 9635 
 
 9576 
 
 9518 
 
 59 
 
 6 
 
 7 
 
 35 
 41 
 
 36 
 42 
 
 568 
 
 79 9459 
 
 9400 
 
 9342 
 
 9283 
 
 9224 
 
 9165 
 
 9106 
 
 9048 
 
 8989 
 
 8930 
 
 
 8 
 
 47 
 
 48 
 
 569 
 
 8871 
 
 8812 
 
 8753 
 
 8694 
 
 8635 
 
 8576 
 
 8517 
 
 8458 
 
 8399 
 
 8340 
 
 
 9 
 
 53 
 
 54 
 
 570 
 
 798281 
 
 8222 
 
 8163 
 
 8104 
 
 8044 
 
 7985 
 
 7926 
 
 7867 
 
 7807 
 
 7748 
 
 
 
 61 
 
 fi9 
 
 571 
 
 7689 
 
 7629 
 
 7570 
 
 7511 
 
 7451 
 
 7392 
 
 7332 
 
 7273 
 
 7213 
 
 7154 
 
 60 
 
 
 
 O 
 
 572 
 
 7094 
 
 7035 
 
 6975 
 
 6915 
 
 6856 
 
 6796 
 
 6737 
 
 6677 
 
 6617 
 
 6557 
 
 
 ~] 
 
 6 
 
 6 
 
 573 
 
 6498 
 
 6438 
 
 6378 
 
 6318 
 
 6258 
 
 6198 
 
 6139 
 
 6079 
 
 6019 
 
 5959 
 
 
 2 
 
 12 
 
 12 
 
 574 
 
 5899 
 
 5839 
 
 5779 
 
 5719 
 
 5659 
 
 5599 
 
 5538 
 
 5478 
 
 5418 
 
 5358 
 
 
 3 
 
 18 
 
 19 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 24 
 
 25 
 
 575 
 
 5298 
 
 5238 
 
 5177 
 
 5117 
 
 5057 
 
 4996 
 
 4936 
 
 4876 
 
 4815 
 
 4755 
 
 
 5 
 
 31 
 
 31 
 
 576 
 
 4694 
 
 4634 
 
 4574 
 
 4513 
 
 4453 
 
 4392 
 
 4331 
 
 4271 
 
 4210 
 
 4150 
 
 61 
 
 6 
 
 37 
 
 37 
 
 577 
 
 4089 
 
 4028 
 
 3968 
 
 3907 
 
 3846 
 
 3785 
 
 3725 
 
 3664 
 
 3603 
 
 3542 
 
 
 7 
 
 43 
 
 43 
 
 578 
 
 3481 
 
 3420 
 
 3359 
 
 3298 
 
 3238 
 
 3177 
 
 3116 
 
 3054 
 
 2993 
 
 2932 
 
 
 8 
 
 49 
 
 50 
 
 579 
 
 2871 
 
 2810 
 
 2749 
 
 2688 
 
 2627 
 
 2565 
 
 2504 
 
 2443 
 
 2382 
 
 2320 
 
 
 955 
 
 56 
 
 
 
 
 2136 
 
 
 
 
 
 
 
 
 62 
 
 
 580 
 
 79 2259 
 
 2198 
 
 2075 
 
 2013 
 
 1952 
 
 1891 
 
 1829 
 
 1768 
 
 1706 
 
 
 63 
 
 64 
 
 581 
 
 1644 
 
 1583 
 
 1521 
 
 1460 
 
 1398 
 
 1336 
 
 1275 
 
 1213 
 
 1151 
 
 1089 
 
 
 
 
 
 
 582 
 
 1028 
 
 0966 
 
 0904 
 
 0842 
 
 0780 
 
 0718 
 
 0656 
 
 0594 
 
 0533 
 
 0471 
 
 
 1 
 
 6 
 
 6 
 
 583 
 
 0409 
 
 0346 
 
 0284 
 
 0222 
 
 0160 
 
 0098 
 
 0036 
 
 9974 
 
 9912 
 
 9849 
 
 
 2 
 
 13 
 
 13 
 
 584 
 
 78 9787 
 
 9725 
 
 9662 
 
 9600 
 
 9538 
 
 9475 
 
 9413 
 
 9351 
 
 9288 
 
 9226 
 
 
 3 
 4 
 
 19 
 25 
 
 19 
 26 
 
 585 
 
 9163 
 
 9101 
 
 9038 
 
 8976 
 
 8913 
 
 8851 
 
 8788 
 
 8725 
 
 8663 
 
 8600 
 
 63 
 
 5 
 
 32 
 
 32 
 
 586 
 
 8537 
 
 8474 
 
 8412 
 
 8349 
 
 8286 
 
 8223 
 
 8160 
 
 8098 
 
 8035 
 
 7972 
 
 
 6 
 
 38 
 
 38 
 
 587 
 
 7909 
 
 7846 
 
 7783 
 
 7720 
 
 7657 
 
 7594 
 
 7531 
 
 7467 
 
 7404 
 
 7341 
 
 
 7 
 
 44 
 
 45 
 
 588 
 
 7278 
 
 7215 
 
 7151 
 
 7088 
 
 7025 
 
 6962 
 
 6898 
 
 6835 
 
 6772 
 
 6708 
 
 
 8 
 
 50 
 
 51 
 
 589 
 
 6645 
 
 6581 
 
 6518 
 
 6454 
 
 6391 
 
 6327 
 
 6264 
 
 6200 
 
 6136 
 
 6073 
 
 64 
 
 9 
 
 MM 
 
 57 
 
 tmmm 
 
 58 
 
 3M 
 
 590 
 
 78 6009 
 
 5945 
 
 5882 
 
 5818 
 
 5754 
 
 5690 
 
 5627 
 
 5563 
 
 5499 
 
 5435 
 
 
 
 65 
 
 66 
 
 591 
 
 5371 
 
 5307 
 
 5243 
 
 5179 
 
 5115 
 
 5051 
 
 4987 
 
 4923 
 
 4859 
 
 4795 
 
 
 
 
 
 
 
 
 592 
 
 4731 
 
 4667 
 
 4602 
 
 4538 
 
 4474 
 
 4410 
 
 4345 
 
 4281 
 
 4217 
 
 4152 
 
 
 1 
 
 2 
 
 7 
 13 
 
 7 
 13 
 
 593 
 
 4088 
 
 4023 
 
 3959 
 
 3895 
 
 3830 
 
 3766 
 
 3701 
 
 3636 
 
 3572 
 
 3507 
 
 65 
 
 3 
 
 20 
 
 20 
 
 594 
 
 3443 
 
 3378 
 
 3313 
 
 3249 
 
 3184 
 
 3119 
 
 3054 
 
 2990 
 
 2925 
 
 2860 
 
 
 4 
 
 26 
 
 26 
 
 595 
 
 2795 
 
 2730 
 
 2665 
 
 2600 
 
 2535 
 
 2470 
 
 2405 
 
 2340 
 
 2275 
 
 2210 
 
 
 5 
 
 33 
 
 33 
 
 596 
 
 2145 
 
 2080 
 
 2014 
 
 1949 
 
 1884 
 
 1819 
 
 1753 
 
 1688 
 
 1623 
 
 1558 
 
 
 6 
 
 39 
 
 46 
 
 40 
 if 
 
 597 
 
 1492 
 
 1427 
 
 1361 
 
 1296 
 
 1230 
 
 1165 
 
 1099 
 
 1034 
 
 0968 
 
 0903 
 
 66 
 
 6 
 
 52 
 
 *rC 
 
 53 
 
 598 
 
 0837 
 
 0771 
 
 0706 
 
 0640 
 
 0574 
 
 0508 
 
 0443 
 
 0377 
 
 0311 
 
 0245 
 
 
 
 
 59 
 
 59 
 
 599 
 
 0179 
 
 0113 
 
 0047 
 
 9982 
 
 9916 
 
 9850 
 
 9784 
 
 9717 
 
 9651 
 
 9585 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. 
 Parts. 
 
 59 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(i-*) 
 
 Diff 
 
 Pro. Parts. 
 
 - 
 
 66 
 
 7 
 13 
 
 20 
 
 26 
 
 33 
 4C 
 46 
 53 
 5S 
 
 67 
 
 7 
 13 
 20 
 27 
 34 
 40 
 47 
 54 
 60 
 
 68 
 
 14 
 
 20 
 27 
 34 
 41 
 
 48 
 54 
 6 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 r-eoo 
 
 601 
 602 
 603 
 604 
 
 605 
 606 
 607 
 608 
 609 
 
 T-77 9519 
 8857 
 8191 
 7523 
 6853 
 
 6180 
 5505 
 4826 
 4146 
 3462 
 
 9453 
 
 8790 
 8125 
 7457 
 6786 
 
 6113 
 5437 
 
 4758 
 4077 
 3394 
 
 9387 
 8724 
 8058 
 7390 
 6719 
 
 6045 
 5369 
 4690 
 4009 
 3325 
 
 932 
 
 8657 
 7991 
 7323 
 6651 
 5978 
 5301 
 4622 
 3941 
 3257 
 
 9254 
 8591 
 7924 
 7256 
 6584 
 
 5910 
 5234 
 4554 
 3873 
 3188 
 
 9188 
 
 8524 
 7858 
 7189 
 6517 
 
 5843 
 5166 
 4486 
 3804 
 3119 
 
 9122 
 8458 
 7791 
 7122 
 6450 
 
 5775 
 5098 
 4418 
 3736 
 3051 
 
 9056 
 8391 
 7724 
 7054 
 6382 
 
 5708 
 5030 
 4350 
 3667 
 
 2982 
 
 8989 
 8325 
 7657 
 6987 
 6315 
 
 5640 
 4962 
 4282 
 3599 
 2914 
 
 8923 
 8258 
 7590 
 6920 
 6248 
 
 5572 
 4894 
 4214 
 353 
 
 2845 
 
 2156 
 1465 
 0771 
 0075 
 9375 
 
 8673 
 7969 
 7261 
 6550 
 5837 
 
 67 
 68 
 69 
 70 
 71 
 
 72 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 79 
 80 
 81 
 
 2 
 3 
 4 
 
 5 
 6 
 7 
 8 
 9 
 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 68 
 
 7 
 14 
 21 
 
 28 
 35 
 41 
 48 
 55 
 62 
 
 70 
 
 7 
 14 
 21 
 28 
 35 
 42 
 49 
 56 
 63 
 
 71 
 
 14 
 21 
 28 
 36 
 43 
 50 
 57 
 64 
 
 610 
 611 
 612 
 613 
 614 
 
 615 
 
 616 
 617 
 618 
 619 
 
 77 2776 
 2087 
 1396 
 0702 
 0005 
 
 76 9305 
 8603 
 7898 
 7190 
 6479 
 
 2707 
 2018 
 1327 
 0632 
 9935 
 
 9235 
 8533 
 7827 
 7119 
 6408 
 
 2639 
 1949 
 1257 
 0563 
 
 9865 
 
 9165 
 8462 
 
 7757 
 7048 
 6337 
 
 2570 
 1880 
 1188 
 0493 
 9795 
 
 9095 
 8392 
 7686 
 6977 
 6266 
 
 2501 
 1811 
 1119 
 0423 
 
 9725 
 
 9025 
 8321 
 7615 
 6906 
 6194 
 
 2432 
 1742 
 1049 
 0354 
 9655 
 
 8954 
 8251 
 7544 
 6835 
 6123 
 
 2363 
 1673 
 0980 
 0284 
 9585 
 8884 
 8180 
 7473 
 6764 
 6052 
 
 2294 
 1604 
 0910 
 0214 
 9515 
 8814 
 8110 
 7403 
 6693 
 5980 
 
 2225 
 1534 
 0841 
 0144 
 9445 
 
 8744 
 8039 
 7332 
 6622 
 5909 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 ' 
 
