idee na Bee i cree seit er ester ae 2 Ee CORNELL UNIVERSITY LIBRARY DEPARTMENT OF COMMERCE AND LABOR COAST AND GEODETIC SURVEY oO. H. TITTIMANN SUPERINTENDENT GEODESY THE EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION UPON THE INTENSITY OF GRAVITY BY JOHN F. HAYFORD Formerly Inspector of Geodetic Work and Chief of the Computing Division AND WILLIAM BOWIE Inspector of Geodetic Work and Chief of the Computing Division Assistant, Coast and Geodetic Survey SPECIAL PUBLICATION No. 10 WASHINGTON: a3 GOVERNMENT PRINTING OFFICE wey 1912 gi fea, CONTENTS. Page Generalistatementsai cosa cel. agen ease ee Gane at oc uated Cum east Mi re aah Ltd 5 TeOStASY: CONN ed shea sve Pe cee ese ue alihus Stein eda apatiaieaiany Lo dee eenl a eRe 2 ened ies aida oo 6 Assumptions as to isostasy..... 2.22.00 002 eee ee ee cee eee cece eee eceeeeeeeeeseeesreeeess 10 Formulas: Sinneaciodec ocasatnacives-smaaumec ke oa esac netnel sah dade semstred videlsaatekteoscudate: Guioe side 12 Division of the surface of the earth into zones and compartments..............2.0-0222.0ec eee e cee eeeeeeeeees 17 Computation of reduction tables for near zones............... 20000202 eee eee eee cee eee cee eee e eee cesses 19 Computation of reduction tables for distant zones...........2. 0.000200 e cece eee cence ee eee cece ee ences 23 Explanation of reduction tables.c;éscocqosesecteci ccmeseca ds tac deeremewe vy sve exec eieaiea' ales Sabdetouin daw eile 28 Reduction tables for lettered zones............. 00020202 c eee ene eee eee nee eee e eee e een eee 30 Reduction tables for numbered zones......... 22.200 eee eee eee een cece teen eee eeeees 44, Special reduction tables for sea stations..............222.0 0.0220 o eee ee eee ee ee eee reese 46 FU seson templates 5.222 cic sictecdsaiics de rane 2 at LAG Oe ie Bat emt ced banat Whe sanmund Sako L ane Oe aes AT Examples of computations of corrections. ..............0.. 20202020 eee ee eee eee eee eee nee eens 48 Corrections for topography and isostatic compensation, separate ZoneS...--.-......022-2 22 eee eee eee 53 Initérpolation:for Outer: 2ON eS! 23 456.02 vevesensnen od sx Sede e 22+ 5 exept eete ds Se oy eee de yer el Seek ee vosies 58 Method of interpolation for outer zones.......-.----.-.-22-2-020 02222 eee eee Beige s tee Sages 60 Criteria of accepted interpolations for outer zones........-..-.. 2.222202 2 eee eee ence eee eee 63 Saving by interpolation for outer zones.........-..2.22- 2222022 e eee eee eee eee eee eect teense 64 Change of sign due to distance 22.0. ee cvs cds hens wanes poss Smueieiimeidie See ed onaionoele See ba dhciamelee ds 26 65 Distant topography necessarily considered.............2.2.2.2. 222020222 cece cece cece eee ete e nee neee 71 Curvatite:must be considered 22. essa. sceccccedes Peseeecuica y Soe SA eeee eae See see eeenioes Sak ysl megie es eis s 71 Principal facts for 89 stations in the United States.................. atta Ma tates alas leita Aah SrA i eaetun a has 72 Correction to Helmert’s formula of 1901..............0..20 2220202 e eee eee eee eee eee eeee 75 Comparison of apparent anomalies by new and old methods...........-... 22. 2.222222 202002 c ee eee eee eee eee 75 Possible relations of anomalies to topography.........-...2.-.22-- 22-2222 eee eee eee eee shee 34 Eaaeeaedunne tes 1T Comparison of Bouguer anomalies with new-method anomalies................-.2..2-2-2020 2022s cece ee eee ee 79° Comparison of free-air anomalies with new-method anomalies................2-2-.-0. 202.22 e cece eee eee eee 80 Test by stations not in the United States...........-.22. 222222222 e eee 81 VISCO UBSLOLN OL: CEN OMS 25555 ve 2 Son avy veuZue ced: S55 ocd aeuesevesonteas Se seve ep eeeaag eset e oe e Gein cacnanns ize oa aS Beans hehe as eds Stubbs ay SR SAA 86. HrrorsOHOMSePVatODN ic iste ec ee c wee tae oA es 2s HE Oten Reds Zr esete oedaneeeeee see 87 Errors of computations... 220 sean ocivinnewudese Seles egioetees bt Pe eee eee teas Ee RU RIES Oe ctemictigeeey ee 88 Nature of apparentianomallesncccciscsaeeeeeccene yo. sy acter des oc se eden dey esas eyeweGee ese eeeeerRe Less 94 The method not subject to hidden errors........... 2.2.2.0 202222 e eee eee eee ee 95° Effects of topography and compensation—why combined............. 2.2.2.2. 2 020 eee eee ee eee eee 97 Regional versus local distribution of compensation.....-..--.....---++-.222 02022022 e eee ee eee eee eee 98 Test of depth of compensation ..o0)c2.sless' ot tgeeieiew es oes oeeaes eyes agen see neem ede te das aende eee ees 103 Graphical comparison of three kinds of anomalies.........--..----- 22-222 0eeeee eee eee eee ec ee eee eee eee 106 Interpretation of anomalies in terms of masses.........--.----- +02 +2202 eee eee eee eee eee eee eee ee 108 Possible relation of new-method anomalies to other things..-........-.-.-..--.-2----2--2- 2222 e eee eee seegieiae's 112 Relation between new-method anomalies and geologic formations...........-...-.-2-2-2-2-2020 eee eee eee eee 113 Discussion of other regional peculiarities. ............---- 20-20-22 eee eee eee ee 117 Hypothesis of horizontal displacement of compensation............---- +. 22-00-2052 cee ee eee e eee e ee eee 121 Comment on Bouguer and free-air anomalies...........-.---- 2-2-2202 c ee ee eee eee eee eee eee eee 122 Comment on Faye method of reduction.............----. 2-2-2222 eee ere eee 125, Summary...-.-----2--- 2-22 ee ee eee ee nent ete tee eee eee eee eee ee 126 BEE eee eB ee ee CONBNMARWNESOS {LLUSTRATIONS. Page . Three unit columns showing ideal depth of isostatic compensation...........2-2.22 0-0-0022 e cece reece eee 7 . Three unit columns showing approximate depth of isostatic compensation as used in computations......... 10 . Showing station and elementary mass at same elevation...........---.-------2200e eee eee eee eee eee eee 15 Showing elementary mass at greater elevation than station..........-.2----- 0-202 eee eee eee eee eee eters 16 . Showing elementary mass at less elevation than station.............-----2---2--e ee eee cece ee eee cere ee eees 17 . Showing topography in land zones—three cases.............0-2- 220-22 e eee eee eee eee eee eee ee eeee 20 . Graphical computation of reduction table for Zone E.................-0-- 20222 eee eee eee eee eee 22 . Graphical representation of values of E for various depths.......-...---.--.----+ 2-20-02 e eee eee ee eee eee 23 . Graphical representation of values of Eg..............2 2-0-2222 eee cee eee eee eee 24 . (a) Template for maps of scale 1/10000 (reduced)...........-.---2- 22-2222 eee e ce ee ee eee ee eee eee eens 48 . (b) Template for maps of scale 1/6018500 (reduced)......-.1.--. 222-022 eee eee eee eee eee eee eee eee eee 48 . Overlapping of corresponding zones for two stations.........-...-.- 222-222 c eee eee eee eee eee tect eeee 58 . Graphical illustration of interpolation...............0-....-. 2202-2 e eee ee ee ee eee eee eeee 60 . Map showing location of gravity stations used in the investigation .......-..--.---.---2---eee eee e eee In pocket . Showing topography and compensation near station...........------- 20022 - eee eee eee eee eee ees 66 . Showing distant topography and compensation...........-...----.---+--+--- waUspe ee 6 Ree Sttaere 2 Srepcreeteds ee 67 . Lines of equal anomaly for new method of reduction ............-...-2-2 22-2202 ee teen e eee eee ee In pocket . Lines of equal anomaly for Bouguer method of reduction ...........-.----.-2-22--2-2 2220-222 e eee ee In pocket . Lines of equal anomaly for free-air method of reduction .......-.....-- 2222-2220 ee eee eee eee eee eee In pocket . Illustration from Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, showing resid- uals of Solution H, all stations, with areas of excessive and defective density, and showing also all gravity stations with new-method anomalies -.._......... 2.22.22 2 222s eee eee cece eee eee eens In pocket 4 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION UPON THE INTENSITY OF GRAVITY. BY Joun F. Hayrrorp,* formerly Inspector of Geodetic Work and Chief of the Computing Division, AND Witu1am Bowis, Inspector of Geodetic Work and Chief of the Computing Division, Assistant, Coast and Geodetic Survey GENERAL STATEMENT. In the United States the assumption of isostasy in a definite and reasonable form has been introduced into the computations of the figure and size of the earth from the observed deflections of the vertical. These computations have shown that the assumption of isostasy is substan- tially correct. They have shown that a close approach to perfect isostatic compensation exists under the United States and adjacent areas. This is important to geology and geophysics. They have also shown that the proper recognition of isostasy in making computations of the figure and size of the earth from observed deflections of the vertical has about doubled the accuracy of such computations by reducing errors of both the accidental and the systematic classes in such work. “This increase in accuracy is important to geodesy. These computations and the investigations of which they form a part have been published in full.t As soon as it was evident that the proper recognition of isostasy in connection with com- putations of the figure and size of the earth from observed deflections of the vertical would produce a great increase in accuracy, it appeared to be very probable that a similar recognition of isostasy in connection with computations of the shape of the earth from observations of the intensity of gravity would produce a similar increase of accuracy. Logically the next step to be taken was therefore to introduce such a definite recognition of isostasy into gravity compu- tations. Moreover, it appeared that if this step were taken it would furnish a proof of the existence of isostasy independent of the proof furnished by observed deflections of the vertical, and would therefore be of great value in supplementing the deflection investigations and in testing the conclusions drawn from them. In other words, the effects of isostasy upon the direction of' gravity at various stations on the earth’s surface having been studied, it then appeared to be almost equally important to investigate the effects of isostasy upon the intensity of gravity. It was evident from the beginning that to properly take into account the possible existence of isostasy in connection with computations of the intensity of gravity a rather extensive revi- sion of formule and methods of computation would be necessary, and that the computations must be thorough and must involve a considerable number of gravity stations if the results were to be convincing. Thus it was realized that the problem was both a large and a difficult one. Partly for this reason, Mr. Hayford, as inspector of geodetic work, recommended frequently from 1900 to 1908 that the Coast and Geodetic Survey confine its energy in geodetic observations and * Now Director, College of Engineering, Northwestern University, Evanston, Tl. + The Figure of the Earth and Isostasy from Measurements in the United States, by John F. Hayford, published in 1909 by the Coast and Geodetic Survey, and Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, by John F. Hayford, published in 1910 by the Coast and Geodetic Survey. [ach of these is a separate publication not included in the annual reports of the survey. They may be obtained by interested parties on application to the Superintendent of the Coast and Geodetic Survey, Washington. D. C. 5 6 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. investigations mainly to deflections of the vertical until that part of the field of investigation had been well covered and reasonably safe conclusions reached, and that then, and not till then, should much energy be expended in gravity observations and the corresponding investigations. This policy was adopted and adhered to. In the summer of 1908 Mr. Hayford began an extensive study of the theoretical side of the investigation, the revision of formule and of methods of computation. Early in 1909 a long, continuous series of gravity observations with the half-second pen- dulum apparatus at various stations in the United States was commenced. This series is still in progress. In this publication there are used 89 stations, including those of this series which are available at this time. : In January, 1909, Mr. Bowie became closely associated with Mr. Hayford at the Coast and Geodetic Survey office and was brought into close touch with the investigation set forth in this publication. In October, 1909, he assumed his present position, and has since that time been in charge of the gravity observations and computations of gravity made in the Coast and Geo- detic Survey, of which many are utilized in this publication. In certain lines he has extended the investigation beyond its former limits. In the preparation of this publication the two authors have cooperated. They are jointly responsible for the opinions expressed and the statement of conclusions reached. Miss Sarah Beall, computer, efficiently supervised much of the computing in connection with this investigation, and especially the computation of the reduction tables, the most diffi- cult part of the work. To her and to the various members of the computing division who assisted, the credit is largely due for the unusual rapidity and success of the computations. In September, 1909, Mr. Hayford presented to the International Geodetic Association at London a paper bearing the same title as the present publication. It has been printed as pages 365-389 of Volume I of the Report of the Sixteenth General Conference of the Inter- national Geodetic Association, held at London and Cambridge in September, 1909. The present investigation is in many respects a counterpart of the previous investigations based on deflections of the vertical, to which reference has already been made. It supplements those investigations, and therefore the three should be studied together to obtain their full force. The computations of the present investigation have been based upon certain assumptions as to the existence of the condition called isostasy which are substantially identical with the assumptions in the previous investigations involving deflections of the vertical. It is important to the reader to understand