AN ATTEMPT TO TEST THE THEORIES OF CAPILLARY ACTION lonUon: C. J. CLAY, M.A. & SON, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 17, PATEKNOSTER Row. CAMBRIDGE: DEIGIITON, HELL, AND CO. LEIPZIG : F. A. BROCKIIAt s. AN ATTEMPT TO TEST THE THEORIES OF CAPILLARY ACTION BY COMPARING THE THEORETICAL AND MEASURED FORMS OF DROPS OF FLUID, BY FRANCIS BASHFORTH, B.D. LATE PROFESSOR OP APPLIED MATHEMATICS TO THE ADVANCED CLASS OF ROYAL ARTILLERY OFFICERS, WOOLWICH, AND FORMERLY FELLOW OF ST JOHN S COLLEGE, CAMBRIDGE. WITH AN EXPLANATION OF THE METHOD OF INTEGRATION EMPLOYED IN CONSTRUCTING THE TABLES WHICH GIVE THE THEORETICAL FORMS OF SUCH DROPS, BY J. C. ADAMS, M.A, F.RS. FELLOW OF PEMBROKE COLLEGE, AND LOWNDEAN PROFESSOR OF ASTRONOMY AND GEOMETRY IN THE UNIVERSITY OF CAMBRIDGE. ^ Camfcrfocj? : AT THE UNIVERSITY PRESS. 1883 Tie writers tfike this opportunity of returning ttieir thanks to tlie Syndics of the University Press for undertaking the expense of printing this ivork. INTKODUCTION. MANY years have elapsed since this work was commenced, and it is even now only partially completed. My object was to test the received theories of Capillary Action, and through them the assumed laws of molecular attraction, on which they are founded. To this end it was proposed to compare the actual forms of drops of fluid resting on horizontal planes they do not wet, with their theoretical forms. After some trials a satisfactory micrometrical instrument was constructed for the measurement of the forms of drops of fluid, but my attempts to calculate their forms as surfaces of double curvature failed entirely, and my undertaking must have ended here, if I had depended upon my own resources. But at this point Professor J. C. Adams furnished me with a perfectly satisfactory method of calculating by quadratures the exact theoretical forms of drops of fluids from the Differential Equation of Laplace, an account of which he has now had the kindness to prepare for publication. After the calculation of a few forms, application was made to the Royal Society for assistance from the Government grant in making the needful calculations. The following extracts from the application (Oct. 27, 1855) will explain the objects of the undertaking. " I have carefully examined all the published "experiments that I could meet with, but these have been generally made with " capillary tubes, and in consequence of the difficulties inherent in this mode of "observation they have not led to consistent and satisfactory results. " It appeared to me that the best test of theory would be obtained by making "careful measures of the forms assumed by drops of fluid resting on horizontal "planes of various solids " "At first I knew of no better mode of arriving at the theoretical forms than that given by geometrical construction, but I am indebted to Mr Adams for a "method of treating th,e differential equation ddz I dz Au u du 2 +-. 7-3-7-2** = b " when put under the form - + - sin <f> = 2 + 2zb* * = 2 + /9 * B. 2 INTRODUCTION. "which gives the theoretical form of the drop with an accuracy exceeding that of "the most refined measurements. Values of |, | and g have been calculated by this "method for values of * at intervals of 2| to 5, from = to = 145, for "values of equal to }, \, 1, 3, 6, 10 and 16. It is however very desirable that "calculations should be made for more numerous as well as for larger values of ft. "I also propose to make accurate measurements of the forms of the common "surfaces of two fluids that do not mix. The form of a drop of fluid (A) will "be taken when immersed in a fluid (), and also the form of a drop of the fluid " () when immersed in fluid (A), and for this purpose a plate-glass cell has been "constructed, so that the observations can be made whether the drops rest on the "bottom, or float in contact with the upper surface. The forms of drops of fluids "(A) and (B) will also be taken when resting on horizontal planes surrounded by the " atmosphere." "The objects of the experiments are "I. To compare the actual forms assumed by drops of fluid when resting on "horizontal planes composed of substances which they do not wet, with their theo- " retical forms. "II. To determine the effects of supporting planes composed of various sub- " stances. " III. To examine the effects of different degrees of roughness of the supporting "planes composed of various substances. "IV. To determine the effects of variations of temperature on the forms of the " drops of fluid from 32 to about 200 F. " V. To examine the mutual action of two fluids that do not mix, and the " effects of variation of temperature on them." The Royal Society voted a grant of 50, the sum applied for. These calcula tions were completed in 1857. And after the calculation of the theoretical forms and volumes of sessile drops had been carried as far as seemed needful, the money in hand was applied to the calculation of theoretical forms and volumes of pendent drops of fluids. The results of these calculations have been printed in Table IV. The delay in the publication of my results has arisen from the interruption of my labours, caused first by my removal in 1857 from College to a country living, and secondly by my appointment in 18G4 to the Professorship of Applied Mathematics to the Advanced Class of Royal Artillery Officers, Woolwich. As no systematic ex periments had then been made since the time of Hutton to determine the Resistance of the Air to the motion of projectiles, and those for round shot only, I was induced to turn my attention to the subject of Ballistics. The Results of my Experiments have beon published under the authority of the Secretary of State for War, ay follows I. Reports on Experiments made with the Bashforth Chronograph to determine the Resistance of the Air to the Motion of Projectiles, 18G5 1870. London, W. Clowes and Sons, &c. &c. INTRODUCTION. II. Final Report on Experiments, &c. &c., 187880. London, W. Clowes and Sons, &c. &c. on And in connection with these Reports I published a Mathematical Treatise the Motion of Projectiles, 1873, and a Supplement to that work, 1881. Immediately after the completion, of these labours I turned my attention to the preparation for publication of a part of my work on Capillary Action, for I cannot now hope to be able to complete the work originally proposed. The Tables II. and III. appear to give all that is required in order to supply the means for filling up the intervals to five places of decimals for all values of /3 under 100, and of <f> under 180. The Table IV. for negative values of /3, although not so complete, will afford considerable assistance, and the deficiencies can be easily supplied by original calcu lation preparatory to interpolation. Table V. gives the theoretical forms of free capillary surfaces of revolution about a vertical axis, which was used in calculating the forms of drops of mercury shewn in the diagrams. Deficiencies may be easily supplied by the help of Table II. by interpolation. As a specimen of the work I proposed to do, I have given diagrams and co ordinates observed and calculated of forms of drops of mercury carefully measured in X863. These shew how correctly the calculated and measured forms of these drops agree, notwithstanding the very considerable variation in their outlines. Also, as I found my measuring instrument in good working order in 1882, I have made numerous measurements of drops of the same kind of mercury of 4, 8, 12, 16, 20 and 24grs. in order to find the values of o and w. The values derived from each particular measurement vary considerably but the mean results for each weight of drop are satisfactory and appear to confirm the received theories of Capil lary Action. But as the Theories of Young, Laplace, Gauss and Poisson lead to the same differential equation, and therefore give the same form of drops of fluid, experi ments of this kind are not capable of deciding whether Poisson is correct in sup posing that a rapid change of density takes place near the free surfaces of fluids. But more definite information on this head may be expected when the values of a and &> at the common surfaces of fluids which do not mix, as well as the effect of variation of temperature on these quantities, have been determined according to the original scheme. Having given examples of the work I proposed to myself in the first instance, I must leave to others the further examination of this important question, for it still appears to me that this is the only way by which we can arrive at any definite results. I take this opportunity to return my best thanks to the Syndics of the University Press for having undertaken the publication of this work. 12 CHAPTER I. THE phenomena which arise from Capillary Action seem to contradict the laws of fluid equilibrium. In consequence, many worthless theories have been proposed with a view to explain apparent anomalies. After long groping in the dark, it was found to be desirable to discover by experiment what were the actual phenomena which required explanation. Hawksbee* found that the height to which a fluid would rise in a capillary tube of given radius was the same for all thicknesses of the tube. From this it was apparent that the attracting force of the tube was situated at or near the inner surface of the tube. But he does not appear to have taken account of the mutual attractions of the particles of the fluid. Jurin b also found that the height of the column of fluid supported by capillary action depended solely upon the interior diameter of the tube at the upper surface of the fluid. From this he con cluded that the column of fluid raised by Capillary Action was supported by the attraction of the periphery or section of the tube to which the upper surface of the fluid cohered or was contiguous. Clairaut was the first to attempt to explain capillary phenomena on right prin ciples, by referring them to the mutual attraction of the particles of the fluid, and to the attraction of the particles of the solid on the particles of the fluid ; and sup posing these attractions to depend upon the same function of the distance, he concludes that even if the attraction of the capillary tube be of a less intensity than that of the water, provided the intensity of the latter attraction be not twice as great as that of the former, the water will still rise in the tube (p. 121). Clairaut supposed that the attraction was sensible only at very small distances (p. 113). Shortly afterwards Segner d introduced the supposition that forces of attraction of both the particles of the solid and of the fluid vanished at sensible distances. He concluded that these forces gave a constant tension to the free capillary surfaces, and Phil. Traru., 1711 and 1712. > Commentarii Soc. Reg. Set. Gottingensis T 1 >> Ibid. 1718 and 1719. 1751. Thforie de la Figure de la Terre, 1743. Chapitre x. THEORETICAL EXPLANATIONS OF CAPILLARY ACTION. 5 thence he tried to calculate the forms of sessile drops of fluid with a view to com pare them with their measured forms. But in his calculations he took into account only the curvature of the vertical sections made by a plane passing through the axis of the drop. His measurements of the actual forms appear not to have been very precise. An important paper on the Cohesion of Fluids was read before the Royal Society by Dr Young a in which he pointed out the necessity of taking into account the curvatures of both of the principal sections of the drop, and clearly propounded the true principles on which the solution of the problem must depend. He arrived at the conclusions (1) that the tension of a free capillary surface would be constant, and (2) that the angle of contact between a given solid and fluid surface would also be con stant. He attempted to derive these hypotheses from physical considerations, but it is not easy to follow his reasoning. Even the editor of his works, Dean Peacock, observes on his Analysis of the Simplest Forms that "In the original Essay, the " mathematical form of this investigation and the figures were suppressed, the reasoning " and the results to which it leads being expressed in ordinary language : even in its " altered form the investigation is unduly concise and obscure " b . And respecting the appropriate angle of contact, Young confesses that "the whole of this reasoning on the " attraction of solids is to be considered rather as an approximation than as a strict "demonstration" . This may in part be urged as a reason why Laplace d did not more fully recognise the value of Young s labours. And although many of their results agreed, the processes by which they arrived at them were very different, except that they were much on a par in respect to the constancy of the angle of contact, which Laplace did not deduce mathematically from his theory. Very good accounts of Laplace s Theory were given by Petit e and Pessuti f , while it was attacked by others, as Young B , Brunacci h , Poisson J and others. Gauss k by a new and striking mathematical investigation obtained the same differential equation to the form of capillary surfaces as Laplace had done, and also supplied the defect of his work by obtaining an expression for the angle of contact of the fluid with the solid. Like Laplace he supposed the fluid to be homogeneous and incompressible. Bertrand has published a Memoir on Capillary Action, with a view to make known the method of Gauss, as well as some simplifications of which it is susceptible. In 1831, Poisson published his important work, the Nouvelle Theorie de I Action Capillaire. He strongly objects to Laplace s Theory because he has omitted in his calculations to take account of a physical circumstance, the consideration of which was essential; that is, the rapid variation of density which the liquid suffers near Dec 20 1804. g Quarterly -Review and Works, Vol. I., p. 436. b Works, Vol. i., p. 420 (note). " Bmgnatelli, T. ., 1816. c Ib 434 N uvelle Tlieorie, 1831. Mec. cei. supp.auXLivre,l80G,1801. * Princip. Gen. Theo. Fig. Fluid. Gott. Journal de Vecole Polytechnique. Cahier xvi, Dove s Repertorium, Bd. v., p. 49. Llouville xiii., p. 185. f Mem. Soc. Ital. T. xiv. 6 THEORETICAL EXPLANATIONS OF CAPILLARY ACTION. its free surface, and near the solid against which it rests, "sans laquelle les phe no- " raenes capillaires n auraient pas lieu " *. But he, in fact, arrives at a differential equa tion of precisely the same form as Young, Laplace and Gauss. It must be confessed that Poisson is probably quite right in supposing a rapid variation of density near the free surface of a fluid, and he has done good service in shewing how this sup posed variation of density near the free surface of fluids may be taken account of in the mathematical treatment of Capillary Action. The reader may be further referred to a Mtmoire sur la Theorie de F Action MoUculaire, par Jean Plana b . Nouvettf Thiorie, p. 5. b Turin Memoires, 2 S6rie, T. xiv. CHAPTER II. EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. MANY attempts have been made in recent times to test by experiment these theoretical explanations of capillary phenomena. For this purpose Haiiy and Tremery* at the request of Laplace made some experiments to determine the elevation of water and of oil of oranges, and the depression of mercury in capillary tubes. Their results appear to have satisfied Laplace that the elevation or the depression of a fluid in capillary tubes varied inversely as the diameter of the tube. A tube of one milli metre in diameter gave a mean elevation of 13 mm> 569 for water, and of 6 nun 738f) for oil of oranges, and a mean depression of 7 mm 333 for mercury. In the Supplement a la Theorie de V Action Capillaire, Laplace found the fol lowing expression for the approximate thickness (</) of a large drop of fluid resting on a horizontal plane b : q+ -.- = A/- sin + * ab V a 2 Soil sin - For comparison, Gay-Lussac measured the thickness of a drop of mercury one decimetre (2Z) in diameter resting upon a perfectly horizontal glass plane, and found it to be 3 mm 378 at a temperature 12 8C. In calculating the value of q Laplace neglects the value of ~r because it is an insensible quantity. He then supposes - = 13 square millimetres, and m = 152 grades = 13G 8 as determined by some previous experiments, and substituting finds q = 3 mra 39064, instead of the measured thickness 3" 378. Gauss merely refers to the results of Laplace, and gives the value of his a* which is equivalent to the - - of Laplace, equal 3 25 square millimetres. f&L Supp. au X Livre, p. 52, 53. b P - 64 - g EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. Poisson obtains the following expression for the approximate theoretical thick ness (*) of a drop of fluid resting on a horizontal plane: .f \ Here the a, and of Poisson are respectively the and *-<* of Laplace. Referring to a previous experiment, Poisson writes a cos = 4 5746 for a tempera ture of 12--8C., and for a first approximation he uses only the first term m (o). Thus & = ( a V2 cos ^ = a 2 (1 -f cos a/), or & cos &/ = (a 2 cos w ) (1 -f cos a> ). And writing for k, 3 mm "378, the experimental thickness of a drop of mercury of radius UIKT*, * a temperature 12 8C., as found by Gay-Lussac, he obtains (3 378) 2 cos ft/ = a 2 cos a/ (1 + cos &> ) = 4 5746 (1 + cos to ), which gives cos a/ = cos 48 nearly, or to = 48 nearly, and a 2 cos G> = a 2 cos 48 = 4 5746 now gives a or A/- = 2 mm G146. In the next place the term - -- only is neglected, because it is insensible : /* Z = Z + (v / 2-l)a = 50 + 1-083 = 51-083; and ^=3 mm -378. Substituting in (o), w is found to be 45 30 , which gives by the help of the equation a* cos a> = 4 5746, a* or - = 6 5262 square millimetres, and a or A/- = 2 mm> 5547. a V a Avogadro b made numerous experiments to clear up some doubtful points relative to capillary action. He carefully examined how far any air or moisture commonly supposed to adhere to the interior of glass tubes might affect the depression of mercury. With this object in view, he exhausted the air, and heated the glass tube when the mercury was not in contact with it, and he found that the depression of the mercury in the tube was precisely the same after as it was before these pre cautions were taken. In order however to determine the capillary constant a 2 or - , for mercury, he made use of a tube of copper 20 nun long, and 2 mm "80 in diameter , well amalgamated Nwvelle TUorie, p. 217. b Accad . p iti e Mat . Torino, T. 40 (1836). e Ibid. p. 221. EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. f) in the interior, and found it to be 5 56 square millimetres*, and, therefore, a or A/- =2 mm 357. Then substituting this value of a 2 in Poisson s b formula h= - - I7 2 *% a. b 8 ^ 3 -" and making h = 4 mm 69, a = radius of tube = O mm 9525, according to Gay-Lussac s ex periment, he obtained b = cos &> = O8440 or &> = 32 5 c = (180 147 5) nearly, instead of 45 5 given by Poisson. Substituting these two values a 2 = 5 5G and &/ = 32-5 in Poisson s expression (o), for the theoretical thickness of a large drop of mercury quoted above, he obtains 3 mm> 235 instead of the measured thickness 3 mm 378. Upon this he remarks that the smallness of this difference which corresponds to considerable differences in the values of a 2 and of cos &> , shews that this observation was little adapted to give, by its combination with the depression of mercury in capillary tubes, exact values of these quantities. Avogadro then determined to measure the depression of mercury in a capillary tube, so that he might obtain a value of &/ determined entirely from his own ex periments. His glass tube had a radius of O mm 80 d . He adopted a depression of 5 mm 125, that being the mean of a great number of careful observations. The tempera ture was between 10 C. and 14 C. This depression is rather less than that found by Gay-Lussac quoted above, when allowance is made for difference in the radii of the tubes with which they experimented. Substituting as before he finds w = 40 21 = (180 - 1 39 39 ). In the next place Avogadro substitutes the value of cos a/ just found = 07021 and a 2 =5-56, in Poisson s formula (o) quoted above, and finds 3 mra> 154 for the thick ness of a large drop of mercury instead of Gay-Lussac s measured thickness 3" 378. Desains 6 has deduced from Danger s experiments f a 2 or - = 67144, which gives a or A /- = 2 mm -5912 and = 37 52 33" =(180 -142 7 27"), which values appeared V a to satisfy best the whole of the experiments. He states however that for different sorts of mercury a or y/i varied from 2 mm 55 to 2 nim -61, and o> from 38 to 45 or from (180 -142) to (ISO 9 -135). Desains also obtained from experiments with large drops of mercury a or ^-2621 and = 41 36 30"=(180-138 23 30"). Still more recently Quincke has made very numerous experiments with a view to determine the capillary constants for a variety of fluids, and also for metals at P. 221. Nouvelle Theorie, p. 147. Acead. Fis. e Mat. p. 223. B. 227 8 EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. Poisson- obtains the following expression for the approximate theoretical thick ness (k) of a drop of fluid resting on a horizontal plane : / w a k = a V2 cos 5- - + Here the a, and of Poisson are respectively the \ and TT-^ of Laplace. Referring to a previous experiment, Poisson writes a cos = 4 5746 for a tempera ture of 12-8C., and for a first approximation he uses only the first term m ( Thus /. j V= (a V2 cos |-J = a 2 (1 4 cos a> )> or & cos a> = (a 2 cos &> ) (1 + cos a> ). And writing for k, 3 mm 378, the experimental thickness of a drop of mercury of radius J-WT*, * a temperature 12 8C., as found by Gay-Lussac, he obtains (3 378) s cos w = a 2 cos &> (1 + cos a/) = 4 5746 (1 + cos a> ), which gives cos &> = cos 48 nearly, or o> = 48 nearly, and a 2 cos = a* cos 48 = 4 5746 now gives a or A/ - = 2 mm G146. In the next place the term -- only is neglected, because it is insensible : 1 = 1 + 0/2-1) a = 50 + 1-083 = 51-083 ; and &=3 mm 378. Substituting in (o), &> is found to be 45 30 , which gives by the help of the equation a* cos a/ = 4 5746, a 2 or - = 6 5262 square millimetres, and a or A/- = 2 mm 5547. Avogadro b made numerous experiments to clear up some doubtful points relative to capillary action. He carefully examined how far any air or moisture commonly supposed to adhere to the interior of glass tubes might affect the depression of mercury. With