 14 
 22 
 
 29 
 
 36 
 43 
 
 50 
 58 
 65 
 
 73 
 
 7 
 15 
 22 
 29 
 27 
 44 
 51 
 58 
 66 
 
 74 
 
 15 
 22 
 
 30 
 
 37 
 44 
 52 
 
 59 
 67 
 
 620 
 621 
 622 
 623 
 624 
 
 625 
 626 
 627 
 628 
 629 
 
 76 5766 
 5050 
 4330 
 3608 
 
 2884 
 
 2156 
 1425 
 0692 
 75 9955 
 9216 
 
 5694 
 4978 
 4258 
 3536 
 2811 
 
 2083 
 1352 
 0618 
 9881 
 9142 
 
 8399 
 7653 
 6904 
 6153 
 5398 
 
 4640 
 3879 
 3115 
 2348 
 
 1578 
 
 5623 
 4906 
 4186 
 3464 
 
 2738 
 
 2010 
 1279 
 0545 
 9807 
 9067 
 
 5551 
 4834 
 4114 
 3391 
 2666 
 
 1937 
 1205 
 0471 
 9734 
 
 8993 
 
 8250 
 7504 
 6754 
 6002 
 5246 
 
 4488 
 3727 
 2962 
 2194 
 1423 
 
 5480 
 4762 
 4042 
 3319 
 2593 
 
 1864 
 1132 
 0397 
 9660 
 8919 
 
 5408 
 4690 
 3970 
 3246 
 2520 
 
 1791 
 1059 
 0324 
 9586 
 
 8845 
 
 5336 
 4618 
 3898 
 3174 
 2447 
 
 1718 
 0985 
 0250 
 9512 
 
 8771 
 
 5265 
 4546 
 3825 
 3101 
 
 5193 
 4474 
 3753 
 3029 
 2302 
 
 1572 
 0839 
 0103 
 9364 
 
 8622 
 
 5121 
 4402 
 3681 
 2956 
 2229 
 
 1498 
 0765 
 0029 
 9290 
 
 8548 
 
 23V4 
 
 1645 
 
 0912 
 0176 
 9438 
 8696 
 
 2 
 3 
 4 
 5 
 
 6 
 
 7 
 8 
 9 
 
 75 
 
 8 
 15 
 
 23 
 30 
 38 
 45 
 53 
 60 
 68 
 
 76 
 
 8 
 
 23 
 30 
 38 
 46 
 53 
 61 
 68 
 
 77 
 
 8 
 15 
 23 
 31 
 
 39 
 46 
 54 
 
 i'2 
 69 
 
 630 
 631 
 632 
 633 
 634 
 
 635 
 636 
 637 
 638 
 639 
 
 75 8473 
 
 7728 
 6979 
 6228 
 5473 
 
 4716 
 3955 
 3192 
 2425 
 1655 
 
 8324 
 7578 
 6829 
 6077 
 5322 
 
 4564 
 3803 
 3038 
 2271 
 1500 
 
 8175 
 7429 
 6679 
 5926 
 5171 
 
 4412 
 3650 
 
 2885 
 2117 
 1346 
 
 8101 
 7354 
 6604 
 5851 
 5095 
 4336 
 3574 
 2809 
 2040 
 1269 
 
 8026 
 7279 
 6529 
 5776 
 5019 
 
 4260 
 3497 
 2732 
 1963 
 1191 
 
 7952 
 7204 
 6454 
 5700 
 4943 
 
 4184 
 3421 
 2655 
 1886 
 1114 
 
 7877 
 7129 
 6378 
 5625 
 4868 
 
 4108 
 3345 
 
 2578 
 1809 
 1037 
 
 7802 
 7054 
 6303 
 5549 
 4792 
 
 4031 
 3268 
 2502 
 1732 
 0959 
 
 ] 
 2 
 3 
 4 
 
 I 
 
 9 
 
 78 
 
 ~8 
 16 
 23 
 31 
 
 39 
 
 n 
 
 55 
 62 
 70 
 
 79 
 
 ~8~ 
 
 
 
 32 
 
 40 
 47 
 55 
 63 
 71 
 
 80 
 
 8 
 16 
 24 
 32 
 
 40 
 
 48 
 56 
 S4 
 72 
 
 640 
 641 
 642 
 643 
 644 
 
 645 
 646 
 647 
 648 
 649 
 
 75 0882 
 0105 
 74 9326 
 8543 
 
 7757 
 
 6968 
 6176 
 5380 
 4581 
 3779 
 
 0804 
 0028 
 9248 
 8465 
 7679 
 
 6889 
 6097 
 5301 
 4501 
 3699 
 
 0727 
 9950 
 9170 
 8386 
 7600 
 
 6810 
 6017 
 5221 
 4421 
 3618 
 
 0649 
 
 9872 
 9092 
 8308 
 7521 
 
 6731 
 5938 
 5141 
 4341 
 3538 
 
 0572 
 9794 
 9013 
 8229 
 7442 
 
 6652 
 5858 
 5061 
 4261 
 3457 
 
 0494 
 9716 
 8935 
 8151 
 7363 
 
 6573 
 5779 
 4981 
 4181 
 3377 
 
 0416 
 
 9638 
 
 8857 
 8072 
 
 7284 
 
 6493 
 5699 
 4901 
 4100 
 3296 
 
 0339 
 9560 
 8778 
 7994 
 7205 
 
 6414 
 5619 
 4821 
 4020 
 3216 
 
 0261 
 9482 
 8700 
 7915 
 7126 
 
 6335 
 5540 
 4741 
 3940 
 3135 
 
 0183 
 9404 
 
 8622 
 7836 
 7047 
 
 6255 
 5460 
 4661 
 3860 
 3054 
 
 | 1-1 CM co rj< o :o > co o> 
 
 81 
 
 8 
 16 
 24 
 32 
 11 
 19 
 i7 
 55 
 3 
 
 82j 
 
 1 
 
 16 
 25 
 33; 
 
 41* 
 
 49; 
 
 i7i 
 56' 
 
 ? 47 
 
 33 
 
 8 
 17 
 25 
 J3 
 t2 
 >0 
 S 
 6 
 5 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 Pro. Parts. 1 
 
 I 
 
 60 
 
TABLE II. 
 
 Log 
 
 Log(l-ar) 
 
 
 ?ro. Parts. 
 
 Diff. 
 
 2 
 3 
 
 : 
 
 6 
 
 ' 8 
 9 
 
 32 
 
 8 
 16 
 25 
 33 
 41 
 49 
 57 
 66 
 74 
 
 33 
 
 8 
 17 
 25 
 33 
 42 
 50 
 58 
 66 
 75 
 
 34 
 
 8 
 17 
 25 
 34 
 
 42 
 
 50 
 59 
 67 
 76 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 f-650 
 651 
 652 
 653 
 654 
 