clearly the meaning of the word isostasy and of certain related phrases, as otherwise he may fail to understand, or may misunderstand, many statements in this publication. These definitions are given below in substantially the same words as were used in connection with the previous investigations. ISOSTASY DEFINED. If the earth were composed of homogeneous material, its figure of equilibrium, under the influence of gravity and its own rotation, would be an ellipsoid of revolution. The earth is composed of heterogeneous material which varies considerably in density. If this heterogeneous material were so arranged that its density at any point depended simply upon the depth of that point below the surface, or, more accurately, if all the material lying at each equipotential surface (rotation considered) was of one density, a state of equilibrium would exist, and there would be no tendency toward a rearrangement of masses. The figure of the earth in this case would be a very close approximation to an ellipsoid of revolution. If the heterogeneous material composing the earth were not arranged in this manner at the outset, the stresses produced by gravity would tend to bring about such an arrangement; but as the material is not a perfect fluid, since it possesses considerable viscosity, at least near the surface, the rearrangement will be imperfect. In the partial rearrangement some stresses will still remain, different portions of the same horizontal stratum may have somewhat different densities, and the actual surface of the earth will be a slight departure from the ellipsoid of revolution in the sense that above each region of deficient density there will be a bulge or bump EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 7 on the ellipsoid, and above each region of excessive density there will be a hollow, relatively speaking. The bumps on this supposed earth will be the mountains, the plateaus, the conti- nents; and the hollows will be the oceans. The excess of material represented by that portion of the continent which is above sea level will be compensated for by a defect of density in the underlying material. The continents will be floated, so to speak, because they are composed of relatively light material; and, similarly, the floor of the ocean will, on this supposed earth, be depressed because it is composed of unusually dense material. This particular condition of approximate equilibrium has been given the name “ isostasy.”’ The adjustment of the material toward this condition, which is produced in nature by the stresses due to gravity, may be called the “isostatic adjustment.” The compensation of the excess of matter at the surface (continents) by the defect of density below, and of surface defect of matter (oceans) by excess of density below, may be called the “isostatic compensation.” Let the depth below sea level within which the isostatic compensation is complete be called the “depth of compensation.” At and below this depth the condition as to stress of any element of mass is isostatic; that is, any element of mass is subject to equal pressures from all di- rections as if it were a portion of a perfect fluid. Above this depth, on the other hand, each element of mass is subject in general to different pressures in different directions—to stresses which tend to distort it and to move it. Consider the relations of the masses, densi- ties, and volumes, above the depth of compen- sation, fixed by the preceding definition. The mass in any prismatic column which has for its base a unit area of the horizontal surface which lies at the depth of compensation, for its edges vertical lines (lines of gravity) and for its upper limit the actual irregular surface of the earth (or the sea surface if the area in question is beneath inland Column Sea Goast Column Ocean Column the ocean) is the same as the mass in any other I:tustration No. 1.—Three unit columns showing ideal depth of similar prismatic column having any other unit i a area of the same surface for its base.* Tllustration No. 1 represents three such unit columns. Let the depth of compensation be called h, and the mean surface density of the solid portion of the earth be called 6. Then the mass of material in a column of unit area at the seacoast is oh, + (density times volume). , Let the elevation above sea level of the irregular surface of the earth over the unit area of an inland column be called H. Then the mass of material in the inland column above sea level is OH. Also, let the density of that portion of the inland column between sea level and the depth of compensation be called 6; Then the mass of material in the column is expressed by the equation Surface of ground Mass in any land unit column =dH +0,h, (1) By definition, at the depth of isostasy, any element of mass is subject to equal pressures from all directions as if it were a portion of a perfect fluid. In order that this may be true, the vertical pressures due to gravity on the various units of area at that depth must all be the * It would be more accurate to use the words “inverted truncated pyramid” instead of “prismatic column.” The latter expression has been selected because it is sufficiently exact for the purpose and corresponds to the allowable approximations actually made in the mathematical part of the investigation. + For the purpose of this demonstration it is assumed that the average density of the earth’s crust below the seacoast between sea level and the depth of compensation is equal to the average density of the solid portion of the earth’s surface (2.67). This assumption ignores the probability that within a depth as great as 114 kilometers (the assumed depth of compensation) there is probably a slight increase in density with increase of depth, due to increased pressure, the density being some unknown function of the depth. This neglect also appears in various other places in this publication. It is shown later, under the heading “ Discussion of errors,”’ that this neglect introduces no appreciable errors into the computation. Tt is justified, therefore, as a means of avoiding unnecessarily long and complicated statements. 8. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. same, and therefore masses of the various unit columns must all be the same. Therefore the mass in a land unit column must be equal to the mass in a seacoast column, or OH +6rh,=6h, (1a) From equation (1a) it follows that 3,— Sa) (2) Pod The difference called 0, between 4 and 4; is expressed by the equation 3,-0- a (2a) 1 AH ae 3 = 95, (3) This difference between the normal density at the surface of the land and also throughout a column at the seacoast on the one hand, and the density of an inland column below sea level on the other hand, is the average compensating defect of density, and this difference multiplied by the depth of compensation is the compensating defect of mass, 0,h,. The total mass in the inland column may also be expressed by the equation (see illustra- tion No. 1), Mass in any land unit column =0H +0h,—6,h, (4) As the mass in each unit column is the same, namely 6A,, it is obvious from equation (4) that OH=6,h, (4a) This equation is a statement in mathematical symbols that in each unit column the com- -pensating defects of mass below sea level must be exactly equal to the mass above sea level which is considered to be the surface excess. Equation (3) indicates that the compensating defect of density is proportional to the elevation of the surface above the sea level as 0 and h, are assumed to be constant. In an ocean unit column the top of the solid portion happens to be below sea level, being a part of the bottom of the ocean. In the ocean column let the depth of the water be called D and the density of the sea water 0. Then the depth of the solid portion of the column will be h,—D. Let the density of this solid portion be called 6,. Then the mass of material in this unit column will be expressed by the equation Mass in any ocean unit column =6,D+0,(h, —D) (46) By definition, this mass must equal the mass of the unit column at the seacoast, hence byD + d4(h, — D) =sh, (4c) From equation (4c) it follows that _dhy—OyD ~ h,—D The difference 0, between the density of the solid portion of the ocean column 6, and the normal density 6 is expressed by the equation _dhy—dyD Oo (4d) Ve ag (4e) — h-D- od The total mass in any ocean unit column may also be expressed by the equation (see illus- tration No. 1), Mass in any ocean unit column =6,,D + (0 +6,)(A,—D) (5) EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 9 As the mass in each unit column is the same, namely 6/,, it follows from equation (5) that D(6—6y) =6,(h, —D) (5a) That is, in the solid portion of each ocean unit column the compensating excess of mass must be exactly equal to the defect of mass in the water portion of the column. Equation (4f) indicates that the compensating excess of density is nearly proportional to the depth of water, as 6 and 6, are assumed to be constant and (h,—D) is approximately constant. In this publication the mean surface density of the solid portion of the earth, 0, is assumed to be 2.67. The density of sea water, 0,, is 1.027. With these values 0—6,,=0.6150. Hence, for oceanic unit columns, equation (5a) becomes (d—0,)D =0.615dD = 0,(h, —D) (5d) and equation (4f) becomes 0.615D b= 05 iy o Note that equation (6) differs from equation (3) only by containing the factor 0.615; in having D, a depth, in the place of H, an elevation; and in having (A,—D) as a denominator instead of h,. As a concrete illustration, consider three unit columns such as are indicated in illustration No. 1, one beneath a mountain summit at an elevation of 3 kilometers, one underlying an area which is at sea level, a portion of the seashore for example, and the third under the ocean at a point where it is 5 kilometers deep. Let the depth of compensation be assumed to be 114 kilometers below sea level, and the mean surface density 6=2.67. In the first column the ratio H to h, being op according to equation (3) the defect of density, 6,, is a of 2.67 or 0.07, and the density of the material below sea level is 2.67—0.07=2.60. In the second column the density of the material is 2.67. In the third column the compensating excess of density of the material underlying the ocean is, by equation (6), pO619)O) _ 93. =0.07 and the density of the material is therefore 2.67 + 0.07 =2.74. Under such a mountain, therefore, if isostasy exists as defined by the stated assumptions, the average density is about 3 per cent less than under the seacoast, and on the other hand, under a portion of the ocean 5 kilometers deep the average density is about 3 per cent greater than under the seacoast, down to the depth of compensation in each case. As a rough approximation it may be stated, on the basis of the preceding paragraph, that beneath areas which lie above sea level the density is defective by about 1 per cent for each kilometer of elevation of the surface. Since much of the land portion of the earth’s surface is at an elevation of less than 1 kilometer and very little of it above the elevation 3 kilometers, the compensating defects of density beneath most land areas are less than 1 per cent of the mean density and exceed 3 per cent only under a few small areas on very high mountains. Similarly, the compensating excesses of density under ocean areas seldom exceed 3 per cent as the depths exceed 5 kilometers (16 000 feet or 2700 fathoms) in but a small portion of the ocean. If the condition of equal pressures, that is of equal superimposed masses, is fully satisfied at a given depth, the compensation is said to be complete at that depth. If there is a variation from equality of superimposed masses, the differences may be taken as a measure of the degree of incompleteness of the compensation. In the above definitions it has been tacitly assumed that g, the intensity of gravity, is everywhere the same at a given depth. Equal superincumbent masses would produce equal pressures only in case the intensity of gravity is the same in the two cases. The intensity of gravity varies with change of latitude and is subject also to anomalous variations which are to some extent associated with the relation to continents and oceanic areas. But even the 10 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. extreme variations in the intensity of gravity are small in comparison with the variations in density postulated. The extreme variation of the intensity of gravity at sea level on each side of its mean value is only 1 part in 400. Even this small range of variation does not occur except between points which are many thousands of kilometers apart. As will be shown later, the postulated variations in mean densities are about 1 part in 30 on each side of an average value. Hence, it is not advisable to complicate the conception of isostasy and introduce long circumlocutions into its definition in order to introduce the refinement of considering the variations in the intensity of gravity. The variation of the intensity of gravity with change of depth below the surface need not be considered, as its effect in the various columns of apaterial considered will be substantially the same. The idea implied in this definition of the phrase “depth of compensation,” that the isostatic compensation is complete within some depth much less than the radius of the earth, is not ordinarily expressed in the literature of the sub- ject,* but it is an idea which it is difficult to Sea level avoid if the subject is studied carefully from any point of view. | Surface of ASSUMPTIONS AS TO ISOSTASY. In the computations of the investigation here published the depth of compensation is assumed to be 113.7 kilometers under every separate por- tion of the earth’s surface. This is substantially the value given in The Figure of the Earth and Isostasy, page 175. It was the best value available at the time the com- putation of the gravity reduction tables pub- lished herein was commenced. A better value, 122 kilometers, became available while these com- on putations were in progress, but too late to be used. Cok rc (See Supplementary Investigation in 1909 of The ILLUSTRATION No. 2.