this object in view, he exhausted the air, and heated the glass tube when the mercury was not in contact with it, and he found that the depression of the mercury in the tube was precisely the same after as it was before these pre cautions were taken. In order however to determine the capillary constant a 2 or - , for mercury, he made use of a tube of copper 20 nun long, and 2"" 80 in diameter , well amalgamated NouvelU TMorit, p. 217. b Accad . p ig . e MaL TorinOj T> 40 (1836) Ibid. p. 221. EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. J) in the interior, and found it to be 5 5G square millimetres*, and, therefore, a or A / - = 2 mm -357. Then substituting this value of a z in Poisson s b formula and making 7i = 4 mtn- 69, a = radius of tube = O mm- 9525, according to Gay-Lussac s ex periment, he obtained b = cos &> = 8440 or w = 32-5 c = (180 - 147 5) nearly, instead of 45 5 given by Poisson. Substituting these two values a 2 = 5 5G and = 32 5 in Poisson s expression (o), for the theoretical thickness of a large drop of mercury quoted above, he obtains 3 mm> 235 instead of the measured thickness 3 mm< 378. Upon this he remarks that the smallness of this difference which corresponds to considerable differences in the values of a 2 and of cos &/, shews that this observation was little adapted to give, by its combination with the depression of mercury in capillary tubes, exact values of these quantities. Avogadro then determined to measure the depression of mercury in a capillary tube, so that he might obtain a value of a/ determined entirely from his own ex periments. His glass tube had a radius of O mm 80 d . He adopted a depression of 5 mm -125, that being the mean of a great number of careful observations. The tempera ture was between 10 C. and 14 G. This depression is rather less than that found by Gay-Lussac quoted above, when allowance is made for difference in the radii of the tubes with which they experimented. Substituting as before he finds = 40 21 = (180 - 1 39 39 ). In the next place Avogadro substitutes the value of cos&> just found = 07G21 and a 2 =5-56, in Poisson s formula (o) quoted above, and finds 3 mm> 154 for the thick ness of a large drop of mercury instead of Gay-Lussac s measured thickness 3 378. Desains 6 has deduced from Danger s experiments f a 2 or - = 67144, which gives a or /l = 2 mm -5912 and &> = 37 52 33" = (180 - 142 7 27"), which values appeared V a to satisfy best the whole of the experiments. He states however that for different sorts of mercury a or ^ varied from 2 mm 55 to 2 nim -61, and from 38 to 4.5 or from (180 -142) to (180* - 135). Desains also obtained from experiments with large drops of mercury a or ^ = 2-621 and - 41 36 30"=(180-138 23 30"). Still more recently Quincke has made very numerous experiments with a view to determine the capillary constants for a variety of fluids, and al a P 221 " P " 227 ^ ,, vi - 117 e Ann. de Ch. Ph. [3] T. u. (1857). f W f I! ;;:; P p ^3 f Ann. de CH. Pk. [3] T. p. 501. c Accad. Jfis. e Mat. p. ^o. B. 10 EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. a temperature just above the melting point. He found that the values of a or decreased for the same drop of mercury a , according to the time it had stood in position. He also found that varied from 38 to 45 ob , or from (180 - 142) to (180 - 135). But other results were obtained far beyond these limits. For the mean value of a or J^ he adopted 2" in "8 c , and some of his experiments gave as * a high a value as 2 mm 9, both of which differ considerably from the previously .received value 2 -6. In 18G8-9 Quincke published 11 the results of some experiments made to deter mine the capillary constants at the common surfaces of two fluids incapable of mixing. In this case he pursued methods of experimenting in some respects similar to those I had suggested in my application to the Koyal Society in 1855. But the value of Quincke s results is very much diminished by the manner in which he carried out his experiments, and by his mode of determining the theoretical forms of sessile drops of fluid. Thus Quincke s method requires the measurement, with great precision, of the height of the vertex of a large drop above/ the largest horizontal section of the drop. But in my experiments I have found that only a rough approximation to this quantity can be obtained directly by the most careful measurement. The theoretical forms of Quincke are much the same as those of ner, for in the calculations of both, one of the two principal radii of curvature is supposed to be infinite. There is also a further objection to the use of large drops of fluid, which Quincke s methods of calculation necessitated, because they change their form slowly when a change in their volume is made. But only a slight change in the volume of a small drop will give a marked change in its form. The favourite method of testing the theories of capillary action has been by tin.- measurements of the heights to which fluids rise in capillary tubes. In cases where the fluid wets the solid, there is only one constant, a, to be determined, as the angle &> = 0. But experiments of this kind are very liable to be vitiated by irregularities in the bore of the tubes, or by impurities adhering to the inner surface of fine tubes, which do not admit of being cleaned. The layer of fluid which lines the tubes must make a sensible reduction in the radii of the finer upillary tubes. And the theoretical expressions for the height of the fluids in these tubes are approximations which are not strictly applicable to tubes of large diameter used in experiments of this kind. Some recent writers on capillary action have disputed the correctness of the arrived at by the earlier experimenters. Thus Simon 6 has concluded from 5 experiments of his own that the elevation of water in capillary tubes is ery far from varying inversely as their diameters, and that the height to which ter rises between parallel plates compared with that which takes place in tubes being as 1 : 2, is as 1 : 3, or rather as 1 to TT. r- Ann. Bd. cv., p. 35 (1858). a Pogg . ^ Bd . CXXXIX Ann. de Ch. Ph. [3] T. xxxvm. (1851). EXPERIMENTAL TESTtt OF THEORIES OF CAPILLARY ACTION. 11 Bede a comes to the conclusion that the depression of mercury and the elevation of water in glass tubes do not respectively vary inversely as the diameters of the tubes exactly, and that the thickness of the substance of the tubes has a sensible effect, or, in other words, that the molecular attractions are not insensible at sensible distances. Wolf b afterwards concluded from his experiments that the elevation of the same fluid in capillary tubes, all circumstances being alike in other respects, depends upon the nature of the tube. Laplace and Poisson considered that the only effect of a change of temperature was to change the elevation of a capillary column according to the change in density. Thus Laplace c says "L e levation d un fluide qui mouille exactement les parois d un " tube capillaire, est, a diverses temperatures, en raison directe de la density du " fluide, et en raison inverse du diametre interieur du tube." And Poisson d obtains for the elevation (h) of a fluid in a capillary tube of radius a Rr dr. He then supposes that by a change of temperature h, p and R are respectively changed into h , p and R , neglecting the change in a. And having found T-=- he remarks " L expeVience montre, en effet, que pour un meme liquide a differentes "temperatures, Felevation du point C croit proportionellement a la densite^ ce qui " donne lieu de croire que la force repulsive de la chaleur, ou du moins, sa " variation, que nous avons neglige e, n a qu une influence insensible sur 1 inte grale Very careful experiments have been carried out by Frankenheim and Sondhauss, and afterwards by Brunner, to determine how far the height of the capillary column depends upon the temperature. Frankenheim 6 found that the height to which water rises in a capillary tube l mm in radius at a temperature t C. is 15 ram -33G- 0-0275U - 000014 2 between -2 5 and 93-4C., and Brunner f . finds it to be 15 mm -33215 -0028G396* from to 82 C. Hence it appears that the elevation of fluids decreases with an increase of temperature much more rapidly than would be expected according of Laplace and Poisson. In the foregoing sketch of the progress of experiments made to determine capillary constants I have given attention chiefly to those where mercury was Savans Etr. Brux. T. xxv. (1853). Ann. de Ch. Ph. [3] T. XLIX. Supp. Th. de V Action Capillaire, p. 39. d Nouvelle TMorie, p. 106. Pogg. Ann. Bd. LXXII. (1847). f Disquisitio Phys. Exp., p. 34, 35 (1846). 22 12 EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. the fluid employed. Every experimenter finds that changes of form are constantly going on in capillary surfaces from one cause or another. Still something more definite is desirable in the results. But as the experiments have been conducted apparently with every precaution, it does not appear probable that any new experi ments of the same kind would lead to better results. When &> is determined by reflection its value must be obtained for a point at a short distance from the junction of the solid and fluid surfaces. The experiments on the thicknesses of large drops of fluid are not satisfactory because the theoretical expression is not exact, and because the thickness of the drop varies so slowly in large drops. Also the approximate theoretical thickness is given in terms of two unknown quantities a and to. During the time when I was able to use the Cambridge University Library, I made copious extracts from numerous papers on this subject, but it does not appear necessary for me to allude further to them in this place, especially as the late Professor Challis has published a very good and elaborate report on Capillary Action*. For numerous references to the works of early writers on the subject, reference may be made to the articles " Capillaritat," "Cohasion" and "Tropfen" in Gehler s Physikalisches Worterbuch. Recent experiments will be found referred to in Fortschritte der Physik 1845, &c. and in Jahresbericht, 1847, &c. von Liebig, Kopp, u. Will. See also the article on Capillary Action in the 9th edition of the Encyclopaedia Britannica by the late Professor Clerk Maxwell. Brit. Ass. Report, 1834. CHAPTER III. ON THE CALCULATION OF THE THEORETICAL FORMS OF DROPS OF FLUID, UNDER THE INFLUENCE OF CAPILLARY ACTION, WHEN SUCH DROPS ARE BOUNDED BY SURFACES OF REVOLUTION WHICH MEET THEIR RESPECTIVE AXES AT RIGHT ANGLES. WE have already stated that various methods of obtaining the differential equa tion to the surface of fluid under the action of capillary forces have been given by Laplace and other writers on Capillary Action. The form of the equation obtained by these different methods is, however, in all cases the same. Perhaps the simplest way of obtaining the equation in question is to consider the fluid to be in equilibrium under the action of gravity and of a uniform surface tension. Let T be this uniform tension, R and R the principal radii of curvature at any point of the surface of the fluid, p the fluid pressure at that point. 1 1 p Then R + R ~T If z be the vertical coordinate of the point measured downwards, <r the density of the fluid, and g the force of gravity, then p = gcrz + C, where C is a constant. When two different fluids are separated by the capillary surface, p is the dif ference of the pressures in the two fluids at their point of meeting, and <r is the difference of the densities of the fluids. When a drop rests upon or hangs from a horizontal plane surface, the remaining surface of the drop being free, this free surface will evidently be one of revolution about a vertical axis, and it will meet the axis at right angles. Take the axis of revolution as the axis of z, and the point in which it meets the free surface as the origin. 14 CALCULATION OF FORMS OF DROPS. Let x be the horizontal and z the vertical coordinate of any point in a meridional section of the surface of the fluid, p the radius of curvature of the meridional section at that point, and (f> the angle which the normal to the surface makes with the axis of revolution. Then the length of the normal terminated by the axis is , and we have sin <f> mm ijf j.v . . sin and the above found equation becomes 1 sin <f> _ C + gcrz p x ~~T~ Let 6 be the radius of curvature at the origin, so that at that point we have both p = b, and limit \-^~. r )=6, Vsm <f>J Hence _=_ Wh and the equation becomes 1 , sin 6 2 flfo- 1 E = _ _1_ y_ . p x b ^ T J + ir 2+y ^- y- be called ft which is an abstract number. Also let , be the length of o ^ IlCriCllOIicU S6CL1OI1 TnpnQnT*Arl ff/\>v- 4-T- " i L wns id eratio the on S in and terminated at the Then j n < = /a cos dz = p sin , = For the sake of s i mplicity , we will ^ ,_ ,_ f ^ .^^ ^ f . ? o b b CALCULATION OF FORMS OF DROPS. 15 which amounts to taking the quantity b as the unit of length, and we may at any time re-introduce the quantity b by writing x z p s . b b 6 a 6 lnsteac * f x > z > P an( J * Thus simplified, our equation becomes p Also when < = 0, we have z = 0, p = 1 and limit ( * ) = 1, hence the form of the \siu <p/ curve depends on the single parameter . The magnitude of the curve, or its scale, is proportional to b. The same equation is applicable to the case of hanging drops, but in that case 2 is to be measured upwards from the vertex, and /3 will be negative. Since _ . . . _ , \,dx dz and sin rf> = %L the above equation is equivalent to ~ + jl + (-} \ -^- = (2 +&z\ jl + dx { \dxj } xdx v (. a differential equation of the 2nd order. The two arbitrary constants which enter into the integral of this equation are to be determined by the condition that when x = 0, z = 0, and r-= 1. xdx We are unable either to find the general relation between x and z, by means i this equation, or to express these two quantities in terms of a third variable. We may, however, as in all cases where the differential equation to a curve is given, develope the increments of the coordinates in series proceeding according i<> ascending powers of the increment of the quantity chosen as the independent varial>l<-. Thus we can trace a small portion of the curve starting from a known point, and then we may make the point which terminates this portion a new starting point for tracing another small portion, and so on successively until any required portion of the curve has been traced. For instance, suppose the given equation to be d 2 u ,. (dy \ -j-y = j I -7 , y, tj, where / denotes any function of the quantities H , y and t. at 16 CALCULATION OF FORMS OF DROPS. Then by repeated differentiations of this equation, and by substitution of the value of ^ in the successive results, we may find the general values of the higher rfr differential coefficients Jf W in terms of -J*., y and t. Hence if, for a given value t. of t, we know that y = y and -]- = [-2 ] , suppose, we can find the values of ^ and the higher differential coefficients of y, which corre spond to t = t . /d*y\ fd*u\ n Let these values be denoted by H^J , ^J , &e. Therefore if t, = t + Bt , and if y l and t-j) be the values of y and ^ which correspond to t = t lt we have by Taylor s theorem The increment & must be taken so small as to render these series convergent. The values of y^ and ( -j- ) being thus known, we may find ( -A ) , (-^} , &c., by vflc/j \dt J l \ut /j the same formulae as before ; and then if Hiid if y t and f-^J be the values of y and -JJ which correspond to = < x , we may simi larly find y t and f -^-J , and the same process may be repeated as often as we pl ease. ^ A similar process may be employed if we have any number of simultaneous ifferential equations, and the same number of dependent variables, such as, for instance, the following: dx %-/(*, y, t), CALCULATION OF FORMS OF DROPS. 17 The method fails if any of the differential coefficients employed become infinite in the interval over which the integrations extend, and therefore the independent variable should be so chosen that no infinite or very large values of the differential coefficients will be introduced. The intervals adopted should be so small that a few of the terms of the series will suffice to give the results with all the accuracy that is desired. After a few points of the curve, in the neighbourhood of the starting point, have been determined by the foregoing or some equivalent method, it will usually be found more convenient to determine other points of the curve in succession by making use of a series of successive values of the differential coefficient which is given immediately by the differential equation, rather than by employing the values of the successive differential coefficients of higher orders which are found by means of the several derived equations. To fix the ideas we will suppose, with especial reference to our present problem, that the given differential equation is one of the first order, say dy / / ,\ J =-/&). Let ... _,, t_ z , t_ y t_ lt t , t lt &c. be a series of values of the independent variable t, forming an arithmetical progression with the common difference w. Let . .. y y y_ 3 > y,y y^t y& y^ < ^ c * denote the corresponding values of y, and let be the corresponding values of q, or of -~ , and suppose a> to be so small that the successive differences of these values of q soon become small enough to be neglected. Let t = t + nw, and suppose that we have already found the values of 2/-4 y~s> y~v y~i U P ^ y* and therefore also those of ... g_ 4 ,. q_ z> ?_ 2 , ?_ t up to q , and that the successive differences of these quantities are taken according to the following scheme : n q -4 q_ t _3 q 3 AY 2 ... &c. A ? _ 2 A ?_ t 2 o A q A q Q &c. A<7_i A 3 <7 1 <?_! A 2 ft q .j B. lg CALCULATION OF FOEMS OF DROPS. Then the general value of q found by the ordinary formula of interpolation, for any value of n, will be provided that n be taken between limits for which this series remains convergent. Hence the general value of y will be y = fadt = a) fydn, or, substituting the above value of q, and adding a constant to the integral so as to make y=y when =0, where all the integrals are supposed to vanish when n = 0. If, in particular, we put rc = 1, and substitute the several values of the definite integrals we shall have, by changing the signs throughout, 1 A 1 , 1 19 . 3 A5 863 _ &c __ 24192 3628800 1036800 Similarly, putting n = 1 and substituting the values of the definite integrals f l n(n + l)j / 1 n(r + l)(n + 2) J J T^~ ** J - 1.2.3 ~ dn > &C " we shall have 5257 ,17( )017 A 8 2082753 | ** 17280 ^ + 3628800 ^ + 7257600 q + C \ It will usually be found expedient to choose o> so small as to render it unnecessary to proceed beyond the fourth order of differences. The series last found gives the value of y l in terms of quantities which are supposed to be already known, that is, the value of the variable y which was tously known for values of the independent variable extendino- as far as t = t now becomes known for the value t = t + <o, or at the end of an additional interval a> CALCULATION OF FORMS OF DROPS. 19 It will be remarked, however, that the coefficients of the series above found for y y_ l} after the first two terms, are much smaller and diminish much more rapidly than the corresponding coefficients of the series for y l y . Hence by taking into account the same number of terms of the series in the two cases, the value of y y_j will be determined with much greater accuracy than that of y^ y . In what has gone before, the successive values of y up to y are supposed to be already known, and therefore the equation which gives the value of y y_ t may be regarded as merely supplying a verification of former work. If, however, we suppose that the value of y is only approximately known, while the successive values as far as y_ v have been found with the degree of accuracy desired, we may use the equation for y y_^ to give the corrected value of y , in the following manner. Suppose that (y ) is an approximate value of y , and let y = 0/o)+ 7 ?> where 77 is so small that its square may be neglected. Also let (<? ) be the corresponding approximate value of q found from the equation by putting y = (y a ] and t = t . Then we may put q = (<? ) + krj, where k denotes the value of the partial differential coefficient ~ or , found dy dy by substituting (T/ O ) for y and t for t after the differentiation. Let A(g ), A 2 (^ ), A 3 (g ), A 4 (q ), &c. denote the values of the successive dif ferences formed with the approximate value (g ) and the known values q_ v q_ v &c. which immediately precede it, then we have A= A + * &c. = &c. But, by the equation before obtained, - \ A?o - ^ ^\~ 21 AS ?o - 720 A *?o ~ leo ^ " 60480 275 A7 33953_ .* 8183 , _ & ) 92 q ~ 3628800 q 1036800 " j Or, substituting for y ol q a , A^ , A 2 ^, &c. their values in terms of r, and known quantities, II l 19 c I -2~i2"24~726~ 32 o CALCULATION OF FOKMS OF DROPS. Hence if e denote the excess of the quantity -&c 12" w/ 24 - over the quantity (yj-y.v we shall have ,111 J9.-&C ~ 2 ~ 12 24 720 e w h_l_l_JL- 19 --& c which detennines 77, and therefore y = (#o) + 7 ?> and ?o = (<?o) + ^ both become known. If in finding e we stop at the term involving A 4 (q ), we shall have and 251 ke i 251 k ^m^ It will be observed that the coefficient of a>k in the denominator of these expressions is the same as that of A 4 (? in the expression for y l y a . This is no mere coincidence, as it is easy to shew that, generally, the coefficient of any term &)A r ^ , in the expression for y l y v is equal to the sum of the coefficients of the terms involving coq , a>bq , aA*j , &c. . . . a> A r ^ in the expression for y y_^ Hence if in finding e we also include the term involving A 5 (fl ), we shall similarly have e 17 = 95 and 95 An approximate value of y may always be found from the series of values y-v y.v 2/-t. y_, previously calculated, by taking the successive differences of four or five of the last terms of the series, and assuming that the last difference so found remains constant. CALCULATION OF FORMS OF DROPS. 21 The numerical operations will be greatly facilitated by the use of Tables which exhibit the values of 19 4 3 . 