 655 
 656 
 657 
 658 
 659 
 
 1-742974 
 2165 
 1352 
 0536 
 739717 
 
 8895 
 8069 
 7239 
 6406 
 5569 
 
 2893 
 2084 
 1271 
 0455 
 9635 
 
 8812 
 7986 
 7156 
 6323 
 5486 
 
 2812 
 2002 
 1189 
 0373 
 9553 
 
 8730 
 7903 
 7073 
 6239 
 5402 
 
 2731 
 1921 
 1108 
 0291 
 9471 
 
 8647 
 7820 
 6990 
 6155 
 5318 
 
 2650 
 1840 
 1026 
 0209 
 9389 
 
 8565 
 7737 
 6906 
 6072 
 5234 
 
 2570 
 1759 
 0945 
 0127 
 9306 
 
 8482 
 7654 
 6823 
 5988 
 5150 
 
 2489 
 1678 
 0863 
 0045 
 9224 
 
 8399 
 7571 
 6740 
 5904 
 5066 
 
 2408 
 1596 
 0782 
 9963 
 9142 
 
 8317 
 7488 
 6656 
 5821 
 4982 
 
 2327 
 1515 
 
 0700 
 9881 
 9059 
 
 8234 
 7405 
 6573 
 5737 
 
 4898 
 
 2246 
 1434 
 
 0618 
 9799 
 8977 
 
 8151 
 7322 
 6489 
 5653 
 4813 
 
 82 
 
 83 
 
 84 
 85 
 86 
 
 87 
 88 
 89 
 
 90 
 91 
 
 92 
 
 93 
 
 94 
 
 95 
 96 
 
 97 
 
 98 
 
 99 
 100 
 
 2 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 85 
 
 9 
 17 
 26 
 34 
 43 
 51 
 60 
 68 
 77 
 
 8687 
 
 9 9 
 17 17 
 
 2626 
 3435 
 4344 
 5252 
 6061 
 6970 
 77178 
 
 660 
 661 
 662 
 663 
 664 
 
 665 
 666 
 667 
 668 
 669 
 
 73 4729 
 3886 
 3038 
 
 2187 
 1333 
 
 4645 
 3801 
 2953 
 2102 
 1247 
 
 0389 
 9526 
 8660 
 7791 
 6917 
 
 4561 
 3716 
 2868 
 2017 
 1161 
 
 0303 
 9440 
 8574 
 7703 
 6830 
 
 4477 
 3632 
 2783 
 1931 
 1076 
 
 0216 
 9353 
 8487 
 7616 
 6742 
 
 4392 
 3547 
 2698 
 1846 
 0990 
 
 0130 
 9267 
 8400 
 7529 
 6654 
 
 4308 
 3462 
 2613 
 1761 
 0904 
 
 0044 
 9180 
 8313 
 7442 
 6567 
 
 4223 
 
 3378 
 2528 
 1675 
 0818 
 
 9958 
 9094 
 8226 
 7354 
 6479 
 
 4139 
 3293 
 2443 
 1590 
 0732 
 
 9872 
 9007 
 8139 
 7267 
 6391 
 
 4055 
 3208 
 2358 
 1504 
 0647 
 
 9785 
 8921 
 8052 
 7180 
 6303 
 
 3970 
 3123 
 2273 
 1418 
 0561 
 
 9699 
 8834 
 7965 
 7092 
 6216 
 
 0475 
 729613 
 
 8747 
 7878 
 7005 
 
 1 
 
 2 
 
 4 
 5 
 
 6 
 
 7 
 8 
 9 
 
 88 
 
 9 
 18 
 26 
 35 
 
 44 
 53 
 
 62 
 70 
 79 
 
 89 
 
 9 
 18 
 27 
 3d 
 45 
 53 
 62 
 71 
 80 
 
 90 
 
 9 
 18 
 27 
 36 
 
 45 
 54 
 63 
 72 
 81 
 
 670 
 671 
 672 
 673 
 674 
 675 
 676 
 677 
 678 
 679 
 
 726128 
 5247 
 4363 
 3474 
 2582 
 
 1686 
 0786 
 71 9882 
 8974 
 8062 
 
 6040 
 5159 
 4274 
 3385 
 2493 
 
 1596 
 0696 
 9791 
 8883 
 7971 
 
 5952 
 5071 
 4185 
 3296 
 2403 
 
 1506 
 0605 
 9701 
 8792 
 7879 
 
 5864 
 4982 
 409? 
 3207 
 2314 
 
 1416 
 0515 
 9610 
 8701 
 
 7788 
 
 5776 
 4894 
 4008 
 3118 
 2224 
 
 1326 
 0425 
 9519 
 8610 
 7696 
 
 5688 
 4805 
 3919 
 3029 
 2134 
 
 1236 
 0334 
 9428 
 8518 
 7604 
 
 5600 
 4717 
 3830 
 2939 
 2045 
 
 1146 
 0244 
 9338 
 8427 
 7513 
 
 5512 
 
 4628 
 3741 
 2850 
 1955 
 
 1056 
 0154 
 9247 
 8336 
 7421 
 
 5424 
 4540 
 3652 
 2761 
 1865 
 
 0966 
 0063 
 9156 
 8245 
 7329 
 
 5335 
 4451 
 3563 
 2671 
 1776 
 
 0876 
 9973 
 9065 
 8153 
 7238 
 
 I 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 91 
 
 9 
 18 
 27 
 36 
 46 
 55 
 64 
 73 
 82 
 
 92 
 
 9 
 18 
 28 
 37 
 46 
 55 
 64 
 74 
 83 
 
 93 
 
 9 
 19 
 
 28 
 37 
 
 47 
 56 
 65 
 74 
 
 84 
 
 680 
 681 
 682 
 683 
 684 
 
 685 
 686 
 687 
 688 
 689 
 
 71 7146 
 6226 
 5302 
 4373 
 3441 
 
 2505 
 1564 
 0619 
 70 9670 
 8716 
 
 7054 
 6134 
 5209 
 4280 
 3348 
 
 2411 
 
 1470 
 0524 
 9575 
 8621 
 
 6962 
 6041 
 5116 
 4187 
 3254 
 
 2317 
 1375 
 0429 
 9479 
 8525 
 
 6870 
 5949 
 5024 
 4094 
 3161 
 
 2223 
 1281 
 0335 
 9384 
 8429 
 
 6778 
 5857 
 4931 
 4001 
 3067 
 
 2129 
 1186 
 0240 
 9289 
 8334 
 
 6686 
 5764 
 
 4838 
 3908 
 2973 
 
 2035 
 1092 
 0145 
 9194 
 
 8238 
 
 6594 
 5672 
 4745 
 3815 
 
 2880 
 
 1941 
 0997 
 0050 
 9098 
 8142 
 
 6502 
 5579 
 4652 
 3721 
 
 2786 
 
 1847 
 0903 
 9955 
 9003 
 8046 
 
 6410 
 
 5487 
 4559 
 3628 
 2692 
 
 1752 
 
 0808 
 9860 
 8907 
 7950 
 
 6318 
 5394 
 4466 
 3535 
 
 2598 
 
 1658 
 0714 
 9765 
 8812 
 
 7854 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 94 
 
 9 
 19 
 
 28 
 38 
 
 47 
 56 
 66 
 75 
 
 85 
 
 95 
 
 10 
 19 
 29 
 38 
 
 48 
 57 
 67 
 76 
 
 86 
 
 96 
 
 10 
 19 
 29 
 38 
 
 48 
 58 
 67 
 77 
 86 
 
 690 
 691 
 692 
 693 
 694 
 
 695 
 696 
 697 
 698 
 699 
 
 70 7758 
 6796 
 5830 
 4859 
 
 3884 
 
 2904 
 1920 
 0931 
 69 9938 
 8940 
 
 7662 
 6700 
 5733 
 4762 
 
 3786 
 
 2806 
 1821 
 0832 
 9838 
 8840 
 
 7566 
 6603 
 5636 
 4664 
 
 3688 
 
 2708 
 1722 
 0733 
 9739 
 8740 
 
 7470 
 6507 
 5539 
 4567 
 3590 
 
 2609 
 1624 
 0634 
 9639 
 8640 
 
 7374 
 6410 
 5442 
 4469 
 3492 
 
 2511 
 1525 
 0534 
 9539 
 8540 
 
 7278 
 6314 
 5345 
 4372 
 3394 
 
 2413 
 1426 
 0435 
 9439 
 8439 
 
 7182 
 6217 
 5248 
 4274 
 3296 
 
 2314 
 1327 
 0336 
 9340 
 8339 
 
 7085 
 6120 
 5151 
 4177 
 3198 
 
 2216 
 1228 
 0236 
 9240 
 8239 
 
 6989 
 6024 
 5054 
 4079 
 3100 
 
 2117 
 1129 
 0137 
 9140 
 8138 
 
 6893 
 5927 
 4956 
 3981 
 3002 
 
 2018 
 1030 
 0037 
 9040 
 8038 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 97 
 
 10 
 19 
 29 
 39 
 
 49 
 
 58 
 68 
 
 78 
 87 
 
 98 
 
 99 
 
 10 
 20 
 29 
 39 
 49 
 59 
 69 
 78 
 
 88 
 
 10 
 20 
 30 
 40 
 50 
 59 
 69 
 79 
 
 RQ 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff 
 
 Pro. Parts. 
 
 ol 
 
TABLE II. 
 
 Log 
 
 Log (1 -x) 
 
 Diff. 
 
 Pro. Parts. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10(1 
 
 10 
 20 
 30 
 40 
 
 50 
 60 
 70 
 80 
 90 
 
 101 
 
 10 
 20 
 30 
 40 
 51 
 61 
 71 
 81 
 i 91 
 
 T-700 
 701 
 702 
 703 
 704 
 
 70.5 
 706 
 707 
 708 
 709 
 
 1-69 7938 
 6930 
 5919 
 4902 
 3881 
 
 2855 
 1824 
 0789 
 68 9748 
 8703 
 
 783-5 
 
 683C 
 5817 
 480C 
 377S 
 
 2752 
 1721 
 0685 
 9644 
 
 8598 
 
 7737 
 > 6729 
 5716 
 4698 
 3676 
 
 2649 
 1618 
 0581 
 9540 
 8493 
 
 7636 
 6627 
 5614 
 4596 
 3574 
 
 2546 
 1514 
 0477 
 9435 
 
 8388 
 
 7535 
 6526 
 5513 
 4494 
 3471 
 
 2443 
 1411 
 0373 
 9331 
 
 8283 
 
 7435 
 6425 
 5411 
 4392 
 3369 
 
 2340 
 1307 
 0269 
 9226 
 
 8178 
 
 7334 
 6324 
 5309 
 4290 
 3266 
 
 2237 
 1204 
 0165 
 9122 
 8073 
 
 7233 
 6223 
 5208 
 
 4188 
 3163 
 
 2134 
 1100 
 0061 
 9017 
 7968 
 
 7132 
 6121 
 5106 
 4086 
 3061 
 
 2031 
 0996 
 9957 
 8912 
 7863 
 
 7031 
 6020 
 5004 
 3983 
 2958 
 
 1928 
 0893 
 9853 
 8808 
 7758 
 
 101 
 102 
 103 
 
 104 
 105 
 
 106 
 107 
 108 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 102 
 
 10 
 20 
 31 
 41 
 
 51 
 61 
 71 
 
 82 
 92 
 
 103 
 
 10 
 21 
 
 Si 
 
 52 
 62 
 72 
 82 
 93 
 
 710 
 711 
 712 
 713 
 714 
 
 715 
 716 
 717 
 718 
 719 
 
 68 7653 
 6597 
 5537 
 4472 
 3401 
 
 2326 
 1245 
 0159 
 67 9068 
 7971 
 
 7547 
 6491 
 5431 
 4365 
 3294 
 
 2218 
 1136 
 0050 
 8958 
 7861 
 
 7442 
 6386 
 5324 
 
 4258 
 3186 
 
 2110 
 1028 
 9941 
 8849 
 7751 
 
 7336 
 6280 
 5218 
 4151 
 3079 
 
 2002 
 0920 
 9832 
 8739 
 7641 
 
 7231 
 6174 
 5111 
 4044 
 2972 
 
 1894 
 0811 
 9723 
 8630 
 7531 
 
 7126 
 6068 
 5005 
 3937 
 2864 
 
 1786 
 0703 
 9614 
 8520 
 7421 
 
 7020 
 5962 
 4898 
 3830 
 2756 
 
 1678 
 0594 
 9505 
 8411 
 7311 
 
 6914 
 5856 
 4792 
 3723 
 2649 
 
 1570 
 0485 
 9396 
 8301 
 7201 
 
 6809 
 5749 
 4685 
 3616 
 2541 
 
 1461 
 0377 
 
 9286 
 8191 
 7090 
 
 6703 
 5643 
 
 4578 
 3508 
 2433 
 
 1353 
 
 0268 
 9177 
 8081 
 6980 
 
 109 
 110 
 
 111 
 
 112 
 113 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 104 
 
 10 
 21 
 31 
 42 
 52 
 62 
 73 
 83 
 94 
 
 105 
 
 2] 
 32 
 42 
 53 
 63 
 74 
 84 
 95 
 
 720 
 721 
 722 
 723 
 724 
 
 725 
 726 
 
 727 
 728 
 729 
 
 67 6870 
 5763 
 4650 
 3532 
 2409 
 
 1280 
 0145 
 66 9005 
 7860 
 6708 
 
 6759 
 5652 
 4538 
 3420 
 2296 
 
 1167 
 0032 
 8891 
 7745 
 6593 
 
 6649 
 5540 
 4427 
 3308 
 2183 
 
 1053 
 9918 
 8777 
 7630 
 6477 
 
 6538 
 5429 
 4315 
 3196 
 2071 
 
 0940 
 9804 
 8662 
 7515 
 6362 
 
 6427 
 5318 
 4204 
 3083 
 1958 
 
 0827 
 9690 
 8548 
 7400 
 6246 
 
 5087 
 3922 
 2751 
 1574 
 0391 
 
 9202 
 8007 
 6806 
 5599 
 4386 
 
 6317 
 5207 
 4092 
 2971 
 1845 
 
 0713 
 9576 
 8433 
 
 7285 
 6130 
 
 6206 
 5096 
 3980 
 2859 
 1732 
 
 0600 
 9462 
 8319 
 7170 
 6015 
 
 6095 
 4984 
 3868 
 2746 
 1619 
 
 0486 
 9348 
 8204 
 7054 
 5899 
 
 5984 
 4873 
 3756 
 2634 
 1506 
 
 0373 
 9234 
 8089 
 6939 
 5783 
 
 4621 
 3454 
 2280 
 1101 
 9916 
 
 8725 
 7528 
 6324 
 5115 
 3899 
 
 5873 
 4762 
 3644 
 2521 
 1393 
 
 0259 
 9120 
 7974 
 6824 
 5667 
 
 114 
 115 
 116 
 117 
 118 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 106 
 
 11 
 21 
 32 
 
 42 
 
 53 
 64 
 74 
 85 
 95 
 
 107 
 
 11 
 21 
 
 Of 
 
 43 
 54 
 
 64 
 75 
 86 
 
 9ti 
 
 730 
 731 
 732 
 733 
 734 
 
 735 
 736 
 737 
 
 738 
 739 
 
 665551 
 
 4388 
 3220 
 2045 
 0865 
 
 65 9678 
 8486 
 
 7287 
 6083 
 
 4872 
 
 5435 
 
 4272 
 3102 
 1927 
 0746 
 
 9559 
 8366 
 7167 
 5962 
 4751 
 
 5319 
 4155 
 2985 
 1809 
 0628 
 
 9440 
 8247 
 7047 
 5841 
 4629 
 
 5203 
 4038 
 2868 
 1692 
 0509 
 
 9321 
 
 8127 
 6927 
 5720 
 4508 
 
 4970 
 3805 
 2633 
 1456 
 0272 
 
 9083 
 7887 
 6686 
 5478 
 4265 
 
 4854 
 3688 
 2516 
 1338 
 0154 
 
 8964 
 7768 
 6565 
 5357 
 4143 
 
 4738 
 3571 
 2398 
 1219 
 0035 
 
 8844 
 7648 
 6445 
 5236 
 4021 
 
 4505 
 3337 
 2163 
 
 0983 
 9797 
 
 8605 
 7408 
 6204 
 4994 
 3777 
 
 119 
 120 
 
 121 
 122 
 
 Diff. 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 108 
 
 11 
 22 
 32 
 43 
 54 
 65 
 76 
 86 
 97 
 
 109 
 
 ~7I 
 
 22 
 33 
 44 
 
 55 
 65 
 
 76 
 
 87 
 
 98 
 
 
 
 
 
 1 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 9 
 
 11 
 
 1 
 2 
 3 
 4 
 6 
 6 
 7 
 8 
 9 
 
 111 
 
 i 11 
 
 2 22 
 3 33 
 4 44 
 
 5 56 
 6 67 
 
 7 78 
 8 89 
 9 100 
 
 112 
 
 11 
 
 22 
 34 
 45 
 56 
 67 
 78 
 90 
 101 
 
 113 
 
 11 
 23 
 34 
 45 
 
 57 
 68 
 79 
 90 
 102 
 
 114 115 116 117 
 
 118 
 
 12 
 24 
 35 
 
 47 
 59 
 71 
 
 83 
 94 
 
 106 
 
 119 120 12 
 
 ~J2 12 1 
 
 24 24 2' 
 36 36 3( 
 48 48 41 
 60 60 6 
 71 72 7( 
 83 84 81 
 95 96 9' 
 107 108 10< 
 
 I 122 
 
 2 12 
 
 t 24 
 > 37 
 J 49 
 [ 61 
 J 73 
 > 85 
 1 98 
 ) 110 
 
 123 
 
 124 125 
 
 126 
 
 13 
 
 25 
 38 
 50 
 63 
 76 
 88 
 101 
 113 
 
 127 
 
 13 
 25 
 38 
 51 
 
 64 
 76 
 89 
 102 
 114 
 
 11 12 12 12 
 23 23 23 23 
 34 35 35 35 
 46 46 46 47 
 57 58 58 59 
 68 69 70 70 
 80 81 81 82 
 91 92 93 94 
 103 104 104 105 
 
 12 
 25 
 37 
 49 
 62 
 74 
 86 
 98 
 111 
 
 12 13 
 25 25 
 37 38 
 50 50 
 
 62 63 
 
 74 75 
 87 88 
 99 JOO 
 112 113 
 
 62 
 
TABLE II 
 
 Log 
 
 LogO-*) 
 
 Diff. 
 