—Three unit columns showing approximate depth of isostatic compensation as used in computations. Figur e of the Earth, p- 77. ) The mean surface density of the earth—that is, the mean density of the solid portion of the earth for the first few miles below the surface— is assumed in this investigation to be 2.67. The phrase ‘of the solid portion of the earth” is inserted in the preceding sentence to indicate that the ocean, with a density of only 1.027, is excluded from this mean. The computations concerned in this investigation were actually made on the assumption indicated in illustration No. 2 instead of those indicated in illustration No. 1 and used on pages 7-9. This slight change was made to simplify and facilitate computations and is justified by the fact that the errors so introduced are negligible, as shown later under the heading “Discussion of errors.’”’ In illustration No. 1 and in the corresponding text, the compensation is assumed to extend everywhere to a depth of 113.7 kilometers below sea level. In illustration No. 2 and in the actual computations, the compensation is assumed to extend everywhere to a depth of 113.7 kilometers measured downward from the solid surface of the earth—that is, from the land surface in land areas (above sea level) and from the ocean bottom in oceanic areas (below sea level). For land areas, in computing the direct effect of the topography, the portion above sea level was assumed to have the density 0 as indicated in illustration No. 2, but in computing the effect of the isostatic compensation the density was assumed to be d—0, * See, however, a reference to Pratt’s Hypothesis in Helmert’s Héhere Geodisie, II Theil, p. 367. } For the data and considerations upon which this value is based, see The Solar Parallax and its Related Constants, by William Harkness, Washington, Government Printing Office, 1891, pp. 91-92; see also The Figure of the Earth and Isostasy from Measurements in the United States, p. 128. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 11 above sea level as well as below, 6, being computed from formula (3). The seacoast column is the same in the two illustrations. Upon the assumption indicated in illustration No. 2 and used in actual computations for oceanic compartments, formula (6) becomes 0.615 D i (6a) 3,=0 In the computations of this investigation the compensation under each separate portion of the earth’s surface is assumed to be uniformly distributed with respect to depth from the surface down to the depth of compensation, 113.7 kilometers. In other words, the compen- sating defect or excess of density under a given area is assumed to be, at all depths less than the depth of compensation, exactly equal to the 6, of equations (3) and (6), which was defined as being the average defect (or excess) of density. Elsewhere * it has been assumed temporarily for investigation purposes that the compen- sating defect (or excess) of density varies with respect to depth, being for example greatest near the surface and diminishing uniformly to zero at the depth of compensation, its average value being 06,. In the pringival computations of this investigation the isostatic cotapenwation is assumed to be complete under every separate portion of the earth’s surface, however small the area considered. That is, equations (3) and (6) are assumed to be true los every separate unit of area even though a very small unit be chosen, as for example, 1 square foot. The authors do not believe that any one of these assumptions upon which the computa- tions are based is absolutely accurate. The mean surface density is probably not exactly 2.67 and the actual surface density in any given area probably does not agree exactly with the mean. The depth of compensation is probably not exactly 113.7 kilometers, and it possibly is some- what different under different portions of the earth’s surface. The compensation is probably not distributed uniformly with respect to depth. It is especially improbable that the com- pensation is complete under each separate small area, under each hill, each narrow valley, and each little depression in the sea bottom. It is exceedingly improbable, for example, that as each ton of material is eroded from a land area, carried out of a river mouth, and deposited on the ocean bottom, that corresponding changes of isostatic compensation occur at the same time under the eroded area and under the area of deposition at just such a rate as to keep the compensation complete under each. The authors believe that the assumptions on which the computations are based are a close approximation to the truth. They believe also that the quickest and most effective way to ascertain the facts as to the distribution of density beneath the surface of the earth is to make the assumptions stated, to base upon them careful computations for many observa- tion stations scattered widely over the earth’s surface, and then to compare the computed values with the observed values of the intensity of gravity in order to ascertain how much and in what manner the facts differ from the assumptions. In this investigation, accordingly, the intensity of gravity at many observation stations has been computed on the assumptions stated. These computed values have been compared with the observed values at these stations. The differences between the observed and the computed values, the residuals, are due to two classes of errors. In the first class are errors in the observations and in the computations. In the second class are errors in the assumptions. The average and maximum magnitudes of the errors of the first class are fairly well known. The magnitude and character of the residuals which may be produced by them are fairly well known. It is shown in this publication that the residuals, differences between observed and computed values of the intensity of gravity, are larger than may be accounted for by the first class of errors. Therefore it is certain that the second class of errors are of appreciable size. In other words, it is certain that the assumptions are appreciably in error. But, as the residuals are but little larger than may be accounted for by the first class of errors, it is certain that the assumptions are nearly correct. * The Figure of the Earth and Isostasy from Measurements in the United States, pp. 156-163. 12 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. The residuals contain evidence not only as to the extent but also as to the manner in which assumptions depart from the truth. To read and interpret this evidence precisely is exceedingly difficult, because of the fact that the residuals are small. If the residuals were large, it would be clear that the assumptions were far from the truth, and it would be easy to see in which direction the truth lay. In the actual case it is difficult to ascertain in what way the assump- tions should be changed to make them a closer approximation to the whole truth, while still remaining a statement of general laws applicable to the whole United States. FORMULE. It was desired to compute the intensity of gravity at any selected station on the earth upon the assumptions as to isostasy which have been stated. It was necessary to select the formule and methods of computation. The computations may be most conveniently made in two parts. First, the intensity of gravity may be computed on an ideal earth having the same size and shape as the ellipsoid of revolution which most nearly coincides with the sea-level surface of the real earth, and having no topography and no variations in density at any given depth below the surface. To convert the real earth into this ideal earth all material on the real earth above sea level must be removed, the water of the ocean must be replaced by material of den- sity equal to the mean surface density of the real earth, and all variations in density at any given depth in the real earth must then be removed by taking out or injecting enough material in each part to make the density conform accurately to the mean density in the real earth at that depth. In this ideal earth the density will increase with increase of depth in the same manner as it does upon an average in the real earth, but in the ideal earth all masses lying at the same depth will have the same density, whereas in the real earth such masses have densities which are known to differ slightly from each other. This computation was made by using Helmert’s formula of 1901,* namely, + 7o=978.046(1 +0.005 302 sin 26—0.000 007 sin 724) (7) The symbol ;, stands for the required value of gravity at a station on the ideal earth above described in the latitude ¢. On such an ideal earth the value of gravity at the surface would be a function of the latitude only, as expressed by this formula. The numerical value of 7, com- puted from formula (7) is both the acceleration of gravity in centimeters and the attraction of gravity in dynes on a unit mass (1 gram) at the station expressed in the centimeter-gram- second system. The form of this formula is fixed by theory. The three constants which it contains, namely, 978.046, 0.005 302, and 0.000 007, were computed from a large number of observations of gravity at stations scattered widely over the earth’s surface. New and better values of these constants may be obtained by further research and the use of more observations, but at the beginning of this investigation the formula as written was believed to be the best representation available s * Der normale Theil der Schwerkraft im Meeresniveau, von F. R. Helmert, S. 328-336, Sitzungsberichte / der K6niglich Preussischen / Akademie der Wissenschaften / zu Berlin, / Jahrgang 1901 / Erster Halbband, Januar bis Juni. See also Bericht tiber die relativen Messungen der Schwers kraft mit Pendelapparaten fiir den Zeitraum von 1900 bis 1903, unter MAtwirkung von F. R. Helmert erstattet von E. Borrass, S. 133-136, Verhand- lungen / der vom 4 bis 13 August 1903 in Kopenhagen abgehaltenen / Vierzehnten Allgemeinen Conferenz der / Internationalen Erdmessung / Redi- girt vom stindigen Secretar H. G. van de Sande Bakhuyzen. / II. Theil: Spezialberichte. See also The Figure of the Earth and Isostasy from Measurements in the United States, p. 172, for some comments upon this formula. + After the manuscript of this publication was completed, a letter addressed to the Superintendent, of which the following is a translation, was received from Dr. Helmert: : ) PoTsDAM, October 31, 1911. es Bowie sent to me a small brochure for which I offer my best thanks to the sender and to you. Permit me to make a remark in regard to my formula. In 1901 I did indeed give: -yo= 978.046 (1+-0.005302 sin 26—0.000007 sin 2 2 4) This formula is based on the value of g in the Vienna system (Sterneck). The American values of g are, however, referred to Potsdam. The constant 978.046 must, therefore, be modified by the application of —0.016 by which correction it is referred to Potsdam, as I have several times stated in my reports. I therefore request that in your investigations in North America you will use the value 7o= 978.030 (1-++0.005302 sin 26—0.000007 sin 2 2 6) as being my improved formula. f I know that your scientists think that the value 978.038 is more suitable for the United States. That value, of course, may be used. I only wanted to emphasize that, in so far as my work is concerned, the value of g in the United States is not 978.046, but 978.030. Tt is clear that the values of gravity in the United States, used in this publication, are hased upon Potsdam, as shown on p. 73, and that, there- fore, the position taken by Dr. Helmert in this letter is correct. The only manner in which this change ultimately affects the conclusions reached. in this publication is shown on p. 75. . EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 13 of gravity at sea level on the ideal earth described in the preceding paragraph. During the progress of this investigation a small correction to the constant 978.046 was derived, as shown later in this publication. The formula, with this small correction applied, is believed by the authors to be the best available at present for the purpose for which it is intended. The Helmert formula of 1901 corresponds to a value of 298.3+0.7 for the reciprocal of the flattening of the earth. This is in fair agreement with the best value now available for this quantity as derived from observed deflections in the United States, namely, 297.0+0.5.* The stations at which observations of gravity were made are situated on the real earth, not the ideal earth, and are in general above sea level, not at sea level. The second portion of the computation of the intensity of gravity at any observation station must therefore take ac- count of the topography which exists upon the real earth, take account so far as is possible of the variations in density beneath the surface of the real earth, and take account of the effect of the elevation of the observation station above sea level. The correction for elevation was computed by the formula —0.000 308 6 H in which H is the elevation of the station above sea level in meters. This correction of the attraction upon a unit mass (1 gram) at the station is in dynes and reduces from sea level to the actual station. It takes account of the increased distance of the station from the attract- ing mass, the earth, as if the station were in the air at the stated elevation and there were no topography on the earth. This is an old formula and needs no comment other than that it has been adopted in this simple form by Dr. Helmert as being sufficiently accurate. The real difficulty of the investigation was encountered when an attempt was made to compute the effect, upon the attraction at a given station, of the topography which exists upon the earth and of the isostatic compensation of that topography which is assumed to exist be- neath the surface of the earth. For this purpose new formule and new methods of computa- tion were found to be necessary. It was desired to compute the effect upon the attraction at each station of all the topography of the world and of the isostatic compensation of that topography. It was desired to do this with sufficient accuracy to insure that all constant errors in the computed effects would cer- tainly be less than 1 part in 200 and all accidental errors in the separate parts of the computa- tion less than .0002 dyne. This, it was believed, would insure that the computed total correction for any station would ordinarily be in error in so far as the computation alone is concerned by less than 0.003 dyne. In order to make this computation with the specified degree of accuracy with a minimum expenditure of time and energy the formule and methods of computation about to be given were selected and used. This publication contains full information as to the degree of success with which the computations were made, both as to accuracy and rapidity. This degree of success is the proper measure of the excellence of the formule and methods of computation selected. The attraction of any elementary mass, dm, acting upon a mass of 1 gram at the station of observation is, in dynes, kdm D (8) in which k is the gravitation constant and D is the distance from the station to the elementary mass. In order to get the result in dynes all quantities in this formula must be expressed in the centimeter-gram-second system. : The general expression for Newton’s law of gravitation is MM. F=k3/? Te z (8a) * See Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, pp. 60, 77. Tt See p. 651 of “ her die Reduction der auf der physichen Erdoberfliiche beobachteten Schwerebeschleunigungen auf ein gemeinsames Niveau Von F. R. Helmert in Sitzungsberichte der K6niglich Preussischen Akademie der Wissenschaften 1903 Erster Halbband. 