5 8G3 720 160 60480 " for given values of A 4 g, A 5 <?, A 6 <?, &c. Such Tables have been formed by Mr Bashforth for this purpose, and are given at the end of this Chapter. Having made these preliminary observations on the general method of finding successive small portions of a curve by means of its differential equation, we will now proceed to apply the method to the problem under consideration, viz. to the tracing of the curve formed by a meridional section of a drop of fluid, by means of the equation above found 1 sin <b - + -- =2 + &z. p x First, suppose < to be taken as the independent variable. The above equation may be regarded as giving p as a function of the co ordinates x and z, and these latter quantities are to be found by the integration of the equations dx dz Also x and z vanish with <, and p is initially = 1. We will first find the form of the curve in the neighbourhood of the origin by developing p and the coordinates x and z in series of ascending powers of </>. Instead of employing the general method described at the outset, it will be found more convenient, in this particular case, to proceed as follows : Assume, as we evidently may do, p = 1 + Itf + btf + bjt + M>" + where & 2 , l v &c. are constants to be determined, then dx -, 1 , , _! __ A* CALCULATION OF FORMS OF DROPS. Substitute the assumed value of p and integrate, therefore 155 * - <wVo * + *" - - + &c, &c. Similarly and therefore + *c., &c. Also, we find - (V - 46A , > _ _ ftA + CALCULATION OF FORMS OF DROPS. sin $ /sn Also " and and from above + JPr/ i\rr* cvC.j vx\- Hence, by performing the division indicated, we may find + &c., &c. Substitute these expressions in the equation x 24 CALCULATION OF FOEMS OF DROPS. and equate the coefficients of corresponding powers of </>, and we shall find successively b- /j /j._ 5760 p 128 p 9216 l 6799 -- 8960 18432 "" 92160 81920 Pl = _ __ J_ 1469 14515200 p ^ 36288 p 442368 p 104513 4 _ 4882031 5529600 p 88473600 ^ which gives the value of p in terms of <f>, as far as < 10 . Again, substituting these values of b t , 1 4 , &c., in the expressions for - , x and *, we shall obtain ^ --R __ ?Lfl- 401 ff 8431 8960 P 30720 p 92160 P ~ 737280 /_233_ 17 1517 U4515200 P 725760 p f 2211840^ 7409 ff ,522091 2764800 P * 88473600 to the 10th order in WNW*^ " ( 39916800 145IT200 - 443821 22118400 88473600 to the llth order, and CALCULATION OF FORMS OF DROPS. 25 V40320 49 67,,, + 7Q70Q P 9 -C 3628800 241920 ^ 184320 H 691 200 p 819200 1 269 7993 35j>J_ n ~ i 7/1,1 Qf>A.(\i\ r i oao* rnoA P ~OAO 41^^ 479001600 174182400^ 139345920^ 5308416 724007 4 4882031 265420800 P 1061683200 P to the 12th order. It is hardly necessary to remark that in these expressions the coefficient of each power of <f> thus found is exact, and not merely approximate. Also if s denote the length of the arc of the curve measured from the origin, 1 3 ( 1 1 53 - -- 80640 165888 829440 737280 233 _1 _**, _ 1469 *s, 104513 P p ^ p r + - - If] <f>" 1 t i c i A n " / /\ r\ " I V159667200^ 399168^ "4866048^ 60825600 443821 88473600 to the llth order in <. In order that the terms in these series which involve higher powers of <j> may be insignificant, $ must not exceed a certain limiting value which will, of course, depend on the value of 8. The larger the value of j3, the smaller will be this limiting O value of (f). To find the values of the coordinates for larger values of 9, we must proceed step by step according to the method described above, 9 being taken for t, and x and z in turn taken for y, the value of 9 being increased at each step by a given *l small quantity. 9 Let o> be the circular measure of the interval between two consecutive values of 9, then must be so chosen that the series above found will give sufficiently accurate values of the coordinates throughout several, say four or five such intervals. Suppose ...9_ 5 , 9_ 4 , 9_ 3 , 9- 2 > 9-!- 9o to be a series of consecutive values of 9, with the common difference &>, and let B. CALCULATION OF FORMS OF DROPS, ^.o be the corresponding values of the coordinates, and ...p-*, P-4, P-S P-*> P-*> P the corresponding radii of curvature. The equations to be integrated are dx , -=pcos^ dz . , -=psm^ 1 sin0 _ b> Q where - + - - z f p*. jj *>u Suppose that the values of the coordinates, and consequently those of the radius of curvature, have been calculated for the successive values of up to $_ and we wish to fiud the values of the same quantities for < = < In the first place, we may obtain an approximate value of p in the following manner. Tabulate the calculated values of logp, and form their successive differences according to the following scheme: log p b Alogp_ 4 Alogp_ 3 A 3 logp_. A 2 logp_ 2 A \ogp_, Alog/3_ 2 AMogp., logp_ A 2 log p., A log p_, lo gP-, If o> is taken sufficiently small, the differences as we proceed to higher orders will rapidly diminish, and it will generally be easy by inspection of the two or three last calculated fourth differences, to fix upon an approximate value of the fourth A*logp immediately succeeding. Call this approximate value A 4 log(p ), and by successive additions form A 3 log(p ), A*log(/> ), Alog(/j ) and Iog(p ), thus _ t A logp., Alogp_, A 3 log(p ) AMog(p ) Alog(p ) CALCULATION OF FORMS OF DROPS. 27 Form the values of ...cfo_ 5 , dx^, dx_ s , dx_ 3 , dx_ v dz - 5 , dz_ 4 , dz_ 3 , dz_ 2 , dz_ v and of their successive differences, according to the following scheme : cos 0_ 5 = dx_ cos <_ 4 = dx cos <_ 3 = rf^. cos </>_ 2 = c?^_ cos >_ = dx and sn _ 5 = _ sin $_ 4 = yp_ 3 sin <_ 3 = rf^_ w/D_ 2 sin <_ 2 = c?^ 2 c^ Aefc., ty/j^j sin <^>_ t = c?^_j If p were known, we might similarly form dx = cop cos ^ and dz = (ap sin , and the successive differences , A 4 c?a; , &c., , AV , &c., and then we should have, by what has been already proved, ^o ~ #-1 = d3c o ~ g Arf;r o ~ J2 ^ dx ~~ 24 A3 ^ ~ 720 1 1 1 19 and z - z_ t = dz - ^ Mz - A V* - ^ 3 and when X Q and z had thus been found, we should have the equation / o x o in verification of the value which had been used for p . 42 og CALCULATION OF FORMS OF DROPS. Now, let (dr.) and (dz ) be approximate values of dx and dz respectively, given by (d-x ) = ca (PO) cos <f) , (dz t ) = < (po) sin fa, and let the successive differences found by employing (dxj instead of dx v and (c instead of dz , be denoted by Hi HI \*"0 J \0 \ O/ respectively, and suppose that (a; ) and (a ) are given by the equations 00 - *_, = (dx ) - \ A (^ ) - 1 A 2 (ete.) - ^ A s (^ ) - ^ A 4 (^ ) - &c., () - ., = (dz ) - \ A (efe ) - ^ A 2 (& ) - i A 3 (dz ) - WQ A 4 (dz ] - &c. Also let [p ] be found from the equation and suppose that this gives [p ] = (p ) (1 -f e), where e is a very small known quantity. Then if the true value of p = (p 9 ) (1 + ??), the correction of the value of (efo ), and therefore also that of the values of A (dx^, A 2 (dx \ A 3 (dx \ A 4 (dz? ), &c. will be rja) (p ) cos ^> , and the correction of the values of (dz ), A (dz ), A 2 (c?^ ), A 3 (c/^ ), A 4 (da,,), &c. will be <7> (Po) sin 0o- Hence if we stop at the terms which involve differences of the 4th order, we shall have . , 251 #o - (*) = ^ (Po) cos fa and Hence, since 1 + 8 and we find 251 72l) ^ ^ s ^ u ^ I 7~~y^ + & I neai ly CALCULATION OF FORMS OF DROPS. 29 but ^ )( 1 -^nearly, and _L = JL (1 _ e) nearl Hence T~\ [ e ??] = R^TJ ^w (PO) s i n $o ^ ? + $ nearly, and therefore 77 = - _ - , nearly. Hence 77 is found, and therefore the values of x and z , which were required, become known. In practice, the following slight modification of the above process will be found convenient. Suppose the assumed value of Iog(/o ) to be increased by 100 units of the last place of decimals employed, then while calculating the values of (dx \ (dz ), (x ), (z ) and the consequent value of [p ], note at the side of the work, the changes which would be severally caused in each of these quantities by such an augmen tation of log (p ). It may be remarked that the changes in (# ) and (z ) will be 251 times the changes in (dx Q ) and (dz ) respectively, when we stop at terms 251 involving A 4 , and that ^^ may be conveniently put under the form Now suppose that an increase of 100 units in log(/> ) causes a diminution of //, units in log [p ], and that the excess of log[/3 ] above log (p ) is X of the same units, then the correction to be applied to the assumed value log(/? ) will be 100 X - - such units, 100 + JJL and the correction to the value of log [p ] will be ^ such units, 100 and the proportionate changes required in the values of (da- ), (dz ), (a- ) and (Z Q ) will be at once found. If in findino- fa;) and (z) we include the terms which involve differences of O \ \ O/ the 5th order, the fraction \ , which occurs in the above, should be replaced by 720 95 288 3 V 96, We may, of course, change the value of to whenever the more or less rapid rate of diminution of the successive differences shews that it is expedient to increase or diminish the interval. It is only necessary, by selection from or interpolation between the values already calculated, to find the coordinates for a few val separated from each other by the newly chosen interval. 32 CALCULATION OF FORMS OF DROPS. This circumstance makes it necessary, when /3 is negative, to choose a different independent variable. Suppose now that s, the length of the arc measured from the vertex, is taken as the independent variable. The equations to be integrated are d<b = - ds, P dx = cos cf) ds, dz = sin <j) ds, where the value of - in terms of x, z and < is given, as before, by the equation 1 sin <f> p x Also to determine the constants of integration, we have, when s = 0, 05=0, z = Q, </> = 0, and - = 1. We must first find the form of the curve in the neighbourhood of the vertex, by developing <, x and z in series of ascending powers of s. We have already found s as well as as and z in series of ascending powers of </>, and by means of Lagrange s theorem it is easy to transform these series so as to obtain the required series in powers of s. From the expression of s in terms of <f>, we find by transposition, 8+ 5S 2011 P - 80640 165888 829440 737280 233 _J__*, 1469 _fl.+ 104513 g , I " P p p "^ p T ___ _ 59667200 P 399168 p 4866048 p "^ 60825600 p T 88473600 - &c., or <f) = s + F(<j>), suppose, which is in the proper form for the application of Lagrange s theorem. Hence, we have CALCULATION OF FORMS OF DROPS. 33 fl 1* fs Z Also if in the values of x, z, -j-r , -,- in terras of 6, we change <h into s. and dtp cLcp denote the results by f , f . (dx\ , (), (.), (^j and we have, by the same theorem, In this way, we obtain 487 _ 80(J40 5806080 5806080 737280 /J^3_ 7 17539 271 1 \ r V159er67200 p T 2851200 p r 851558400 p 47308800 p r 88473600 P / + &c., <fec. _ X ~ S 1 Q + JL a* Z_ ^ fc80 P 725760 p 82944 p ) 362880 4480 725760 .s, n s u p p 39916800 118800 319334400 15966720 2703360 &c., &c. 2 T^O T l ,- , ~T * /< I O T I fv~{\(\ _ -. ____ - 40320 64512 107520 73728 , ( i __ 19 *,_!i 8 i 9 ^_^ + . 3028800 403200 P + 58060800 P 7257600 p r 7372800 , 1 _2477_ 2917 2 1264267 479001600 1916006400 P 121651200 p 30656102400^ 42137 &, 1 P + 68124672UO M 1061683200 -f &c., &c. B. CALCULATION OF FORMS OF DROPS. Also, we have 1 _d<f> p ds I 629 487 -_ p _ 645120 045120 81920 ,/_233_ 7 17539 271 11 ^ 10 V14515200 p 259200 p 77414400 p 4300800 P 88473600 p J + &c., &c. As before, it may be remarked that the coefficient of each power of .9 thus found is exact, and not merely approximate. We may also find these series for <f>, x and z in terms of s independently, in the following manner : Assume, as we evidently may do, - = 1 + c/ + c/ + c/ + c/ + c 10 .s 10 + &c., therefore < = I - - = s + - c/ + ^ c/ + ^ c/ + - c/ + -- c 10 s" + &c., since < and s vanish together. Hence we may find cos (b = i - s ( 1 1 JL V40320 ~ 360 2 + 36 2 30 4 15 7 i i _ _ 3028800 mlO 2 432 2 162 2 600 4 ~ 30 * 50 4 ~42 C6+ 21 1,11 1 &c., &c., and /I 1 \ . . / 1 1 2160 ~ 2 108 2 162 2 120 " 4 15 24 14 " G / 1_ _1_ _1_ 2 _ 1 1 _1 V399168UO 120960 2160 324 c * + 3600 4 90 ^ JL .A _J J_ JL J_ + 90 2 f4 + 50 4 168 Cs + 21 C A + 18 8 11 ^ 10 &c., &c. CALCULATION OF FORMS OF DROPS. 35 And therefore " C ,362880 3240 2 324 270 " ~ 1735 " 63 ( / __ 1_ 1 JL_ 2 1 J^ J^ V39916SOO 166320 2 + 4752 c * 1782 2 + 6600 4 330 550 *~ 462 + &c., &c., and 1 . /I 1 N , / 1 1 I =2 a 124- 12 C > +(720-36 C ^30 V40320 / 1 _ __ _ ^ V3628800 21600 Z 1080 2 f620 2 1200 4 150 -- 140 Ca / _ 1_ J_ 1 2 _ 1 8 1_ V479001600 1451520 2 25920 Ca 3888 C * + 43200 4 _ 2 _ 2 __ _ 1080 ^ 4 1080 2 C * + 600 4 2016 6 1 52 _L 1 + 216 C " ~ 132 C + &c., &c. Hence, we may find by division 4 8 41 2 52 2 16 172 4725 2 ~ 14175 * ~ 567 2 ~ 4725 4 " 4725 63 6 9 4 32 2 __ 532 3 __52_ J^ J^ 93555 C ~ 66825 ^ 467775 2 155925 4 3465 ^ 567 24 , 4_ J>96 8 1 ( jj, 1925 4 1485 C 10395 C2C(5 2 97 C>! 1 1 10 " &c., &c. 36 CALCULATION OF FORMS OF DROPS. Substitute these expressions in the equation p x and equate the coefficients of corresponding powers of s, and we shall find successively, 3 c = 8 /9 2 4- ft 3 6 5760 p 1920 p r 9216 p __p + __ /9 .___, Q ., o 8 8960 p 645120 p 645120 P r 81920 p : 7539 271 _ P + 10 P ,3 _ _ _ _ 14515200 259200 77414400 4300800 88473600 which agree with the coefficients of the several powers of s in the value of - which has been already found in another way. Also by the substitution of these coefficients in the expressions for x and z given above, we shall obtain the same values of x and z as those which have been before found. By means of the above series, we may determine the values of x, z and $ for given values of s in the neighbourhood of the origin. As in the case where </> was taken as the independent variable, in order that the terms of these series which involve higher powers of s may be insignificant, s must not exceed a certain limiting value which will, of course, depend on the value of 8. The larger the value of 8, the smaller will be this limiting value of s. In order to find the values of x, z and < for larger values of s, we must proceed step by step, as in the former case, the value of s being increased at each step by a given small quantity, suppose. The interval &> should be so chosen that the series above found will give sufficiently accurate values of x, z and <j> throughout several, say four or five such intervals. The process to be followed is exactly similar to that explained before, except that in this case there are three quantities x, z and < to be determined by inte gration instead of the two x and z. It is this circumstance only which makes it preferable to employ < as the independent variable in the case where this method is applicable, viz. when 8 is a positive quantity. The present method is equally applicable whether 8 be positive or negative. CALCULATION OF FORMS OF DROPS. 37 Now, suppose ...s_ 3 , s_,, s_ 3 , s_ 2 , s_ lt s to be a series of consecutive values of s, with the common difference to, and let ...$_ 5 , <_ 4 , $_ 3 , $_ 2 , t^, <f> be the correspond ing values of <f>, and the corresponding values of the coordinates, and P_5> p_ 4 , p_ 3 , p_ 2 , the corresponding radii of curvature. The equations to be integrated are <ty = l ds p dx dz -;- = sm <b, ds 1 sin <f> where - + - - == 2 + /S^. p a; Suppose that the values of <, x and 0, and consequently also the corresponding values of the radius of curvature p, are known for the successive values of s up to ._,, and we wish to find the value of each of these quantities for s = $ . In the first place, we may obtain an approximate value of in the following P manner. Tabulate the calculated values of - , and form their successive differences accord- P ing to the following scheme : P- 5 A-i- A 3 ?-4 A 2 P-3 -, P-* A 4 - P-4 A A 3 - p ~* P-s A- ^ T P- 2 A 4 ^ P- A A 3 - ^ P-a A 2 - P-i 38 CALCULATION OF FORMS OF DROPS. If o) is taken sufficiently small, the differences as we proceed to higher orders will rapidly diminish, and it will generally be easy by inspection of the two or three last calculated fourth differences to fix upon an approximate value of the fourth difference A 4 immediately succeeding. Po Call this approximate value A 4 ( J , so that the approximate is distinguished from the true value by being inclosed in a parenthesis, and by successive additions form f l \ \ i l \ i i l \ ^ A" - , A and 1 , thus W W W -i L. A 2 - Vo 1 Pi A8/l\ r-i A 3 - P-2 A _ Vpo/ Po Form the values of ... dx_ 5 , dx_4, dx_ 3 , dx_ 2 , dx_^ ... dz_,, dz_ 4 , dz_ 3 , dz_^, dz_ lt and of their successive differences, according to the following scheme P-5 w P-4 w cos <_ 5 = o cos <_ 4 = w cos <>_ = &) cos <i = dx , ft) COS _ = CALCULATION OF FORMS OF DROPS. 39 and co sin <>_. = dz_ 5 A 2 cfe dz (osm<j>_ t = dz_ 4 AVz_ 3 3 A 4 cfe_ 2 co sin <j)_ 3 = cfe_ 3 AV2 . AV* eo sin <^>_ 2 = dz_ z co sin <_j = tfe_j If -- and ^> were known, we might similarly form Po d(j) = - , dx Q = co cos (f) and dz Q = co sin ^ Po and the successive differences A 2 <r , A 3 dx , A 4 dx 0) &c., Afi 7 ^\ ft 7 /\ ft y t\7T* * _ ( ! , i^ Lv(Vj|| . * f .. , Utfuaj and then we should have, by what has been already proved, A rffc - A^ - Ay0 - &c., j7 - &c., - A^^ - A 2 c^ - 2j A 3 ^ - and when < , ^ and z were thus found, ought to agree with its assumed value, and the values of <, x and z should satisfy the equation which thus affords a verification of the value which was used for . Po Now, let (d(f> ) be an approximate value of dcf> , given by 1 Wo) = and let the successive differences found by employing (d(j) ) instead of d<f) be denoted by and suppose that (< ) is given by the equation 1 1 1 in - - A (d<p ) --- A 2 (Jc/> ) - 57 _ 1 w - I 40 CALCULATION OF FORMS OF DROPS. Also, let (dx ) and (dz Q ] be approximate values of dx Q and dz respectively, given by (dx ) = co cos (< ), (dz ) = ft> sin ((/> ), and let the successive differences found by employing (dx^) instead of dx , and (dz ) instead of dz , be denoted by A (dx.\ A ((fe ), A s (cfe ), A 4 (fro). &c., and A (dz^ t A 2 (<fe ), A 8 (ek ), A 4 (d* ), &c., respectively, and suppose that (a- ) and (z ) are given by the equations 0*o) - *-i = (^o) - g A W - i 2 Ai W - 24 A(^ ) - ^ A 4 (^ ) - &c., (^ ) - ^ = (^ ) - 1 A ((fo c ) - 1 A 2 (^ ) - i A 3 (^ ) - A 4 ^ - &c. Also let be found from the equation and suppose that this gives Vo where e is a very small known quantity. Then, if the true value of =( ) + the correction of the value of (dd> a ], and Po W therefore also that of the values of A (d(f> ), A 2 (cZ< ), A 3 (<f</> ), A 4 (d$ ), &c., will be wrj. Hence, if we stop at the terms which involve differences of the 4th order, we shall have ^ /A \ 251 9o ~ (9o) = 251 Wherefore cos <f> cos (^> ) 0)77 sin 251 and sin fa = sin (< ) + 0)77 cos Hence the correction to be applied to the values of (<fo- ), A (cLr ), A 2 (cy ), 4 (^ ), &c., will be 251 8 . ,, ~ 720 w ^ Sm ^ ) CALCULATION OF FORMS OF DROPS. 41 and the correction to be applied to the values of (dz Q }, A (dz u ), A 2 (cb ), A 3 (W2 ), A 4 (Wz ), &c., will be 251 2 72Q o>n cos Whence if, as before, we stop at the terms which involve differences of the 4th order, we shall have 251 sm , x /25i *o - (*) = ( Hence, since !-+= Po ^o and _Po we find 1 1 ) 251 sin (cfr ) ff 251 251 or 1 Ml sin (< ) J/251 y . /. \1_251 cos Po LPoJ (^o) 2 1\720 / j 720 (a n + P ^720 J ^7 cos (</> ) ; but 1 /1\ = 77, by supposition. V/ or which gives ?;. Whence the values of , </> , .-r and ^ become known. P 251 If terms involving differences of the 5th order be included, the coefficient 95 in the above expressions must be replaced everywhere by As in the former case, the following slight modification of the above process will be found convenient in practice. B. 6 42 CALCULATION OF FORMS OF DROPS. Suppose [ ] the assumed value of to be increased by 100 units of the last place W P of decimals employed, then while calculating the values of (d<j> ), (< ), (dx ), (ok ), (# ), (z ) and the consequent value of , note at the side of the work the changes which would be severally caused in each of these quantities by such an augmentation of [ ) \rO As before, it may be remarked that if we stop at the terms involving A 4 the 251 changes in (< ), (# ) and (z ) will be j times the changes in (d</> ), (dx ) and (dz ) 251 respectively, and that =^-r may be conveniently put under the form l 12 ^ 251 If we also include the terms involving A 5 , the coefficient ^ must be replaced 95 1 Or - Now suppose that an increase of 100 units in I ) causes a diminution of u, units W in , and that the excess of above [ 1 is X of the same units, then the cor- LPoJ LPoJ W rection to be applied to the assumed value [ ] will be W 100 X such units. wi LPoJ such units, 100 + and the correction to the value of - | will be L * X//. ~ 100 + yU, and the proportionate changes required in the values of (d</> ), (< ), (dx ), (dz u ), (#) and (z ) will be at once found. A numerical example of the method, when s is taken as the independent variable, is given hereafter. As before, we may, if it is found convenient, increase or diminish the interval between the successive values of s. It may be remarked, as before, that when, by means of the appropriate series, we have found the values of - - for a sufficient number of small values of s, we can P form the corresponding value of d<j>, and thence derive the corresponding values of <j> by integration, and again by means of these we can find the corresponding values of dx and dz, and thence derive by integration the corresponding values of x and z without employing the series for those quantities, unless we choose to do so as a means of verification. CALCULATION OF FORMS OF DROPS. 43 In the foregoing investigation, </> is supposed to be expressed in the circular measure, but in order to find cos $ and sin </> from the tables, < must be converted into degrees, minutes and seconds. This can be readily done by means of special tables for the purpose. Or we may, if we choose, express d(f> at once in seconds by multiplying by 20G2G4-8, the number of seconds in the unit of circular measure, and thence find the number of seconds in < by integration, without passing through the circular measure. Thus if $" denote the number of seconds in </>, and if 7 denote the number 20G2G4 S, we shall have w l j l a<p = - 7 as = - 7&>, where log 7 = 5 3144251. In conclusion, it may be worth while to say a few words in order to point out the distinction between the method of integration above explained and that which is commonly known under the name of " Integration by Quadratures." In this latter method, we have to find y from the equation where f(t) is a known function of t. If we regard q as the ordinate of any point of a curve corresponding to the abscissa t, then y will be the area included between the curve, the axis of t, the ordinate q, and some fixed ordinate. In this case the values of q can be found, a priori, for any given values of t, whereas in the more general case already treated of, where q is a function of y as well as of t, the unknown quantities y and q must be found simultaneously, and therefore we can only proceed step by step. As the simpler case is included in the more general one, we may, of course, still employ the same formula of integration that we have already obtained, but it will be more advantageous to use a slightly different one. If we denote the successive values of q by "?,,-2> ?