 Pro. Parts. 
 
 X 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 128 
 
 13 
 
 26 
 38 
 51 
 
 64 
 
 77 
 9(1 
 102 
 115 
 
 129 
 
 13 
 26 
 39 
 52 
 
 65 
 77 
 90 
 103 
 !16 
 
 1-740 
 741 
 
 742 
 743 
 744 
 
 745 
 746 
 747 
 
 748 
 749 
 
 T-65 3655 
 2432 
 1203 
 64 9967 
 
 8725 
 
 7477 
 6222 
 4960 
 3692 
 2417 
 
 3533 
 2310 
 1080 
 9843 
 8601 
 
 7351 
 6096 
 4834 
 3565 
 2289 
 
 3411 
 
 2187 
 0956 
 9719 
 8476 
 
 7226 
 5970 
 4707 
 3437 
 2161 
 
 0878 
 9589 
 8292 
 6989 
 5679 
 
 4361 
 3037 
 1706 
 0367 
 9021 
 
 7668 
 6307 
 4939 
 3564 
 2181 
 
 0790 
 9392 
 7986 
 6572 
 5150 
 
 3289 
 2064 
 0833 
 9595 
 8351 
 
 7101 
 5844 
 4580 
 3310 
 2033 
 
 3167 
 1941 
 0709 
 9471 
 
 8226 
 
 6975 
 5718 
 4454 
 3183 
 1905 
 
 0621 
 9330 
 8032 
 6727 
 5416 
 
 4097 
 2771 
 1438 
 0098 
 8751 
 
 3045 
 1818 
 0586 
 9347 
 8102 
 
 6850 
 5592 
 4327 
 3055 
 1777 
 
 2922 
 1695 
 0462 
 9223 
 7977 
 
 6724 
 5465 
 4200 
 2928 
 1649 
 
 2800 
 1572 
 0339 
 9098 
 
 7852 
 
 6599 
 5339 
 4073 
 2800 
 1521 
 
 2677 
 1449 
 0215 
 8974 
 
 7727 
 
 6473 
 5213 
 3946 
 2673 
 1392 
 
 2555 
 1326 
 0091 
 
 8850 
 7602 
 
 6347 
 5086 
 3819 
 2545 
 1264 
 
 123 
 124 
 
 125 
 126 
 
 127 
 
 128 
 
 129 
 130 
 131 
 
 132 
 133 
 
 134 
 135 
 
 136 
 137 
 
 138 
 
 139 
 
 140 
 141 
 142 
 
 143 
 
 144 
 145 
 146 
 
 147 
 
 148 
 149 
 150 
 151 
 152 
 
 Diff. 
 
 1 
 
 2 
 3 
 4 
 5 
 
 (3 
 7 
 8 
 9 
 
 130 
 
 13 
 26 
 39 
 52 
 
 65 
 78 
 91 
 104 
 117 
 
 131 
 
 13 
 
 26 
 39 
 52 
 
 66 
 79 
 92 
 105 
 
 118 
 
 750 
 751 
 752 
 753 
 754 
 
 755 
 756 
 757 
 
 758 
 759 
 
 641136 
 63 9847 
 8552 
 7250 
 5941 
 
 4625 
 3303 
 1972 
 0635 
 62 9291 
 
 1007 
 9718 
 8422 
 7120 
 5810 
 
 4493 
 3170 
 1839 
 0501 
 9156 
 
 0750 
 9459 
 8162 
 6858 
 5547 
 
 4229 
 2904 
 1572 
 0233 
 
 8886 
 
 0492 
 9201 
 7902 
 6597 
 
 5284 
 
 3965 
 2638 
 1305 
 9964 
 8616 
 
 0363 
 9071 
 7772 
 6466 
 5153 
 
 3833 
 2505 
 1171 
 
 9829 
 8481 
 
 0234 
 8941 
 7642 
 6335 
 5021 
 
 3700 
 2372 
 1037 
 9695 
 8345 
 
 0105 
 
 8812 
 7511 
 6204 
 
 4889 
 
 3568 
 2239 
 0903 
 9560 
 8210 
 
 9976 
 8682 
 7381 
 6073 
 4757 
 
 3435 
 2106 
 0769 
 9426 
 
 8075 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 132 133 
 
 13 13 
 26 27 
 40 40 
 53 53 
 66 67 
 79 80 
 92 93 
 106 106 
 119 120 
 
 760 
 761 
 762 
 763 
 764 
 
 765 
 766 
 767 
 768 
 769 
 
 62 7939 
 6580 
 5214 
 
 3840 
 
 2458 
 
 1069 
 61 9672 
 8268 
 6855 
 5435 
 
 7804 
 6444 
 5077 
 3702 
 2320 
 
 0930 
 9532 
 8127 
 6714 
 5293 
 
 7532 
 6171 
 4802 
 3426 
 2042 
 
 0651 
 9252 
 7845 
 6430 
 5008 
 
 7396 
 6034 
 4665 
 3288 
 1903 
 
 0511 
 9111 
 7704 
 
 6288 
 4865 
 
 7261 
 5898 
 4528 
 3150 
 1765 
 
 0372 
 8971 
 7563 
 6146 
 
 4722 
 
 7125 
 5761 
 4390 
 3012 
 1626 
 
 0232 
 8830 
 7421 
 6004 
 4579 
 
 3146 
 1705 
 0256 
 8799 
 7333 
 
 5859 
 4377 
 
 2886 
 1386 
 
 9878 
 
 6989 
 5624 
 4253 
 
 2873 
 1487 
 
 0092 
 8690 
 7280 
 5862 
 4436 
 
 3003 
 1561 
 0111 
 8652 
 7186 
 
 5711 
 
 4228 
 2736 
 1236 
 9726 
 
 6852 
 5487 
 4115 
 2735 
 1347 
 
 9952 
 8549 
 7138 
 5720 
 4293 
 
 6716 
 5351 
 3977 
 2597 
 1208 
 
 9812 
 8409 
 6997 
 5578 
 4150 
 
 
 134 
 
 13 
 
 27 
 40 
 54 
 
 67 
 80 
 94 
 107 
 121 
 
 135 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 14 
 
 27 
 41 
 54 
 
 68 
 81 
 95 
 108 
 122 
 
 770 
 771 
 
 772 
 773 
 
 774 
 
 775 
 776 
 
 777 
 778 
 779 
 
 61 4007 
 2571 
 1127 
 60 9674 
 8213 
 
 6744 
 5267 
 3781 
 
 2287 
 0784 
 
 ~6~ 
 
 3864 
 2427 
 
 0982 
 9528 
 8067 
 
 6597 
 5119 
 3632 
 2137 
 0633 
 
 T" 
 
 3720 
 2283 
 0837 
 9383 
 7920 
 
 6450 
 4971 
 3483 
 1987 
 0482 
 
 3577 
 2138 
 0692 
 9237 
 
 7774 
 
 6302 
 4822 
 3334 
 1837 
 0331 
 
 3433 
 1994 
 0547 
 9091 
 7627 
 
 6154 
 4674 
 3184 
 1687 
 0180 
 
 3290 
 1850 
 0401 
 8945 
 7480 
 
 6007 
 4525 
 3035 
 1536 
 0029 
 
 2859 
 1416 
 9965 
 8506 
 7039 
 
 5563 
 4079 
 2586 
 1085 
 
 9575 
 
 2715 
 1271 
 
 9820 
 8360 
 6892 
 
 5415 
 3930 
 2437 
 0934 
 9424 
 
 1 
 2 
 3 
 
 4, 
 5 
 6 
 7 
 8 
 9 
 
 136 
 
 14 
 27 
 41 
 54 
 
 68 
 82 
 95 
 109 
 122 
 
 137 
 
 14 
 27 
 41 
 55 
 69 
 82 
 96 
 110 
 123 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 _ 
 
 1 
 
 2 
 3 
 
 4 
 5 
 
 6 
 7 
 8 
 9 
 
 138 139 
 
 14 14 
 28 28 
 41 42 
 55 56 
 69 70 
 83 83 
 97 97 
 110 111 
 124 125 
 
 140 
 
 14 
 
 2S 
 42 
 56 
 70 
 84 
 98 
 112 
 12f 
 
 | 141 
 
 "lil 
 28 
 42 
 56 
 71 
 85 
 99 
 113 
 127 
 
 142 143 144 145 146 
 
 14 14 14 lo! 15 
 28 29 29 29 29 
 43 43 43 44 44 
 57 57 58 58 58 
 71 72 72 73 73 
 85 86 86 87 88 
 99 100 101 102 102 
 1J4 114 115 116i 117 
 128 129 130 131 131 
 
 1471 148 149 150 
 
 151 
 
 15 
 30 
 45 
 60 
 
 76 
 91 
 106 
 121 
 136 
 
 152! 153 
 
 15 15 
 
 30 31 
 46 46 
 61 61 
 76 77 
 91 92 
 106 107 
 122 122 
 137 138 
 
 154 
 
 15 
 31 
 
 46 
 62 
 
 77 
 92 
 108 
 123 
 139 
 
 155 
 
 16 
 31 
 47 
 62 
 78 
 93 
 109 
 124 
 140 
 
 15 15 15 15 
 
 29 30 30 30 
 44 44 45 45 
 59 59 60 60 
 74 74 75 75 
 88 89 B9 90 
 103 104 104 105 
 118 118 119 120 
 132 133 134 135 
 
TABLE II. 
 
 Log 
 
 X 
 
 
 
 Log 
 
 (1-*) 
 
 
 
 Diff. 
 