14 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. in which m, and m, are two masses each of dimensions infinitesimal in comparison with the distance D between them and F is the attraction between the two masses. Newton’s law of gravitation is frequently expressed merely in the form of a proportion, F being stated to be pro- portional to oe The gravitation constant, k in formula (8a), is the factor by which the product of two masses divided by the square of their distance asunder must be multiplied in order to express the force exerted by those masses on one another. The gravitation constant is not a mere numeral. Its dimensions are shown by the exponents in (L**M“T-*) if L, M, T denote the units of length, mass, and time, respectively. That is, the gravitation constant is the cube of a distance divided by the product of a mass and the square of a time. Formula (8) is merely the special case of formula (8a) which is pertinent to the problem in hand. The value adopted in this investigation for k in the centimeter-gram-second system is 6673 (10). The basis of this adopted value is as follows, as stated by Dr. R. S. Woodward :* In spite of the superb experimental investigations made particularly during the past quarter of a century by Cornu and Baille (Comptes rendus, LX:XVI, 1873), Poynting (The Mean Density of the Earth, by J. H. Poynting, London, Charles Griffin & Co., 1894), Boys (Philosophical Transactions, No. 186, 1895), Richarz and Krigar-Menzel (Sitzungsberichte, Berlin Academy, Band 2, 1896), and Braun (Denkschriften, Math. Natur. Classe, Vienna Academy, Bd. LXIV, 1897), it must be said that the gravitation constant is uncertain by some units in the fourth significant figure, and possibly even by one or two units in the third figure. The results of the investigators mentioned for the gravitation constant are, in C. G. S. units, as follows, the first result having been computed from data given by MM. Cornu and Baille in the publication referred to: Cornu and Baille (1873) 6668 (10-1!) Poynting (1894) 6698 (10711) Boys (1894) 6657 (107!) Richarz and Krigar-Menzel (1896) 6685 (10! Braun (1897) 6658 co Regarding these as of equal weight, their mean is 6673 (107) with a probable error of +5 units in the fourth place, or 1/1330th part. This is of about the same order of precision as that deduced by Prof. Newcomb from astronomical data. The uncertainty in the adopted value is, however, within allowable limits for the present, investigation. The vertical component at the observation station of the attraction expressed in formula (8) is, in dynes, sin f kame (9) in which f is the angle of depression, below the horizon of the station, of the straight line from the station to the elementary mass. This vertical component is all that is concerned in this investigation. The integral of all such vertical components at the station, corresponding to all the elementary masses which together constitute the earth, is the vertical force due to gravitation which acts on a mass of one gram placed at the station. This vertical force expressed in dynes is necessarily numerically equal to the acceleration (both being expressed in the centimeter-gram-second system) which would be produced by gravitation acting upon any mass at the station left free to fall. They are, of course, affected by the centrifugal force due to the earth’s rotation, but this effect need not be considered in the discussion of these formule. The term, gravity, is used in its generally accepted sense; that is, it is the resultant of the earth’s gravitation and the centrifugal force due to the earth’s rotation. * See p. 153 of an address entitled “‘The Century’s Progress in Applied Mathematics,” by R.S. Woodward, Bulletin of the American Mathe- matical Society, 2d Series, Vol. VI, No. 4, pp. 133-163. In this address and in another by the same author entitled ‘‘ Measurement and Calcu- lation,” published in Science, new series, Vol. XV, No. 390, pp. 961-971, June 20, 1902, are given excellent statements of the nature of the gravi- tation constant and the importance of determining its value accurately. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 15 In general it will be found that throughout this publication the attraction (expressed in dynes) is dealt with directly by preference rather than its numerical equivalent, the acceleration (expressed in centimeters and seconds). This preference is due to the belief that thereby circumlocutions are avoided and greater clearness secured in the conceptions. If the station and the elementary mass, dm, are at the same elevation referred to sea level 0 b=5 and _ O D=2r sin 5 (see illustration No. 3), in which @ is the angle at the center of the earth subtended between the station and the elementary mass and r is the radius of the earth. If absolute accuracy were desired it would be necessary to use for r the average radius of curvature, between the station and the mass considered, of the equipotential surface in which they both lie. This average radius depends upon the elevation above sea level and also, since the sea-level surface is an ellipscid of revolution (not a sphere), it depends upon the latitude of the station and the azimuth of the line from the station to the mass under consideration. But with sufficient accuracy for this investigation r is assumed to be constant with the value 637 000 000 centimeters in this and similar formule. This is equivalent to assuming, in deriving these formule, that the station is on the surface of a spherical earth having the radius stated. Under the heading “Discussion of errors”’ it will be shown that this assumption is far within the allowable limits of approximation. By substituting these values of @ and D in (9) there is obtained as the formula for the vertical component of the attraction in dynes upon a unit mass at the station, due to an elementary mass which is at the same elevation as the station, sin : kdm——, 4r? sin? 5 =kdm E (10) The single symbol F is used to represent that portion of the formula si g im 5 4r? sin? 5 which depends simply upon the direction and distance of the elementary mass from the station, because later it is most convenient to deal with E separately from k and dm. ILLUSTRATION No. 3.—Showing station and ele- mentary mass at same elevation. To divide both the numerator and denominator of (10) by sin 5 would simplify the expression, but by so doing the close analogy between (10) and the more complicated expressions (15) and (16) would become less obvious. In each of the illustrations Nos. 3, 4, and 5, S represents the gravity station, and the circle represents the intersection of the level surface which lies at the elevation of the station with a plane defined by the station, the center of the earth (C), and the elementary mass considered. 16 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. B is the location of the elementary mass dm, £ is the angle between the horizon of the station (SH) and the straight line from the station to B, and D is the distance from the station to B. In illustrations Nos. 4 and 5, D, is the distance from the station to a point A at the same ele- vation as the station and in the same vertical line as B, the location of the elementary mass. Illustration No. 4 represents the case in which the elementary mass, dm, is higher than the station, the difference of elevation being 4. In the triangle SAB, from the law of proportional sines, h cos : : 2 (11) ao also, in this triangle, according to plane trigonometry, D?=D2+h? +2D,h sin § (12) From illustration No. 4 it appears that a=3 (13) and B=Pa-Be (14) By substituting from formule (11), (12), (13), and (14) in (9) there is obtained as the formula for the vertical com- ponent of the attraction in dynes upon a unit mass at the sta- ticn, due to an elementary mass which is higher than the station, 6 h cos sin{ 5— Se Di +h+2D,4 sin 5 (15) kdm =kdm E, De +h? +2D,h sin : Here again a single symbol, E,, is taken to represent that por- tion of the formula which depends simply upon the direction and distance of the elementary mass from the station. Illustration No. 5 represents the case in which the elementary mass, dm, is lower than the station, the difference of elevation being h. By the same process as that used above it may be shown that the vertical component of the attraction in dynes upon a unit mass at the station, due to an elemcntary mass which is lower than the station is ILLUSTRATION No. 4.—Showing elementary mass at greater elevation than station. h cos g hes ta 2 sin 5 +sin™ Sa 4/ D2 + —2D,hsin5 (16) kdm 9 =kdmE, D?2+h?—2D,A sin 3 in which £, is used to represent that part of the formula which depends simply upon the direc- tion and distance of the elementary mass from the station. It is important to note that the only approximation made in deriving formule (10), (15), and (16) is that to which attention has already been called, namely that the radius of curvature EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 17 concerned at each station is 637 000 000 centimeters. In every other respect the derivation of the formule is exact regardless of the distance of the attracting mass from the observation station. The attracting mass may even be located at the antipodes of the station. These formule were used in connection with all attracting masses which are so far from the station that the curvature of the sea level surface must be taken into account in order to insure that the errors of computation of the effects are less than 1 part in 200. For masses near the station, the well-known formula for the attraction of a mass having the form of a right cylinder upon a point outside the cylinder and lying in its axis produced was utilized.* This formula, in a convenient form for the present purpose, for the attraction in dynes upon a unit mass, (1 gram) at the station, is k2nd{ VP +h — Vet (h+t)?+¢} (17) in which & is the gravitation constant, 0 is the density of the material, c is the radius of the cylinder, ¢ is the length of an element of the cylinder, and h is the distance from the attracted point, the station, to the nearest end of the cylinder. For a mass which has the form of a cylindrical shell, that is, the difference of two concentric right cylinders of the same length having different radii, c, and c, formula (17) becomes hond{ Vee +h? — yep +h?— yee + h+t+ yer+ hte} (18) This is the attraction in dynes upon a unit mass (1 gram) at the station. The formule (17) and (18) are exact if applied to cylinders and cylindrical shells. The justification of the radical departure from past prac- tice represented by formule (10), (15), and (16), and by the ee ee introduction of the gravitation constant into formule (17) and (18) is the success attained thereby in securing quick and accurate computations. The reader is therefore requested to suspend judgment until the remainder of this publication has been read and the degree of success has been compared with that obtained by the use of any other formule with which comparison is made. DIVISION OF THE SURFACE OF THE EARTH INTO ZONES AND COMPARTMENTS. In order to apply formule (10), (15), (16), (17), and (18) to the computation of the effect of the topography and the isostatic compensation, the whole surface of the earth was divided into zones by circles, each having the station at its center, and each zone was divided into equal compartments by radial lines. The division adopted is shown in the following table. Tllus- trations Nos. 10a@ and 10), page 48, show the shapes of certain compartments. * For two statements of this formula see A Treatise on Attractions, Laplace’s Function and Figures of the Earth, by John TH. Pratt, third edition, p. 46, and Traité de Mécanique Céleste, F. Tisserand, Tome II, pp. 71-72. 15593°—12——2 18 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. Deslenotion colnet of nee aes of Compartments Meters Meters A 0 2 1 B 2 68 4 C 68 230 4 D 230 590 6 : E 590 1 280 8 F 1 280 2 290 10 G 2 290 3 520 12 H 3 520 5 240 16 I 3 240 8 440 20 J 8 440 12 400 16 K 12 400 18 800 20 L 18 800 28 800 24 M 28 800 58 800 14 N 58 800 99 000 16 0 99 000 166 700 28 ° 7 VW ° / ut 18 1 29 58 1 41 18 1 17 1 41 13 1 54 52 1 16 1 54 52 2 ll 53 1 16 2 11 53 2 33 46 1 14 2 33 46 3 03 05 1 13 3 03 05 4 19 138 16 12 4 19 13 5 46 34 10 ll 5 46 34 7 51 30 8 10 7 51 30 10 44 6 9 10 44 14 09 4 8 14 09 20 41 4 7 20 41 26 41 2 6 26 41 35 58 18 5 35 58 51 04 16 4 51 04 72 #13 12 3 72 13 105 48 10 2 105 48 150 56 6 1 150 56 180 1 For the numbered zones it was found to be more convenient to use the radii of the zone in degrees and minutes of a great circle than in meters. The inner radius of zone 18 is the same as the outer radius of zone O, that is, on a sphere of the adopted size, radius 637 000 000 centi- meters, 1° 29’ 58’’ of a great circle (the inner radius of zone 18) has a length 166 700 meters (the outer radius of zone O). Zone A commences at the station, and zone 1 ends at the antipodes of the station. All the zones together cover the earth completely. Zone A, with a single compartment, is a circle about the station with a radius of 2 meters. Similarly, zone 1, with a single compartment, is a circle about the antipodes of the station with a radius of 29° 04’ (3240 kilometers). Zones 18 to 14 each have a single compartment. All other zones have from 2 to 28 compartments each, the number of compartments being even in each case. For each zone a special reduction table was prepared in the manner indicated hereafter under the heading, ‘‘Computation of reduction tables.’”’ This table for each zone gives the relation between the mean elevation of the surface of the ground in each compartment of that zone and the effect of topography and the isostatic compensation in that compartment upon the vertical component of the attraction at the station. In making the arbitrary selection of radii of zones and of the number of compartments in each zone, it was necessary to consider the effect of the size and shape of the compartment; first, upon the time required to complete the computations; second, upon the accuracy of the computations in so far as it depends upon the accuracy of the estimates made by the computer of the mean elevation within each compartment; and, third, upon the accuracy of certain necessary assumptions in the computation. The larger the compartments are made, the smaller will be the number of compart- ments, and therefore the smaller the number of estimates of mean elevation to be made, one for each compartment. But as the compartments are made larger, the time required for each EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 19 estimate becomes greater. For with a large compartment it is necessary to estimate the mean elevation more closely to secure a given degree of accuracy than with a small compartment; to estimate to the nearest hundred feet, for example, instead of to the nearest thousand feet. Also, the larger the compartment the greater the total range of elevations within the compart- ment, and therefore the greater the time necessary to secure an estimate of the mean to a given degree of accuracy. Hence the adoption of compartments either too large or too small would have made the time required for the computation greater than would otherwise have been necessary. There are 317 compartments in all, 199 in the 15 lettered zones near the station, and 118 in the 18 numbered zones, all of which are more than 166 kilometers from the station. It is believed that the size and shape of each compartment has been so fixed that the error of computation for any compartment is ordinarily less than 0.0002 dyne, and is of the accidental class. The basis of this belief will be indicated in connection with the topic, ‘Discussion of errors.” Jt is known that notable success has been attained in securing rapid computation. With the experience now available, a better selection of radii of zones, and of numbers of compartments in each zone could be made. But such a new selection would make it necessary to recompute the reduction tables. It is not probable that the improvement would be sufficient to warrant this recomputation. COMPUTATION OF REDUCTION TABLES FOR NEAR ZONES. For zone A, comprising the surface of the earth in a circle around the station with a radius of 2 meters, the reduction table was computed by formula (17). The effect of the topography in this zone, if the station is on land, is the effect of a cylinder of material having the density, 0, assumed to be the mean surface density of the earth, namely, 2.67, having a radius c=200 centimeters and a length, t, equal to the elevation of thestation. In the formula h=o for this case, as the station is at the suid of the cylinder in question. The eleva- tion of the surface of the ground in all parts of this small zone is assumed in the computation to be the same as the elevation of the station. In the computations it was necessary, of course, to express all distances in centimeters to conform to the adopted value of k, which is expressed in the centimeter-gram-second system. (See p. 13.) The attraction computed is evidently a vertical force, as the station lies in the axis of the cylinder, which is vertical. The effect of the corresponding isostatic compensation was computed from the same formula (17) with the same values of ¢ and A, but with ¢=11 370 000 centimeters, the assumed depth of compensation (it should be remembered that compensation is assumed to begin at the surface of the ground and at the bottom of the sea, see page 10), and with a value of 6, from formula (3), page 8, substituted for 6, namely, 0, =n? 67-35 B00" in which Z is the elevation of the surface above sea level (assumed to ie the same as the elevation of the station). The isostatic compensation is thus treated as a cylinder of material of a negative density 0,, or, in other words, as a negative mass just equal to the positive mass which would exist in this zone above sea level if the actual density of all material in the zone above sea level were 2.67. For a land compartment the computed effect of the topography is positive, an increase in the downward attraction upon a unit mass (1 gram) at the station. The computed effect of the isostatic compensation is negative, a decrease in the downward attraction upon a unit mass at the station. The difference of the two is the resultant effect of the combined topography and isostatic compensation. This resultant effect was computed for various assumed values of the elevation of the station above sea level, and then the reduction table for zone A written as shown on page 30. An inspection of the table will make it clear that as soon as a few of the tabular values had been computed the remainder could be safely interpolated with the required degree of accuracy. 20 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. To apply formula (17) to a station at sea, such as those occupied by Dr. Hecker on the Atlantic and Pacific,* it is necessary in computing the effect of the topography to substitute for 0 in formula (17) the value (0—6,) (see pp. 8-9, and illustration No. 2), the defect of density of sea water in comparison with solid earth. The value of h is zero, the station being assumed to be at sea level. The mass thus considered is a mass which is the difference between that actually contained in the cylinder of radius 200 centimeters extending from the station at sea level down to the bottom of the ocean, and the mass which would fill this same cylinder if solid earth with a density of 2.67 were substituted for the sea water. To apply formula (17) to a station at sea, in computing the effect of the isostatic com- pensation, it is necessary to substitute for the 0 of formula (17) the f iret, Gabe value of 6, computed by formula (6a), page 11, namely 0.615 D 0.615 D Qo 7 2.8777 370 000 in which D is the depth of the water. In this case the h of formula (17) is not zero but equal to D, as the upper limit of the compensa- tion is at the ocean bottom at a distance D below the station. a For an oceanic compartment the computed effect of the topog- Sesundt Coad raphy (in this case submerged topography, or hydrography) is negative, a decrease in the downward attraction upon a unit mass at the station. That is, the attraction is less than it would be if in the compartment from the ocean bottom to sea level material of density 2.67 were substituted for the sea water which is actually in this space. The computed effect of the isostatic compensation is positive, an increase in the downward attraction upon a unit mass at the station, for the compensation is in this case an excess of density and of mass. The resultant effect is in this case again a numerical difference. Similarly formula (18). was used in computing the reduction tables for zones B to O inclusive. It was used separately for the topography and the isostatic compensation, and the results were also combined. ‘The values for the 6 of formula (18) were the same as have been stated already in connection with the application of formula (17). In using formula (18) to compute the effect of the topography E at in land zones three cases arise. : First, when the mean elevation of the surface of the ground in the zone is the same as the elevation of the station, h is zero in 7 formula (18). (See illustration No. 6.) In this case the attracted point, the station, is in the plane of the upper end of the cylindrical shell considered. This cylindrical shell contains all the material in the zone, from the actual surface of the ground down to sea level, the inner and outer radii of the shell being the same as the inner and outer radii of the zone, and the length of an element of the cylindrical shell being the mean elevation of the surface of the ground. Second, when the station is above the mean elevation of the surface of the ground in the zone, as indicated in the second case in illustration No. 6, h is the difference of elevation between the station and the mean surface of the ground in the zone, and in other respects this case is similar to the first one. For any land zone the computed effect of the topography in either the first or the second case is always positive, an increase in the downward attraction at the station. ee Third Case a t t — ILLUSTRATION No. 6.—Showing topog- raphy in land zones—three cases. * Bestimmung der Schwerkraft / auf dem / Atlantischen Ozean / Sowie in / Rio de Janeiro, Lissabon und Madrid / Mit Neun Tafeln / von O. Hecker; Berlin, 1903. Bestimmung der Schwerkraft / auf dem / Indischen und Groszen Ozean / und / An Deren Kiisten | Sowie Erdmagnetische Messungen / Mit Zw6lf Tafeln. / von Prof. Dr. O. Hecker; Berlin, 1908. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 21 Third, when the station is below the mean elevation of the surface of the ground in the zone, as indicated in the third case in illustration No. 6, the cylindrical shell containing the topography is considered broken into two separate cylindrical shells, one above the other, indicated as shell A and shell B in the illustration, and formula (18) is applied separately to the two shells. Shell A extends from sea level to the elevation of the station, and its effect is com- puted exactly as was that of the shell in the first case. Shell B contains the remainder of the material in the zone above sea level. It extends from the level of the station up to the mean elevation of the surface of the ground in the zone. In this shell c, and c, have the same values as in shell A, h is zero, the station being in the plane of the lower end of the shell, and ¢ is the difference between the elevation of the station and the mean elevation of the surface of the ground in the zone. The effect of the material in shell B is an upward attraction at the station. Hence the resultant effect at the station of the topography in this case is the difference of the separate effects of shell A and shell B. This resultant effect will evidently be positive, a down- ward attraction, if shell A is longer than shell B, and will be negative if shell B is the longer. If the station is at an elevation exactly one-half of the mean elevation of the surface of the ground in the zone, shell A and shell B are of equal lengths, and the resultant effect is zero. For oceanic zones the first and second cases arise, but never the third case. Hence for oceanic zones the computed effect is always negative, the downward attraction at the station being always less than it would be if material of density 2.67 were substituted for sea water. In applying formula (18) to the computation of the effect of the isostatic compensation for land zones all three of the cases described above arise. Hence, in the third case, the effect of the compensation was obtained by computing separately the effects of two shells correspond- ing toshell A andshell B. In computing the effect of the compensation the length of an element of the shell is 11 370 000 centimeters (the depth of compensation) in the first and second cases, 11 370 000 centimeters minus the difference between the elevation of the station and the mean elevation of the ground in the zone in shell A of the third case, and simply the difference between the elevation of the station and'the mean elevation of the surface of the ground in the zone in shell B of the third case. In all these cases, including both shells in the third case, the value to be used for 6 in formula (18) is that computed from formula (3), page 8, in which the mean elevation of the surface of the zone is to be used for H and the assumed depth of com- pensation for f,. As shell A is always much longer than shell B in connection with the com- pensation, its effect always predominates, and the computed effect of the compensation for these zones is always negative, a decrease of downward attraction at the station. In applying formula (18) to the computation of the effect of the isostatic compensation for oceanic zones the second case is the only one which arises, and the computed effect of the compensation is always positive, an increase in the downward attraction at the station. The value to be used for 6 in formula (18) is computed from formula (6a), page 11. For zones B to O the combined effect of topography and compensation is not always a numerical difference of the separate effects. In a few rare cases for land zones, namely, when shell B of the third case happens to be longer than shell A, the effects at the station of the topography and its compensation are both negative, and their combined effect is the numerical sum. To avoid circumlocutions a few paragraphs just preceding this have been worded as if the mean elevation for the whole of each zone was dealt with in the computation. In zone F, see page 18, which is divided into 10 equal compartments, the effect of the topography or of the compensation in any one compartment upon the vertical component of the attraction at the station is evidently exactly one-tenth of that computed for the whole zone from formula (18), provided the elevation of the surface of the ground is the same throughout the zone. The actual practice was to use formula (18) in computing the effects for a whole zone at once, then to divide