-! In, &C., and if the corresponding values of y be denoted by y-*i y-i> #> &c -> and if, in a notation similar to that already employed, the successive differences of the quantities q be represented as in the following scheme : q *~ l " 3 A 5 A 7 A 9 Ag^, AV +! A 5 g )1+ , 3 AVy,, 44 * A 9 g n+ 62 44 CALCULATION OF FORMS OF DROPS. then, as is before proved, we shall have y n ~~ y n -\ = fydt between the limits t = t Q + (n 1) w and t = t a + nw I . 1 1 19 . 3 A5 863 g. - A j. - A 275 33953 2492 " 8628800 Q 3183 1036800 A If we transform this series into one which contains only such differences as ; in the above scheme, occur in the same horizontal lines as q n and Ay M , we shall find that the coefficients of the successive differences ultimately diminish much more rapidly than before. When the differences of higher orders than the 9th are neglected, it is readily shewn that the above series is equivalent to 191 60480 ~ 12 191 1209 60 7 q + 3 24 72U 2497 3628800 2497 A <, ) ^ 7257600 ^ B+ * ;C J Similarly, by repeated applications of this formula, we have y^ ~ Vn-, = < {^i - \ A?^ - ^ A j. + i A g. + A^, i+1 - 191 6 191 7 60480 ^ t+2 + 120960 ^ &c., &c. _ 3628800 1 + + 24 A ?" 191 A6 191 2497 _ _ A v/ -I __ A n -I __ \ 60480 y " !+4 ^ 120960 ^ m+4 3628800 Adding all these equations, and observing that A + A + &c. + Ar = &c. &c. &c. _. 7257600 A 11 ^ m+3 " ^ + 720 11 1440 2497 _ _ _ A n _ <VP I y " +5 " &c. - A we obtain n y m = fadt between the limits = t + ?mo and t = t + nca, 24 720 CALCULATION OF FORMS OF DROPS. 45 2497 497 ) + 3628800 (A ^ ~ A?<7 - ) ~ 7^7666 (A ^ " A ^ ~ &c j in the first line of which expression 9Wi + ? m+a + &c. + q n - (q n - q m ) may be replaced by (? + 2.) + <7 m+1 + q^ + &c. + g n _ r Also, by substituting for the differences of odd orders in the series for y n y_, viz. by putting A <7= ? -?-!> A flu-A a^-A fr, &c. &c., we obtain (2. + (7^) - 2l ( A ^-i + A ^) + 14-40 (A ^ +Z + A4 ^ +l) 191 2497 120960 - 7257600 - - ~ and similarly, by substituting for the differences of even orders in the series for y*-y n > viz - b y putting &c. &c., we obtain /-i 1 1 c\ t \^^ In I ^"^ /n4-ty I c* A 24 v j m+iy 24 When, by means of the method before explained, we have found a series of successive values of q, viz. <? m . ?> &c - 9-i. <?> together with the differences of odd orders which are immediately contiguous to the horizontal lines through q m and q n , we may advantageously employ the formula just obtained in verification of the value of y n -y m previously found. 46 CALCULATION OF FORMS OF DROPS. Weddle s approximate formula for the area of a curve which is divided into 6 portions by 7 given equidistant ordinates*, viz. has likewise been found to afford a convenient means of verification. NOTE. In the reference made at the top of p. 31 to Bertrand s paper, the page referred to should be 208 instead of 185, the latter being the page at which the paper begins. EXAMPLE OF THE METHOD, WHEN < IS TAKEN AS THE INDEPENDENT VARIABLE. Suppose that j3 = 6, and that the values of x and z, and also that of p, have been already calculated for values of < at intervals of 2| from to 32^, and that we wish to find the values of the same quantities for = 35. Here &> = the circular measure of 2-|, = 0-04363323, log co = 8-6398174. In the first place calculate a table giving the logarithms of w cos <, &> sin ^> and sin <f) for values of $> at intervals of 2|. Thus for = 35 the calculation will be &) 8-6398174 8-6398174 cos< 9-9133645 sin 97585913 8-5531819 8-3984087 The following is a portion of the table. TABLE A. lo w cos log (a) sin log (sin </>) 30 8-5773480 8-3387874 9-6989700 321 8-5658466 8-3700339 97302165 35 8-5531819 8-3984087 97585913 37| 8-5392841 8-4242645 9-7844471 40 8-5240714 8-4478849 9-8080675 * Boole s Finite Differences, p. 39. CALCULATION OF FORMS OF DROPS. 47 Next, collect in a table the values of log p for the successive values of < up to $ = 32, and find their differences to the 4th order, thus logp TABLE I. A A* A 3 A 4 25 30 2502 1869 1449 1049) - 633 420 - (400) 9-8858160 198034 9-8660126 - 1245 199279 9-8460847 + 624 -198655 9-8262192 + 2073 - 196582 9-8065610 (+ 3122) (- 193460) 35 (97872150) In order to find an approximate value of log p for < = 35, assume a value for the 4th difference immediately following those already found in the table, and by means of this form successively the approximate values of the 3rd, 2nd and 1st differences and of the log p for < = 35. These values are placed in parentheses to indicate that they are only approximate. In the above case, we have assumed the next 4th difference to be 4UO. Collect in a table the values of dx = pco cos < for the successive values of < up to = 32^, forming them by means of the known values of log p and the logarithms in the 1st column of Table A. Add to this table the approximate value of dx for <f> = 35, formed by means of the approximate value of logp just found, and find the differences of these quan tities to the 4th order, thus TABLI s II. 4- dx &dx A 8 efcc tfdx AUc 03099194 + 2802 -968 - 194464 + 2314 25 02904730 + 5116 -861 - 189348 + 1453 971 02715382 + 6569 - 585 - 182779 + 868 30 02532603 + 7437 (- 462) - 175342 (+ 406) 321 02357261 i (+ 7843) (- 167499) 35 (-02189762) 48 CALCULATION OF FORMS OF DROPS. For <f> = 35 the calculation will be log (p) 97872150 log (&) cos 0) 8-5531819 log 8-3403969 + 100 (das ) -02189762 + 50 3 The change of (dx) in units of the last decimal place, which would be produced by an increase in log (p) of 100 units in the last decimal, is placed at the side. Similarly, collect in a table the values of dz = pa sin < for the same values of 0, forming them by means of the logarithms in the 2nd column of Table A. Add to this table the approximate value of dz for < = 35, and find the differences, as before, to the 4th order, thus dz TABLE III. Mz tfdz tfd 22J -01283728 - 13145 + 79 + 70770 + 1416 25 -01354498 - 11729 68 + 59041 + 1348 27^ -01413539 - 10381 - 86 + 48660 + 1262 30 -01462199 - 9119 (- 136) + 39541 (+ 1126) 32J -01501740 (- 7993) (+ 31548) 35 (-01533288) For </> = 35 the calculation will be log(p) 97872150 log (w sin 0) 8-3984087 + 100 log 8-1856237 (dz) -01533288 + 35 3 the change of (dz) for an increase of 100 units in log (p) is placed at the side. Collect in two other tables the successive values of x and z which have been already computed, and form the differences of these quantities to the 4th or 5th orders, by which means any error of consequence that may have crept into the work will at once become apparent. CALCULATION OF FORMS OF DROPS. From the values of (dx), (dz) and their differences, and from the known values of x and z for < = 32, find the approximate values of x and z for = 35, thus For < = 32|, x -45780303 For <f> = 32|, z -12246288 < = 35, (da:) 02189762 $ = 35, (dz] "01533288 - i A (dx] 83749,5 50,3 _ 1 A (<fe) - 15774 35,3 - TU A 2 (cfo) - G53,6 2,5 _ ^ A s (efe) 666,1 1,8 - 2 A! W ~ 16,9 - ,2 _ J T A (cfe) - 46,9 - ,2 -T*&A 4 (rf*0_ + 12,2 3) 52,6 -7>&A 4 (cfe) 3,6 3)^9 </> = 35, (x) -48053156 17,5 ( z ] -13764424,8 12^3 6_ 6 ft (z) 82586549 74 In order to prevent an accumulation of small errors, the quantities involving A, A 2 , A 3 and A 4 are carried to one place of decimals beyond those which are ultimately retained. At the side are calculated the changes of (x) and (z), in units of the 8th place of decimals, which would be required if log (p) were increased by 100 units of the 7th place of decimals. These changes are found by multiplying the corresponding changes of (dx) and (dz) already found by 251 = !ii + !fi-.! 720 3 \ r 20 { 12 Next, from (x) and (z) find ^ and log [p] by the formula 1 _ _ f . sin <f> , . i p: = A + p (z) p-r- , p being here = b. Table (A), log sin 35 97585913 2 + 6 (z) = 2 8258655 7,4 log (a?) 9-6817219 1,5 subtract 11936290 -_4,_ log 00768694 -_1,5_ 1 R N. 1-1936290 - 4, H log i. 0-2127831 3,0 or log [p] 9-7872169 - 3,0 log (p) 97872150 Difference 19 The numbers placed at the side are the changes which would be caused by the increase in log (p) before mentioned. B. 7 50 CALCULATION OF FORMS OF DROPS. It should be remarked that in this last calculation the quantity 2 + 6 (z) is only carried to 7 places of decimals, whereas in the above calculation 6 (z) was given to 8 places. Consequently the change of 2 + 6 (z] or that of 6 (z) before found must be divided by 10, in order to reduce it to units of the 7th decimal. We see that the value thus found for log [p] exceeds the assumed value of log (p} by 19 units in the 7th decimal place. Hence the correction to be applied to log (p} will be in 100 19 x -^ = 18,4 such units, and the corrected value of log p for </> = 35 will be 9-7872168, which may now be added to the numbers in Table I. Similarly, the corrections to be applied to the values of (dx) and (dz) will be 50,3 x 184 = 9,3 and 35,3 x 184 = 6,5 respectively, in units of the 8th decimal, so that the corrected values will be etc = -02189771 and <fe= -01533294, which may be added to the numbers in Tables II. and III. The successive differences of log (p) will of course require the same correction as log(p) itself, and similarly for the differences of (dx} and (dz). Also, the corrections to be applied to the values of (x) and (z) will be 17,5 x -184 = 3,2 and 12,3 x "184 = 2,3 respectively, in units of the 8th decimal, so that the corrected values will be x = -48053159 and z = "13764427, which may be added to the Tables of the collected values of x and z respectively. The provisional values of log (p}, (dx) and (dz} and of their respective differences, which in the foregoing example have been inclosed in parentheses, may in the actual work be merely written in pencil, so that, when they have served their purpose, they may be easily effaced, and then replaced by the corrected values written in ink. The Volume V of the portion of the drop corresponding to </> = 35, is at once found by the formula 15 P CALCULATION OP FORMS OF DROPS. 51 thus, 1-6322367 P ^i 11936290 0-4386077 log 9-6420762 a; 2 log 9-3634438 TT log 0-4971499 9-5026699 ft log 07781513 log 8-7245186 V -05302963 In this as well as in the former part of the work of this example, b is supposed to be unity, so that in the general case the quantities above denoted by D T* P \> p, x, z and V will be replaced by j-, j, -= and ^ respectively. The following shews the application of the formula of correction 77 = , 251 l+ W> to this example. log x, 9-68172 251 2-39967 720 2-85733 9-54234 cos <f> 9-91336 (cr ) 2 9-36344 0-54992 N. 3-5475 6- 9-5475 log 0-97989 ( Po Y 9-57443 sin </> 975859 <u 8-63982 ff^ 9-54234 8-49507 031266 _! 1-031266 Hence ^=:___. nearly, which agrees very well with the result found by the other method. 72 52 CALCULATION OF FORMS OF DROPS. EXAMPLE OF THE METHOD, WHEN s IS TAKEN AS THE INDEPENDENT VARIABLE. Suppose that it is required to calculate the theoretical form of a pendent drop of fluid, where /3 = 5. As in the former example, we will suppose that 6 = 1. First, putting j3 = 5 in the general formula for - , we obtain for points near the origin - = 1 - 0-1875 s 2 + 0-01692,7083 s 4 - 0-00248,2096 s 6 + 0-00028,3075 s 8 - 0-00003,3537 s 10 + &c., which series is sufficient to give - to 9 or 10 places of decimals, provided s do not exceed 4. The value of - is the same for corresponding positive and negative values of s, so that if we put s 0, s = + 01, s = 2, s = + 3 and s = + 4 in succession, we may obtain s P o i- 0-1 0-99812,67 0-2 0-99252,69 0-3 0-98326,03 + 0-4 0-97042,33 Also tj>=l-ds = 8- 0-0625 s 3 + 00338,54166 s 5 - 0-00035,4585 s 1 + 0-00003,1453 s 9 - 00000,3049 s 11 + &c. Similarly, putting /3 = 0*5 in the series for x and z respectively, we obtain x = s- 0-1666 s 5 + 0-02083,3 s 5 - 00244,9157 s 7 + 00029,4734 s 9 - 0-00003,5199 s n + &c., z = 0-5 s 2 - 0-05729,16 s 4 + 0-00716,14583 s 6 - 0-00085,03747 s 8 + 0-00010,17165 s 10 - 0-00001,21847 s 12 + &c. From which we may find s (f) (in circ. measure). < (in deg. &c.) x z 0-0 0" 0-0 0-0 0-1 0-09993,753 5 43 33 596 + 0*09983,354 00499,428 0-2 + 019950,108 11 25 50 051 + 19867,330 01990,879 0-3 0-29832,065 17 5 33 051 G 29555,009 0-04454,110 0-4 +0-39603,409 224127*896 38954 ; 273 0-07856,212 CALCULATION OF FORMS OF DROPS. 53 Now, let us suppose that the values of p, <, a; and z have been already calcu lated for s = 0"l, s = 2 and s = 3, and that we wish to find the values of the same quantities for s = 4 by the foregoing method of integration. Here we have <a = O l. From the given values of -- we may find the corresponding values of (< = -, and their successive differences, as shewn in the following Table : s (> - 0-3 0-09832,603 92,666 -0-2 0-09925,269 -36,668 55,998 -0,597 -01 0-09981,267 -37,265 +0,396 18,733 - 0,201 +,006 0-0 0-1 -37,466 +0,402 - 18,733 +0,201 -,006 0-1 0-09981,267 -37,265 +0,396 - 55 ; 998 +0,597 (-,018) 0-2 0-09925,269 -36,668 (+0,378) - 92,666 (+0,975) 0-3 0-09832,603 (-35,693) (-128,359) 0-4 (0-09704,244) If we supply another line of differences by supposing the 6th diffei ence to be constant, we shall obtain the quantities included in parentheses in the above Table. and the corresponding assumed value of .- .- for s = 4 will be 97042,44. From the values of (J<j>) and its differences, and the known value of < for s = 3. find the approximate value of <f) for s = 4, thus For s = 0-3, <f> -29832,065 5 = 0-4, d<f> -09704,244 - A d<f> + 64,179,5 100 in units of 8th decimal place - ^A"efc - ,040,6 - 7 V 9 oAU - ,010 -A 5 + ,000,4 (<) -39603,413 _33 or 22 41 27" 903 0" 068 The changes placed at the side correspond to an increase in ^- of 100 units in the 7th decimal place. As the interval &> is rather large, we have taken into account the terms in A 9 . 54 CALCULATION OP FORMS OF DROPS. With this value of (<) we calculate the corresponding values of (dx) and (dz), thus logcos(<) 9-9650125 -0,6 log sin (<) 9 5863199 +3,4 log a) 81] log &) 9 8-9650125 (dx) -09225980 -1,2 8-5863199 (dz) -03857624 +3,0. The small quantities at the side are the changes of the quantities opposite to which they stand, in units of the last decimals respectively employed, which would be caused by an increase of 0" 06S in (</>), or by an increase of 100 units in the 7th decimal place of (P) With these values of (dx) and (dz) and the previously calculated values of dx and dz for s=01, s = 2 and s = 3, we may form the following Tables: s dx Acfo tfdx A efo Sdx tfdx -0-3 09558,313 243,344 -0-2 09801,657 - 94,895 148,449 - 3,660 -01 09950,106 98,555 + 2,427 49,894 - 1,233 + 39 o- 1 - 99,788 + 2,466 49,894 + 1,233 -39 o-i 09950,106 98,555 + 2,427 - 148,449 + 3,660 (-181) 0-2 09801,657 - 94,895 ( + 2,246) - 243,344 (+ 5,906) 0-3 09558,313 (- 88,989) (- 332,333) 0-4 (-09225,980) s dz 0-3 --02939,154 0-2 - 01981,803 01 - -00997,712 0- -0 01 + -00997,712 0-2 -01981,803 0-3 -02939,154 0-4 (-03857,624) tfdz 957,351 984,091 26,740 -13,119 13,621 -502 997,712 - 13,621 502 997,712 - 13,621 502 -13,621 + 502 984,091 -13,119 (476) - 26,740 (+978) 957,351 (- 12,141) (-38,881) (918,470) CALCULATION OF FORMS OF DROPS. 55 Hence we find x and z for s = 4, thus For s = 3, x= -29555009 For s = 3, .z = -04454110 For , s = 0-4, efo = -09225980 For s = 4, dz -03857624 fAcfo 166166,5 - ^ ^ dz - 459235 T ^AUc 7415,7 ^ i. 2 dz 3240,1 - ^ A efce - 246,1 - ^ i s dz 505,9 - 7 y>_A 4 cb - 59,3 ~ T|^ *dz 25,8 -TfoA 6 ^ 3,4 -^ ?dz 8,9 0) -38954269 -0,4 0) 2) -07856210 -#s -03928105 0,5 The changes in (x) and (z) are found by multiplying those in (dx) and (dz) respectively, by -^ or 1 nearly. Next, with these values of (</>), (#) and (z), find p^ by the formula -, 8 bein here = -0 5. logsin(() 9-5863199 +3,4 2 + /3 (5) =1-9607189,5 -0,05 logO) 9-5905551 0,0 subtract "9902955 7,7 9 9957648 -M * o^^ -w N. -9902955 7,7 [p] but ri- =0-9704244 .-. difference = 9,5 It will be seen that in the above we have expressed the change of 2 + /3 (z) in units of the 7th decimal place instead of the 8th decimal as before, so that the number found before has been divided by 10. The value thus found for r^ is less than the assumed value of -r-r by 9,5 [p] (p) units in the 7th decimal place. Hence the correction to be applied to T-T will be \P) 100 9,o n n -^, = - 9 such units, 107,7o and the corrected value of : - will be 9704235. P Similarly, the correction to be applied to the value of ($) will be - -09 x " 068 = - "-006, so that the corrected value of (/> will be 22 41 27" 897. 56 CALCULATION OF FOBMS OF DROPS. Also the corrections to be applied to the values of (a?) and (z) respectively, will be . - -09 x - 0,4 and - 09 x 1,0 in units of the 8th decimal place, both of which are insensible. These results agree very well with the more accurate ones found before by the use of series. The Volume V is found by the formula TT /sin 1 F _ thus, 1 p log loff 0-9902954 0-9704235 _ _ 0-0198719 8-2982394 9-1811102 IT 0-4971499 7-9764995 9-6989700 log log V 8-2775295 -0189465 Application of the formula of correction 95 sin 95 to this example. lo 9-51833 8-51833 cos(< ) 9-96501 - i og 9-99570 8-51833 8-51409 2 8-48334 9-59055 8-89279 078125 002134 000502 080761 7-02818 2 log 0-30103 7-32921 Hence = _ _ 1-081 w) , log cos(c/> ) 9-96501 ~/8 9*69897 6-70064 which very nearly agrees with the result found before by the other method. TABLES FOR FACILITATING THE CALCULATION OF J.*y-i- O . K 726 A 160 A - B. 58 TABLES. Table shewing the value of A 4 which corresponds to each unit in the value of -L _/ A A - - 720 37-8947 A 4 . O I 2 3 456 7 8 9 o I 2 3 o 379 758 H37 38 76 114 4i7 455 493 796 834 872 1175 1213 1251 152 189 227 531 568 606 909 947 985 1288 1326 1364 265 303 341 644 682 720 1023 1061 1099 1402 1440 1478 o i 2 3 4 5 6 1516 1895 2274 : 554 1592 1629 : 933 J 97i 2008 2312 2349 2387 1667 1705 1743 2046 2084 2122 2425 2463 2501 1781 1819 1857 2160 2198 2236 2 539 2 577 2615 4 5 6 7 8 9 2653 33 2 34ii 2691 2728 2766 3069 3107 3145 3448 3486 3524 2804 2842 2880 3183 3221 3259 3562 3600 3638 2918 2956 2994 3 2 97 3335 3373 3676 3714 3752 7 8 9 10 1 1 12 3789 4168 4547 3827 3865 3903 4206 4244 4282 45 8 5 4623 4661 3941 3979 4oi7 4320 4358 439 6 4699 4737 4775 4055 4093 4131 4434 4472 459 4813 4851 4888 10 i.i 12 J 3 14 15 4926 535 5684 4964 5002 5040 5343 538i 54i9 5722 5760 5798 5078 5116 5154 5457 5495 5533 5836 5874 5912 5192 5229 5267 13 5571 5608 5646 14 5949 5987 6025 15 16 i7 18 6063 6442 6821 6101 6139 6177 6480 6518 6556 6859 6897 6935 6215 6253 6291 6594 6632 6669 6973 7011 7048 6328 6366 6404 6707 6745 6783 7086 7124 7162 16 17 18 19 20 21 7200 7579 7958 7238 7276 73M 7617 7655 7693 7996 8034 8072 7352 7389 7427 7731 7768 7806 8109 8147 8185 7465 753 754i 7844 7882 7920 8223 8261 8299 19 20 21 22 2 3 24 8337 8716 9095 8375 8413 8451 8754 8792 8829 9133 9171 9208 8488 8526 8564 8867 8905 8943 9246 9284 9322 8602 8640 8678 8981 9019 9057 9360 9398 9436 22 2 3 24 25 26 27 9474 9853 10232 95 12 9549 9587 9891 9928 9966 10269 i37 I0 345 9625 9663 9701 10004 10042 10080 10383 10421 10459 9739 9777 9815 10118 10156 10194 10497 10535 I0 573 25 26 27 19 The units of ^-^ A 4 are placed at the top of the Table, and the tens at the side. TABLES. 59 Table shewing the value of A 5 which corresponds to each unit in the value of A 5 = A 5 160 53-3333 O I 2 3 456 7 8 9 o I 2 3 o 533 1067 1600 53 107 160 587 640 693 II2O U73 1227 1653 1707 1760 213 267 320 747 800 853 1280 1333 1387 1813 1867 1920 373 427 480 907 960 1013 1440 1493 1547 1973 2027 2080 o i 2 3 4 5 6 2133 2667 3200 2187 224O 2293 2720 2773 2827 3 2 53 337 33 6 2347 2400 2453 2880 2933 2987 3413 3467 35 20 2507 2560 2613 3040 3093 3147 3573 3 62 7 3680 4 5 6 7 8 9 3733 4267 4800 3787 3840 3893 4320 4373 4427 4853 4907 4960 3947 4ooo 4053 448o 4533 4587 5013 5067 5120 4107 4160 4213 4640 4693 4747 5*73 5227 5280 7 8 9 to 1 1 12 5333 5867 6400 5387 5440 5493 5920 5973 6027 6 453 6 57 6 5 6 5547 5600 5653 6080 6133 6187 6613 6667 6720 577 5/6o 5813 6240 6293 6347 6773 6827 6880 10 1 1 12 13 14 15 6933 7467 8000 6987 7040 7093 7520 7573 7627 8053 8107 8160 7147 7200 7253 7680 7733 7787 8213 8267 8320 7307 7360 7413 7840 7893 7947 8373 8427 8480 J 3 14 15 16 17 18 8533 9067 9600 8587 8640 8693 9120 9173 9227 9 6 53 977 976o 8747 8800 8853 9280 9333 9387 9813 9.867 9920 8907 8960 9013 9440 9493 9547 9973 10027 10080 16 i7 18 The units of r A A 5 are placed at the top of the Table, and the tens at the side. 60 TABLES. Table shewing the value of A 9 which corresponds to each unit in the value of 863 60480 A = 70-0811 A a . O I 2 3 456 7 8 9 o I 2 3 701 1402 2102 7O 140 210 771 841 911 1472 1542 1612 2173 2243 2313 280 350 420 981 1051 II2I 1682 1752 l822 2383 2453 2523 491 561 631 1191 1261 1332 1892 1962 2032 2593 2663 2733 o i 2 3 4 5 6 2803 3S4 4205 2873 2943 3013 3574 3 6 44 37H 4275 4345 4415 3084 3154 3224 3784 3854 3925 4485 4555 4625 3294 33 6 4 3434 3995 4065 4135 4695 4766 4836 4 5 6 7 8 9 4906 5606 6307 4976 5046 5116 5677 5747 5817 6377 6447 6518 5186 5256 5326 5887 5957 6027 6588 6658 6728 5396 5466 5536 6097 6167 6237 6798 6868 6938 7 8 9 10 ii 12 7008 7709 8410 7078 7148 7218 7779 7849 79 J 9 8480 8550 8620 7288 7359 7429 7989 8059 8129 8690 8760 8830 7499 75 6 9 7639 8199 8270 8340 8900 8970 9040 10 ii 12 r 3 14 IS 9III 98ll I05I2 9181 9251 9321 9881 9952 10022 10582 10652 IO722 9391 9461 9531 10092 10162 10232 10792 10863 10933 9601 9671 9741 10302 10372 10442 11003 n73 IIJ 43 13 14 15 The units of 863 are placed at the top of the Table, and the tens at the side. TABLES. 