 Pro. Parts 
 
 
 
 1 
 
 2 
 
 8969 
 7446 
 5915 
 4375 
 2826 
 
 1268 
 9700 
 8123 
 6537 
 4941 
 
 3 
 
 4 
 
 5 
 
 8513 
 6988 
 5454 
 3911 
 2359 
 
 0798 
 9228 
 7648 
 6059 
 4461 
 
 6 
 
 
 7 
 
 8 
 
 9 
 
 2 
 
 3 
 
 4 
 5 
 
 6 
 
 7 
 8 
 9 
 
 156 
 
 16 
 31 
 
 47 
 62 
 
 78 
 94 
 109 
 125 
 140 
 
 157 
 
 16 
 31 
 47 
 63 
 
 79 
 94 
 110 
 126 
 141 
 
 158 
 
 1-780 
 781 
 
 782 
 783 
 784 
 
 785 
 786 
 787 
 788 
 789 
 
 1-59 9272 
 
 7752 
 6222 
 4684 
 3136 
 
 1580 
 0014 
 58 8439 
 6855 
 5261 
 
 9120 
 7599 
 6069 
 4530 
 2981 
 
 1424 
 
 9857 
 8281 
 6696 
 5101 
 
 8817 
 7294 
 5762 
 4221 
 2670 
 
 1111 
 9543 
 7965 
 6378 
 4781 
 
 8665 
 7141 
 5608 
 4066 
 2515 
 
 0955 
 9385 
 7807 
 6219 
 4621 
 
 8361 
 6835 
 5300 
 3757 
 2204 
 
 0642 
 9070 
 7490 
 5900 
 4300 
 
 8209 
 6682 
 5146 
 3602 
 2048 
 
 0485 
 8913 
 7331 
 5740 
 4140 
 
 8056 
 6529 
 4992 
 3447 
 1892 
 
 0328 
 8755 
 7173 
 5581 
 3979 
 
 7904 
 6375 
 4838 
 3292 
 1736 
 
 0171 
 8597 
 7014 
 5421 
 3819 
 
 2206 
 0585 
 8953 
 7311 
 5660 
 
 3998 
 2326 
 0644 
 8951 
 7247 
 
 5534 
 
 3809 
 2073 
 0326 
 8569 
 
 6800 
 5019 
 3228 
 1424 
 9609 
 
 152 
 153 
 154 
 155 
 156 
 
 157 
 158 
 159 
 160 
 161 
 
 162 
 163 
 164 
 165 
 166 
 
 167 
 
 16 
 32 
 47 
 63 
 
 79 
 95 
 111 
 
 126 
 142 
 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 159 
 
 16 
 32 
 48 
 64 
 80 
 95 
 111 
 127 
 143 
 
 160 
 
 16 
 32 
 48 
 64 
 80 
 96 
 112 
 128 
 144 
 
 161 
 
 ~16 
 32 
 
 48 
 64 
 
 81 
 97 
 113 
 129 
 145 
 
 790 
 791 
 792 
 793 
 794 
 
 795 
 796 
 797 
 798 
 799 
 
 58 3658 
 2045 
 0422 
 57 8789 
 7147 
 
 5494 
 3831 
 2158 
 0475 
 568781 
 
 3497 
 
 1883 
 0259 
 8625 
 6982 
 
 5328 
 3664 
 1990 
 0306 
 8611 
 
 3336 
 1721 
 0096 
 8462 
 6817 
 
 5162 
 3497 
 1822 
 0137 
 8441 
 
 3175 
 1559 
 9933 
 
 8298 
 6652 
 
 4996 
 3330 
 1654 
 9968 
 
 8271 
 
 6563 
 4845 
 3116 
 1376 
 9625 
 
 7863 
 6089 
 4304 
 2508 
 0700 
 
 3014 
 1397 
 
 9770 
 8133 
 6487 
 
 4830 
 3163 
 1486 
 9799 
 8100 
 
 2852 
 1235 
 9607 
 7969 
 6322 
 
 4664 
 2996 
 1318 
 9629 
 7930 
 
 2691 
 1072 
 9444 
 7805 
 6156 
 
 J4498 
 2829 
 1149 
 9460 
 7760 
 
 2530 
 0910 
 9280 
 7640 
 5991 
 
 4331 
 2661 
 0981 
 9290 
 7589 
 
 2368 
 0747 
 
 9117 
 
 7476 
 
 5825 
 
 4165 
 2494 
 0812 
 
 9121 
 
 7418 
 
 16 
 16 
 17 
 17 
 
 17 
 17 
 17 
 17 
 17 
 
 17 
 17 
 17 
 18 
 
 18 
 
 18 
 18 
 18 
 18 
 18 
 
 18 
 19 
 19 
 19 
 19 
 
 Dif 
 
 8 
 9 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 2 
 
 3 
 4 
 5 
 
 6 
 
 8 
 
 9 
 
 1 
 3 
 4 
 
 F. 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 162 
 
 16 
 32 
 49 
 65 
 81 
 97 
 113 
 130 
 146 
 
 163 
 
 16 
 33 
 49 
 65 
 82 
 98 
 114 
 130 
 147 
 
 164 
 
 16 
 33 
 49 
 66 
 
 82 
 98 
 115 
 131 
 
 148 
 
 800 
 801 
 802 
 803 
 804 
 
 805 
 806 
 807 
 808 
 809 
 
 56 7077 
 5362 
 3636 
 1899 
 0151 
 
 55 8392 
 6622 
 4841 
 3048 
 1243 
 
 6906 
 5189 
 3462 
 1725 
 
 9976 
 
 8216 
 6445 
 4662 
 2868 
 1062 
 
 6734 
 5017 
 3289 
 1550 
 
 9800 
 
 8039 
 6267 
 4483 
 2688 
 0881 
 
 6392 
 4672 
 2942 
 1201 
 9449 
 
 7686 
 5911 
 4125 
 2328 
 0518 
 
 8697 
 6865 
 5020 
 3163 
 1294 
 
 9412 
 7518 
 5611 
 3691 
 1758 
 
 6220 
 4500 
 2769 
 1026 
 
 9273 
 
 7509 
 5733 
 3946 
 2147 
 0337 
 
 8515 
 6681 
 4835 
 2976 
 1106 
 
 9223 
 7328 
 5419 
 3498 
 1564 
 
 6049 
 4327 
 2595 
 0852 
 9097 
 
 7332 
 5555 
 3766 
 1967 
 0155 
 
 5877 
 4155 
 2421 
 0677 
 8921 
 
 7154 
 5376 
 3587 
 1786 
 9973 
 
 5705 
 3982 
 2247 
 0502 
 
 8745 
 
 6977 
 5198 
 3407 
 1605 
 9791 
 
 7966 
 6128 
 4278 
 2417 
 0542 
 
 8656 
 6756 
 4844 
 2919 
 0981 
 
 1 
 
 2 
 3 
 4 
 
 5 
 
 6 
 7 
 8 
 9 
 
 165 166 
 
 17 17 
 33 33 
 50 50 
 66 66 
 83 83 
 99 100 
 116 116 
 132 133 
 149j 149 
 
 167 
 
 17 
 33 
 50 
 67 
 84 
 100 
 117 
 134 
 150 
 
 810 
 811 
 812 
 813 
 814 
 
 815 
 816 
 
 817 
 818 
 819 
 
 54 9427 
 7599 
 5759 
 3907 
 2043 
 
 0166 
 53 8277 
 6375 
 4460 
 2533 
 
 9245 9063 
 7416 7232 
 5575 5390 
 372 1! 3535 
 1856 1668 
 
 9978 9789 
 8087 7897 
 6184 5993 
 4268 4076 
 2339 2146 
 
 8880 
 7048 
 5205 
 3349 
 1481 
 
 9601 
 
 7708 
 5802 
 3884 
 1952 
 
 8332 
 6497 
 4649 
 2790 
 0918 
 
 9034 
 7137 
 
 5228 
 3305 
 1370 
 
 8149 
 6312 
 4464 
 2603 
 0730 
 
 8845 
 6947 
 5036 
 3112 
 1176 
 
 7783 
 5944 
 4093 
 2230 
 0354 
 
 8466 
 6566 
 4652 
 2726 
 
 0787 
 
 ! 168 
 
 1 17 
 
 2 34 
 3 50 
 4 ! 67 
 5 84 
 6 101 
 7 118 
 8 134 
 9 151 
 
 169 
 
 17 
 34 
 51 
 68 
 85 
 101 
 118 
 135 
 152 
 
 170 
 
 17 
 34 
 51 
 68 
 85 
 102 
 119 
 136 
 153 
 
 
 
 
 1 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 
 
 171 172 
 
 173 174 
 
 175 176 
 
 177 17S 
 
 179 180 
 
 181 
 
 18 
 
 2 183 184 
 
 185 
 
 186 
 
 187 188 
 
 189 
 
 19 
 38 
 57 
 76 
 95 
 113 
 132 
 151 
 170, 
 
 1 
 
 1 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 17! 17 
 34 34 
 51 52 
 68 69 
 86J 86 
 103 103 
 120 120 
 137 138 
 154 155 
 
 17 17 
 35 35 
 52 1 52 
 
 69 70 
 87 87 
 104 104 
 121 122 
 138 139 
 156 157 
 
 18 18 
 35 35 
 53 53 
 70 70 
 88 88 
 105 106 
 123 123 
 140 141 
 158 158 
 
 18 
 35 
 53 
 71 
 
 89 
 106 
 124 
 142 
 159 
 
 18 
 36 
 53 
 71 
 89 
 107 
 125 
 142 
 160 
 
 18 
 36 
 54 
 72 
 90 
 107 
 125 
 143 
 161 
 
 18 
 36 
 54 
 72 
 90 
 108 
 126 
 144 
 162 
 
 18 
 36 
 54 
 72 
 91 
 109 
 127 
 145 
 163 
 
 1 
 
 3 
 5 
 
 7 
 
 9 
 10 
 12 
 14 
 
 16 
 
 S 18 
 S 37 
 5 55 
 3 73 
 1 92 
 9 110 
 7 128 
 3 146 
 1 165 
 
 18 
 37 
 55 
 74 
 92 
 110 
 129 
 147 
 166 
 
 19 
 37 
 
 8 
 
 93 
 111 
 
 130 
 148 
 167 
 
 19 
 37 
 56 
 74 
 93 
 112 
 130 
 149 
 167 
 
 19 
 
 37 
 56 
 75 
 94 
 112i 
 131 
 150 
 168 
 
 19 
 38 
 56 
 75 
 94 
 113 
 132 
 150 
 169 
 
 64 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log (1-ar) 
 
 Diff. 
 
 Prd. Parts. 
 
 
 190 
 
 191 
 
 192 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 9 
 
 19 
 38 
 57 
 76 
 
 95 
 114 
 133 
 152 
 171 
 
 19 
 
 38 
 57 
 76 
 
 96 
 115 
 134 
 153 
 172 
 
 19 
 38 
 58 
 77 
 96 
 115 
 134 
 154 
 173 
 
 1 T 820 
 821 
 
 822 
 823 
 824 
 
 825 
 826 
 827 
 828 
 829 
 
 ~-53 0592 
 52 8638 
 6671 
 4691 
 2696 
 
 0688 
 51 8666 
 6629 
 4579 
 2514 
 
 0397 
 8442 
 6474 
 4492 
 2496 
 
 0486 
 8463 
 6425 
 4373 
 2306 
 
 0203 
 8246 
 6276 
 4293 
 2296 
 
 0285 
 8260 
 6220 
 4167 
 2099 
 
 0008 
 8050 
 6078 
 4094 
 2095 
 
 0083 
 8056 
 6016 
 3961 
 1891 
 
 9807 
 7708 
 5594 
 3464 
 1319 
 
 9159 
 6982 
 4790 
 2581 
 0356 
 
 9812 
 7853 
 5881 
 3894 
 1895 
 
 9881 
 7853 
 5811 
 3754 
 1683 
 
 9617 
 7657 
 5683 
 3695 
 1694 
 
 9679 
 7649 
 5606 
 3548 
 1476 
 
 9422 
 7460 
 5484 
 3496 
 1493 
 
 9476 
 7446 
 5401 
 3341 
 1267 
 
 9226 
 7263 
 5286 
 3296 
 1292 
 
 9274 
 7242 
 5195 
 3135 
 1059 
 
 9030 
 7066 
 5088 
 3096 
 1091 
 
 9071 
 7038 
 4990 
 2928 
 0851 
 
 8834 
 6869 
 4889 
 2896 
 0889 
 
 8869 
 6834 
 4784 
 2721 
 0642 
 
 8549 
 6441 
 4318 
 2179 
 0025 
 
 7855 
 5669 
 3467 
 1248 
 9013 
 
 195 
 197 
 198 
 199 
 201 
 
 202 
 204 
 205 
 206 
 208 
 
 209 
 211 
 213 
 214 
 216 
 
 217 
 219 
 220 
 
 222 
 224 
 
 225 
 227 
 229 
 230 
 232 
 
 234 
 236 
 238 
 240 
 241 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 193J 194| 
 