the result by the number of compartments in that zone (10 for zone F) to obtain the effect of each compartment. These effects for separate compartments were then tabulated in the reduction tables, and in using these tables the mean elevation for each compartment. was used, not the mean elevation for the whole zone. 22 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. It was not found necessary to compute each separate value in the reduction tables for zones B to O. For each of these tables a few scattered values, in each of several selected columns, were computed. For each selected column the points so computed were plotted on cross-section paper, using the assumed mean elevation of the compartments as abscisse and the computed values as ordinates. When the number of plotted points was sufficient to enable one to do so with the required degree of accuracy, a curve was drawn through these points to represent all the required values corresponding to the column in question. The intermediate values for the column were then scaled from the curve and entered in the table, together with the computed values. After the values in a few columns of the table had been so obtained, it obviously became possible to interpolate the values for the remaining columns with the required degree of accuracy. The vertical differences in the columns, filled in from the computations and curves, served as checks in making these interpolations. Illustration No. 7 shows the curves used as indicated above in connection with the reduction table for zone E. On each curve the computed points are indicated by small circles. The curves were drawn by eye, using a draftsman’s flexible ruler. The shape of each curve and its position relative to the other curves furnish a sensitive check for detecting errors in the plotted values due to the computations or plotting. As the computations of the reduction tables by formula (18) could be made much more easily than by formale (10), (15), and (16), it was desired to extend the use of formula (18) to as many zones as possible. It was found that out to zone L the errors secured by the use of formula (18) in the manner already described were within allowable limits. It appeared that when formula (18) was applied to zone O the principal error arose from the fact that a point in the middle of this zone which is at the same elevation as the station lies 4500 feet below the horizontal plane of the station on account of the curvature of the sea-level surface. It appeared that possibly this particular error could be eliminated and a very close approximation to the truth obtained by using for the A of formula (18) not the difference of elevation between the station and the mean surface of the ground in the zone, but instead the difference of elevation between the station and a point 4500 feet below the mean elevation of the zone. This would have the effect of making the second correction in the table zero if the mean surface of the com- partment lay in the horizontal plane of the station. Accordingly, the column headed “Station above compartment, 800 feet,’ in the reduction table for zone O, was computed with a value 4500 +800 = 5300 feet for h, the next column with a value 4500 + 1600 =6100 feet for h, and so on. Similarly the values in the column headed ‘‘Station below compartment, 800 feet,” were computed with the value 4500 — 800 =3700 feet. The corrections in the column headed “Station at same elevation as compartment”’ are applicable to compartments in which the mean elevation of the surface of the ground is the same as that of the station. These values were computed with h = 4500 feet in formula (18). Similar modifications to take account of the curvature approximately were made in the tables for zones M and N, but for zones nearer the station it appeared that such changes would not amount to as much as 0.0001 dyne, and they were therefore not computed. After computations for zone O were made by formula (18), using the modified method indicated in the preceding two paragraphs, in which method the curvature of the sea-level surface is taken into account in part, certain values of the table were also computed by formule (10), (15), and (16), which are exact, the curvature being fully taken into account. This test showed that the tabular values as computed by formula (18) by the method described are each within 0.0002 dyne, and are in error by less than 1 part in 200 on an average. This made it certain that the errors in zones M and N, and other zones nearer the station than zone O, are well within the adopted limits. The test in zone O also indicated that for the next larger zone the adopted limits of error might be exceeded if formula (18) were used, even with the modifi- cation described. Therefore formule (10), (15), and (16) were used for all zones beyond O. ’ . ‘W SNOZ HOS 379V_L NOILONGAY JO NOLLVLNdWOD TVWOIHdvuy—"Z "ON NOILVYLSNTN AW Wins WoIsmTOse-S:11 Hi Ad OBLNIMe ARG UF ssaunpsoduiad jo Woupiaa Ubapy e 8s 8 f 8 wo = aay pouiaae YL Ay $0 Stn seu Ur woUaLIE) wo x2 oh AJANNS “IVDIN07030'S N BML AG GBLNINd “SH1daq SNOINVA Os FZ 4O SANIVA JO NOILVLNASSudaYy WOlHdvVYOH—'S "ON NOILVYLSNTII Srepauultwa? OODLE Jo Syptantxis UI yaog “95,18 YUM SYIDAP SNOLIDA LOf 7 J0 SONfBA BUMOYS BAIT) 0000! 00S 00002 00062 (920) 7 Jo sanyy EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 23 COMPUTATION OF REDUCTION TABLES FOR DISTANT ZONES. To use these formule in computing the effect of the topography within a given zone for a land area it is necessary to integrate the expression kdmE to include all elementary masses within that zone between the surface of the ground and sea level. E is understood in this statement to be the E of formula (10), the EF, of formula (15), or the E, of formula. (16) for each elementary mass according to whether it is at the same level as the station, higher than the station or lower than the station, respectively. Since kdmE is the vertical component of the attraction in dynes upon a unit mass at the station due to an elementary mass, dm, the integral stated is evidently the vertical component of the attraction due to all the elementary masses which combined constitute the material lying above sea level in the zone in question. In the integration k is a constant, and the sum.of all the elementary masses, dm, is the total mass m, which is known in terms of the volume and density. No difficulty was encountered in dealing with these quantities. But the expression for £ is a function of h and 6, which can not, so far as the writers know, be directly integrated with respect to these quantities by calculus. There- fore, an integration by numerical computation was made. The vertical component of the attraction in dynes upon a unit mass at the station due to all the topography within any zone lying entirely in a land area was therefore expressed as the integral of kdmE or km (average value of F for the zone) (19) in which it is understood that the various values of EH, of which the average is taken, must correspond to equal elementary masses, of which the sum is m, the total mass represented by the topography in the zone. Similarly the vertical component of the attraction in dynes upon a unit mass at the station due to the isostatic compensation of the topography within any zone lying entirely in a land area is also represented by formula (19). The negative mass involved is m, the values of E are those fixed by the direction and distance of the compensation from the station, and h is made to vary to cover the whole range occupied by the compensation, namely, from sea level down to the depth 113.7 kilometers below that surface. The effect of the topography and the effect of its compensation might have been computed separately from formula (19), but it was believed that greater rapidity would be secured without loss of accuracy by combining and dealing directly with the resultant difference of the effects of the topography and its compensation.- Accordingly, the actual process followed is that described in the following paragraphs. The computation will be described first for land zones haying an elevation of 100 feet and for the station assumed to be at sea level. The modifications introduced for other elevations, for ocean zones, and for assumed positions of the station above sea level will be stated later. For a selected value of 6, EH was computed by formule (10) and (16) for several equally spaced values of h, varying from zero to the depth of compensation. Let the required mean value of an infinite number of such equally spaced values, covering the depth of compensation, be called E,. By successive trials with increasing numbers of equally spaced values of h it was ascertained how many values were necessary in order to secure the required degree of accuracy in the mean value, £,, corresponding to the selected value of 6. As E varies con- tinuously according to a law which may be graphically expressed by a smooth and regular curve, it was not difficult with the numerical values at hand to make certain that one had secured the required degree of accuracy. [Illustration No. 8 is an example of such a curve, which corresponds to @=1° 55’; that is, to compensation which lies in a part of zone 16. (See page 18.) The values of h are plotted as abscissz and the corresponding values of EF as ordinates. The small circles each represent a computed value of EH. A smooth curve has been drawn by eye through these computed points. It is evident that as the curve is nearly a straight line between successive computed points but little change would be secured in the mean by com- puting more points. This is still more clearly and precisely shown in the following table, corresponding to illustration No. 8, and showing the computed values of H and their first and second differences. 24 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. Values of E for various depths, with 6=1° 58’. Depth (10%) | Difference | ghecond | Centimeters 3684 1/1611 370 000 11 026 +7342 — 60 2/1611 370 000 18 308 +7282 —133 3/1611 370 000 25 457 +7149 —209 4/16 X11 370 000 32 397 +6940 —255 5/1611 370 000 39 082: +6685 —332 6/16 X11 370 000 45 435 +6353 | —354 7/16 X11 370 000 51 434 +5999 —409 8/1611 370 000 57 024 +5590 —454 9/1611 370 000 62 160 +5136 —462 10/1611 370 000 66 834 +4674 —477 11/1611 370 000 71 031 +4197 —499 12/1611 370 000 74 729 +3698 —433 13/1611 370 000 77 994 +3265 —485 44/1611 370 000 80 774 +2780 —473 15/16 X11 370 000 83 081 +2307 —431 11 370 000 84 947 +1876 Mean 52 568 If an infinite number of points were computed on the curve shown in illustration No. 8 and the mean taken, instead of the mean of the finite number of points there shown, the change in the computed mean would be represented by the average ordinate included between the curve and the series of chords joining the computed points which are shown in the illustration. As a convenient rough guide it was assumed that this average ordinate would usually be less than one-eighth of the average second difference shown in the preceding table. That this ratio, one-eighth, is a reasonably safe assumption in such a case may be verified either by trial or by geometry, assuming the short portion of the curve between successive points to be an arc of a circle. A similar process of reasoning was followed to obtain the mean value of EF corresponding to the topography for the same selected value of 6. Let Ey be the required mean value of an infinite number of equally spaced values covering the range from zero to the arbitrarily selected elevation, 100 feet. After E had been computed from formule (10) and (15) it usually appeared that in order to secure a sufficiently exact value of Ey it was necessary to compute but two values, one for h=o and one for h=100 feet, the mean of these two being sufficiently accurate. It will be shown later how topography of a greater elevation than 100 feet was dealt with. Keeping in mind that the negative mass, which is the isostatic compensation, is necessarily exactly equal to the positive surface excess of mass, which is the topography, formula (19) as applied to the topography and the compensation combined may be written km(E7p— Eo) or km Ep (20) Ex being written for H,— Eo. As this process of computing Ez, corresponding to a selected value of 6, is slow it was impor- tant to use good judgment in selecting the various values of @ for which the computation was to be made. It was desired to obtain a sufficiently accurate value of Ep for every possible value of 6 by computing a moderate number of values for selected values of 6. At first Ep was computed for 6=180°, that is for the antipodes of the station, and for 0=90°, midway between the station and its antipodes. Then the computation was made for a few more values of 6 at large intervals. It soon became evident that Ep varies quite slowly and at nearly a uniform rate if. is near 180°, but that for small values of 6, Ez varies at a large and rapidly changing rate. Therefore, to secure a given degree of accuracy in interpolated values it was evidently necessary to compute Ep for closely spaced small values of 0, but only for widely separated large values of 0. The following table shows the various values of Hz actually computed. The values of 0 shown in this table were selected by inspection by a step by step process, computations being made first for two values only of @ as already stated, then for values spaced at intervals of 30°, then at intervals of 10° for smaller values only, at intervals of 5° for still smaller values, and so on. All values of Ep intermediate between those shown in the table were obtained by interpolation. Illustration No. 9 shows a part of this table expressed graphically. ABAUNS TYOISOIOIS'S 0 FHL AB OBLNTHd ol "Y3 40 SANIVA JO NOILLVLINSSSHdSY IVOIHAVY—'6G “ON NOILVYLSNTH 6 f0 Sanjoy 8 256 06 001 orl oOo o& a z (pe) "7 40 Sampo 061 EF FE cT OF TOP OGR., AP HY AN D Is OSTATIC ee PE NS. ATI ON ON GRA VI TY 25 8 ° Ey 20 180 : ) 160 00 461 Fg (10) 150 00 at a 0 1. 2 E 140 a + 62 8466 = Tp (10%) 130 0 bean = 62. 166 | 00 A 5.5 6 05 = 0 110 0 ae ~ Ga f 0 105 a A. 9805 = 64. ae - . 5552 E 100 a i ae 1425 = Pee ae bd 5591 ar 'p (1020) 9500| 4 75.2134 | — 7 1043) - Sa 1 ap z ee ee | 2 i) ae ae 0 3. 7 . 05 | oe 30 1 80 00 a ao a i = 2 10 15 : a Bp (10 af + os: 196° — 84. 