61 Table shewing the value of A T which corresponds to each unit in the value of 275 , = _1_ 24192 87-9709 O I 2 3 456 7 8 9 o I 2 3 O 880 1759 2639 88 176 264 968 1056 1144 1847 1935 2023 2727 2815 2903 35 2 440 528 1232 1320 1408 2III 2199 2287 2991 3079 3167 616 704 792 1496 1583 1671 2375 2463 2551 3255 3343 343 1 o i 2 3 4 5 6 35 T 9 4399 5278 3607 3695 3783 4487 4574 4662 5366 5454 5542 3871 3959 4047 4750 4838 4926 5630 5718 5806 4135 4223 4311 5014 5102 5^0 5894 5982 6070 4 5 6 7 8 9 6158 7038 7917 6246 6334 6422 7126 7214 7302 8005 8093 8181 6510 6598 6686 7390 7478 7565 8269 8357 8445 6774 6862 6950 7653 774i 7829 8533 8621 8709 7 8 9 10 ii 12 8797 9677 10557 8885 8973 9061 9765 9853 994i 10644 I0 732 10820 9 X 49 9 2 37 93 2 5 10029 10117 10205 10908 10996 11084 9413 9501 9589 10293 10381 10469 11172 11260 11348 10 1 1 12 275 The units of A 7 are placed at the top of the Table, and the tens at the side. *J T -i I/ *J 62 TABLES. Table shewing the value of A 8 which corresponds to each unit in the value of 33953 1 3628800 106-87715 A 8 . O I 2 3 456 7 9 o I 2 3 o 1069 2138 3206 107 214 321 1176 1283 1389 2244 2351 2458 3313 3420 3527 428 534 641 1496 1603 1710 2565 2672 2779 3 6 34 374i 3848 748 855 962 1817 1924 2031 2886 2993 3099 3954 4061 4168 o i 2 3 4 5 6 4275 5344 6413 4382 4489 4596 5451 5558 5664 6520 6626 6733 4703 4809 4916 577i 5878 59 8 5 6840 6947 7054 523 5 J 3 5237 6092 6199 6306 7161 7268 7375 4 5 6 7 8 9 7481 8550 9619 7588 7695 7802 8657 8764 8871 9726 9833 9940 7909 8016 8123 8978 9 8 5 9 J 9i 10046 10153 10260 8230 8336 8443 9298 9405 9512 10367 10474 10581 7 8 9 33953 The units of ~ . A are placed at the top of the Table, and the tens at the side. Table shewing the value of A 9 which corresponds to each unit in the value of 8183 1 1036800 1267017 A 9 . O I 2 3 456 7 8 9 o I 2 3 o 1267 2534 3801 127 253 380 1394 1520 1647 2661 2787 2914 3928 4054 4181 507 634 760 1774 1901 2027 3041 3168 3294 43 8 4435 45 6 * 887 1014 1140 2154 2281 2407 3421 3548 3674 4688 4815 4941 o i 2 3 4 5 6 5068 6335 7602 5195 5321 5448 6462 6588 6715 7729 7856 7982 5575 572 5828 6842 6969 7095 8109 8236 8362 5955 6o8 2 6208 7222 7349 7475 8489 8616 8742 4 5 6 7 8 9 8869 10136 11403 8996 9123 9249 10263 10390 10516 11530 11657 11783 9376 953 9 62 9 10643 10770 10896 11910 12037 12163 9756 9883 10009 11023 11150 11276 12290 12417 12543 7 8 9 8183 X I X X The units of -A 9 are placed at the top of the Table, and the tens at the side. CHAPTER IV. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. (C Z THE coordinates j and j for the curves represented by Laplace s differential equation were calculated by the method of Professor Adams for values of <, 5, 10", 15 175, 180, and for values of & 1, , f, i; | , 1 ; H, 2, 2; 3, 4, 5, 6, 7, 8 ; 10, 12, 14, 16; 20, 24, 28, 32; 40, 48, 56, 64, 72, 80, 88, 96 and 100. For /3=1 the calculations were made by Professor Adams, for ft = 10 by Professor W. G. Adams, and for the values of /?, |, , 3, 6, 16 and 32 by myself. The calculations for the remaining positive values of fi were made by Dr C. Powalky, who was recommended for the work by the late Professor Encke. T* Z Afterwards the values of - and corresponding to 6 = 5, and for the suc- b b cessive values of ft, 0, 8, 16, 24, 32, 40... 88 and 96 were arranged in order and differenced. Then the values of .- and -,- corresponding to the same value of <4, and b o to the values of (3, 3G, 44... 92 were found from the above by interpolation, arranged in order with the values of the same quantities for the values of {3, 0, 4, 8, &c. T 2 already calculated, and the whole differenced. Next the values of r and ~ corre sponding to < = 5 and to the values of /3, 18, 22, 26, 30, 34, 38, 42, &c. were found from the above by interpolation, arranged in order with the values of the same quantities for values of /3, 0, 2, 4, 6, 8, 10, &c. already found, and the whole IT differenced. And in the same manner the values of -j and j corresponding to $ = 5" were found by interpolation for the values of /?, 9, 11, 13, 15, 17, 19, &c., arranged in order with the values of the same quantities already found for the remaining integral values of (3 up to 100, and the whole differenced. Further, the same process was gone through for values of <f>, 10, 15, 20.... 175 and 180. Finally the results were arranged as they have been given in Table II., with </> for their argument, and differenced to test the accuracy of the work. 64 COMPARISON OF CALCULATED AND MEASURED FORMS OJF DROPS. Y Values of j- z were calculated by the formula already given (p. 30) for the same T* Z values of /3 for which j and j were calculated, and the results have been given in Table III. Values of | and | corresponding to values of <f>, 15, 30, 45, 60, 90, 120, 135, 140, 145 and 150 were taken from Table II., and, the intermediate values having been supplied by interpolation, the results have been given in Table V. This Table is useful in calculating the theoretical forms of drops corresponding to given values of /3. x z V Other Tables were formed by interpolation of values of r , and inter- 6 6 6 mediate to those given in the above-mentioned Tables, but they are at once too extensive and yet too incomplete for publication in their present form. In this way X 2 Tables of values of j , j and p- were formed corresponding to values of /S, O O, 01, 0-2, 0-3 49-8, 49-9, 50 0, for values of tf>, 135, 136, 137, 138 153, 154", 155, which were extremely useful, as will be explained hereafter, in deducing the capillary constants from the measured forms of drops of mercury. But these Tables would have been still more convenient if they had been formed so as to five CC ^ V^ CO 2 log - , log and log p instead of ^ , ^ and p . /V f* The values of y and j vary so rapidly for low values of /3 and high values of </> that they could be obtained by interpolation correct only to four places of decimals from /3 = to ft = T9. Beyond that they were calculated to five places V of decimals, while the values of ^ were calculated, unfortunately, to only four places of decimals throughout. The rectangular coordinates of points in the outlines of drops of mercury were measured by the help of a microscope moveable on vertical and horizontal slides by micrometer screws. The microscope was focused by motion along a third slide parallel to the line of collimation. These three slides were parallel to three rectangular axes, two of which were horizontal. There was found to be a difficulty in arranging the cross lines of the microscope, so as to obtain a correct outline of the drop of fluid, because it would be difficult to judge when the crossing of ihe micrometer lines was exactly on the contour of the drop. Much labour was expended on the construction of a position micrometer, where the intersection of the middle of one micrometer line with a side of the other line was to be the centre, about which they turned. At each observation the micrometer COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 65 lines were to be turned so that the above-mentioned side of the micrometer line became a tangent to the contour of the drop, while the other line was a normal at that point. But this could not be considered a satisfactory arrangement, because there would be some error of excentricity, and the microscope would be liable to be slightly disturbed by the application of the force necessary to bring the cross lines into their proper position. This contrivance was therefore abandoned in favour of a more simple arrangement suggested to me by the late Professor W. H. Miller. This was to use two pairs of parallel and equidistant spider lines, one pair being horizontal and the other pair vertical, so as to form by their intersection a small square in the centre of the field of view (Fig. 1). This arrangement appears to be quite satisfactory, for it is easy to judge when a small arc of the contour of a drop of fluid passes through the middle of this small central square. The screws used to measure the vertical and horizontal coordinates of points in the contour of a drop of fluid were originally formed of one piece of metal. Great care was taken to obtain a screw of uniform pitch throughout, of about 53 turns to the inch. When however the screws were mounted, it was found that although each one was tolerably uniform, the two screws differed sensibly in their pitch. By the use of a micrometer ruled to the one-hundredth of an inch, the exact rate of both screws at every point was determined. In this way tables of the value of every turn were calculated for both the vertical and horizontal screws. In the following measure ments of the forms of five drops of mercury made in 1863, the original readings of the screws are given as they were entered in the observing book, as well as their values in inches obtained by the help of the above-mentioned Tables. The coordinates of numerous points in the contours of these five drops of mercury, which vary considerably in size, were measured, because it was desired to find whether the theoretical forms would agree satisfactorily with the true forms of drops of mercury. In Fig. 2, the theoretical forms of these drops are given on a large scale, and an attempt has been made to indicate by a cross the position of some of the points measured. UHI7BRSIIT B. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. No. I Original readings in turns of the screws Readings converted into inches The same when the origin of coordinates is at the vertex Calculated z" *," A- * z *. *; z *1 *, <t> z X inch inch inch inch inch inch inch in 13-282 23-416 30-288 : 0-2491 0-4410 575 0-0925 0-0650 0-0645 I44 48 0-0925 O O( 13-300 23-398 30-340 1 2495 4406 5715 0921 0654 6 55 T irV -n 13-400 13-500 23-270 23-170 30^40 3 54o 25 J 3 2532 4382 43 6 3 5733 5752 0903 0884 0678 0697 0673 0692 140 o 135 "09 1 1 "0892 "O o 13-600 23-072 30-625 255 1 "4345 5768 0865 07IS 0708 13-800 22-970 30-772 2588 4326 5796 0828 734 0736 I20 /"jQ *y "~> 14*000 22-879 30-876 2626 4308 5816 0790 0752 0756 wo o 14-200 22-797 30-950 2663 4293 583 753 0767 0770 14-400 22735 31-003 2701 4281 5840 0715 0779 0780 14-600 22*680 3 I<0 33 2738 4271 5845 0678 0789 0785 oo-o r\C\ *7 ^ 14-960 22-672 31-064 2806 4269 5851 0610 0791 0791 VJU \J OO2j O 15-200 22-687 3i 34 2851 4272 5845 0565 0788 0785 15-400 22-712 3 993 2888 4277 5837 0528 0783 0777 15-600 22*762 30-952 2926 4286 583 "0490 0774 0770 15-800 22-840 30-890 2963 4301 5818 0453 7S9 0758 16*000 22-925 30-826 3001 4317 5806 0415 743 0746 l6 2OO 23-010 30-720 339 4333 5786 0377 0727 0726 fin-n *C\ ^ *7 (~\ 16-400 23-128 30-590 3076 435 6 5762 0340 0704 0702 UU \J j7 "O i6 6oo 23-250 3 447 3 TI 4 4378 5735 0302 0682 0675 16-800 23-418 30-274 SJS 1 4410 5702 0265 0650 0642 A r"-o 17-000 23-620 30-090 3189 4448 5667 0227 0612 0607 45 0230 "O 17-200 23-860 29-865 3226 4493 5625 0190 0567 5 6 5 -?o-n 17*600 24 437 29-278 33 01 4602 5514 0115 0458 454 3 "OI 2O "O iS ooo 2 5 39 28-316 3376 4782 5333 0040 0278 0273 T C-0 18-214 *#* #** 3416 5060 5060 oooo oooo "OOOO I 5 33 "O Weight 4-57 grs. {3 = 2-334 a = 119-6 o = U4-48 b = 0-09878 in. Temp. 40 F. Error in calculation of V= + O OOO 035 cubic inch. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 67 No. II Original readings in turns of the screws Readings converted into inches The same when the origin of coordinates is at the vertex Calculated " < *," 4 < * Z X, *. (rf) z X inch inch inch inch inch inch inch inch 13-023 24-710 34-330 0*2443 0*4654 0*6467 0*1054 0*0907 0*0906 I -j- " 2 o 0*1054 0-09065 13-100 24*603 34-464 2457 4634 6492 1040 0927 0931 145; 1044 _ o 0921 13-200 24*489 34-596 2476 4612 6SI7 1021 0949 -0956 140^0 IO2O 943 13-300 24-356 34-678 2495 4587 -6532 IOO2 0974 0971 I 35 IOO9 0963 13-400 24*260 34-762 2 5 X 3 4569 6548 0984 0992 0987 13*600 13*800 24-146 24-040 34-904 35 010 2551 2588 "4547 4527 6575 6595 0946 0909 1014 1034 "1014 1034 I20*0 0938 1018 14-000 23-975 35*086 2626 45IS 6609 0871 1046 1048 14*400 23-867 35-I78 2701 4495 6626 0796 1066 1065 I4-73 2 23-848 35-203 2763 4491 6631 734 1070 1070 90*o -734 1070 15*000 23-866 35 -I 9 2 2813 "4495 6629 0684 1066 1068 15-400 23-950 35-105 2888 45 10 "6613 0609 1051 1052 15-800 l6 20O 24-072 24-270 34-989 34*782 2963 3039 "4533 4571 6591 -534 0458 1028 0990 1030 0991 6o f *o 0463 993 i6 6oo 24-534 34-5 15 3114 4620 6501 0383 0941 0940 -,8-78 17-000 24-906 34*162 3189 4691 6435 0308 0871 0874 45 " 3 X 5 Oo i o 17-400 17-800 25-361 33-078 3264 3339 4776 4893 6345 6230 0233 OI 5 8 0785 0668 0784 0669 3o o 0170 0687 18-000 26-368 32*688 3376 4966 6157 *OI2I 0595 0596 l8 2OO 26-867 32*165 3414 5060 6058 0083 0501 0497 o 18*400 27 54o 31*478 345 1 5187 59 2 9 0046 374 0368 X 5 " -0051 "394 18*600 28*637 30-348 3489 "5394 "57J6 *ooo8 0167 0155 18*642 -X-** *#* "3497 556i 556! *oooo *oooo *oooo Weight 9-523 grs. = 6-44 a=12G-2 u=148*28 6 = 0*1597Gin. Temp. 37 F. Error in calculation of V= +0*000 044 cubic inch. 92 68 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. No. Ill Original readings in turns of the screws Readings converted into inches The same when the origin of coordinates is at the vertex Calculated z" x" *." Z < X 3 Z *, *> <t> z X inch inch inch inch inch inch inch inch 12-570 22-563 34*683 0-2358 0-4249 0-6533 O II27 0-1151 0-II33 I4i 0> 7i o l 127 0*1 142 12-900 13-000 13-400 22.282 22*232 22*024 35 5 35 112 35-326 2420 2438 4196 4187 -4147 "6602 ! 6614 6654 1065 1047 0972 1204 "1213 -I2 53 I2O2 1214 -1254 i4o oo i 3 5-oo I2O 0p OO 1121 I IOI -IO26 1150 1171 1228 14-300 21-863 i -2682 4117 6683 0803 1283 1283 90 n> oo O8l 3 1283 i6 - ooo 17-000 iS ooo 18-580 22.418 23-358 25^225 ##* 34-9 28 32-132 3001 3189 3376 3485 4222 4399 475 1 -5400 6579 "6401 6052 5400 0484 0296 "0109 OOOO 1178 IOOI 0649 oooo 1179 "IOOI 0652 oooo 6o-oo 4 C i 5 -oo 0527 0367 O2O6 0065 1202 1078 0865 0516 Weight 14-725 grs. 0=11-0 a = 118-2 w=141-71 6-0-21572in. Temp. 39 F. Error in calculation of V= + O OOO 057 cubic inch. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. No. IV Original readings in turns of the screws Readings converted into inches The same when the origin of coordinates is at the vertex Calculated z *i" *." z *, < Z x l x * <A z X inch inch inch inch inch inch inch inch 12-985 19-918 33-865 0-2435 0-3750 0*6379 0*1168 0*1311 0-1318 i4o*o 0-1168 0-I3I5 13*000 13-100 19-900 19-812 33-9i8 34-010 2438 2457 3747 3730 6389 6406 1165 1146 1314 -I 33 I 1328 1345 i 3 5-oo 1148 1337 13-200 19725 34-093 "2476 3714 6422 1127 1347 1361 13-300 19*642 34-167 2495 3698 6436 1108 -1363 -1375 13-400 i3 5 !9"553 19-500 34 23o 34-30 25*3 "2532 3681 3670 6448 6461 1090 1071 1380 -I 39 I 1387 "1400 I2O*OO 1072 " T 394 13-600 19-440 34-362 2551 3660 6472 1052 1401 1411 13-800 19-360 34 4i2 2588 3645 6482 1015 1416 1421 14*000 19-294 34-486 2626 "3633 6496 0977 1428 1435 14-200 19-243 34 532 2663 3 6 23 6505 0940 1438 1444 14-400 i9 -I 95 34-548 2701 3614 6508 0902 1447 1447 14*600 14*800 19-182 19-200 34 5 6 34 5 6 2738 2776 3611 36l5 6510 6510 0865 0827 145 1446 1449 1449 9O oo 0856 145 15*000 19*226 34-545 2813 3620 6507 0790 1441 1446 15-200 19*277 34 52o 2851 3629 6502 0752 "1432 1441 15-400 19-307 34-45 6 2888 ^3635 6490 07*5 "1426 1429 15-600 i9"353 34-388 2926 3644 6477 0677 1417 1416 15-800 1 9 445 34-322 "2963 3661 6465 0640 "1400 1404 i6 ooo I9-539 34-228 3001 3679 6447 0602 1382 1386 6o oo *or 6 7 T l67 l6 2OO 19*652 34-107 "3039 3700 6424 0564 1361 1363 U U / 1 J U / 16-400 19*760 33-975 3076 3721 -6399 0527 1340 1338 i6 6oo 19*913 33-832 3 ll 4 3749 6373 0489 1312 1312 16*800 20*050 33-687 3151 3775 6345 0452 1286 1284 17-000 17*200 20*240 20*442 33H96 33-282 3189 3226 3811 3 8 49 6309 6269 0414 0377 1250 "1212 1248 1208 45" "0402 1239 17*400 20*653 33-036 3264 3889 6223 0339 1172 1162 17*600 20*902 32-792 "33 01 3936 6177 0302 -1125 1116 17*800 18*000 21*204 21-548 3 2 5 l6 32-178 3339 3376 3993 -4058 6125 6061 0264 0227 1068 -1003 1064 1000 3o-oo 0234 "1016 18*200 21-927 31-786 3414 4129 5987 0189 0932 0926 18*400 22*342 31-320 "345 1 4208 5899 0152 0853 0838 18*600 22-880 30-816 3489 4309 5804 0114 0752 0743 18-700 18*800 23-170 23"5 J 7 30 494 3 I 34 3507 "3526 4363 4429 5744 5676 "0096 0077 0698 0632 0683 0615 i5 n *oo 0078 0631 18*900 23"95o 29*762 3545 45 I0 5606 0058 05 5 1 545 19-000 24-436 29-205 3564 4602 5501 0039 0459 "0440 19-100 25" l6 5 28-522 3582 4739 5372 *OO2I 0322 0311 19-210 #** **# 3603 5061 5061 oooo oooo "OOOO 0*00 OOOO oooo Weight 19 77 ffrs. 13=17-5 o=116-9 w=140-00 6 = 0-27358 in. Temp. 38" F. Error in calculation of V + 0*000 012 cubic inch. 70 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. No. V Original readings in turns of the screws Readings converted into inches The same when the origin of coordinates is at the vertex Calculated jt - r / x" z x i x a 2 *1 X 2 < z X inch inch inch inch inch inch inch inch 17732 21-555 36-516 0-3326 0-4059 0-6878 0-1174 0-1411 0*1408 i39 4i 0-1174 0-14095 1 7 - 8oo 21-492 36-603 "3339 -4047 6895 1161 "1423 -1425 T -> r--^ 17*900 21-398 36-702 "3357 4029 6914 -1143 1441 1444 *35 o 1156 .1429 iS ooo 21-333 36782 -3376 4017 6929 1124 -1453 -I 459 l8 2OO 1 8 400 21-189 21-068 36-910 37-005 -3414 345 1 "399 3967 6953 6971 1086 1049 "1480 *S3 1483 1501 I2O OO 1082 1486 18*600 20*986 37-094 3489 395 2 6987 ion 1518 -I 5i7 19-000 20-910 37-200 3564 "3937 7007 0936 1533 -I 537 !9 336 20-868 37 2I 4 3627 3929 7010 0873 "1541 "1540 ^_o ^Q f O 19*800 20-905 37 -I 94 37*4 3936 -7006 0786 1534 1536 90 oo OOOO I 54Q 20*209 2 1 OO6 37"095 3789 "3955 6988 0711 1515 1518 20*600 2I OOO 21-141 21-368 36-938 36-736 3864 "3939 3981 4024 6958 6920 0636 0561 1489 1446 1488 145 6o*oo 5 8l 1459 21-400 21-638 3 6 443 4014 4075 686 5 0486 -I 395 " J 395 , _o 2 I 800 22"OOO 36-084 4089 -4143 6797 0411 " I 3 2 7 1327 45 o 0417 "L33* 22 2OO 22*460 35- 6 4o 4164 4230 6713 0336 1240 -1243 22 6OO 23-031 35- 5 2 4239 "4337 6603 0261 " I: 33 "33 _ _n._ , _ / 23 OOO 23-788 34 3 12 "4314 4480 6463 0186 0990 -993 30 oo 0247 1 106 23-400 24-786 33-256 4389 4668 6264 01 I I 0802 "794 -r r- ^^- ~ o r 23-800 26-546 3! 57o "4463 5000 -5946 0037 0470 "0476 15 oo OOOO 0707 2 3 995 *#* *## 4500 5470 5470 oooo OOOO oooo o O Weight? /3=24-023 o=119-9 w=139-41 J = 31646in. Temp. 49 F. The theoretical forms of these five drops of mercury have been drawn to a large scale in Flo*. 2. where the measured points are indicated by small crosses. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 71 The agreement between theory and experiment appears to be so far satisfactory. And if on more exact comparison any slight discrepancy between theory and experi ment should become apparent, it will be known that, this is not due to any error in the calculated forms. In adapting a theoretical form to the measured form of a drop of mercury, it would be sufficient to secure its passing through the vertex A (Fig. 4) and the two points B, C, for which < = 90, if it was possible to measure AO correctly. But this can be accomplished practically only with sufficient accuracy to give a rough first approximation to the value of /3, by finding OC+AO and referring to Table I. This value of /3, if erroneous, must be corrected by trial till a curve is found from Table II., which passes through D and E, the extremities of the base, or till two curves are found for consecutive values of /3, one of which falls outside, and the other within DE. Then by proportional parts the exact value of /3 required can be found. Let BC=2R, DE=2r, and AN = H. The following example will explain how the values of the capillary constants are obtained by means of these quantities. For the drop No. V. 2E = 03081 inch, H= 01174 inch, and 2r = 2819 inch. Having found by the help of Table I. and by trial that the proper value of $ lies between 24 and 24 1, we proceed to find b the radius of curvature at the vertex corresponding to /3 = 24 0. From Table II., when </> = 90, we find that I = R = Olo40_o = . 48692 and therefore 5 = _1|| _| ) which g ives log ft = 9-50020. That will suffice to secure a curve which passes through the vertex A and has the correct width BC. We wish in addition to secure a curve which passes through the two points D, E at the base of the drop, or through two points d, e near the base. log r = 914907 log H= 9 06967 log b = 9-50020 log b = 9-50020 log ~ = 9-64887 log p = 9-56947 (s : t/ and therefore J = 44552 and ~ = 37108. 6 6 And to find the theoretical form of this drop we use the manuscript Tables z H above referred to a . For ft = 24 the Table gives -r, = 37108 = p corresponding to = 139 36. And for the same value of d>, =0-4460S-(H)0050=0-44558. Hence 6 f -fi = and -?