 "iSps 
 
 39 39 
 
 58 58 
 77 78 
 
 97 97 
 116 116 
 135 136 
 
 154 155 
 174 175 
 
 195 
 
 20 
 39 
 59 
 78 
 98 
 117 
 137 
 156 
 176 
 
 830 
 831 
 832 
 833 
 834 
 
 835 
 836 
 837 
 838 
 839 
 
 51 0434 
 50 8339 
 6229 
 4105 
 1964 
 
 49 9808 
 7637 
 5449 
 3246 
 1026 
 
 0225 
 8129 
 6018 
 3891 
 1749 
 
 9592 
 7419 
 5230 
 3024 
 0803 
 
 0016 
 7918 
 5806 
 3678 
 1534 
 
 9375 
 7201 
 5010 
 2803 
 0580 
 
 9598 
 7497 
 5381 
 3250 
 1104 
 
 8942 
 6764 
 4570 
 2360 
 0133 
 
 9388 
 7286 
 5169 
 3036 
 
 0888 
 
 8725 
 6545 
 4350 
 2138 
 9909 
 
 9179 
 7075 
 4956 
 
 2822 
 0673 
 
 8507 
 6326 
 4129 
 1916 
 9686 
 
 8969 
 6864 
 4744 
 2608 
 0457 
 
 8290 
 6107 
 3909 
 1693 
 9462 
 
 8759 
 6653 
 4531 
 2394 
 0241 
 
 8073 
 5888 
 3688 
 1471 
 9238 
 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 196 197 198 
 
 20 20 20 
 39 39 40 
 59, 59 59 
 78 79 79 
 98 99 99 
 118 118 119 
 137 138 139 
 157 158 158 
 176 177 178 
 
 840 
 841 
 842 
 '843 
 
 844 
 
 845 
 846 
 
 847 
 848 
 849 
 
 48 8789 
 6536 
 4265 
 1978 
 47 9673 
 
 7350 
 5009 
 2651 
 0274 
 46 7878 
 
 8564 ; 8340 8115 
 6309 6083 5856 
 4037 3809 3581 
 1748 1518 1288 
 9441 9209 8978 
 
 7117 6883 6650 
 4774: 4539 4304 
 2414 2177 1939 
 0035 9796 9557 
 7637i7397 7156 
 
 7890 
 5629 
 3352 
 1058 
 8746 
 
 6416 
 4068 
 1702 
 9318 
 6914 
 
 7664 
 5403 
 3124 
 0827 
 8513 
 
 6182 
 3832 
 1464 
 9078 
 6673 
 
 7439 
 5175 
 2895 
 0597 
 
 8281 
 
 5948 
 3596 
 1227 
 8839 
 6432 
 
 7213 
 4948 
 2666 
 0366 
 8049 
 
 5713 
 3360 
 0989 
 8599 
 6190 
 
 6988 
 4721 
 2437 
 0135 
 7816 
 
 5479 
 3124 
 0751 
 8359 
 5948 
 
 6762 
 4493 
 2207 
 9904 
 7583 
 
 5244 
 
 2887 
 0512 
 
 8118 
 
 5706 
 
 o 
 3 
 
 4 
 
 5 
 
 6 
 
 199 200 
 
 20 20 
 40 40 
 60 60 
 80 80 
 100 100 
 119 120 
 139 140 
 159 160 
 179 180 
 
 201 
 
 20 
 40 
 60 
 80 
 101 
 121 
 141 
 161 
 181 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 8 
 9 
 
 1 
 
 i 
 
 6 
 
 I 
 
 9 
 
 202 203 
 
 204 
 
 205 
 
 206 207 
 
 208 
 
 20 
 
 > 210J 211 212 
 
 213 
 
 214 
 
 215 
 
 216 217 
 
 218 
 
 219 
 
 220 
 
 20, 20 
 40 41 
 61 61 
 8) 81 
 101 102 
 121 122 
 141 142 
 162 162 
 182 183 
 
 20 
 41 
 61 
 82 
 102 
 122 
 143 
 163 
 184 
 
 21 
 41 
 
 62 
 
 82 
 
 103 
 123 
 144 
 164 
 185 
 
 21 
 
 41 
 62 
 
 82 
 
 103 
 124 
 144 
 165 
 
 185 
 
 21 
 41 
 62 
 83 
 104 
 124 
 145 
 166 
 186 
 
 21 21 
 42 4i 
 62 6C 
 83 $4 
 104 10; 
 1251 12 
 146 14( 
 166 16< 
 187 18fc 
 
 21 
 ! 42 
 
 I 84 
 105 
 > 126 
 i 147 
 
 168 
 J 189 
 
 21 
 42 
 
 63 
 84 
 
 106 
 127 
 
 148 
 169 
 190 
 
 21 
 42 
 
 64 
 
 85 
 
 106 
 127 
 148 
 170 
 191 
 
 21 
 43 
 64 
 85 
 
 107 
 128 
 149 
 170 
 192 
 
 21 
 43 
 64 
 86 
 107 
 128 
 ISC 
 171 
 193 
 
 22 
 
 43 
 65 
 86 
 108 
 129 
 I 151 
 | 172 
 194 
 
 22 
 43 
 65 
 86 
 
 108 
 130 
 151 
 173 
 194 
 
 22 
 43 
 65 
 87 
 109 
 130 
 152 
 174 
 195 
 
 22 
 44 
 65 
 87 
 109 
 131 
 153 
 174 
 196 
 
 22 
 44 
 66 
 88 
 110 
 131 
 153 
 175 
 197 
 
 22 
 
 44 
 66 
 88 
 110 
 132 
 154 
 176 
 198 
 
 ; 221 222 
 
 223 
 
 224 225, 226 
 
 227 
 
 228 229 230 
 
 231 
 
 232 
 
 231 
 
 \ 234 235 236 237 j 238 
 
 239 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 22 22 
 44 44 
 66 67 
 88 89 
 111 111 
 133 133 
 155 155 
 177 178 
 199 200 
 
 22 
 45 
 67 
 89 
 112 
 134 
 156 
 178 
 201 
 
 22 
 
 45 
 67 
 9G 
 115 
 134 
 157 
 
 m 
 
 205 
 
 23 
 4J 
 68 
 90 
 
 113 
 
 135 
 158 
 180 
 203 
 
 23 
 45 
 
 68 
 90 
 
 113 
 136 
 158 
 181 
 
 203 
 
 s 
 
 68 
 91 
 
 114 
 
 136 
 159 
 
 182 
 204 
 
 21 
 
 4( 
 6* 
 9 
 
 ll< 
 
 13' 
 161 
 
 18! 
 20. 
 
 } 23 
 > 46 
 } 69 
 92 
 1 115 
 
 r 137 
 
 ) 160 
 I 183 
 
 3 206 
 
 23 
 
 46 
 69 
 92 
 
 115 
 
 138 
 161 
 184 
 
 207 
 
 23 
 46 
 69 
 92 
 
 116 
 139 
 162 
 185 
 
 208 
 
 23 
 
 46 
 70 
 93 
 116 
 139 
 162 
 186 
 209 
 
 2i 
 4' 
 71 
 9J 
 
 ir 
 
 !4( 
 
 ie; 
 
 18< 
 
 21( 
 
 \ 23 
 ' 47 
 ) 70 
 J 94 
 
 r 117 
 ) 140 
 J 164 
 > 187 
 ) 211 
 
 24 
 
 47 
 71 
 94 
 
 118 
 141 
 
 165 
 188 
 212 
 
 24 
 47 
 71 
 94 
 
 118 
 142 
 165 
 189 
 212 
 
 24 
 47 
 71 
 95 
 119 
 142 
 166 
 190 
 213 
 
 24 
 48 
 71 
 95 
 119 
 143 
 167 
 190 
 214 
 
 24 
 48 
 72 
 96 
 120 
 143 
 167 
 191 
 215 
 
TABLK II. 
 
 Log 
 
 X 
 
 LogO -a?) 
 
 Diff. 
 
 Pro. Parts. 
 
 
 240 241 242 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 24 
 41 
 7i 
 96 
 12C 
 144 
 I6b 
 192 
 216 
 