4071 e et aa + e782 = 88 : 65 HA 0h soo | 88. a4 | 305 8 0 08 - 50.8 = 07. = ae a: : = : = ga) Hee “2s hn 3 0 Fis. ait = eS 00 4 3.2 —10 . 892 - 1 75 45 763. = il 8 = 362. 57 12 22 = =~ bale 8 3 +7 9 77 3 .6 00 a 5.1 1 68 ite 1 0 85 = 12 2 = 89 56 12 18 es . 476 8 + 8 2 32 4 25 00 at 7.0 12 72 = 15 07 ee .8 = 55 0 129, 150 ae rae 8. 00 t 831.2 rag 45e 4 a +131, 120 eo a = 7. 45 + aoe a =| 1000. b 2S FO ag ae — 130. 0 : mae: 73 eG - 1432. = ee 00 a 3.4 13 88 ae 2 0 ae 83 2 15 2 = 29 52 0 135 710 Se arale 715 lL 4 pet B74. 2 +18 | ae a4. BGS — 2886 7 ong ral 599 2, ona 00 + 8.0: 13 05 = 3 00 41 = 16 9 = 24, 50 140 80 zs 6.8 . 005 6 4 +9 8 97 6 oil 00 + 5 13 48 = 3. 5 a 74. _ 18 «1 = 80, 49 14 45 eae 134 63 10 2 05 7 9 00 ae 3.1 14 84 os 3 0 a 08 19 3 = 43 48 14 i ee . 270 61 10 9 25 8 .6 00 ae 5.7 144 23 s, 3. 5 ae 46 = 20 9 = 14 47 14 84 a: 4 418 6 0 10: 61 8 Ll 00 ais 8.5 14 70 =< 3 0 ae 86 es 22 3 ss 93 46 151 70 es 7.2 574 . 54 1 13 9 9 00 + 4 15 32 = es 5 30 = ae .3 el vee 4 1 7 = 0. TA 5 + 9 ea ae sr ze “iis | 3 05 5 8 ie | me | ae 43 0 160. 1 = . 278 = 1 30 1246 — 8 = 1345 iG Lee ie — 168. 08 eae 52 +128 — Foe 1693 40 0 171 6 i . 666 5 02 15 1304 3 6 i 1911 39 00 +175. 928 —Ne. 46 = Poe 5 10 +132 a - a ll ae 92 ero ae 5 + 4 = 3 66 38 180. 6 1a 56 05 134 0 ee 00 + 13 178 3 3 5.6 5 + 4 = 388 60 37 0 184 - —18 717 6 70 00 1366 40 A = a 36 re +189. 569 = 172 = 014 4 55 +1388 = ae = 2466 35 to +194. 240 a1) 7. 877 -— 6 387 4 50 +1412 a es oy 2577 34 00 199, 876 ee ~ 13H 4 45 +143 - 4320 | - 269 00 + 37 198 1 < 7.2 4 + 6 Ss 482 6 33 204. 6 = .12 7 46 40 146 4 Se 282 00 ay a) 203 3 _ eo 4 41 0 a 658 1 32 0 210 —20 . 716 8 39 35 2 486 484: — 2954 ee 4216. 72 =o ats me 4 30 per = F058 = as 30 a 4993, 92 oe 00 _— 9. 883 4 25 ae - ae = 3946 29 5 +230. S 330. 76 — 10. oe ia Tie = 5503 = 8 28 0 4-238. 54 =e See 415 1598 = a = 359 ort de 04 237 ae 4 +1 eee ae 27 246 —2 - 76 12 10 628 6 =e 00 + . 06 46 Se 4 +1 ae ana 26 254. -2 -10 13 05 660 6 = ee ae 66 55 ae 4 +1 = oa 25 0 263 —26 .10 14 00 ab 694 658 = 203 23 00 {84 65 a ~ 17:08 3 50 ae — 137 Se 59 00 396 82 ae = ist 3 45 1801 — T601 a ; Lo 3 +1 ee: 5 BL Ot +308. 02 ae 20 40 840 79 — 5236 0 + . 02 a 13. — . 72 3 +1 - 86 19 80 za 8 88 5% i = Ses = $88 a fou 08 ey = 3862 3 25 1868 = 0385 = eon 18 00 oe ae = 85 7 a0 = 108 = 1 17 oe ee =. 3 os aa = 1a = sn 16 3 404. 9 —45 . 60 = 54, 00 ate 26 _ 1268 - 557 . 98 oe =D 2 2 5 9 ne Han a = 8:8 4 ts = as = 88 15 0 442. 6 504, 47 = 68. 49 9 45 49 16 o 15462 _]j 570 aa a eee — 74.14 a 1356s — 16567 ee 14 2b 472. _5A . 04 - 80.4 35 ak 562 = 1782 a 231 0 +4 0 ae 74 8 9 23 264 1 9 a 131 13 3 88. 57 i 7. 0 ats 2 = 921. 76 0 +5) 2 ie 1.9 9 59 22 272, 9 4 = 142 13 0 05 59 ss 5.7 5 Be 6 =, 073 17 0 Le 6 — 598.7 10. 0 a 281 aa a 12 3 24.2 628 ae 2 : +2 : Soa 16 a2 12 00 ioe 860 6 aes 2 10 429) ay = 18174 toe ae 10.0 Ae tus - Bon a ; ao. —155. 0 -3 = “ee “13.0 155 39m = 84855 = 2 . 198, 50 6 s 83) = 56 a 3 1 +3 70 28 aa iB] ie = a = ae 5.0 1 35 +4014 = Bune ~ BB oe 1 30 +4212 ae = ae 1 25 cr = bei = 43048 7 4 = a ee vag | — ssa7 a= 96711 = aes es eae —10 ee 48 26 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. It was desired that each value of E, used in the computation, whether obtained directly or by interpolation, must be correct within 1 part in 200. Having sufficiently accurate values of Ez for each separate value of @ the next step is essentially an integration with respect to @ as the variable. The area of any zone lying between the limiting values of 0, 6,, and 6,, on the surface of the sphere which is being considered, one having a radius of 637 000 000 centimeters (see p. 15) is 2zxr* (cos 0,—cos 6,) , (21) or (6.283186) (637000000)? (cos 6,—cos 6.) Hence for this zone formula (20) becomes kd [(6.283186) (637000000)? (cos 6,—cos 0,)] (mean value of Ez for the zone) (22) in which for m there has been substituted its value in terms of density and volume, namely, OH (area). With the numerical values before one it is not difficult to determine that for zones of a moderate width the average value of E, for the zone is with sufficient accuracy the mean of its values at the two edges of the zone corresponding to 0, and 6,. Therefore, formula (22), which is an expression for the required vertical component of the attraction in dynes upon a unit mass at the station due to the combined effect of both the topography and its isostatic compensation lying in a zone, may be evaluated by making separate numerical computations for separate narrow zones and adding the values. By examination of the table showing values of Ep it is evident that the separate zones which may be used in this process are wide near the antipodes and decrease in width as 6 becomes smaller. The actual widths used did not exceed the following limits and were occasionally less. Limits of widths of subzones. 6g Limit of width of subzone o + o +7 ov 180 00 to 72 00 2 00 72 00 to 20 00 1 00 20 00 to 10 30 0 30 10 30 to 5 40 0 15 5 40 to 1 25 0 05 It was known from a reconnoissance of the problem that for all distant zones (beyond 6=1° 29’ 58’’) the value of the attraction computed from formula (22) would be nearly propor- tional to H. Therefore, as a time-saving device, it was decided to determine such widths for the selected zones and fix the number of compartments in each zone so that an attraction of 0.0001 dyne for any one compartment would correspond to a value for H of either 100, 1000, or 10 000 feet in that compartment. In that case the computation would consist simply of estimating the mean elevation within the compartment in feet and moving the decimal point a certain number of places to the left to obtain the attraction in dynes. The arbitrarily selected unit of elevation corresponding to 0.0001 dyne was 10 000 feet for zones 1 to 6 (see tables on pp. 44-46), 1000 feet for zones 7 to 13, and 100 feet for zones 14 to 18. The number of compartments in each zone was arbitrarily fixed as shown in the same tables. By formula (22) the width of the zone was computed which would satisfy the condition that the attraction in one compartment corresponding to a unit of elevation was exactly 0.0001 dyne. For example, for zone 3 having 10 compartments it must be 0.0010 dyne for the zone if the mean elevation in the zone is 10 000 feet. No difficulty was found in making this computation. An example of the actual arrange- ment of the numerical work is shown below for zone 12 having 10 compartments. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 27 Computation of limit of zone 12. [em. =kd H=0.00543061 for H=1000 feet.] 6 to Oe Cos 61—cos 6 a ie Ep (10m) | Area xX kmX Ep o + Ws oO 4+ Vt 5 46 34 to 5 40 00 0. 0001905 0. 000004857 | 0. 00000002638 2198 0. 00005798 5 40 00 to 5 35 00 . 0001425 - 000003633 . 00000001973 2310 . 00004558 5 35 00 to 5 30 00 . 0001405 - 000003582 . 00000001945 2413 . 00004693 5 30 00 to5 25 00 - 0001383 © . 000003526 . 00000001915 2522 . 00004830 5 25 00 to 5 20 00 . 0001363 . 000003475 . 00000001887 2636 . 00004974 5 20 00 tod 15 00 . 0001341 . 000003419 . 00000001857 2758 . 00005122 5 15 00 to 5 10 00 . 0001321 . 000003368 . 00000001829 2888 . 00005282 5 10 00 to 5 05 00 - 0001299 . 000003312 . 00000001799 3024 . 00005440 5 05 00 to 5 00 00 . 0001278 . 000003258 . 00000001769 3170 . 00005608 5 00 00 to 4 55 00 . 0001257 . 000003205 . 00000001741 3330 . 00005798 4 55 00 to 4 50 00 . 0001236 . 000003151 . 00000001711 3504 . 00005995 4 50 00 to 4 45 00 . 0001215 . 000003098 . 00000001682 3690 . 00006207 4 45 00 to 4 40 00 . 0001194 . 000003044 . 00000001653 3888 . 00006427 4 40 00 to 4 35 00 . 0001173 . 000002991 . 00000001624 4097 . 00006654 4 35 00 to 4 30 00 . 0001151 . 000002934 . 00000001593 4321 . 00006883 4 30 00 to 4 25 00 . 0001131 . 000002884 . 00000001566 4564 . 00007147 4 25 00 to 4 20 00 . 0001109 . 000002827 . 00000001535 4822 . 00007401 4 20 00 to 4 19 12 . 0000176 . 000000449 . 00000000244 4987 . 00001217 ; Sum=0,00100034 The change for 1’’ is about 0.000 000 25, therefore, the-inner limit of zone 12 is, to the- nearest second, 4° 19’ 13’’.. With that limit the above sum becomes 0.001 000 34 — 0.000 000 25 =0.001 000 09. : The basis for the arbitrary decisions as to unit elevations and number of compartments in each zone will be indicated under the topic ‘‘Discussion of errors.’”’ It suffices to state here that the selection was guided by the desirability of making the computations as rapidly as pos- sible subject to the chosen standard of accuracy. Errors of judgment in one direction would make the computation slow, and in the opposite direction would make the computation too inaccurate. The limits of zones 1 to 18 computed as indicated above are shown in the reduction tables on pages 44-46, as well as on page 18. To apply formula (22) to the computation for oceanic zones it was necessary merely to take into account the fact that the defect of density represented by sea water is 0Q—d,=0.615 0. (See p. 9.) Therefore, if the unit of elevation is 10 000 feet for a land compartment, correspond ing to an attraction of 0.0001 dyne, it will be for an oceanic compartment to produce the same effect 10 000 feet 0.615 =16 260 feet =2710 fathoms. Hence the unit of depth shown for zones 1 to 18 in the reduction tables on pages 44-46. The attraction computed from formula (22) for a given compartment is not strictly propor- tional to H as assumed for a first close approximation. The limits of A used in computing Fp must correspond to H. For a land compartment, as H is made greater EH, becomes smaller, as it is an average value covering larger values of fh in formula (15). Also as H, the assumed elevation, is made greater EH, tends to become smaller, for the isostatic compensation is assumed to commence at the solid surface of the ground (above sea level) (see illustration No. 2, page 10), and to extend to a depth of 113.7 kilometers measured from that level. The limits of h used in formule (15) and (16) must be fixed accordingly. Similar modifications must be inserted for oceanic compartments, the compensation commencing in this case at the ocean bottom, not at the sea level. As Hpand £,, and their difference Ep, vary slightly for different values of H, the computed attractions in formula (22) are not strictly proportional to H as they would be if Ey were independent of H. This departure from strict proportionality was found 28 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. upon investigation to be inappreciable for zones 1 to 13. For zones 14 to 18 a few special com- putations were made to evaluate the corrections for departure from proportionality shown in the reduction tables on page 44. These special computations were made by using the proper limiting values of h as indicated above, comparing the computed values with the values based on the assumption of proportionality and the original computations with 4 =100 feet, and tabu- lating the differences as shown in the reduction tables. A few computations only were necessary because the corrections were small, and regular in their variation. It was also assumed in order to secure a first close approximation to the attraction required that the station is at sea level. In general the station lies above sea level and, therefore, to secure exact results the values of h, used in formule (15) and (16) in computing E; and E-, must be differences of elevation between the station and the elementary mass, not merely the elevation of the elementary mass as was assumed in the first approximation. To secure the corrections for elevation shown in the reduction tables on page 44 a few special computations were made on the exact basis, compared with the first approximation, and the differences tabulated as corrections for elevation of the station. The corrections for elevation were found to be negligible for zones 1 to 13, and to be small as shown in the reduction tables for zones 14 to 18. Because the corrections are small and their variations regular but few special computations were necessary. EXPLANATION OF REDUCTION TABLES. The complete reduction tables for all the zones are given in the following pages. All tabular values are the vertical components of the attraction upon a unit mass at the station expressed in units of the fourth decimal place in dynes. It is equally true that these are corrections in units of the fourth decimal place of centimeters, to the acceleration of gravity, expressed in the centimeter-gram-second system. These tables cover the whole of the earth’s surface, from the station of observation to its antipodes. By their use one may quickly compute the effect upon the attraction of gravity, at any station on the earth, of all the topography of the earth and of its isostatic compensation assumed to be complete and uniformly distributed, with respect to depth, down to a limiting depth of compensation of 113.7 kilometers. The radii of the zones A to O are given in meters, while those for zones 18 to 1 are in degrees, minutes, and seconds of an arc of a great circle. The first column of each table from A to O contains values for the mean elevation of the compartment as read from the maps. The second, third, and fourth columns contain the corrections for the. topography, the compensation, and the algebraic sum of the corrections for topography and the compensation respectively. These values are computed upon the assump- tion that the station is at the same elevation as the compartment. For zone A the elevation of the zone is necessarily that of the station, as its radius is only two meters. In the tables for zones B to O corrections for the elevation of the stations above or below the compartments are shown, The corrections for the topography and compensation, the station being at the same eleva- tion as the compartments, are shown separately in columns 2 and 3 for the zones out to O, in order that certain comparisons may be made between the effects of the assumption of complete local isostatic compensation and of regional isostatic compensation complete within a stated distance from the station. (See pp. 98-102.) For the regular computations of the combined effect of topography and compensation, one correction is taken from column 4 of each table from zone A to zone O. For zone A this is the only correction. For zones B to L, inclusive, a second correction must