/ = - 0-00006, < = 139-36. " See note a on next page. 72 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. Again, corresponding to ft" = 241 we find in the same manner as before log b" = 9-50070, which gives p = 44501 and p=0 37066. And for & = 241 b the Table gives ^=0-37066 corresponding to </> = 139 5S, and ^ ; = 0-44481. Here we have ^ * =0; --^ = + 0-00020; <4 = ) 00 The required value of /S therefore falls between 24 and 241, and its exact value may be found as follows : P = 24-0 gives log b = 9 50020 ; error in = - 00006 ; and f = 139 36 " = 24-1 gives log I" = 9 50070 ; *= + 0-00020; and f = 139-58 Diff. + 0-1 + 0-00050 + 0-00026 + 0-22 Hence by taking proportional parts we must find such values of S/3 , 8 log 6 cc and 4> as will make the error in 7-, vanish. b & = 24-0 gives log b = 9 50020; error in |, =-0 00006 ; and = 139 36 8/3 =+ -023 5 log b = + 0-0001 2; Hence j3 = 24-023 log 6= 9-50032; ft +0-00006 ; 0-05 = 139-41 Hence 6 = 0-31646 inch, a=-,= ^-^ = 119 94, and iJ -= 01291 inch. Also J _ v CC the volume =6 3 x 0-2072 = 0-0065667 cubic inch, and the corresponding weight is 22 535 grains. IT / Nearly the same results might be arrived at by using the values of j and j in Table II. for ft = 24 and ft = 25, only the differences would correspond to a difference of 1 instead of 01 in the value of- ft, and to a difference of 5 instead of 1 in the value of 6. NOTE a . /3=24-0 X z <t> b b A A 138 44747 ,n Q 36942 , 139 44608 J4Q 37065 1 ion 140 44468 37185 + iJU &c. &c. &c. NOTE b . t-su X z b b A A 138 139 140 44700 _ 139 44561 JJJ 44421" 36874 36997]" 37117 + 120 &c. &c. &c. COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 73 It is evident that the above calculations would have been facilitated if the Tables referred to in the note had been calculated for log x - , log 2 and log ? rather x z V than for r > r arid j-g, as has been already remarked. The coordinates at the points of the theoretical curve at which the tangent is inclined to the horizon at angles of 15, 30, 45, 60, 90, 120,- 135 &c., are found by the help of Table Y. for values of , O O, 01, 2, 3 46 5, 46 6, 467. For instance for <f> = 135, /3 = 24-0 ; y = 0-45156; | = 36554 v b + 8/3 = 0-023 gives -11 -15 X =24-023; ?T = 0-45145 ; ^ = Q-36539 and 6 = 0-31646 inch. Therefore x = 6 x 45145 = 1429 inch, and = 6x0-36539 = 0-1156 inch. DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY IN CONTACT WITH GLASS. The great impediment to the exact determination of capillary constants arises from the changes that usually take place at capillary surfaces when left undisturbed for some time. All careful experimenters have recognised this difficulty. It seemed therefore best to place a drop of mercury in position and to take measures of 2R, 2r and H as opportunity offered. Drops weighing 4, 8, 12, 1C, 20 and 24 grains were used, because it was expected that, if a and w were not really constant for mercury resting on glass, some indication of the manner in which they varied would thus be made manifest. The mercury was obtained as being pure from a leading philo sophical instrument maker about 1862. When any experiment was to be made, a sufficient quantity was taken from this store, and after having been used, it was treated as waste. Also the same glass plate table was used in all the experiments. The glass plate was cleaned with blotting paper or with the pith of the stalk of the artichoke. And after this, either the same or a fresh drop of mercur} T was placed in position and vibrated. In the following tables of experiments the operation of cleaning the glass and replacing the same drop of mercury is indicated by a dotted line But a change in the mercury used is denoted by a line across the table. The reading of the thermometer is given and also the time duritig which the drop had been in position when the measurement was made. The experiments were carried on in. a small workshop built in a garden apart from other buildings. The B. 10 74 DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. observing table rested on supports driven into the ground which were independent of the brick floor. There were public roads, used chiefly for light traffic, on two sides at the distances of 50 and 60 yards. The slow changes in the forms of drops of fluid appear to arise, (1) from some small change that takes place in the tension of the enveloping surface, (2) from changes of temperature between night and day, and (3) from slight tremors arising from passing vehicles, &c. The calculation of the capillary constants was carried on as the experiments were made. After all had been completed the reductions of the instrumental observations into inches were carefully examined, and the calculations of all the 145 experiments were repeated, so that the results given in the following tables may be considered to be quite correct. The variation in the value of the capillary constants deduced from drops of mercury of the same size was much greater than was expected. But, when the mean values of to and a derived from each form of drop were compared, the agree ment was surprisingly close. Hence so far as these experiments go the form of sessile drops appears to be that indicated by the Theories of Young, Laplace, Gauss and Poisson. Finally the values of a, o>, and V were calculated from the mean values of 2.R, 2r, and H for each size of drop of mercury. The results are given on the last page for comparison with the means of the values of a and &> derived from each experi ment for each size of drop of mercury. In order to carry out the original scheme, as sketched in the Introduction, many more experiments should be made, particularly for the purpose of finding the effects of variation of temperature on the values of the capillary constants. The calculations for negative values of /3 should also be greatly extended, so that the intervals between them might be readily filled up by interpolation, as we have done in the case of positive values of /3. The measuring instrument in its present form appears to be satisfactory. The microscope descends in a vertical direction by its own weight and is raised by the screw. A screw of about 50 turns to the inch is very suitable for experimenting with mercury. But a quicker motion will become desirable when experiments are made with a drop of one fluid immersed in another fluid, as the drops may be then much larger. All documents connected with these calculations now in my possession will be carefully preserved, and every assistance will be afforded to any person who may under take the completion of the work. MINTING VICARAGE, Oct. 1883. DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. 75 DROP OF 4 GRAINS OF MERCURY. No. of Observa tion J? H r a a> N/- V a Temp F Error in V Hours in position I 24 25 26 27 28 29 3 31 117 nS 119 I2O 121 122 I2 3 I2 4 !25 126 127 128 I2 9 I 3 IS* I 3 2 J 33- Means inch "07460 inch 0*09050 inch 0-05990 * 9 2 S 117-00 J 45 43 inch 0-1308 m. metres 3 32I 60 cubic inch + "OOOOO2 ? 7535 7 6 35 07610 09150 08950 08940 06040 "06170 06185 06165 I 922 2> 394 2-316 II4-54 130-70 128-52 J 45-65 146-67 145-48 1321 1237 1248 3-356 3-142 3-169 61 58 59 + 000039 + 000045 + 000036 12 18 ? 07600 08950 2*292 127-90 145-68 1251 3-176 61 + -000034 07475 07570 09040 08870 05965 06070 06050 06045 2-005 2 45 7 120-03 J35-42 146-70 148-11 1291 1215 3-279 3-087 60 58 + 000005 + 000013 \ *3 07580! 08930 07565 08895 2*400 2-424 132-84 134-18 148-52 148-47 1227 1221 3-II7 3-101 60 62 + "OOOO22 + "000014 8 22 07525 07595 07585 08950 06115 2-126 I2 3 59 144-68 1272 3-23I 63 + OOOOIO I r 3 15 08760 08790 06140 06160 2-670 2-549 142-61 138-41 14773 146-56 Il84 I2O2 3-008 3 053 62 64 4- "000005 + -000007 07585 08875 06090 2-472 !35 47 !47 97 1215 3-086 66 + "000017 3 07535 7545 07620 08795 08800 05940 05965 2-638 2-633 143-56 143-09 151-40 150-96 1180 1182 2-997 3-003 64 67 "000009 - "000005 34 36 08760 "06140 2 755 I44-77 148-57 1175 2-985 66 + "000014 21 07380 07430 07475 07490 07470 08850 08810 08770 08760 08925 05860 05960 05955 05975 05970 2T57 2-263 2-481 2-525 2-172 129-85 132-60 139-85 141-02 127-39 148-16 147-20 149-17 149-16 i47 32 1241 1228 1196 1191 1253 3-I52 3-119 3-038 3-025 3T83 63 62 61 63 61 "000048 - "000037 - "000030 - -000027 - "OOOOII O 24 4 8 7 1 88 07460 08970 08940 08880 08880 08850 05965 2-075 123-57 14677 1272 3-23I 61 000008 o 07480 75 I 5 07545 07550 06035 06040 06045 06040 2-104 2-305 2-391 2-467 124-15 !3i 33 !3373 !36 53 145-62 147-09 I47-94 148-49 1269 1234 1223 I2IO 3-224 3-I34 3-106 3-074 61 59 57 58 "000005 "OOOOO2 + "000007 f -000005 38 61 96 in 0-07531 0*08890 0-06041 132-03 I47-52 0-1233 3-I3I 102 76 DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. DROP OF 8 GRAINS OF MERCURY. Xo. of Observa R H r p a 0) V l\ Temp. f Error in y Hours in tion position inch inch inch inch m. metres cubic inch 2 0x39980 0*10100 0-08535 5-226 127-96 144-45 0-1250 3-I75 60 + OOOOOI ? 3 IOOOO "10090 08535 5-370 I2 9 57 145-10 "1242 3-156 60 + 000017 ? 4 09905 10360 08345 4H56 117-69 145-89 1304 3-3II 58 + 000030 ? 32 09915 "IO2IO 08380 4-878 124-28 146-11 1268 3*222 62 + 000004 I 33 09990 10070 08400 5-700 134-61 148-62 1219 3-096 61 4- OOOOOS 18 34 09990 lOIIO 08410 5-534 132-23 148*09 1230 3-124 62 + 000017 22 J 34 -09945 10295 08315 4-892 12374 148*18 1271 3^29 58 + 000037 5 135 IOOIO 10180 08350 5-52I 131-53 149-83 1233 3 >:[ 32 57 + 000043 21 136 10015 10180 08340 132-05 150-25; 1231 3-126 57 + 000046 29 137 09940 10290 08320 4^74 123-59 147-94 1272 3-231 58 + 000034 12 138 -09975 10265 08330 5-101 126-22 148*82 " I2 59 3*197 58 + 000045 22 139 09950 10230 08335 5-079 126-52 148-19 " I2 57 3 -I 94 56 + 000026 O 140 09960 J.O2OO 08335 5-228 128-51 148-67 1248 3-169 58 + 000023 IO 141 09960 IOI50 08390 5-280 129-29 147-51 1244 3-I59 55 + 000013 I 3 142 09980 IOI20 08420 5-423 130-89 147-44 1236 3-140 56 4- OOOOl6 2 3 143 09985 IOIOO 08445 5-464 1 3 I 3S 147-06 1234 3"!34 57 4- OOOOI4 37 144 09985 10090 08435 5-522 132*20 147-39 1230 3-124 58 + OOOOII 46 145 09965 IOI2O 08430 5 3 21 129-77 4675 1241 3-I53 58 + 000008 61 Means 0*09969 0-IOI76 0-08392 128-44 r 47 57 0*1248 3-i7i DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. 77 DROP OF 12 GRAINS OF MERCURY. No. of Obser vation R H r ft a (D V 11 a Temp. F Error in V Hours in position 5 6 7 inch 0-11720 JI 735 11763 inch 0*10900 10920 10785 inch O lOIIO 10230 10288 8-728 3-331 9-044 124-99 121-50 126-53 I47-27 144-48 I44 43 inch 0*1265 1283 I2 57 m. metres 3 2I3 3 259 3-I93 59 59 59 cubic inch "OOOOOI + "OOOOII - 000013 ? 2 20 35 36 11700 i 1810 I IO2O IO7OO IOO2O IOII5 8-267 10-592 12170 136-68 148-32 I50-66 1282 1210 3 256 3-073 62 64 + OOOO22 "000009 \ 38 37 33 11830 11810 IO8OO 10785 IOI05 IOII5 10-183 10-073 J 33 39 !33 7 I50-9I I50 20 1225 1226 3 IIO 3-H4 63 64 + -000035 + 000016 2 9 39 40 4i 42 11710 11810 11770 11770 II030 10750 10790 10760 10050 IOI 35 !oi35 10160 8 2OO 10-215 9-642 9-726 120*94 134-08 130-85 i3i 47 M7-83 149-89 148-58 148-10 1286 1221 1236 1233 3-266 3-102 3-140 3-133 62 63 62 64 + -000032 + 000006 -000005 "000015 i 12 J 3 16 43 44 45 46 47 11740 11675 11710 11740 11765 <I0 955 10915 10787 10820 10790 10005 10025 10035 10040 10025 8-925 8-588 9 59 9^46 9-969 126-11 124*84 131-22 130-81 J 33 35 150-17 148-15 149-54 150-04 151-26 <I2 59 1266 1235 1237 1225 3-I99 3 2I5 3-136 3" T 4i 3-111 63 63 63 64 65 + 000027 - -000025 - "000043 - -000014 - -000009 2 7 21 45 J 45 Means o"ii754 0-10844 O lOIOO 128-85 148-74 0*1247 3-166 DROP OF 16 GRAINS OF MERCURY. No. of Obser vation R H r ft a 10 \ /: a Temp. F Error in V Hours in position 8 9 inch 0-13298 13290 inch 0-11305 11283 inch 0-11645 11642 14703 14*809 I27-57 128*19 148-23 148-20 inch 0*1252 1249 m. metres 3-180 3-I73 59 59 cubic inch + "OOOOIO + "OOOOI2 6 7 10 1 1 13185 -I32I5 11370 11290 "535 "575 13-448 14-228 124*10 127*08 147-70 147-90 1270 1255 3-225 3-I87 55 5 1 - -000031 "000033 o 2 48* 49* 50* " I 3 2 75 "!3378 " I 3375 "11420 11250 11245 11675 11750 11760 I3-355 15-584 *5 S3 I23-77 129*73 I2 9 57 146-32 i47 94 i47 6 3 1271 1242 1242 3-229 3-I54 3-I56 67 66 66 + 000014 + 000028 + -000021 1 3 6 5i* 52* 133*5 13338 11285 11240 "745 11725 4-529 !5 3" 126*50 129*38 146-25 I47-57 I2 57 1243 3 -I 94 3-I58 65 63 *OOOOO8 000007 1 1 21 53 54 13225 -I 34i5 II 5 2 5 11115 11718 11823 11-853 16-922 115*62 134-30 143-46 147-72 13*5 "I22O 3-34I 3-100 62 62 + 000053 + 000037 47 55 56 13338 " I 33 2 5 11240 II2IO 11695 11665 15-472 15-758 130*06 131*48 148-31 148-89 "1240 1233 3 !5 3 -I 33 63 63 + 000028 + 000009 16 J 9 Means 0-13306 O II29I 0-11689 127-49 147-39 -1253 3-183 * Weight of this drop was 16*12 grains. 78 DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. DROP OF 20 GRAINS OF MERCURY. No. of Obser vation JS H r a 0) V 11 a Temp. F Error in V Hours in position 12 inch 0-14655 inch 0-11425 inch 0*13040 24-243 ^33 1 147*93 inch 0*1226 m. metres 3 II4 5i cubic inch - "000033 12 13 14 15 16 i7 14405 14420 i43 2 5 14420 -I 4435 11910 Il88o 12 100 11810 11820 12710 12740 12605 12680 12705 17-063 !7"373 15-018 18-438 18-387 n6 94 117-71 III IO 121*11 120*69 I47-47 147-28 147-16 149-04 148-76 *I 3 o8 -1304 1342 1285 1287 3-3 2 2 3"3n 3-408 3-264 3-270 5i 5i 5i 5i 5 2 - 000013 - "000016 + "OOOOO2 "000046 - "000031 2 3 ? ? ? 57 53 59 -I 45 6 3 1459 14803 11725 11695 11370 12980 13023 "13200 19-386 19-836 26-497 121-60 122-44 13572 i457i I 45 5 148-00 1283 1278 1214 3-258 3-246 3-083 61 61 62 + "OOOO2O + -OOOO27 + "000054 1 i 47 60 61 i445 8 14508 11785 11665 12720 12788 18-967 20-595 I22 II I26-03 149-11 149*26 1280 "1260 3-251 3-200 62 63 "000036 -000048 i 13 62 63 64 65 14528 I 4543 14520 I 45 I 3 11730 11685 11695 11695 12785 12785 I2 755 12790 20*184 20-937 20-671 20-330 I24 50 126-38 126-03 I25-I8 i49"58 150-17 150*27 i49 25 1267 *I2 5 8 1260 1264 3-219 3 -I 95 3-200 3-211 61 63 64 63 + "OOOOOI "000008 - "000023 "000029 4 19 24 29 66 67 68 14650 14775 14790 11580 11380 11360 >I 345 I3I93 13220 22-031 25-840 26-186 127-50 I 34 7i !35 24 146-98 i47 38 147-17 1252 1218 1216 3-181 3-095 3-089 63 61 60 + "000027 + 000037 + "000040 i 12 23 69 70 7i 14805 14758 14768 11380 11480 11472 13160 -I 3233 13223 26-769 23-662 24-065 136-29 129-80 130*62 148-99 145-48 146-04 *I2II 1241 1237 3-077 3-I53 3 -I 43 61 64 63 -r "OOOo6l + 000063 + "O00068 13 25 36 72 73 74 75 76 14618 14615 14675 14648 14640 "535 1153 11382 "SOS H550 12905 12935 13028 13043 13015 2 3" I 75 22-958 25-358 22-982 22-481 I 3 I 5 I 3 54 135-42 130*02 128-86 149*82 149-04 148-94 i47 3i !47 59 >I2 35 1238 1215 1240 1246 3-I38 3-144 3-087 3 -1 5 3 l6 4 65 65 66 66 67 OOOO2O 000023 000046 "OOOOI I OOOOOO 3 7 47 49 53 Means 0>I 4593 O Il62I i 2 935 126-95 148*05 0*1256 3-I9 1 DETERMINATION OF CAPILLARY CONSTANTS OP MERCURY. DROP OF 24 GRAINS OF MERCURY. 7 ( J No. of Obser vation R H r ft a to V /! a Temp. f Error in V Hours in position 18 19 20 inch 0-16085 16055 16060 inch O l 1800 11840 11820 inch 0-I4550 1455 US? 33 54i 32 OOO 3 2 2 33 127-29 125-23 125-47 i45 2 9 i44 3 6 144-07 inch 0-1254 1264 1263 m. metres 3-I84 3-210 3-207 54 55 58 cubic inch + "000241 + -000231 + 000225 22 23 35 21 15870 11825 14085 32-264 130-29 I5i-i5 I2 39 3 -I 47 56 -t- -000072 i7 22 23 15638 I5 6 45 I2OIO II925 J 3745 !373 28-249 29-889 125-08 128-08 152-70 I 53 63 1265 1250 3-212 3 i74 60 60 -000050 "000078 8 3 2 77 78 79 80 15835 I 597 !59 2 3 I 59 2 3 11830 H583 II582 II685 14243 14380 H453 !4455 30-584 37-232 34-856 32-75 126-26 134-9 132-01 128-58 146-41 i47 49 144-44 i43 97 I2 59 1218 1231 1247 3 i97 3-093 3-126 3-168 67 65 65 66 + "000024 + "000014 - "000039 + "000016 i 14 24 38 81 82 83 84 15845 15893 15908 15898 II770 II7IO 11615 "Il6lO 14215 14248 14210 14208 32-230 34 2i7 37-078 36-928 128-97 131-48 J3572 i35 67 147-64 148-28 150*00 149*82 1245 I2 33 1214 1214 3-163 3*133 3-083 3-084 66 65 63 63 "OOOOOI + "000008 OOOO2O - -000033 3 8 !9 3 1 8q 86 87 88 89 90 9 1 15675 15840 15908 -I 5945 !5938 1579 15855 II950 II7IO 11675 II570 I 1670 II9OO II825 14013 14280 14323 i43 2 5 14343 14320 1433 27-527 32-457 34 352 37-58o 34-99 1 27-462 30-109 123-08 129-43 i3i-45 135-84 I 3^ 99 121 lS 125-10 ^T S 1 146-13 146-95 148-28 147-22 143-09 144-80 1275 1243 1234 1213 1231 1285 1264 3-238 3 i57 3 >:[ 33 3-082 3-127 3-263 3-212 64 64 65 63 62 61 61 - -000061 -000038 4- "000005 OOOOIO + 000032 + 000009 + 000037 i 16 24 60 72 86 87 92 93 94 15683 15660 15640 II947 11895 II930 I 395 13985 13990 28-350 28-357 27-340 124-51 124-94 123*28 149-24 148-05 147-28 1267 1265 1274 3-219 3 2I 4 3-235 61 61 61 -QOOO49 OOOOgS "OOOIOO i 15 i7 95 96 97 98 -I 5795 !58o5 15780 -I 575 11805 II750 Il8l 5 H775 14040 14065 14053 14065 3 2 35 2 33 373 3I-695 3I-59 2 130-00 I3I-55 129-10 129-42 150*46 i5o-37 149-76 148-88 1240 1233 1245 1243 3-i5i 3 -I 32 3-161 3-I58 61 64 64 62 -000018 -000037 - -000033 -000081 6 26 3 1 45 99 15820 11830 14220 30-500 126-35 146-60 1258 3-196 63 + -OOOOI2 4 IOO 15760 11845 14120 29-962 126-35 i47 47 1258 3-196 6 5 000040 i 101 IO2 103 15785 15820 1583 11810 "755 -II 755 14163 i4i95 14200 30-681 32-I45 3 2> 333 127-24 129-23 129-39 147-21 i47 53 147*66 1254 1244 1243 3-184 3-160 3-I58 67 61 62 - -000033 - 000027 - "OOO02O 7 2 2 3 1 IO4 I5 1 06 107 1 08 top no III 15745 15785 1573 15810 15820 15785 15770 15765 11790 11785 11850 H795 11740 11740 -II 775 11760 14095 I4H5 14120 14250 14200 14210 14220 14225 30-860 3 I- 339 29-209 3o 5!7 32-372 3 I 39 I 30-263 3 348 128-21 128-40 i25 44 126-54 129*62 128*49 12673 126-96 i47 94 147-76 146-69 I45-75 147-48 146-34 !45 54 I45-37 1249 1248 1263 1257 1242 1248 1256 "55 3-172 3-170 3- 20 7 3"!93 3-155 3-169 3-J9 1 3-188 62 63 63 64 61 61 61 63 "000076 "000040 "000063 -OOOO2O - -000033 - 000072 - -000066 OOOoSl 1 12 26 36 5 74 98 146 112 JI 3 114 H5 116 !577o 15850 15820 15830 ^SS 11870 11710 11710 11700 11725 14120 14250 14220 14210 14260 29-761 33 077 32-692 33-261 32-668 125-82 130-32 130-16 130-95 129-63 147-61 147-13 147-11 147-66 146-82 1261 1239 1240 1236 1242 3-202 3 -I 47 3 <I 49 3 r 39 3^55 64 61 58 61 61 000013 -000025 000051 -QOO045 - -000019 i 10 34 59 81 Means 0-15825 0-11778 0-14199 128-52 147-46 1248 3-169 80 DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. SUMMARY OF MEAN KESULTS FOR EACH WEIGHT OF DROP OF MERCURY. Weight of Drop Laplace s a Error in a CO Error in CO J\ Error 5/j Error Grains inch inch m. metres m. metre 4 132-02 + 3 3! I 47 52 - 0-27 0-1233 -0015 3"i3i -0-038 8 i28 44 0-27 I47-57 O*22 1248 3-171 + O OO2 12 128-85 + 0*14 148-74 + 95 1247 OOOI 3-166 -0-003 16 127-49 1*22 M7-39 0*40 -I2 53 + 0005 3-183 + 0*014 20 126-95 1*76 148-05 + 0*26 1256 + -0008 3-191 + O O22 24 128-52 O lp 147-46 -"33 1248 3-169 Means 12871 i47 79 0-1248 3-169 VALUES OF a, w, &c., DEDUCED FROM THE MEAN VALUES OF R, If, AND r FOR EACH SIZE OF DROP OF MERCURY. Weight of Drop R H r * a CO A Error in V Grains 4 8 12 16 20 24 Means inch 0*07531 09969 "754 13306 " I 4593 15825 inch 0*08890 10176 10844 11291 11621 11778 inch 0*06041 08392 IOIOO 11689 >I2 935 14199 2-334 9-328 14*681 21-433 31*796 131*96 128*41 128*88 127*32 126-87 128*55 I47-5I 148-76 147*40 148*07 147-46 inch 0-1231 1248 1246 >I2 53 1256 1247 m. metres 3*127 3-164 3-I83 3-189 3-168 cubic inch + 0*000003 + OOOO21 + OOOOO2 f "000025 + OOOOOI 000013 128*67 J 4779 0*1247 3-167 The forms of these six drops are given in Fig. 3 on a large scale. (-} W< = 90 ft o I 2 "3 4 5 6 7 8 9 o I 2 I OOOOO 15466 24507 02180 16546 25248 04149 17576 25967 05942 18562 26666 07589 19508 27345 09115 20418 28006 10542 21294 28650 11880 22138 29278 13140 22953 29890 14333 23742 30488 3 4 5 1-31072 36278 40615 31643 36745 41012 32201 37204 4M03 32748 37656 41789 33283 38100 42169 33807 38535 42544 34320 38963 42914 34824 39386 43278 353i8 39802 43638, 35803 40211 43993 6 7 8 i 44344 47621 5 55o 44690 47928 50827 45032 48232 51101 45369 48533 S^i 45702 48830 51640 46032 49124 51906 46358 49415 52169 46679 49703 52430 46996 49988 52689 473io 50270 52946 9 10 ii 1-53200 55 621 57852 53452 55851 58065 53702 56080 58277 53949 56307 58488 54194 56533 58698 54437 56758 58906 54678 56981 59112 "549*7 57202 59317 55*54 57421 5952 55389 57638 59722 12 13 14 i 599 2 3 61856 63667 60122 62042 63842 60320 62227 64016 60517 62411 64189 60712 62594 64361 60906 62776 64532 61099 62957 64702 61290 63136 64871 61480 633M 65039 61669 6349 1 65206 15 16 17 1-65372 66984 68512 65537 67140 68661 65701 67296 68809 65864 67451 68956 66027 67605 69102 66189 67758 69248 66350 67910 69393 66510 68062 69537 66669 68213 69681 66827 68363 69824 18 J 9 20 1-69966 71353 72678 70108 71488 72807 70249 71623 72936 70389 71757 73064 70528 71890 73192 70667 72023 73319 70805 72155 73446 70943 72287 73572 71080 72418 73698 71217 72548 73823 21 22 2 3 173947 75165 76336 74071 75284 76451 74194 75403 76565 74317 75521 76679 74440 75639 76792 74562 75756 76905 74684 75873 77017 74805 75989 77129 74926 76105 77241 75046 76221 77352 24 25 26 1-77463 78550 79600 77574 78657 79703 77684 78764 79806 77794 78870 79908 77903 78975 80010 78011 79080 80112 78119 79185 80213 78227 79289 80314 78335 79393 80415 78443 79497 80515 2 7 28 2 9 1-80615 81598 82550 80715 81695 82643 80814 81791 82736 80913 81887 82829 81012 81983 82922 81110 82078 83015 81208 82173 83107 81306 82268 83199 81404 82362 83291 81501 82456 83383 3 3i 3 2 i ^3474 84371 85242 83565 84459 85328 83656 84547 85414 83746 84635 85499 83836 84722 85584 83926 84809 85669 84015 84896 85754 84104 84983 85838 84193 85070 85922 84282 85156 86006 33 34 35 1-86090 86915 87719 86173 86996 87798 86256 87077 87877 86339 87158 87956 86422 87239 88035 86505 87320 88113 86587 87400 88191 86669 87480 88269 86751 87560 88347 ;868 33 87640 88425 c. (I) 11 (-} W^ = 90 ft o I *2 3 4 5 6 7 8 9 36 37 38 ; 1-88503 89268 90015 88580 89344 "90089 88657 89419 90163 88734 89494 90236 888ll 89569 90309 .QQQQQ OOOOO 89644 90382 88964 89719 90455 89040 89793 90528 89116 89867 90600 89192 89941 90672 39 40 4i 1-90744 9*457 9 2I 54 90816 9J5 2 7 92223 90888 9*597 92292 90960 91667 92361 91031 W37 92429 9II02 91807 9 2497 9 ri 73 91877 92565 91244 91947 92633 91315 92016 92701 91386 "92085 92769 42 43 44 1-92836 9354 94158 92904 93570 94223 92971 93636 94288 93038 93702 94352 93105 93768 94416 93172 93833 94480 93239 93898 94544 93306 93963 94608 93372 94028 94672 93438 94093 94735 45 . 46 47 1-94798 95426 96042 94861 95488 96103 94924 95550 96164 94987 95612 96225 95050 95 6 74 96285 95 IX 3 95736 96345 95176 95798 96406 *95 2 39 95859 96466 95302 95920 96526 95364 95981 96586 48 49 5 1-96646 97239 97821 96706 97298 97879 96766 97357 97937 96826 97415 97994 96885 97473 98051 96944 97531 98108 97003 97589 98165 97062 97647 98222 97121 97705 98279 97181 97763 98336 5i 52 53 i 9 8 393 98954 99506 98450 99010 99561 98507 99066 99616 98563 99I2I 99671 98619 99176 99725 98675 99231 99779 98731 99286 99833 98787 99341 99887 98843 99396 99941 98899 9945 r 99995 54 55 56 2 00049 00582 OIIO7 OOIO3 00635 OII59 00157 00688 OI2II OO2II OO74O 01263 00264 00793 01314 00317 00845 01366 00370 00898 01418 00423 00950 01470 00476 01003 01521 00529 01055 01572 57 58 59 2 Ol623 02132 02633 01674 02183 02683 01725 02234 02733 01776 02284 02782 01827 02334 02831 01878 02384 02880 01929 02434 02929 01980 02484 02978 02031 02534 03027 02081 02583 03076 6a 6 1 62 2-03125 03610 "04088 03174 03658 04135 03223 03706 04183 03271 03754 04230 03320 03802 04277 03368 03850 04324 03417 03898 04371 03465 03945 04418 03514 03993 04465 03562 04040 04512 63 64 65 2-04559 05023 05480 04606 05069 05525 04652 05H5 05571 04699 05160- 05616 04745 05206 05662 04792 05252 05707 04838 "05298 05752 04885 05343 05797 04931 05389 05842 04977 05434 05887 66 67 68 2-05932 06 37 8 06817 05977 06422 06860 O6O22 06466 06904 06067 06510 06947 06111 o6554 06990 06156 06598 07034 06200 06642 07077 06245 06686 07120 06289 06729 07164 06334 06773 07207 69 70 ?i 2-07250 07678 oSlOO 07293 07720 08142 07336 07763 08184 07379 07805 08226 07422 07848 08267 07465 07890 08309 07508 07932 08351 07550 07974 08392 07593 08016 08434 07635 08058 08475 (2) (-} \2/(/> = 90 ft o "I *2 3 *4 5 6 7 8 *9 72 2-08517 08558 08600 -08641 08683 08724 08765 08806 08847 08888 73 08929 08970 09011 09051 09092 09133 09173 09214 09254 09295 74 9335 09375 09416 09456 09496 09536 09576 09616 09656 "09696 75 2-09736 09776 09816 09855 09895 09935 09975 10014 10054 -10093 76 10133 10173 IO2I2 10252 10291 10330 10369 10408 10447 -10486 77 10525 10564 10603 10641 10680 10719 10758 10796 10835 10873 78 2-10912 10950 10989 11027 11066 11104 11142 11180 11218 11256 79 11294 11332 II370 11408 H445 11483 11521 -II 559 11596 -11634 80 11672 11710 II747 11785 11822 11860 11897 H934 11972 12009 81 2*12046 12083 I2I20 12157 12194 12231 12268 12305 12341 12378 82 12415 12452 12488 12525 12561 12598 12634 12671 12707 12744 83 12780 12816 12852 12889 12925 12961 12997 i3 33 13070 13106 84 2-13142 13178 I32I4 13250 13285 13321 13357 13393 13429 13465 85 13500 13536 I357 1 13607 13642 13677 13712 13748 13783 13818 86 13853 13888 13923 13958 13993 14028 14063 14098 -I 4i33 14168 87 2-14203 14237 14272 14307 i434i 14376 14411 14445 14480 MSU 88 14549 14583 14618 14652 14687 14721 !4755 14790 14824 14858 89 14892 14926 14960 14994 15028 15062 15096 1513 15164 15197 90 2-15231 15265 15298 15332 15366 !5399 15433 15466 1550 15533 9 1 15567 15600 15633 15667 15700 15733 15766 15800 15833 15866 92 -I 5 8 99 I 593 2 15965 15998 16031 16064 16097 16129 16162 16195 93 2*16228 16260 16293 16326 16358 16391 16424 16456 16489 16521 94 16554 16586 16619 16651 16684 16716 16748 16780 16813 16845 95 16877 16909 16941 16973 17005 17037 17069 17101 17132 17164 96 2-17196 17227 i7 2 59 17291 17322 17354 17386 17417 17449 17480 97 17512 17543 I 7575 17606 17638 17669 17701 17732 17763 17795 98 17826 17857 17888 17919 17950 17981 18012 18043 18074 18105 99 2-18136 18167 18197 18228 18259 18290 18320 18351 18382 18412 100 18443 (3) 112 II /3= 0-125 0-25 0-50 075 I O < X 1 z ~b X I z I X ~b z 1 X ~b z ~b X ~b z ~b 5 10 15 08715 17357 25855 00380 01518 03402 08714 17348 25827 00380 01517 03397 08711 17332 25774 00380 01515 03386 08709 17316 25720 00380 01513 3375 08707 17300 25668 00380 01511 03365 20 25 30 34139 42141 49798 06014 09328 I33I4 34076 42022 49600 05997 09288 13232 33952 41790 49217 05964 09210 13075 33830 41564 48849 05932 09134 12925 337H 41344 48495 05900 09060 12781 35 40 45 57049 63838 70115 T7933 23142 28893 56749 63413 69545 17786 22899 28517 56173 62608 68478 17506 22441 27819 55628 61854 -67493 17242 22017 27183 55109 61146 66579 16993 21623 26599 5 55 60 75834 8^955 85445 35133 41807 48855 75104 80055 84371 34582 41033 47807 73752 78407 82427 33573 39637 45946 72522 76928 80706 32669 38408 44335 71394 75587 79161 31851 37312 42919 65 70 75 89278 92430 94889 56216 63826 71621 88033 91027 93348 54840 62067 69425 85807 88545 90647 52436 59042 65707 83859 86396 88332 50389 56511 62646 82127 84502 86305 48613 54342 60056 80 85 90 96644 97694 98042 79537 87508 9547i "94995 95974 96297 76850 84281 91656 92126 92998 93283 72372 78984 8549! 