 1- 24 
 \ 4b 
 ! 72 
 96 
 121 
 14 
 168 
 193 
 217 
 
 24 
 48 
 73 
 97 
 121 
 145 
 169 
 194 
 218 
 
 T850 
 851 
 852 
 853 
 854 
 
 855 
 856 
 857 
 858 
 859 
 
 F-46 5463 
 3030 
 0577 
 45 8104 
 5612 
 
 3099 
 0566 
 448012 
 5437 
 2841 
 
 5221 
 278i 
 033C 
 7856 
 5361 
 
 2847 
 0311 
 7755 
 5178 
 2580 
 
 4978 
 2541 
 0084 
 7607 
 5111 
 
 2594 
 0057 
 7499 
 4919 
 2319 
 
 9696 
 7052 
 4385 
 1696 
 8983 
 
 6248 
 3488 
 0704 
 7896 
 5064 
 
 4735 
 2296 
 9837 
 7359 
 4860 
 
 2341 
 
 9802 
 7242 
 4660 
 2057 
 
 4492 
 2051 
 9590 
 7110 
 4609 
 
 2088 
 9547 
 6984 
 4401 
 1796 
 
 424 
 1806 
 9345 
 686C 
 4358 
 
 1835 
 9291 
 672? 
 4141 
 1534 
 
 4006 
 1560 
 9096 
 6611 
 4106 
 
 1582 
 9036 
 6469 
 3882 
 1272 
 
 3762 
 1315 
 8848 
 6362 
 3855 
 
 1328 
 8780 
 6212 
 3622 
 1010 
 
 3518 
 1069 
 8600 
 6112 
 3603 
 
 1074 
 
 8524 
 5954 
 3362 
 0748 
 
 3274 
 
 0823 
 8352 
 5862 
 3351 
 
 0820 
 8268 
 5695 
 3101 
 0485 
 
 24 
 24 
 
 24 
 24 
 
 25 
 
 25 
 25 
 25 
 25 
 26 
 
 26 
 26 
 26 
 27 
 
 27 
 
 27 
 27 
 28 
 28 
 28 
 
 28 
 29 
 29 
 29 
 
 29 
 
 30 
 30 
 30 
 30 
 31 
 
 3 
 
 5 
 7 
 9 
 1 
 
 3 
 5 
 
 7 
 9 
 2 
 
 4 
 6 
 
 8 
 
 1 
 
 I 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 242 
 
 24 
 4 
 73 
 97 
 123 
 146 
 170 
 194 
 219 
 
 244 
 
 24 
 4S 
 73 
 98 
 122 
 146 
 171 
 195 
 220 
 
 245 
 
 ~25 
 
 49 
 74 
 98 
 123 
 147 
 172 
 196 
 221 
 
 860 
 861 
 862 
 863 
 864 
 
 865 
 866 
 867 
 868 
 869 
 
 44 0223 
 43 7583 
 4920 
 2236 
 42 9528 
 
 6797 
 4042 
 1263 
 41 8460 
 5632 
 
 9960 
 7317 
 4653 
 1966 
 9256 
 
 6522 
 3765 
 0984 
 
 8178 
 5348 
 
 9433 
 6786 
 4117 
 1426 
 8711 
 
 5973 
 3211 
 0425 
 7614 
 4779 
 
 9169 
 6520 
 3849 
 1155 
 
 8438 
 
 5698 
 2933 
 0145 
 7332 
 4494 
 
 8905 
 6254 
 3581 
 0884 
 8165 
 
 5422 
 
 2655 
 9865 
 7049 
 4209 
 
 8641 
 5988 
 3312 
 0614 
 
 7892 
 
 5147 
 2377 
 
 9584 
 6766 
 3923 
 
 8377 
 5721 
 3043 
 0342 
 7618 
 4871 
 2099 
 9304 
 6483 
 3638 
 
 8112 
 
 5455 
 2774 
 0071 
 7345 
 
 4595 
 1821 
 9023 
 6200 
 3352 
 
 7848 
 5188 
 2505 
 9799 
 7071 
 
 4318 
 1542 
 8741 
 5916 
 3066 
 
 5 
 
 8 
 
 2 
 5 
 
 8 
 
 3 
 5 
 
 8 
 
 1 
 4 
 
 7 
 9 
 3 
 
 o 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 | 246 
 
 25 
 49 
 74 
 9fl 
 123 
 148 
 172 
 197 
 221 
 
 247 
 
 25 
 49 
 74 
 99 
 124 
 148 
 173 
 198 
 222 
 
 248 
 
 25 
 50 
 74 
 99 
 124 
 149 
 174 
 198 
 223 
 
 870 
 871 
 
 872 
 873 
 874 
 
 875 
 876 
 877 
 878 
 879 
 
 41 2779 
 40 9901 
 6996 
 4066 
 1108 
 
 398124 
 5112 
 2071 
 38 9003 
 5905 
 
 2493 
 9612 
 6704 
 3771 
 0811 
 
 7824 
 4809 
 1766 
 8694 
 5593 
 
 2206 
 9322 
 6412 
 3476 
 0514 
 
 7524 
 4506 
 1460 
 
 8385 
 5282 
 
 1918 
 9032 
 6120 
 3181 
 0216 
 
 7223 
 4202 
 1154 
 8076 
 4970 
 
 1631 
 
 8742 
 5827 
 2886 
 9918 
 
 6922 
 3899 
 0847 
 7767 
 4657 
 
 1343 
 
 8452 
 5534 
 2590 
 9619 
 
 6621 
 3595 
 0541 
 7457 
 4345 
 
 1055 
 8161 
 5241 
 2295 
 9321 
 
 6320 
 3291 
 0234 
 7147 
 4032 
 
 0767 
 7870 
 4948 
 1998 
 9022 
 
 6018 
 2986 
 9926 
 6837 
 3719 
 
 0479 
 7579 
 4654 
 1702 
 8723 
 
 5716 
 2682 
 9619 
 6527 
 3405 
 
 0190 
 7288 
 4360 
 1405 
 8423 
 
 5414 
 2377 
 
 9311 
 
 6216 
 3092 
 
 ~<T 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 249 
 
 25 
 50 
 75 
 100 
 
 125 
 149 
 J74 
 199 
 224 
 
 25 
 
 2 
 5 
 7 
 10 
 12 
 150 
 175 
 200 
 22o 
 
 251 
 
 25 
 50 
 75 
 100 
 
 126 
 151 
 176 
 201 
 226 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 Diff. 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 25! 
 
 2. 
 5( 
 7< 
 10 
 12< 
 15 
 17( 
 20i 
 22' 
 
 2 253 
 
 i 25 
 ) 51 
 > 76 
 I 101 
 
 5 127 
 I 152 
 i 177 
 
 I 202 
 
 1 228 
 
 254 
 
 25 
 51 
 
 76 
 102 
 
 127 
 152 
 17S 
 
 203 
 
 229 
 
 255 
 
 26 
 51 
 
 77 
 102 
 
 128 
 153 
 179 
 204 
 
 230 
 
 256 
 
 26 
 51 
 
 77 
 102 
 
 128 
 154 
 179 
 205 
 230 
 
 257 
 
 26 
 51 
 77 
 103 
 129 
 154 
 180 
 206 
 231 
 
 25 
 
 8 259 
 
 260 
 
 26 
 52 
 78 
 104 
 
 130 
 156 
 182 
 
 208 
 234 
 
 261 
 
 26 
 52 
 78 
 104 
 
 131 
 157 
 183 
 
 209 
 235 
 
 262 
 
 26 
 52 
 79 
 105 
 131 
 157 
 183 
 210 
 236 
 
 263 264 
 
 26 26 
 53 53 
 79 79 
 105 106 
 132 132 
 158 158 
 184 185 
 210 211 
 237 238 
 
 265 
 
 27 
 53 
 80 
 106 
 133 
 159 
 186 
 212 
 239 
 
 266 
 
 27 
 53 
 80 
 106 
 133 
 160 
 186 
 213 
 239 
 
 267 
 
 27 
 53 
 80 
 107 
 134 
 160 
 187 
 214 
 240 
 
 268 
 
 ~27 
 54 
 80 
 107 
 134 
 161 
 188 
 214 
 241 
 
 269 
 
 270 
 
 27 
 54 
 81 
 108 
 135 
 162 
 189 
 216 
 243 
 
 2 
 5 
 
 7 
 10 
 
 12 
 15 
 18 
 
 20 
 23 
 
 6 26 
 2 52 
 7 78 
 3 104 
 9 130 
 5 155 
 1 181 
 S 207 
 2 233 
 
 27 
 54 
 81 
 108 
 135 
 161 
 188 
 215 
 242 
 
 1 
 2 
 3 
 4 
 
 5 
 
 6 
 
 7 
 8 
 9 
 
 271 
 
 272 
 
 273 
 
 274 
 
 275 
 
 276 
 
 27' 
 
 1 278 
 
 279 
 
 280 
 
 281 
 
 28S 
 
 5 283 
 
 284 
 
 285 
 
 286 
 
 287 
 
 288 
 
 289 
 
 27 
 54 
 81 
 106 
 136 
 16J 
 19C 
 2K 
 244 
 
 27 
 
 [ 54 
 R2 
 5 109 
 ! 136 
 ! 163 
 > 190 
 218 
 [ 24-5 
 
 27 
 55 
 82 
 109 
 
 137 
 164 
 191 
 218 
 246 
 
 27 
 55 
 82 
 110 
 
 137 
 164 
 192 
 219 
 247 
 
 28 
 55 
 83 
 110 
 138 
 165 
 193 
 220 
 248 
 
 28 
 55 
 83 
 110 
 
 138 
 166 
 193 
 221 
 
 248 
 
 2* 
 5i 
 
 s; 
 in 
 
 13< 
 166 
 19^ 
 22S 
 24< 
 
 * 28 
 > 56 
 5 83 
 111 
 ) 139 
 5 167 
 [ 195 
 i 222 
 ) 250 
 
 28 
 56 
 84 
 112 
 140 
 167 
 195 
 223 
 251 
 
 28 
 56 
 84 
 112 
 140 
 168 
 196 
 224 
 252 
 
 28 
 56 
 84 
 112 
 141 
 169 
 197 
 225 
 253 
 
 2t 
 56 
 8i 
 IK 
 14] 
 16< 
 19; 
 226 
 254 
 
 j 28 
 i 57 
 > 85 
 \ 113 
 142 
 ) 170 
 198 
 226 
 \ 255 
 
 28 
 57 
 85 
 114 
 142 
 170 
 199 
 227 
 256 
 
 29 
 57 
 86 
 114 
 
 143 
 171 
 200 
 228 
 257 
 
 29 
 57 
 86 
 114 
 143 
 172 
 200 
 229 
 257 
 
 29 
 57 
 86 
 115 
 144 
 172 
 201 
 230 
 258 
 
 29 
 58 
 86 
 115 
 144 
 173 
 202 
 230 
 259 
 
 29 
 58 
 87 
 116 
 145 
 173 
 202 
 231 
 260 
 
TABLE II. 
 
 Log 
 
 X 
 
 Log(l-aO 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 : 290 291 292 
 
 29 29 29 
 58 58 58 
 87 87 88 
 116 116 117 
 145 146 146 
 174! 175 175 
 203 ] 204 204 
 232 233! 234 
 261 262 263 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 1-880 
 881 
 882 
 883 
 884 
 
 885 
 886 
 887 
 888 
 889 
 
 1-38 2778 
 37 9620 
 6433 
 3214 
 36 9964 
 
 6682 
 3367 
 0018 
 35 6637 
 3220 
 
 2463 
 9303 
 6112 
 2891 
 963? 
 
 6352 
 3033 
 9682 
 6297 
 
 2877 
 
 2149 
 
 8985 
 5791 
 2567 
 9310 
 
 6021 
 2700 
 9345 
 5956 
 2533 
 
 1834 
 8667 
 5470 
 2242 
 
 8983 
 
 5691 
 2366 
 9007 
 5615 
 
 2188 
 
 1518 
 8349 
 5149 
 1918 
 8655 
 
 5360 
 2031 
 8670 
 5274 
 1844 
 
 1202 
 803C 
 
 4827 
 159S 
 8327 
 
 5028 
 169? 
 8332 
 4933 
 1498 
 
 0887 
 7711 
 4505 
 1268 
 7998 
 
 4697 
 1362 
 7993 
 4591 
 1153 
 
 7681 
 4172 
 0626 
 7043 
 3422 
 
 9761 
 6061 
 2321 
 8540 
 4716 
 
 0571 
 7392 
 4183 
 0942 
 7670 
 
 4365 
 1026 
 7655 
 4249 
 
 0808 
 
 0254 
 7073 
 3860 
 0616 
 7341 
 
 4032 
 0691 
 7316 
 3906 
 0462 
 
 9938 
 6753 
 3537 
 0290 
 7011 
 
 3700 
 0355 
 6976 
 3563 
 Olio 
 
 316 
 319 
 322 
 
 325 
 
 328 
 
 331 
 334 
 338 
 341 
 345 
 
 348 
 352 
 356 
 360 
 364 
 368 
 372 
 376 
 380 
 384 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 293 
 
 29 
 59 
 88 
 117 
 
 147 
 
 176 
 205 
 234 
 
 264 
 
 294 
 
 29 
 59 
 88 
 118 
 147 
 176 
 206 
 235 
 265 
 
 295 
 
 30 
 59 
 
 89 
 ,118 
 I 148 
 I 177 
 207 
 236 
 i 266 
 
 890 
 891 
 892 
 893 
 894 
 
 895 
 896 
 897 
 898 
 899 
 
 34 9769 
 6282 
 2758 
 339197 
 5599 
 
 1962 
 32 8286 
 4570 
 0814 
 31 7015 
 
 9422 
 5931 
 2403 
 8839 
 5237 
 
 1596 
 7916 
 4196 
 0436 
 6633 
 
 9074 
 5580 
 2049 
 8481 
 4875 
 
 1230 
 [7546 
 |3822 
 : 0057 
 6251 
 
 8726 
 5228 
 1694 
 8122 
 4512 
 
 0864 
 7176 
 3448 
 9678 
 5868 
 
 8378 
 4876 
 1338 
 7763 
 4149 
 
 0497 
 6805 
 3072 
 9299 
 5484 
 
 8030 
 4524 
 
 0982 
 7403 
 
 3785 
 
 0129 
 6433 
 2697 
 8920 
 5100 
 
 7331 
 3819 
 0270 
 6683 
 3057 
 
 9393 
 5689 
 1945 
 8159 
 4331 
 
 6982 
 3466 
 9913 
 6322 
 2693 
 
 9025 
 5317 
 1568 
 
 7778 
 3946 
 
 6632 
 3112 
 9555 
 5961 
 2328 
 
 8656 
 4944 
 1191 
 7397 
 3561 
 
 1 
 2 
 3 
 4 
 5 
 6 
 
 296 
 
 30 
 59 
 89 
 118 
 148 
 178 
 207 
 237 
 266 
 
 297 
 
 30 
 59 
 89 
 119 
 149 
 178 
 208 
 238 
 267 
 
 298 
 
 30 
 60 
 89 
 U9 
 149 
 179 
 209 
 238 
 268 
 
 
 
 
 1 2 
 
 3 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 8 
 9 
 
 
 29! 
 