89685 90478 90736 68744 74756 80641 87559 88291 88539 65708 71259 76671 95 IOO I0 5 97698 96677 95001 1-03363 I III2I I-I8686 9598i 95047 935 2 4 98920 1-06015 1-12889 93007 92197 90886 9 l8 45 98002 1-03920 90488 89763 88595 86358 91870 -97145 88302 87640 86576 81909 86944 91747 no H5 I2O 92695 89793 86333 i 26ooo 1-33008 1-39656 9 r 443 88841 85759 1-19492 1-25777 1-31699 89109 86902 84306 1*09562 1-14892 1-19880 87018 85067 82782 1*02151 1-06863 1-11255 "85144 83377 81312 96293 1*00561 I-0453 1 I2 5 130 35 82360 77923 73082 1-45895 1-51678 1-56962 82243 78345 74122 1-37220 1-42301 1-46912 81366 78127 74637 1-24498 1-28722 I "3 2 53 I 80201 77365 74318 1-15309 i -19006 1-22333 78983 76428 73686 1-08189 1-11520 i i45 I 4 140 M5 150 67902 62460 56842 1-61710 1-65888 1-69469 69635 64953 60151 1*51026 1-54620 1-57681 70949 67117 63197 1-35912 1-38855 1-41354 71103 67765 64350 1-25280 1-27843 1*30020 70794 67793 64720 1-17165 1-19469 1-21428 i55 160 165 5H50 45497 40013 172434 1-74778 1-76511 553io 50514 4585 1*60203 1-62192 1-63665 59246 55321 51480 1-43412 1-45039 1-46252 60904 57472 54096 1-31815 1-33238 1-34304 61615 58516 55458 1-23045 1-24330 1-25296 170 175 1 80 34830 30080 25864 1-77663 1-78298 1-78487 41402 37245 33439 1-64653 1-65203 1-65372 47773 44247 40941 1-47076 1-47542 1-47688 50818 47672 44690 I-3503 2 I-35448 1-35579 52476 "49599 46853 i 25958 1-26338 1-26459 (4) II 0= i-5 2 2 5 3*o 3*5 <#> X 1 z ~b X I z ~b X I z ~b X b z b X b z ~b 5 IO IS 08703 17268 25564 00380 01507 03344 08699 17236 25462 00379 01502 03324 08695 17204 25363 00379 01498 03305 08691 17173 25265 00379 01494 03286 08687 17142 25170 00379 01490 03268 20 25 3 33479 40923 47826 05838 08920 12511 33255 40523 47203 05779 08787 12262 33039 40143 46619 05723 08662 12030 32830 39780 46071 05668 08543 11814 32628 39434 45553 05615 08430 11613 35 40 45 54M4 59848 64928 16533 20908 25560 53260 58681 63469 16117 20274 24658 52446 57621 62161 15739 19707 23863 51691 56651 60978 i539i 19194 23155 50988 55756 59898 15071 18726 22517 5 55 - 60 69385 73229 76478 30420 35427 40522 67636 71206 74203 29203 33851 38552 66089 69435 72231 28146 32503 36888 64703 67862 70492 27216 > 3 I 33 35454 63448 66448 68939 26387 30294 34199 65 70 75 79152 81277 82879 45655 50780 55857 76656 78596 80052 43261 47939 52552 74510 76305 77649 41262 4559 2 49847 72629 74308 7556i 39556 43604 47573 70956 72539 73718 38073 41886 45622 80 35 90 83987 84630 84838 60850 65724 70453 81055 81635 81822 57070 61466 65717 78572 79104 79275 54004 58038 61931 76420 76915 77074 5 I 442 55I9 1 58803 74523 74989 75137 49255 52771 56154 95 IOO I0 5 84640 84067 8315 75009 79371 83516 81645 81133 80314 69802 73702 77401 79 ir 3 78646 77899 65665 69224 72596 76924 76491 75801 62262 65556 68674 74998 74593 73948 5939i 62472 65386 no "5 I2O 81917 80402 78634 87427 91089 94487 79216 77868 76297 80886 84143 87162 76900 75674 74246 75768 78730 81474 74878 73745 72428 71605 74340 76873 73086 72027 70799 68122 70674 73 39 I2 5 130 i35 76644 74465 72128 97612 i 00453 1*03006 74531 72598 70525 89935 92456 94720 72642 70886 69004 83994 86283 88339 70948 69328 67590 79198 81310 83207 69417 67906 66284 75209 77182 78949 140 145 15 69663 67105 64482 1-05264 1-07229 1-08901 68339 66068 63738 96723 98467 99952 67018 64954 62834 90159 9*744 93 95 65758 63851 61893 84887 86350 87599 64574 62792 60963 80^19 81884 83050 155 1 60 165 61826 59167 56532 1-10284 1-11387 I - I22l8 61375 i 59303 56647 i - oii83 i o2i66 1-02910 60681 58518 56364 94217 95H3 95793 59901 57898 559 01 88636 89467 90097 59101 57223 55353 84020 84798 8539 170 175 180 53949 5M39 49026 I I2792 I-I3I23 ri3229 54329 52069 49885 1-03425 1-03723 1-03819 54240 52164 50151 96265 96538 96630 539 2 7 5 X 994 50116 90535 90790 90872 5350 51684 > 59 1 4 85802 86041 86117 (5) II /3 = 4-0 4 5 5*o 5 5 6-0 t X ~b z "b X I z I X b z I X lb z b X b z b 5" 10 15 08683 17112 25076 00378 01486 03249 08679 17082 24984 00378 01482 03231 08675 17052 24894 00378 01478 03213 08671 17022 24806 00378 01474 03196 08667 16992 24719 00377 01471 03179 20 25 3 32432 39102 45064 05564 08323 11423 32241 38784 44600 05515 08221 11244 32057 38479 44159 "05468 08123 11075 31878 38186 43738 05422 08030 10916 31704 3793 43336 05377 07940 10764 35 40 45 50329 54928 58905 14773 18298 21938 49710 54156 57986 14495 17903 21409 49127 53434 57133 14236 !7537 20922 48576 52755 56335 13993 17196 20472 48053 52116 55588 13764 16878 20055 5 55 60 62302 65165 67536 25642 29370 33^87 61249 63992 66258 24967 28538 32091 60275 62912 65086 24349 27781 31191 59370 61911 64005 23781 27089 30371 58525 60981 63001 23257 26454 29621 65 70 75 69453 70953 72069 36766 40383 439*9 68089 69518 70580 35602 39050 42414 66840 68208 69223 34555 37854 41071 65689 67004 67978 33606 36773 39860 "64625 65891 66829 32739 3579 38761 80 85 90 72832 73271 734II 47355 50676 53869 71306 71722 71856 45682 48837 51868 69917 70314 70441 44i9 2 47203 50095 68643 69024 69M5 42853 45737 48508 67468 67835 67952 41640 44414 47076 95 IOO 105 73279 72898 72290 56922 59825 62569 71730 71369 70792 54766 57518 60119 70322 69977 69428 52858 55482 57960 69032 68701 68176 5H53 53665 56036 67842 675 2 5 67021 49617 52030 54307 no "5 1 20 71478 70483 69326 65M7 67550 69775 70023 69081 67984 62563 64842 66948 685 95 67798 66753 60287 62457 64464 67475 66616 65617 58262 60337 62258 66348 65523 64564 56444 58436 60280 125 J 3 135 68026 66604 65078 71817 73672 75338 66753 65405 63960 68882 70639 72218 65581 64297 62920 66306 67979 69483 64496 63269 61950 64020 65620 67059 63487 62309 61043 61971 63508 64889 140 145 15 63467 61790 60065 76814 78101 79201 62433 60844 59207 73616 74837 75881 61466 59951 58391 70816 71979 72973 6^559 59109 57616 68334 69448 70399 59707 58314 56879 66114 67184 68098 1.55 160 165 5 8 3io 56540 54772 80115 80849 81407 57541 55861 54180 76748 77446 77974 56802 55i9 8 53593 73801 74466 74972 5 6 094 54555 53oi7 71192 71828 72315 554i6 53937 52457 68860 69473 69940 170 175 1 80 53021 51300 49623 81796 82022 82096 52515 50875 49278 78345 78561 78632 52001 50433 48902 75326 75532 75600 51489 49983 48511 72655 72853 72917 509 8 5 49535 48115 70267 70458 70520 (6) II /3= 6-5 7 7 5 8-0 S 5 <A X b z ~b X b z ~b X b z 1) X b z ~b X b z 1) 5 10 15 08663 16963 24634 00377 01467 03163 08659 16934 2455 00377 01463 03147 08655 16906 24467 00377 01459 03131 08651 16877 24386 00376 01456 03115 08647 16849 24306 00376 01452 03100 20 25 3 3 31534 37630 42951 05334 07854 10620 31369 37367 42583 05292 07771 10482 31208 37112 42229 05251 07692 1035* 3*05* 36866 41889 05211 07615 10225 30897 36628 41562 05172 0754* 10105 35 40 45 47556 1 5 I 5 11 54884 13548 16579 19666 47083 50939 54220 13344 16299 19302 46631 50395 5359i 1315 16034 18960 46199 49877 52995 12966 15784 18638 45785 49382 52428 12790 15546 *8334 5 3 55 60 57733 60 1 1 1 62065 22771 25868 28930 56988 59 2 95 61189 22318 25322 28291 56285 58526 60366 21994 24813 27697 55619 57801 59590 21497 24338 27M3 54987 571*5 58857 21123 23893 26624 65 73 75 63635 64857 65762 31943 34890 37756 62710 63892 64768 31209 34061 36834 61842 62989 63837 30528 33293 35983 61024 62139 62963 29895 32581 35*92 60252 61337 62139 29303 31917 34455 80 85 90 66379 66733 66846 40534 43209 45774 65364 6^706 65815 395*9 42105 44584 64415 64745 64851 38583 41088 43488 63524 63844 63947 377i6 40147 42476 62685 62995 63096 36911 39273 4*536 95 IOO i5 66738 66434 65949 48222 50547 5 2 74o 65712 65418 64949 46948 49*93 5*3" 64753 64467 64013 45777 47950 50001 63851 63574 63134 44696 46804 48792 63002 62732 62306 43693 45742 47673 no 115 120 65301 64506 63582 54799 56717 58492 64323 63556 62664 53 2 99 55151 56865 63407 62665 61802 51923 537i6 55375 62546 61827 60990 5o657 52395 54003 61734 61036 60223 49485 5i*73 52735 I 2 5 130 *35 62545 61410 60192 60121 61601 62930 61663 60567 5939 58437 59866 61151 60834 59773 58633 56896 58279 59522 60051 59022 57917 55479 56820 58027 593*1 583*1 57238 54*69 55472 56646 140 HS 15 58904 57563 56179 64111 65142 66023 58147 56851 555*4 62291 63286 64138 5743 56175 54880 60627 61590 62415 5675 55532 54276 59097 60032 60832 56104 54920 53699 57684 5 8 593 5937* 155 160 165 54769 53342 $1914 66758 67349 67800 5415 5 2 77i 51389 64848 65420 65856 53558 52222 50882 63103 63657 64080 52993 51696 50394 61500 62038 62449 5245* 51190 49924 60020 60544 "60944 170 75 1 80 50492 49091 47719 68115 68300 68360 50014 48657 47328 66161 66340 66398 49549 48233 46942 64376 64548 64605 49099 47819 46564 62736 62905 62960 48662 474*6 46193 61223 61389 6i443 (7) II /3= 9 o 9 5 IO*O I0 5 II O < X ~b z b X I z ~b X b z ~b X b z ~b X & z ~b 5 10 15 08643 16821 24228 00376 01448 03085 08639 16793 24151 00376 oi445 03070 08635 16766 24075 00375 01441 03056 08631 16739 24000 00375 01438 03042 08627 16712 23927 00375 01434 03028 20 2 5 3 3 748 3 6 397 41246 05135 07469 09989 30603 36173 40941 05099 07400 09878 30460 35955 40647 05063 7333 09771 30320 35743 40362 05028 07269 09668 30184 35537 40087 04994 07206 09569 35 40 45 45388 48909 51887 12623 15321 18046 45006 48457 S I 37i 12463 15106 17773 44639 48023 50877 12310 14902 i75i4 44285 47606 50404 12163 14707 17267 43944 47205 49949 12022 14520 17032 5^ 55 60 54387 5 6 463 58162 20771 23474 26137 53815 55843 5750 2 20438 23078 25678 53269 55252 56874 2OI2I 22703 25245 5 2 747 54688 56275 19820 22348 24835 52246 54148 55702 !9534 22OII 24446 65 7 75 595 22 60579 61360 28748 31295 33767 58829 59860 60622 28226 30711 33122 58171 59*1* 59921 27734 30l6l 32516 57543 58528 59254 27269 29642 31945 56944 57908 58618 26829 29151 3 J 404 80 85 90 61892 62194 62291 36158 38458 40661 61141 61435 61530 35453 37694 39842 60427 60715 60808 34791 36979 39074 59749 60030 60121 34167 36306 38352 59101 59377 59466 33578 35670 37671 95 TOO i5 62200 61938 61523 42760 44753 46632 61441 61186 60781 41888 43830 45661 60721 60472 60077 4I07I 42965 4475 2 60036 59793 59407 40303 42153 43899 59383 59!45 58768 39579 41388 43095 I 10 IJ 5 1 20 60967 60288 59497 48396 50038 51558 60239 59578 58807 47380 48979 50461 59549 58903 58151 46428 47989 49434 58892 58260 57525 45535 47060 48471 58264 57646 56928 44694 46186 47565 125 130 J 55 58609 57636 56592 5 2 953 54221 55363 57942 56994 55977 51820 53055 54167 57307 56382 55389 50760 51966 53050 56701 55798 54827 49767 5 944 52003 56122 55 2 39 5429 48832 49984 51019 140 145 15 55488 54336 53M7 56373 57258 58015 54900 53778 52618 55153 56015 56753 54339 53 2 43 52111 54013 54854 55575 53801 52731 51624 5 2 944 53767 5447i 53286 52239 5II57 5*940 52744 53433 r 55 1 60 165 51932 573 49470 58648 59158 59547 5M34 50235 49032 57371 57868 58247 50955 49784 48609 56l 7 8 56663 57034 50494 4935 48201 55060 55535 55897 5005 1 48931 47806 54010 54474 54829 170 175 1 80 48240 47025 45832 59820 59982 60034 47831 46645 45480 58514 58671 58722 47436 46277 45138 57294 57447 57497 47053 45920 44804 56152 56301 56350 46683 45573 44480 55078 55224 55272 (8) II = 1 1 5 I2 O 12-5 I3-0 13*5 < X b z 1) X b b X 1 z ~b X b & b X 1) *y I 5 10 15 08623 16685 -3855 00375 01431 03014 08619 16658 23784 00374 01427 03000 08615 16632 23714 00374 01424 02987 08611 16606 23645 00374 01421 02974 08607 16580 23577 00374 01417 02961 20 25 3 3035 1 35337 39820 04961 07145 09474 29921 35M2 39562 04929 07086 09382 29794 3495 2 393 11 04898 07029 09293 29669 34767 39067 04867 06973 09207 29546 34586 38829 04837 06919 09123 35 40 45 43 6l 5 46819 495 12 11887 i434i 16807 43296 46446 49093 H757 14170 16592 42988 46086 48689 11631 14005 16386 42689 45738 48298 11510 13847 16188 42400 45402 47921 H393 13694 15998 5 55 6j 51765 5363 ss^s 19262 21690 24077 5 I 34 53 J 34 54628 19002 21384 23725 50861 52657 54123 18753 21092 23390 5 433 52198 53638 18515 20813 23070 50020 51756 53 1 ? 1 18286 20546 22764 65 70 75 56371 57315 58010 26412 28686 ; 30892 55821 56746 57428 26015 28244 30406 55 2 93 56200 56870 25637 27824 29944 54787 55677 56335 25276 27423 29504 543oo 55*74 55820 2493 1 27040 29083 80 35 90 58483 58753 58840 33020 35068 37027 57892 58157 58241 32492 34498 36418 57326 57586 57667 31990 33957 35839 56782 57037 57H7 3 T 5 13 33442 35289 56259 5 6 5 10 56589 31056 32951 34765 95 TOO I0 5 58759 58526 58157 3^895 4^666 42336 58162 57934 57572 38248 39982 41618 5759 1 57366 57012 37634 39335 40938 57042 56822 56474 3/050 38717 40291 i 5 6 5 J 4 56299 55957 36494 38131 39675 no H5 120 57664 57359 56357 43901 45362 46711 57389 56497 55809 43i5 2 44582 45904 56538 55958 55283 42443 43843 45 J 43 56009 55440 54778 41768 43H2 44414 555 01 54942 54292 41126 42475 43723 I2 5 13 135 1 -55568 54703 53774 4795 1 49079 50091 55036 54189 53279 47119 48223 49216 54525 53695 52802 4633 1 47413 48388 54034 53220 52344 45583 46644 47601 5356i 52762 51902 44871 45913 46853 140 145 *S 52791 51766 50707 5 993 51783 52455 52316 S^n 50273 50099 50870 SI532 5 l8 59 50873 49855 49252 50008 50658 51418 5045 4945 1 48449 49191 49829 50993 50042 49061 47686 48415 49041 155 160 165 49624 48527 47424 53020 53475 53823 49212 48136 47055 52085 52531 52872 48814 47758 46698 51200 5^638 $ 1 97 2 48429 47393 4635 1 50361 5079 1 51119 48057 47039 46015 495 6 4 49986 50308 170 175 1 80 46324 45 2 36 44164 54066 54210 54256 45976 44908 43857 S3 111 53252 53298 45638 44589 43558 52207 5 2 346 5239 1 453 IJ 44280 43267 5 T 35o 51486 5i53i 44993 43979 42983 50535 50669 $07*3 B. (9) 12 II /3 = M . O I, rs 1^ )*0 l v 5 5 K ro </> X b z I X b z I X b z b X b z "b X 1 z b 5 10 15 08604 l6 554 23509 00373 01414 02949 08600 16529 23443 00373 01411 02936 08596 16503 23377 00373 01408 02924 08592 16478 23313 00373 01404 02912 08588 i6453 23249 00372 01401 02900 20 25 30 29426 344io 38598 04808 06866 09042 29308 34237 38374 04779 06815 08963 29192 34069 38155 04751 06765 08887 29078 3395 37942 04723 06716 08813 28967 33744 37734 04696 06669 08741 35 40 45 1 1 -42119 4577 47556 11281 13547 15815 41847 44762 47203 11172 13435 <I 5 6 39 41582 44456 46861 11067 13268 15469 : "41325 44159 46530 10964 13135 15305 41074 438/1 46209 10865 13007 *5*47 5 55 60 49622 51329 52721 18067 20289 22470 49237 50917 "52287 17856 20042 22188 48865 505 1 9 51867 17653 19805 21918 48504 5 OI 34 ! 5 I 46i 17457 19577 21658 48155 49761 51069 17268 19358 21408 65 70 75 53831 54691 55325 24601 26674 28682 53379 54225 54848 24285 26323 28298 52942 53775 54387 23981 25987 27930 52520 53340 53942 23689 25664 27577 52112 52919 535*3 23409 25354 27238 80 85 90 55756 56002 56080 30620 32483 34265 55271 55513 55590 30203 32036 337 8 7 54804 55042 55^8 29804 31607 33330 54353 54588 54662 29422 3"95 32892 53917 54148 54221 29055 30801 3 2 47i 95 IOO io5 56007 55795 55459 35963 37572 39089 555i8 553*0 54980 35456 "37037 38529 55047 54842 54517 3497i 36527 37994 54592 5439 1 54070 345 6 36038 37482 54152 53954 53639 34061 35569 36991 no "5 120 550" 54462 53824 40513 "41839 43066 54539 53999 53372 39928 41232 42439 54084 53553 52936 39370 40652 41839 53645 53122 52515 38836 40097 41265 53220 52705 52107 38324 39566 40716 I 2 5 130 135 53106 52321 5*476 44193 45217 46140 52666 51896 51064 43546 44553 45460 52242 5 I 484 50666 42928 439i8 44811 51832 51085 50280 42337 43312 44191 5*435 50699 49907 41771 42732 43596 I4O *45 153 50582 49648 48684 46959 47676 48291 5 l8 5 49267 48318 46266 46971 47576 49801 48898 47964 45604 46298 46893 49429 48540 47621 4497* 45654 46240 49069 48193 47288 44365 45 37 45614 155 160 165 47697 46696 45689 48805 49220 49537 47348 46363 45372 48081 48489 48801 47009 46040 45o65 4739 47792 48099 46680 45726 44766 46729 47 I2 5 47427 46361 4542i 44475 46096 46486 46784 170 175 180 44684 43687 42707 49760 49892 49935 44383 43403 42438 49021 49*51 49*93 44091 43126 42176 48315 48443 48484 43807 42856 41920 47640 47766 47807 43530 42593 1 41670 46994 47118 47158 (10) II = 16 5 I 7-0 I rs I< 3-o i< 3 5 4> X b z ~b X ~b z b X b z I X b z b X b z b 5 10 i5 08584 16429 23186 00372 01398 02888 08580 16404 23124 00372 oi395 02877 08577 16380 23063 00372 01392 02865 08573 16356 23002 00371 01389 02854 08569 16332 22942 00371 01386 02843 20 25 S^ 28857 33587 3753 1 04669 06622 08671 28750 33433 37333 04643 06577 08603 28645 33282 37!4o 04618 06533 08537 28541 33134 3695 1 04593 06490 08472 28439 32989 36767 04569 06448 08409 35 40 45 40830 "4359 1 45898 10769 12882 14994 40592 433i8 45595 10675 12762 14846 40361 43052 453oi 10584 12646 14703 40135 42794 45 OI 4 10496 12533 14564 399M 42542 44735 10410 12423 1443 5 55 60 "47816 49400 50689 17086 19146 21167 47487 49050 50321 16910 18942 20934 47168 48710 49963 16740 -18745 20709 46858 48379 49616 16575 18554 20493 46556 48057 49279 16415 18369 20284 65 70 75 S 1 ? 1 ? 52511 5309 8 23139 25^56 26913 5 T 334 I 5 2II 7 52696 22879 24769 26599 50962 51735 523^6 22628 24492 26296 50602 51364 5 T 9 2 7 22385 24224 26004 50252 51004 5!559 22151 23966 25723 80 35 90 53495 53722 53795 28702 30422 32067 ! 53387 533H j -53382 28363 30058 31678 52691 5 2 9 r 3 52982 28036 29707 3 I 334 52308 52526 52595 27720 29368 30944 5J936 5215 52219 27415 29041 30596 95 100 I0 5 53727 53532 53222 33634 35ii8 36520 533!5 53123 52818 33223 34685 36067 52916 52727 52426 32827 34269 35631 52529 52343 52046 32446 33868 35211 52153 51970 5 l6 77 32078 33481 348^6 no 115 I2O 52809 52301 51712 37833 3957 40189 52411 51910 5 J 33o 37361 38567 39683 52025 5 r 53i 50959 36907 38096 39196 5 l6 5o 51164 50599 36470 37642 38727 51286 50807 50250 36047 37205 38275 I 2 5 130 135 5i 5i 50325 49545 41228 42176 43025 50679 49962 49194 40707 41641 42478 -5 3i7 49610 48853 40206 41126 41952 49966 49269 48522 39723 40631 41446 49625 "48937 48200 39258 40153 40958 I4O M5 15 i -48720 -47856 46964 43783 44446 45 OI 3 48380 47529 46650 43225 43878 "44437 48050 47212 46344 42689 43331 43884 47730 46903 46047 42172 42807 4335 1 47419 46603 45758 41674 42302 42838 155 160 165 46051 45 I2 4 44192 45489 45873 46167 45750 44836 43916 44907 45285 45575 45456 44555 43648 44347 44720 45006 45*7 l 44282 43386 43808 44176 44459 44893 44016 43I3 1 43289 43653 4393i 170 175 1 80 43261 42337 41426 46375 46496 46536 42998 42087 41188 4578o 45900 45939 42742 41843 40956 45208 45327 45365 42492 41604 40729 44658 44775 44813 42248 41371 40507 44129 44244 44281 (II) 122 II /3= 19-0 19-5 20*0 2I O 22 O < X ~b z 1 X b z b X b z ~b X I z I X b Z b 5 10 15 08565 16308 22883 00371 01383 02832 08562 16284 22825 00371 01380 02821 08558 16261 22768 00370 oi377 02811 08550 16214 22655 00370 01371 02790 08543 16169 22545 00369 01365 02770 20 25 3 28339 3 28 47 36586 4545 06407 08348 28240 32707 36409 04521 06367 08288 28143 32571 36236 04498 06327 08230 27953 32306 359 01 04453 06250 08118 27769 32050 35579 04410 06177 08010 35 40 45 39699 42297 44464 10327 12317 14299 39489 42058 44199 10246 12214 14172 39283 41824 43941 10167 12113 14050 38885 41373 43443 10016 11919 13814 38505 40942 42969 09871 11736 *359* 5 55 60 46262 47745 48951 16260 18190 20081 45977 4744i 48632 16110 18017 19885 45698 47145 48322 15065 17849 19694 45161 46575 47725 15686 17528 *933o 44650 46033 47158 15423 17224 18987 65 70 75 49912 50654 51202 21924 23716 2545 4958i 5 3i4 50855 21705 23474 25186 49259 49984 505*8 21492 23240 24931 48640 49349 49870 21086 22793 24445 48053 48746 49255 20704 22373 23987 80 85 90 51574 51785 5 l8 54 27121 28725 30261 51222 5i43i 5*499 26836 28420 29937 50880 51087 5H53 26561 28126 29623 50223 50426 50489 26037 27565 29026 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33282 34025 34695 140 M5 15 44949 44214 43453 37891 38454 38939 44468 43748 43003 37185 37737 38212 44008 43302 42571 36519 i 37061 37526 , 43567 42874 42157 35888 36420 36876 43*43 42463 41759 35290 358i2 36260 155 160 I6 5 ! 42675 41883 41086 39345 39673 39925 42240 41464 40683 386tO i 38931 39 X 78 41823 41062 40296 379*5 38231 38472 41423 40676 39924 37258 37568 37805 41039 403^5 39566 36635 36940 37173 170 175 180 40287 39494 38712 40101 40206 40240 39900 39123 38356 39351 39454 39487 39528 38766 38014 38643 3 8 744 i 38776 39170 38422 37683 37973 38072 38103 38825 38090 37364 37337 37434 37465 (13) II p= 19-0 i9 5 2O O 2I O 22 O $ X b z b X b z 1 X b z b X ~b z b X 1) z b 5 10 15 ^08565 16308 22883 00371 01383 02832 08562 16284 22825 00371 01380 02821 08558 16261 22768 00370 oi377 02811 08550 16214 22655 00370 01371 02790 08543 16169 22545 00369 01365 02770 20 2 5 3 28339 32847 36586 04545 06407 08348 28240 32707 36409 04521 06367 08288 28143 32571 36236 04498 06327 08230 27953 32306 359oi 04453 06250 08118 27769 32050 35579 04410 06177 oSoio 35 40 45 39699 42297 44464 10327 12317 14299 39489 42058 44199 10246 12214 14172 39283 41824 4394i 10167 12113 14050 38885 41373 43443 10016 11919 13814 38505 40942 42969 09871 11736 I359 1 5 55 60 46262 47745 48951 16260 18190 20081 45977 47441 48632 16110 18017 19885 45698 47145 48322 15065 17849 19694 45161 46575 47725 15686 17528 19330 44650 46033 47158 15423 17224 18987 65 70 75 49912 50654 51202 21924 23716 2545 49581 50314 50855 21705 23474 25186 49259 49984 505*8 21492 23240 24931 48640 49349 49870 21086 22793 24445 48053 48746 49255 20704 22373 23987 80 85 90 51574 51785 51854 27121 28725 30261 51222 51431 5*499 26836 28420 29937 50880 51087 5H53 26561 28126 29623 50223 50426 50489 26037 27565 29026 49600 49798 49860 25543 27037 28465 95 IOO I0 5 5!7 8 9 51608 51319 31723 33108 ; 34415 5 J 435 5 I2 5 6 50970 31380 32748 34038 51091 S9 I 4 50632 31049 32400 33674 5 43o 50257 49982 30418 3*738 32981 49802 49633 49364 29825 3i"5 3 2 33o no "5 I2O 5933 50460 49910 35640 36783 37839 50589 50123 49580 35248 36376 374i8 5^255 49795 49258 34869 35982 37012 49613 49164 48640 34147 35 2 33 36239 49004 48565 48053 33470 3453i 355M 125 130 135 49 2 93 48614 47887 38810 3969.1 40488 48970 48300 47582 38377 39249 40034 48656 47995 47285 37959 38820 39596 48052 47407 46714 37163 : 38005 38762 47478 46848 46170 36418 37240 379 8 o 140 M5 15 47116 46310 45476 41195 41814 42344 46821 46024 45201 40732 41343 41867 46533 45746 44933 40285 40889 41407 45979 45210 44417 39435 40025 40532 45452 44700 43924 38639 39215 39711 *55 160 165 44622 43756 42882 42789 , 43 48 43423 44358 43502 42639 42307 42661 42933 44100 43 2 54 42402 41841 42191 42460 43602 42775 41944 40955 41299 41560 43128 42319 41506 40125 40460 40717 170 175 1 80 42010 4"43 40290 43618 ! 43732 : 43769 1 41777 40921 40078 43125 43238 43275 4i55 40704 39870 42649 42761 42797 41111 40284 39468 41744 41854 41889 1 40690 39881 39082 40897 41004 41039 (12) II /3 = _ / >3 24 25 26 27 <t> X b z 1) X b z I X b z b X ~b z ~b X b z ~b 5 10 15 08535 16123 22438 00365 01359 02750 08528 16079 22333 00368 oi354 02731 . 08521 16035 22230 00368 01348 02712 08513 >1 599 2 22130 00367 oi343 02694 1 08506 -I 5949 22032 00367 01338 02676 20 25 30 27591 31803 35269 04368 06107 07907 27418 31564 3497i 04327 06039 07808 27250 31333 34684 04288 5974 07713 27087 31110 34407 04250 05911 07622 26928 30894 34MO 04213 05850 07535 35 43 45 38141 40529 42516 09733 11561 J3379 3779 1 40134 42082 09601 H395 *3I77 37455 39755 41666 09475 11236 12985 37i3i 39390 41267 09354 11084 12802 36818 39039 40883 09238 10939 12627 5 55 60 44162 : -45516 46618 I5I73 16937 18662 43696 45023 46102 1J 4936 16664 18355 43249 4455i 45609 14711 16405 18063 , 42821 44099 45!37 14496 16159 17786 42410 43665 44684 14291 -I 5924 17522 65 70 75 47493 48171 1 -48670 20343 21976 23556 46960 47624 48112 2OO02 2l6oi 23148 46451 47101 47579 19678 21246 22761 45963 46601 47070 19370 20908 22395 45495 46121 46582 19077 20586 22047 80 35 90 49008 49201 49262 25078 26540 27937 48443 48632 48692 24638 26070 27438 47905 48089 48148 24221 25626 26966 47389 47570 47628 23826 25204 26519 46895 47073 4713 23452 24803 26094 95 TOO I0 5 49205 4939 48776 29268 30529 31718 48636 48474 48216 28741 29976 3II40 48092 47934 47682 28243 29453 30594 47573 47418 47170 27771 28958 30077 47076 46924 46680 27323 28488 29586 no ri 5 120 48424 47995 47494 32833 33871 34833 47872 4745 r 46961 32232 33 2 49 34I9 1 47345 46931 46452 31665 32662 33585 46840 46434 45964 31127 32105 33010 46356 45958 45496 30616 31577 32465 I2 5 130 135 46932 46315 45651 357i8 36521 37246 46410 45806 45156 35057 35844 36554 459 11 45319 44682 34433 35204 35901 45434 44853 44228 33842 34599 35282 44976 44406 43793 33282 34025 34695 140 J 45 15 44949 44214 43453 37891 38454 38939 44468 43748 43003 37185 37737 38212 44008 43302 42571 36519 37061 37526 43567 42874 42157 35888 36420 36876 43M3 42463 41759 35290 35812 36260 155 1 60 165 42675 41883 41086 39345 39673 39925 42240 41464 40683 386tO 38931 39!78 41823 41062 40296 37915 38231 38472 41423 40676 39924 | 37258 37568 37805 41039 403^5 39566 36635 36940 37 73 170 175 180 40287 39494 38712 40101 40206 40240 39900 39123 38356 3935 1 39454 39487 39528 38766 38014 38643 3 8 744 38776 39170 38422 37683 37973 38072 38103 38825 38090 37364 37337 37434 37465 (13) II p= 28 29 30 31 32 <f> X b z b X b z b X b z b X b g b X b g b 5 10 15 08499 i59 3 7 21935 00366 01332 02659 08401 15865 21841 00366 01327 02642 08484 15824 21748 00366 01322 02626 08477 15784 21657 00365 01317 02610 08470 15744 21568 00365 01312 02594 20 25 3 26773 30684 33881 04177 05792 0745 1 26623 30481 33631 04M3 05736 0737 26476 30283 33388 04109 05681 07292 26332 30090 33!53 04076 05628 07217 26192 29903 32924 04044 05576 07144 35 40 45 36517 38701 40514 09127 10800 12459 36226 38376 40159 09020 10666 12298 35944 38061 39816 089 [7 io53 8 12144 35671 37756 39485 08818 10415 11996 35408 37462 39 l6 5 08723 10296 11854 5 55 60 42015 43248 44249 !4095 15700 17269 41634 42847 43831 13907 15485 17028 41268 42461 43429 13727 15280 16797 40914 42088 43 4i 3555 15083 16576 40573 41729 42667 i33$9 14894 16364 65 70 75 45046 45661 46114 18797 20280 21714 44613 45 2I 9 45664 18529 19987 21396 44199 44794 45231 18274 19708 21093 43709 44384 44815 18029 19440 20803 43413 43989 44413 17795 19184 20526 80 85 90 4642 r 46596 46652 23095 24422 25690 45966 46138 46193 22754 24059 25305 45528 45698 4575 2 22429 23711 24937 45 I0 7 45 2 74 45327 22118 23379 245 8 5 447oi 44865 449 1 7 21820 23061 24248 95 I DO 105 46599 46449 46210 26897 28041 29119 46142 45993 45759 26491 27616 28674 457oi 45555 45325 26103 27209 28250 45277 45*33 44907 25732 26820 27845 44868 44727 44504 25377 26448 27457 no "5 120 45892 455 = 1 45 47 30131 31074 31947 45446 45061 44615 29669 30595 3!453 45 OI 7 44638 44200 29228 30139 30983 44604 44231 43800 28807 29703 30534 44206 43840 43415 28404 29286 30104 125 130 135 44537 43977 43374 3 2 749 33479 34137 44H5 43564 42972 32242 32960 33605 437o8 43166 42584 31758 32464 33099 : 433i6 42783 42210 31296 31991 32616 42937 42413 41849 30854 31538 32154 140 M5 153 42736 42068 41376 34722 35 2 35 35675 42345 41687 41008 34182 34686 35n8 41967 41320 40652 33666 34162 34587 4160^ 40966 40309 33 X 73 33662 34080 41251 40625 39977 32702 33183 33596 155 160 165 40668 39947 39221 36044 36343 36572 40311 39602 38888 35482 35776 36000 39966 39269 38566 34945 35 2 34 35455 39634 38947 38255 34432 34717 34935 39312 38636 37955 33942 34222 34437 170 J 75 180 38493 37770 3/056 36734 36829 36860 38172 3746i 36759 36160 36253 36284 37862 37162 36471 35612 35704 35735 37562 36873 36192 35089 35 J 79 35210 37272 36592 35921 34588 34678 347o8 (14) II P= 33 34 35 36 37 < X b z b X ~b z ~b X b z b X ~b z 1) X b z ~b 5" 10 15 08463 15704 21481 00364 01307* 02578 08456 15665 21395 00364 01302 02563 08449 15627 21311 00363 01297 02548 08442 15589 21229 00363 01293 02534 "08435 *555i 211^8 00362 01288 02520 20 2 5 3 J 26055 29721 32702 04012 05526 07073 25921 29544 32486 03982 05478 07005 2579 1 29371 32276 03953 05431 06939 25663 29203 32073 03924 05385 06875 25538 29039 3*875 03896 05341 06813 35 40 45 35*52 37177 38855 08631 10182 11717 34904 36901 38556 08542 10071 11585 34663 36633 38266 08456 09964 11458 34429 36373 37984 08372 09861 "335 34201 36120 377*0 08291 09761 11216 5 55 60 40243 41382 42306 13229 14712 16160 39924 41046 41956 13076 14537 15964 39615 40721 41618 12928 14368 15775 39315 40406 41291 12785 14206 *5594 39024 40101 40974 12647 14049 15419 65 7 75 43040 43607 44025 17570 18938 20260 42680 43239 4365! 17354 18702 20005 42332 42883 43289 17146 18474 *9759 41995 42539 42939 16945 18255 19522 41668 42205 42600 16752 18044 19294 80 85 90 443 9 . 4447 44522 21534 22757 23925 43931 44089 44140 21260 22464 23615 43565 43721 4377 1 20996 22183 23317 43211 43365 43414 20742 21912 23031 42868 430*9 43068 20497 21651 22756 95 TOO *5 44474 44335 44116 25037 26091 27086 44093 43956 43740 24710 25749 26729 43725 43590 43377 24397 25421 26387 43368 43235 43025 24095 25*05 26058 43023 42892 42685 23805 24801 2574* I 10 "5 120 43821 4346i 43 43 28018 28886 29692 4345 43095 42683 27648 28503 29297 43091 42741 42335 27293 28135 28918 42744 42399 41998 26951 27782 28554 42407 42067 41672 26622 27442 28204 I2 5 130 135 42571 42056 41501 30432 31105 31712 42218 41710 41163 30026 30689 31288 41877 41376 40836 29636 30290 30881 4*546 41052 40520 29262 29907 30489 41226 40738 40214 28903 29538 30112 140 45 *5 40912 40295 39656 32250 32724 33133 40583 39975 39346 31818 32285 32688 40265 39665 39045 3*403 31864 32261 39956 39365 38753 31005 3*459 3*850 39656 39074 38469 30622 31070 3*455 *55 160 165 39000 38335 37664 33473 33749 33961 38699 38043 37382 33023 33295 33504 38407 37760 37108 3259 2 32860 33066 38124 37486 36842 32177 32441 32644 37849 37219 36584 31778 32038 32239 170 175 180 36991 : -36320 35658 34108 34I9 8 34228 36718 36056 35404 33649 33738 33767 36453 35800 35*57 33209 33296 33324 36196 3555 2 349*7 32786 32872 32900 35946 353*1 34683 32380 32465 32492 (15) II ft = 38 39 40 41 42 # X b z 7> X b z ~b X 1 z 1 X ~b z ~b X ~b z !> 5 10 15 08428 I55 r 4 21069 00362 01283 02506 08421 15477 20991 00362 01279 02492 08414 i544i 20914 00361 01274 02478 08407 15405 20838 00361 01270 02464 08400 !5369 20764 00360 01266 02451 20 25 3 25416 28879 31682 03869 05298 06753 25297 28723 3*494 03842 05256 06694 25180 28570 31310 03816 5 2I 5 06637 25065 28421 3H3 1 03790 05175 06582 24953 28275 30956 03765 05136 06528 35 40 45 3398o 35875 37445 -08213 09665 II 101 33764 35637 37187 08137 09571 10990 33554 35405 36936 08063 09481 10882 33349 35 T 79 36691 07991 09393 10777 3315 34959 36454 07922 09308 10676 5 3 = 5 60 38742 39805 40667 12514 13898 1525 38468 395*7 40369 12385 >I 375i 15086 38201 39238 40079 12260 13610 14928 37941 38966 39797 12139 *3473 14775 37689 38702 39523 12022 13340 14627 65 70 75 4i35 2 41882 42271 16565 17841 19374 41045 41568 41952 16384 17644 18861 40747 41263 41643 16210 17454 18656 40457 40967 4134- 16041 17270 18458 40175 40679 41050 158/8 17092 18266 80 85 90 42536 42685 42733 20261 21400 22490 42213 42361 42408 20033 21158 22233 41900 42046 42093 19813 20924 21986 41596 41739 41787 19601 20698 21747 41300 41442 41489 19395 20479 21516 95 TOO !5 42688 42559 42355 23525 24508 25436 42364 42237 42035 23256 24226 25142 42049 41924 41724 22996 23954 24858 41743 41621 41422 22745 23691 24583 41446 41325 41129 22502 23437 24318 no "5 120 42081 41745 41355 26305 27115 27866 41764 41433 41048 26000 26799 27540 41457 41130 40749 25705 26495 27226 41158 40835 40459 25420 26201 26923 40868 40549 40177 25M5 25917 26630 I2 5 130 35 4091 5 4-2434 39916 28556 29183 2975 40614 40139 39627 28222 28841 29400 40321 39852 39346 27899 285!! 29063 40036 39573 39073 27587 28192 28738 39760 39302 38808 27286 27884 28423 140 MS i5 3 39366 38791 38194 30253 30695 3 I0 75 39084 38516 37927 29897 3 334 30709 38810 38249 37667 29554 29985 30356 38543 37989 37414 29223 29648 30015 38284 37736 37168 28903 29323 29685 155 1 60 165 37582 36960 36333 3!394 31651 31849 373 2 3 36708 36088 31023 31278 31473 37071 36464 35851 30666 30918 31111 36825 36226 35620 30321 30570 30761 36586 35994 35395 29988 30234 30423 170 175 1 80 35703 35 76 34456 31989 32072 32099 ; 35467 j 34847 34235 31611 31693 31720 35237 34625 34020 31247 31328 31354 35 OI 3 34408 338n 30895 30975 31001 34795 34197 33606 30556 30634 30660 (16) II P= 43 44 45 46 47 < X 1} z b X !) z ~b X I z I X b z I X I z ~b 5 10 i5 08393 15334 20691 00360 oi26r 02438 08387 15299 20619 00359 01257 02426 08380 15265 20548 00359 01253 02414 08373 15231 20478 00358 01249 02402 08366 15197 20409 00358 01245 02390 20 25 3 24843 28132 30785 03741 05098 06475 24735 27992 30618 03717 05061 06424 24629 27855 30455 03694 05024 06374 24525 27721 30296 03671 04989 06326 24423 27590 30140 03649 4955 06279 35 40 45 3 2 955 34744 36223 07855 09225 10578 32765 34535 35997 07789 09144 10482 32579 34330 35777 07725 09066 10389 32398 34131 35562 07663 08990 10299 32221 33936 35353 07602 08916 IO2II So 55 60 37443 38445 39 2 5 6 11908 13211 14483 37204 38194 38997 11798 13086 >J 4344 36971 37950 38744 11691 12965 14209 36744 37712 38497 11587 12847 14078 36522 37480 38256 11485 T2732 I 395 65 70 75 39901 40399 40766 I57I9 16920 18080 39634 40126 40489 15566 16753 17900 39374 39861 40220 15418 16591 17725 39121 39602 39957 15274 16434 17556 38874 39350 39701 i5 T 33 16281 I739 1 80 85 90 41013 4H53 41199 19196 20268 21292 40733 40872 40917 19003 20063 21075 40460 40598 40643 18816 19864 20865 40195 40332 40376 18634 19671 20661 39937 40072 40116 18458 19484 20463 95 IOO r 5 4H57 41037 40844 22267 23191 24061 40876 40757 40566 22039 22952 23813 40603 40485 40295 21818 22721 23572 40336 40219 40032 21604 22497 23339 40076 39960 39776 21396 22279 23113 no US I2O 40586 40270 3993 24879 25641 26346 40311 39999 39636 24621 25375 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(17) 13 II 0= 48 49 50 52 54 <t> X ~b z 1) X b z ~b X 1 z ~b X b z ~b X b z ~b 5 10 15 08360 15164 20342 00358 01241 02378 08353 iS^i 20276 00357 01237 02366 08347 15099 2O2IO 00357 01233 02355 08333 -I 5035 20082 00356 01225 02334 08320 14972 -I 9957 oo355 01217 02313 20 2 5 3 24323 27461 29987 03627 04921 06232 24225 | 27335 29837 03605 04888 06187 24128 272II 29691 03584 04856 06143 23940 26971 29406 03543 04794 06058 23758 26740 29133 03504 04734 05976 35 40 45 32048 33746 35!48 07543 08843 10126 31879 3356o 34948 07485 08773 10043 3I7I3 33378 34753 07429 08704 09962 31392 33026 34375 07320 08571 09805 31084 32689 34013 07216 08445 09656 5 55 60 063^5 37254 38022 11387 12621 13826 36094 37034 37793 11291 12513 13705 35887 36818 37570 11198 12407 13588 35487 36400 37138 11018 12204 13363 35105 36000 36725 10847 1201 I 13148 65 70 75 38633 39104 3945 1 14997 16133 17231 38398 38865 39207 14865 ^5989 17075 38169 38630 38969 14736 15849 16924 37725 38178 38510 14488 i558o 16634 373oi 37745 38071 T4253 15324 >l6 359 80 35 90 39685 39819 39862 18287 19302 20271 "39439 39572 39614 18121 19126 20084 39199 39331 39373 J 7959 18954 19903 38736 38865 38906 17649 18624 19555 38293 38419 38459 17355 18311 19225 95 IOO 105 39822 39707 39525 21194 22068 22893 39575 3946o 39280 20998 21862 22679 39333 39220 39041 20808 21663 22472 38867 38756 38581 20442 21281 22074 38422 38313 38141 20095 20919 21697 no "5 I2O 39281 38982 38635 23667 24388 25056 39039 38743 38400 23445 24159 24820 38803 3 8 5 10 38170 23230 23937 2459 1 38347 38060 37726 22817 23510 24152 379 11 37630 37302 22426 23106 23736 125 I 3 135 38244 378i5 37353 25670 26230 26734 38012 37588 37I3 1 25428 25982 26481 37786 37366 369*4 >2 5 r 93 2574i 26236 3735 36938 36494 24743 25280 25765 36933 36528 36092 24316 24843 2 53 I 9 I 4 J 45 15 36862 36349 35817 27183 27577 27916 36645 36137 356n 26926 27316 27651 36433 35930 35409 26676 27062 27394 36023 35530 35 l8 26197 26575 26902 3563 35*47 34644 25743 26114 26436 155 1 60 165 35271 347i6 34154 28201 28431 28608 35070 345 20 33964 27934 28161 28337 34873 343 2 9 33778 27674 27898 28073 34492 33958 334i8 27176 27396 27567 34127 33603 33073 26705 26921 27089 170 175 1 80 33591 33030 32475 28733 28807 28831 33406 32851 32301 28461 28534 28557 33225 32675 32131 28196 28269 28291 32875 32335 31801 27688 27760 27782 32540 32009 31485 27207 27278 27300 (i 8) II p= 56 53 60 62 64 <#> X 1 z b X b s 1> X 1 z b X b z b X ~b z ~b 5 10 *5 08307 14910 19836 0^354 01 2 10 O2292 08294 14849 19718 00353 01203 02272 08281 14790 19603 00352 01 196 02252 08268 14732 19491 00352 01189 02233 08256 14675 19382 00351 01182 02215 20 25 3 235 82 26516 28870 03466 04677 05898 23411 26300 28617 03429 04622 05823 23246 26092 28372 03394 04569 05751 23086 25891 28136 03360 04518 05682 22930 25696 27908 03327 04468 05616 35 40 45 30787 3^364 33665 07117 08324 9S 1 S 30502 32052 3333i 07022 08209 09380 30227 3i75 2 33010 06931 08099 09250 29963 31463 32701 06844 07994 09126 29707 31185 32404 06760 07892 09007 So 55 60 34738 356i7 36329 10684 11828 12944 34386 3525 3595 10529 11653 12749 34048 34897 355 8 5 10380 11485 12563 33723 34558 35234 10237 H3 2 5 12386 33409 34231 34897 TOIOI 1 1172 12216 65 70 75 36894 37331 37651 14029 15081 16097 36505 36934 37249 13816 14849 15848 36131 36553 36863 13612 14628 15610 35773 36187 36492 13418 14417 15383 35427 35835 36135 13232 14215 15166 80 85 90 37869 37993 38032 17075 18014 18912 37463 37585 37623 16809 I773 1 18614 37073 37193 3723 16555 17462 18329 36698 36816 36853 16313 17205 18058 36338 36454 36491 16081 16959 17799 95 IOO io5 "37995 37889 37720 19766 20575 21339 37586 37482 373 l6 19453 20249 20998 37194 37092 36929 19*54 19937 20674 36818 36717 36557 18869 19639 20365 36456 36357 36199 18597 19355 20069 no 115 I2O 37494 37218 36896 22055 22723 23342 37094 36823 36507 21702 22359 22967 36711 36444 36133 21366 22012 226lO 3 6 343 36079 35774 21045 21681 22269 35988 35729 35428 20739 21364 21943 125 13 135 36534 36136 357^8 23911 24429 24897 36I5 1 3576o 35339 23526 24035 24495 35783 1 35399 i 34985 23159 23660 24II2 35429 ! 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26459 26431 26403 38652 38627 38601 28239 28209 28179 38132 38107 38080 28716 28685 28654 37587 37562 37538 29134 -29102 -29071 37023 37000 36976 29492 29461 29429 (57) V /3 15 30 45 60 90 X 1 z b X 1) z I X b z b X b z b X b z b + 42-9 43 43 1 20698 20691 20683 02440 02438 02437 30802 3 785 30768 06480 06475 06469 36246 36223 36200 10588 10578 10569 39282 39 2 56 39229 14498 14483 14469 41228 41199 41170 21314 21292 21270 43 2 43 3 43 4 20676 20669 20662 02436 02435 02434 3 752 3 735 30718 06464 06459 06454 36177 36155 36132 I0 559 10549 I0 539 39203 39 r 77 > 39 I 5 I -I 4455 14441 14427 41142 41113 41085 21248 21227 21205 43 5 43 6 437 20655 20647 20640 02432 02431 02430 30701 30684 30668 06449 06444 06439 36109 36087 36064 10530 10520 10510 39 I2 5 39099 39074 T44I3 M399 !4385 41057 41029 41001 21183 21161 21 140 43 8 43 9 44-0 20633 20626 20619 02429 02428 02426 3065 x 3o 6 35 30618 06434 06429 06424 36042 36019 35997 10501 10491 10482 39048 39222 38997 14372 I435 8 14344 40973 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20833 20782 20762 45 6 457 45 8 20506 20499 20492 02407 02406 02405 3 359 3 J 344 30328 06346 06341 06336 35647 35626 35604 I0 335 10326 10317 38596 38571 38546 14130 14117 14104 40482 40456 40429 20742 20721 20701 4 5 9 46*0 46 ! 20485 20478 20471 02404 02402 02401 30312 30296 30281 06331 06326 06322 35583 35562 35541 10308 10299 10290 38522 38497 38473 14091 14078 14065 40402 40376 4035 20681 2o66l 20641 46^2 46-3 46-4 20464 20457- 20450 02400 02399 02398 30265 30249 30234 06317 06312 06307 35520 35499 35478 10282 10273 10264 38449 38424 38400 14052 14040 14027 40323 43297 40271 20621 2O60I 20581 46-5 46-6 467 20443 20436 20430 02396 02395 02394 30218 30202 30186 06302 06298 06293 35457 35436 354i6 10255 10246 10238 38376 38352 38328 14014 14001 13988 40245 40219 40193 20561 20542 2O522 (58) V 8 I 2O 135 : 140* 145 150 p X z X Z X z X s X b b 1 ~b b I b b b b 42-9 43 o 43 * 39930 39903 39876 26375 26346 26320 38575 3855 38524 28149 281 19 28089 38057 38032 38007 28624 28593 28562 375*4 3749 37465 29040 29008 28978 1 -36952 36929 36905 29397 29360 29335 43 2 43 3 43 4 39849 39823 39796 26292 26264 26237 38499 38473 38448 28059 28029 28000 37982 37957 37932 28532 28502 28472 3744i 37417 37393 28947 28917 28886 > -36881 i -36857 1 -36834 29304 29273 29242 43 5 437 39769 39743 39716 26209 26181 26154 38423 38398 38373 27970 27941 2791! 37908 37883 37859 28442 28412 28382 37369 37345 37321 28855 28825 28795 36811 36787 -36764 292 1 1 29180 29*5 43 9 44-0 39689 39663 39636 26126 26099 26072 38348 38323 38298 27882 27853 27824 37834 37810 37786 28352 28323 28293 37297 37273 37250 28764 28734 28704 3674* 36718 -36695 29119 29088 29058 44 * 39610 26045 38274 27795 3776i 28263 37226 28674 36672 29028 44-2 39584 26018 38249 27766 37737 28234 37202 28644 36649 28998 44 3 39557 25992 38224 27738 37713 28205 37*79 28614 36626 28968 44 4 39531 25965 38200 27709 37689 28176 37*55 28585 36603 28938 44 5 395 D 5 25938 38175 27680 37665 28147 37*32 28555 36^80 28908 44-6 39479 25912 ; ^ISI 27652 37641 28118 37*09 28525 36558 28879 447 39453 25885 38126 27623 37617 28089 37086 28496 36535 28849 44-8 39427 25859 38102 27595 37593 28060 37062 28467 28819 44 9 394oi 25833 38077 27567 37569 28031 37039 28438 36489 28789 45 39376 25807 38053 27539 37546 28002 37016 28409 36467 28760 45 * 39350 2578 38029 275 11 37522 27974 36993 28380 36444 28731 45 2 39325 25754 38005 27483 37498 27946 36970 2835* 36421 28701 45 3 39299 25728 37981 27455 37475 27917 36947 28322 36399 28672 45 4 39274 25702 37957 27427 3745 1 27889 36924 28293 36377 28643 45 5 39249 25676 37933 27399 37428 27861 36901 28265 36355 28614 45 6 39223 25650 37909 27372 37405 27833 36878 28236 36333 28585 457 39198 25625 37885 27344 -37381 27805 36856 28207 363** 28557 45-8 39*73 25599 37861 273*7 -37358 27777 36833 28179 36289 28528 45 9 39148 25574 37838 "27289 -37335 27749 36810 28151 36267 28499 46*0 39123 25549 37814 27262 -37312 27721 36788 28123 36245 28470 46 ! 39098 25523 37790 27235 37289 27693 36765 28095 36223 28442 46*2 39 73 25498 37767 -27208 37266 27666 36743 28067 36202 28413 46-3 39048 25473 37743 27181 37243 27638 36720 28039 36180 28384 46-4 39023 25448 37720 27154 37220 27610 36698 2801 1 36158 28356 46-5 38999 25423 37697 27127 37197 27583 36676 27983 36136 28328 46-6 38974 25398 37673 27100 37174 27556 36654 27956 36114 28300 467 28950 25373 ! l 37650 27074 37152 27529 36632 27928 36093 28272 (59) Cambridge : I KINTED BY C. J. CLAY, M.A. AND SON, AT THU UNIVERSITY PRESS. GENERAL RE RETURN TO DESK FROM WHICH BORROWED CIRCULATION DEPARTMENT This book is due on the last date stamped below, or on the date to which renewed. , books are subject to immediate recall. LIC fill $lBRARY LOAM Piec d UCB A/M/ SFF V i1Q^ JUL 29 1977 > r- i* n r i i REC. CIR.AU6 2 77 . y INTERLI&PARY |_O, jf^ciRC APR 2 3 1985 1 FEET 2"3 1979 oftV^S uL-rrrn 1 ^ "R-i i n % % : - { igjjc\. **- m \ * Q CD j Hi ol 0, - fr C -< Q > x PEC.CIR. MAY 1 - 5C; RJl !0 LD21 32m 1, (S3845L)4970 75 General Library University of California Berkeley GENERAL LIBRARY - U.C. BERKELEY BDDDflDES51 QC < THE UNIVERSITY OF CALIFORNIA LIBRARY