 3( 
 6( 
 9( 
 12( 
 
 15( 
 17! 
 20! 
 23! 
 26< 
 
 Jj 300 
 
 301 302 
 
 303 
 
 304 
 
 ~30 
 61 
 91 
 122 
 152 
 182 
 213 
 243 
 274 
 
 305 
 
 306 
 
 307 
 
 308 
 
 309 
 
 310 
 
 311 
 
 312 
 
 313 314 
 
 315 
 
 316 317 
 
 1 
 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 9 
 
 ) 30 
 ) 60 
 ) 90 
 ) 120 
 
 ) 150 
 ) 180 
 ) 210 
 ) 240 
 > 270 
 
 30 
 60 
 90 
 120 
 151 
 181 
 211 
 241 
 271 
 
 30 
 i 60 
 91 
 121 
 151 
 181 
 211 
 242 
 i 272 
 
 30 
 61 
 91 
 121 
 152 
 182 
 212 
 242 
 273 
 
 31 
 61 
 92 
 122 
 153 
 183 
 214 
 244 
 275 
 
 31 
 61 
 92 
 122 
 153 
 184 
 214 
 245 
 275 
 
 31 
 61 
 92 
 123 
 154 
 184 
 215 
 246 
 276 
 
 31 
 62 
 92 
 123 
 154 
 185 
 216 
 246 
 277 
 
 31 
 62 
 93 
 124 
 155 
 185 
 216 
 247 
 278 
 
 31 
 6-2 
 9 
 124 
 15c 
 Iflt 
 217 
 24fc 
 271 
 
 31 
 62 
 93 
 
 r 124 
 
 156 
 
 187 
 218 
 ! 249 
 
 > 280 
 
 31 
 62 
 94 
 125 
 156 
 187 
 218 
 250 
 281 
 
 31 
 63 
 94 
 125 
 157 
 188 
 219 
 250 
 282 
 
 31 
 63 
 94 
 126 
 
 157 
 
 188 
 220 
 251 
 
 283 
 
 32 
 63 
 95 
 126 
 158 
 189 
 221 
 252 
 284 
 
 32 32 
 63 63 
 95 95 
 126 127 
 158 159 
 190 190 
 221 222 
 253 254 
 284 285 
 
 31* 
 
 5 319 
 
 320 
 
 321 322 
 
 323 
 
 324 
 
 325 
 
 326 
 
 327 
 
 328 
 
 32 
 
 I 330 
 
 331 
 
 332 
 
 333 
 
 334 335 j 336 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 35 
 6< 
 91 
 127 
 15! 
 19 
 22i 
 25< 
 28t 
 
 > 32 
 I 64 
 > 96 
 ' 128 
 ) 160 
 191 
 \ 223 
 U 255 
 i 287 
 
 32 
 64 
 96 
 128 
 160 
 192 
 224 
 256 
 288 
 
 32 
 64 
 96 
 128 
 161 
 193 
 225 
 257, 
 289 
 
 32 
 64 
 97 
 129 
 161 
 193 
 225 
 258 
 290 
 
 32 
 
 65 
 97 
 129 
 162 
 194 
 226 
 258 
 291 
 
 32 
 65 
 97 
 130 
 162 
 194 
 227 
 259 
 292 
 
 33 
 
 65 
 98 
 130 
 163 
 195 
 228 
 260 
 293 
 
 33 
 65 
 98 
 130 
 163 
 196 
 228 
 261 
 293 
 
 33 
 65 
 98 
 131 
 164 
 196 
 229 
 262 
 294 
 
 33 
 66 
 98 
 131 
 164 
 197 
 230 
 262 
 295 
 
 35 
 66 
 9< 
 135 
 16L 
 197 
 23C 
 265 
 29t 
 
 ; 33 
 ; 66 
 
 > 99 
 J 132 
 
 > 165 
 198 
 > 231 
 i 264 
 
 j 297 
 
 33 
 66 
 99 
 132 
 166 
 199 
 232 
 265 
 298 
 
 33 
 66 
 100 
 133 
 166 
 199 
 232 
 266 
 299 
 
 33 
 
 67 
 100 
 133 
 
 167 
 
 200 
 233 
 
 266 
 
 300 
 
 33 
 67 
 100 
 134 
 
 167 
 200 
 234 
 267 
 301 
 
 34 
 67 
 101 
 134 
 
 168 
 201 
 235 
 268 
 302 
 
 34 
 67 
 101 
 134 
 
 168 
 202 
 235 
 269 
 
 302 
 
 
 337 
 
 338 
 
 339 340 
 
 341 342 
 
 343 
 
 344 
 
 345 
 
 346 j 347 
 
 34d 
 
 349 
 
 350 
 
 351 
 
 352 353 
 
 354 ! 355 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 34 34 
 67 68 
 101 101 
 135 135 
 
 169 169 
 202 203 
 236 237 
 
 270 270 
 303 304 
 
 34 
 
 68 
 102 
 136 
 170 
 203 
 237 
 271 
 305 
 
 34 
 
 68 
 102 
 136 
 170 
 204 
 238 
 272 
 306 
 
 34 
 
 68 
 102 
 136 
 
 171 
 
 205 
 239 
 273 
 
 307 
 
 34 
 68 
 103 
 137 
 171 
 205 
 239 
 274 
 308 
 
 34 
 69 
 103 
 137 
 172 
 206 
 240 
 274 
 309 
 
 34 
 
 69 
 103 
 138 
 172 
 206 
 241 
 275 
 310 
 
 35 
 69 
 104 
 138 
 173 
 207 
 242 
 276 
 311 
 
 35 
 69 
 104 
 138 
 173 
 208 
 242 
 277 
 3J1 
 
 35 
 69 
 104 
 139 
 174 
 208 
 243 
 278 
 312 
 
 35 
 70 
 104 
 139 
 174 
 209 
 244 
 27S 
 313 
 
 35 
 70 
 105 
 140 
 
 175 
 209 
 244 
 279 
 
 314 
 
 35 
 70 
 105 
 140 
 175 
 210 
 245 
 280 
 315! 
 
 35 
 70 
 105 
 140 
 
 176 
 211 
 246 
 281 
 
 316 
 
 35 35 
 70 71 
 106! 106 
 141 141 
 
 176 177 
 211 212 
 
 246 247 
 282! 282 
 317 318 
 
 35 
 71 
 106 
 142 
 
 177 
 212 
 
 248 
 283 
 319 
 
 36 
 71 
 107 
 142 
 
 178 
 213 
 249 
 
 284 
 320 
 
 67 
 
PROPORTIONAL PARTS. 
 
 
 356 
 
 357 
 
 358 
 
 359 
 
 360 361 
 
 362 
 
 363 
 
 364 
 
 365 
 
 1 
 
 36 
 
 36 
 
 36 
 
 36 
 
 36 36 
 
 36 
 
 36 
 
 36 
 
 37 
 
 2 
 
 71 
 
 71 
 
 72 
 
 72 
 
 72 72 
 
 72 
 
 73 
 
 73 
 
 73 
 
 3 
 
 107 
 
 107 
 
 107 
 
 108 
 
 108 108 
 
 109 
 
 109 
 
 109 
 
 110 
 
 4 
 
 142 
 
 143 
 
 143 
 
 144 
 
 144 
 
 144 
 
 145 
 
 145 
 
 146 
 
 146 
 
 5 
 
 178 
 
 179 
 
 179 
 
 180 
 
 180 
 
 181 
 
 181 
 
 182 
 
 182 
 
 183 
 
 6 
 
 214 
 
 214 
 
 215 
 
 215 
 
 216 217 
 
 217 
 
 218 
 
 218 
 
 219 
 
 7 
 
 249 
 
 250 
 
 251 
 
 251 
 
 252 253 
 
 253 
 
 254 
 
 255 
 
 256 
 
 8 
 
 285 
 
 286 
 
 286 
 
 287 
 
 288 289 
 
 290 
 
 290 291 
 
 292 
 
 9 
 
 320 
 
 321 
 
 322 
 
 323 
 
 324 325 
 
 32rt 
 
 327! 323 
 
 329 
 
 
 366 
 
 367 
 
 368 
 
 369 
 
 370 
 
 371 
 
 372 
 
 373 
 
 374 
 
 375 
 
 1 
 
 37 
 
 37 
 
 37 
 
 37 
 
 37 
 
 37 
 
 37 
 
 37 
 
 37 
 
 38 
 
 2 
 
 73 
 
 73 
 
 74 
 
 74 
 
 74 
 
 74 
 
 74 
 
 75 
 
 75 
 
 75 
 
 3 
 
 110 
 
 110 
 
 110 
 
 111 
 
 111 
 
 111 
 
 112 
 
 112 
 
 112 
 
 113 
 
 4 
 
 146 
 
 147 
 
 147 
 
 148 
 
 148 
 
 148 
 
 149 
 
 149 
 
 150 
 
 150 
 
 5 
 
 183 
 
 184 
 
 184 
 
 185 
 
 185 
 
 186 
 
 186 
 
 187 
 
 187 
 
 188 
 
 6 
 
 220 
 
 220 
 
 221 
 
 221 
 
 222 
 
 223 
 
 223 
 
 224 
 
 224 
 
 225 
 
 7 
 
 2:>r> 257 
 
 258 
 
 258 
 
 259 
 
 260 
 
 260 
 
 261 
 
 262 
 
 263 
 
 8 
 
 9 
 
 293 294 
 329 1 330 
 
 294 
 331 
 
 295 
 332 
 
 296 
 333 
 
 297 
 334 
 
 298 
 335 
 
 298 
 336 
 
 299 
 337 
 
 300 
 338 
 
 
 376 
 
 377 
 
 378 
 
 379 
 
 380 
 
 381 
 
 382 
 
 383 
 
 384 
 
 385 
 
 1 
 
 38 
 
 38 
 
 38 
 
 38 
 
 38 
 
 38 
 
 36 
 
 38 
 
 38 
 
 39 
 
 2 
 
 75 
 
 75 
 
 76 
 
 76 
 
 76 
 
 76 
 
 76 
 
 77 
 
 77 
 
 77 
 
 3 
 
 113 
 
 113 
 
 113 
 
 114 
 
 114 
 
 114 
 
 115 
 
 115 
 
 115 
 
 116 
 
 4 
 
 150 
 
 151 
 
 151 
 
 152 
 
 152 
 
 152 
 
 153 
 
 153 
 
 154 
 
 154 
 
 5 
 
 188 
 
 189 
 
 189 
 
 190 
 
 190 
 
 191 
 
 191 
 
 192 
 
 192 
 
 193 
 
 6 
 
 226 
 
 226 
 
 227 
 
 227 
 
 228 
 
 229 
 
 229 
 
 230 
 
 230 
 
 231 
 
 7 
 
 263 
 
 264 
 
 265 
 
 265 
 
 266 
 
 267 
 
 267 
 
 268 
 
 269 
 
 270 
 
 8 
 
 301 
 
 302 
 
 302 
 
 303 
 
 304 
 
 305 
 
 306 
 
 306 
 
 307 
 
 308 
 
 9 
 
 338 
 
 339 
 
 340 
 
 341 
 
 342 
 
 343 
 
 344 
 
 345 
 
 346 
 
 347 
 
 END OK TAUI.K H. 
 
 
 
GENERAL LIBRARY 
 UNIVERSITY OF CALIFORNIA BERKELEY 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 This book is due on the last date stamped below, or on the 
 
 date to which renewed. 
 Renewed books are subject to immediate recall. 
 
 SFebs 
 
 ^RLIBRARY^ 
 
 DEC 9 - 1937 
 
 Uf- <''ALiF.. 8ER 
 
 JAN 3 2003 
 
 LD 21-100m-l, '54(1887816)476 
 
YC 23596 
 
 
 
 THE 
 
 UNIVERSITY OF CALIFORNIA LIBRARY