be applied, as indicated. to take account of the difference of elevation of the station and of the mean surfaces of the ground in the compartment. To the correction based upon the assumption that the station is at the same elevation as the compartment, taken from the fourth column of the table, is added alge- braically the correction for elevation of station above or below the compartment in order to obtain the total effect of topography and compensation. Thus, in zone E, if the mean elevation of the surface of the ground in a compartment is 2000 feet, the first correction is +0.0016 dyne, EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 29 and if in this case the elevation of the station is 3000 feet, 1000 feet above the compartment, the second correction is +0.0007 dyne, and the total effect of both the topography of this compart- ment and its isostatic compensation is to increase the vertical component of the attraction on a unit mass at the station by 0.0023 dyne. It is understood that, for zones B to L, inclusive, the second correction, namely, for station above or station below compartment, is zero if the elevation of the station is the same as the mean elevation of the surface of the ground in the compartment. This fact is used in interpolation if necessary. For example, in the case just cited in zone E, in which the mean elevation of the surface of the ground in the compartment is 2000 feet, if the station happens to be 100 feet above the compartment, the second correction would be + 0.0001 dyne, since the table indicates it to be +0.0002 if the station is 200 feet above, and it is understood to be zero if the station is at the same elevation as the compartment. Similarly, if in this case the station happens to be 100 feet below the compartment, the second correction would be —0.0001 dyne since the table shows it to be — 0.0002 dyne if the station is 200 feet below the compartment. For zones C to O the first column of the tables contains elevations in both fathoms and in feet. Those in fathoms are depths below sea level and are marked minus. The values in the second, third, and fourth columns, corresponding to depths in fathoms, are computed on the supposition that the station is at sea level and in the following columns, headed ‘‘Station above compartment,” the station is assumed to be at the stated distances above sea level. Hence, for all water compartments, there will be two corrections in the regular computations, one from the fourth column and one from the proper column beyond the fourth. Thus, in zone E, if the mean depth in the water compartment is 200 fathoms, and the elevation of the station above sea level is 600 feet, the two corrections are —0.0004 dyne and —0.0002 dyne, and the total effect of both topography in this compartment and its isostatic compensation is to decrease the vertical component of the attraction on a unit mass at the station by 0.0006 dyne. ' For zones M, N, and O, as already explained in connection with the computation of the tables (p. 22), the second correction does not necessarily become zero when: the station is at the same elevation asthe compartment. Instead it has the value shown in the tables for these zones in the extra column headed ‘‘Station at the same elevation as compartment.’’ In taking out the second corrections for these three zones this extra column must be carefully noted, one must take the second correction from it when the station and compartment happen to be at the same elevation, and one must use the values in this column to control interpolations when the station and compartment are nearly at the same elevation. Thus, if the mean elevation of the surface of a compartment in zone M is 12 000 feet the second correction is +0.0001 if the station is also at the elevation 12 000 feet, it is between + 0.0001 and +0.0002 if the station is less than 700 feet above the compartment, and it is between +0.0001 and —0.0002 if the station is less than 700 feet below the compartment. : For zones 18 to 14 three corrections are applied. The first is read directly from the map, being 0.0001 dyne for each unit of elevation, the unit in each case being 100 feet, as indicated in the heading of this table. The second is taken from the second column of the table, using the first correction as an argument in entering the table. It takes account of the slight departure of the actual correction from being strictly proportional to the elevation. The third correction is taken from the last part of the table and takes account of the correction due to the elevation of the station above sea level. Thus, in zone 17, if the correction as read from the map is —0.0100 dyne, the elevation of the zone (the zone has but one compartment) being 10 000 feet, then the correction for departure from proportionality is +0.0001, and if the elevation of the station above sea level is also 10 000 feet the correction for its elevation is + 0.0003 and the total effect of topography and compensation of this zone, upon the vertical component of the attrac- tion upon a unit mass, at the station is — 0.0100 + 0.0001 + 0.0003 = — 0.0096 dyne. Similarly, if zone 17 is all upon the ocean, and the average depth of the water is 2710 fathoms (or 100 of the specified units of depth) the correction as read from the map is +0.0100 dyne, the correction for departure from proportionality is +0.0001, and if the station is at the elevation of 5000 30 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. feet the correction for elevation is — 0.0001 and the total effect of the topography in this zone and its isostatic compensation is + 0.0100 +0.0001—0.0001=0.0100 dyne. The unit of elevation for zones 13 to 7 is 1000 feet, and for zones 6 to 1 is 10 000 feet. These large units of elevation make it easy to estimate quickly the mean elevation within each compartment with the required degree of accuracy. Note that for zones 13-1 there are no corrections for elevation of station and for departure from proportionality. The reduction tables thus far described are believed to cover all cases which will arise when the gravity station is on land. But in order to provide for the computation of the effects of topography and isostatic compensation on the attraction at a gravity station on a vessel at sea, such as those occupied by Dr. Hecker on the Atlantic and Pacific Oceans, the two supple- mentary tables for use in connection with gravity stations at sea were prepared. These tables are computed on the supposition that the observation station is at sea level, since the correction for the small elevation above sea level to which the station is limited on board a ship would be less than 0.0001 dyne in every case. But one correction is to be taken out from these tables for each compartment. This correction is to be taken from the first table if that can be done without using any of the values marked with an asterisk. Otherwise it is to be taken from the second table in order to avoid large errors of interpolation which otherwise would occur on account of the large second differences in the first table. For the remaining zones 18 to 1 no such sea tables are necessary, as the regular tables pre- pared for land stations cover all cases which will arise. REDUCTION TABLES FOR LETTERED ZONES. Zone A, [Inner radius, zero; outer radius, 2 meters. One compartment.] Correction for— Elevation of selon Topogra- and com- Topog- Compen- | phy and partment | raphy sation compen- sation Feet 0 0 0 5 +1 0 +1 10 +2 0 +2 100 +2 0 +2 1 000 +2 0 +2 2 000 +2 0 +2 3 000 +2 0 +2 000 +2 0 +2 5 000 +2 0 +2 6 000 +2 0 +2 7 000 +2 0 +2 8 000 +2 0 +2 9 000 +2 0 +2 10 000 +2 0 +2 11 000 +2 0 +2 12 000 +2 0 +2 13 000 +2 0 +2 14 000 +2 0 +2 15 000 +2 0 +2 For zone A the correction to gravity is a function only of the elevation of the station, for all land stations, as shown by the above table. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 31 Zone B, {Inner radius, 2 meters; outer radius, 68 meters. Four compartments.} Correction for elevation of station— a ; Correction for— : ation of Above compartment Below compartment compart- ment re Topogra- Se oa Bompen- | 25feet | S0fect | 75feet | loo fect | 125 feet | 25 feet | 50 feet | 75 feet | 100 feet | 125 feet sation Feet 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 20 ed 0 +1 0 0 0 0 0 30 +2 0 +2 0 0 -1 -1 -1 —3 40 +3 0 +3 0 0 -1 -1 -1 —3 50 +3 0 +3 0 -1 -1 -1 oe —3 —6 60 +4 0 +4 0 -1 -1 -1 —2 -3 -6 70 +4 0 +4 0 -1 -1 —2 —2 —3 —6 80 +5 0 +5 0 -1 -1 —2 —2 -3 —6 -9 90 +5 0 +5 -1 —1 —2 —2 -3 —3 —6 -9 100 + 6 0 +6 —1 -l —2 —2 —3 —3 —6 -9 —12 150 +8 0 +8 -1 —2 -2 —3 —4 —2 —5 —8 -1l —14 200 +10 0 +10 -1 —2 —3 —4 -5 —2 -5 —7 —10 —12 300 +12 0 +12 -1 —2 —4 —5 —6 —2 —4 -6 —8 —10 400 +14 0 +14 -1 3 -—4 —5 —6 —2 —4 —6 -—8 -9 500 +14 0 +14 -1 —3 —4 —5 —7 —2 —3 —5 -—7 — 8 1 000 +16 0 +16 —2 —3 5 —6 —7 —2 -3 —5 —6 -7 2 000 +17 0 +17 —2 —3 —5 —6 —7 —2 —3 —5 —6 —-7 3 000 +17 0 +17 —2 3 —5 —6 —7 —2 -3 —5 —6 -7 4 000 +17 0 +17 —2 —3 —5 —6 —7 —2 —3 —5 —6 -7 5 000 +17 0 +17 —2 —3 —5 —6 —7 —2 = —5 —6 -7 6 000 +17 0 +17 —2 -3 —5 —6 —7 —2 —3 —5 -—6 —-7 7 000 +17 0 +17 —2 —3 —5 -6 -—7 —2 -3 —5 — 6 —7 8 000 +17 0 +17 —2 —3 —5 —6 —7 —2 -3 5 — 6 -7 9 000 +18 -1 +17 —2 —3 —5 —6 —7 —2 3 —5 —6 -7 10 000 +18 -1 +17 —2 -3 —5 —6 —7 —2 -3 —5 —6 -7 15 000 +19 -1 +18 —2 —3 —5 —6 —7 —2 —3 —5 —6 -7 It is assumed that the mean elevation for any compartment in this zone will never be negative (below sea level) for any gravity station on land. Lone C. [Inner radius, 68 meters; outer radius, 230 meters. Four compartments.] Correction for elevation of station— Correction for— Above compartment Below compartment Mean elevation of com- Topog- partment Goi raphy | Topog- au and 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500 | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500 raphy | Pes? | com- | feet | feet | feet | feet | feet feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet | feet pensa- tion - Fathoms — -9 0 -9 — 40 —.4 0 -4 0 0 0 0 Feet 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 +1 0 +1 0 0 0 0 0 0 0 0 0j— 1] -1] -—2 150 +2 0 +2 0 0 0 0 0 0 0 0/—1/—1] -1] -3 | -4 200 +4 0 +4 0 0 0 0 0 0 0/—1/—1]/—1| —2] —4| -6|-8 250 +6 0 + 6 0 0 0 0 0 0 0j/—1]—1)-— 2} —2); —4 | —7 |—10 |—12 300 8 0 +8 0 0 0 0 0 0 0j—1/—1j— 2} —2| —5 | —8 |—11 |—14 |—16 350 t10 0 +10 0 0 0 0 0 0} —-1)/— 1 |— 2 |— 2) —2| —5 | —8 |—12 |—15 |—18 |—20 400 +12 0 +12 0 0 0 0 0 0] —1/— 2|— 2j— 3 | —2] —5 | —8 |—12 |—16 |—19 |—21 |—24 450 +13 0 +13 0 0 0 0 0 0} —-1j—2/— 2)/-— 3] —2] —4] —7 |—11 |—15 |—19 ;—22 |—25 |—26 500 +15 0 +15 0 0 0 0 0 0] —1j/— 2|/— 3|— 4] —2|-—4 | —7 |—11 |-15 |—19 |—23 |—26 |—28 | —30 16 0 +16 0 0 0 0 0 o0| —-1)— 2|-— 3|— 4] —2] —4 | —6 |—10 |—14 |—18 |—22 |—26 |—28 | —31 m tis 0 +18 0 0 0 0 0] —1| —2/— 3/— 4 |— 5 | —2] —4 1 —6 |—10 |—14 |—18 |—22 |—26 |—29 | —32 700 +20 0 +20} 0 0 o| —1) -1] -2| —3 |— 4 j— 5 /— 6} —1] —3 | —6 |— 9 |—13 |—17 |~20 |—24 |—28 | —31 800 +22 0 +22 o| —1| —1| —2/ —2] -3] —4/-— 5 |— 6 |- 7 | —1] —3 | —5 j— 9 |—12 |—16 |—19 |—22 |—26 | —29 900 +24 0 +24 o| —1/ —2| —2) —2] —4] —5 |-— 6 |— 7 |- 8| —1] —3 | —5 |— 8 jJ—11 |—15 |—18 |—21 |—24 | —27 0 26 o| —-1} —2| —2) -—3| ~-4| -6/—7/— 8 |- 9| —1] —3] —5 |— 8 j—11 |—14 |-17 |—20 |—23 | —26 1 200 13 0 +28 —1|—2| —2] —3} —4] —5| -6|/— 7 |- 9 |—11 0} —2) —4 |— 7 |—10 |—13 |—15 |—18 |—20 | —23 1 400 +30 0 +30; —1] —2| —2| —3| —4] —5 | —6 |— 8 |—9 j-11 0| —2 | —4 |— 6 |— 9 J—11 | 14 |—16 |—18 | —21 1 600 +32 0 4+32|—-1] —2| —3| —3| —4] —6 | —7 |—_9 |-10 |—12 0 | —2 | —4 |— 6 |— 8 |—10 |—13 |—15 |—17 | —20 2 000 +34 0 +34] —1 | —2| —3| —4]} —5 | —6 | —8 |—10 |—11 |—13 0] —1] —3 |— 5 |— 7 |—10 |—12 |-14 |—16 | —18 0 36 | —1| —2| —3} —4} —5 | —7 |] —8 {—10 |—11 |—13 0| —1) —3 |— 5 |J— 7 J— 9 |—12 j—18 |-15 | —17 3 O00 tes 0 $38 —1|-—2/ —3] —4| —6) —7 | —9 |-10 |—12 |-14 0} —1,; —3 |— 5 |— 7 |— 9 |—12 |—14 |-15 | —16 5 000 +41 -1 +40] —1] —2| —3] —4| —6 | —7 | —9 |—10 |—12 |—14 0} —1; —2 |— 4 |— 6 jJ— 8 |—10 |—12 |-14 | —15 15 000 +44 —2 +42} —1/ —2] —3 | —4| -6] —7 | —9 |—11 |—12 |-14 0} —1 | —2 |— 4 |J— 6 |— 8 |-10 |—12 |-18 } —15 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. A —UOT}EIS JO UOTJVAa 10J UOT}DEIIOD —J0J WOT}0A110;) e— | oe— | st— | ot— | 1— | 1— ] or— | 8 — | 4-] — | e—] e— fI- | 1— | os—| or— | zt— | st— | t1— | at— | or— | 8— | 9— | s— | e— J o— | - J] t] set fe | Tet | 000 aT ze— | oe— |} si— | ot— | s1— | at— | or—] 8 -—}2-|s—-—]e—]e— Ji— }t— | o—] st—] or— | st—] et—] m—-] or— | s— | 9- | | e- | e- | T-]t- [9st | s— | Ber | 000 OF #— | Ie— | e1— | 41—| st—] et— } n—|6—|s—]s—l|e—Je- |e— | 1- | ot—|si—]| or—|gr—] er- | u—-]6—|4—-]o-jr—]e-]e-| I~] it~] oet |e | Bet | 0006 se— | ee— | oz— | st— | o1— | #I— | ai— | or— | 8 — J 9—]%—]e- Je- |I— | e1—| st— | ot— | st— | et— | 1- | 6 - | 2-]9- | e-]e- |e] IH IH] et | IH fee | 008 se— | es— | oc— | st— | ot— | a1— | zi— | or— | s — ]9 — | #—| — J e— | i— | er— | si— | ot— | ot— | et | 11- ] 6 — | z- | 9- Je] oe | I- | TH] eet [I~ | eet | OOO 92— | e2~ | Te— | st— | ot | Fr— | at— J or— |] s —]9-—]%—]e- Je— Jr— J or— J ai—)or- | w— | et- | u- | 6 -s2- | 9- | e—-|e-]e-] I~] to] s+ JIT [eer | Onl g %— | | we | eI— | 2I— | s1— | at— | o— | 8 -—]|9-—]s—le- | 2-— Jr | st— | zt— | st—]at—| et- | or—|s—]z-]s-je-]e-]e— | I— fi] ost | To St | 008 G g3— | se— | ec— | oc— | st— | o1— | t— | u—-|}6—]}2—]s—]e- |e- Ji— | st—| ot—| st—]er— | et-]or-|s—|e-|s-]e-]e-]e-]t Jo | ot |I- [Ost | o008 ee— | 92— | ¥— | Te— |-or— | oI— | et— | 1— J or— |2-—]¢—]e- Js— | r- | at—]|ot—] st fat} u-je—]|s—je-|s-je-]e- |e ]i jo feet JIT [set | Oe te~ | ge— | se— | ee— | er— | 2t— | s1— | et— J or— | 2-]s—]e- | e— |1- | or—|st—|er—]a—|or—|s—]2—|e-|e—]e-]e-]IH] tH jo [set |I~ [ert | OOF ze— | 6c— | 92— | e@— | oz— | zt— | 1— | ct— | or— }2-—]s—]e— J2— |i- | or—]u-]er—|m—-|o-|s—]2—]s—-]e—]e-]e-] I]t /o Jet [TH |e | ORE ee— | og— | 2e— | e— | t2— | st— | st— | at— | o— | 8 —} 9 - | s— |2— J i— |st—]m—l|er-|m—-|6—-|s—|9-—|s-l]e-]e-fe-|I—|1t~ so [ert | I~ | br | 0098 ve- | 1e— | se— | se— | ze— | or— | or— | er— | m—|s—f9—]a— Je- |r |st—jer—ja-|u-j6é-js-]9-|s-]e- ie fI-]i jt jo jet [IK [er | OE se— | ee— | 62— | 9e— | ee— | Oc— | AI— | I— | TI— | 6 —| 9 —] F— J e— | I~ | FI-]sI- | - | oI- | 6-]4—]9-|S-]e— |e] I-]/I-}I-]o fire JI c+ | 006 € ze— | ¥8— | Og— | 22— | ¥e— | I2— | sI— | st— | et— J or— | 2 —] g— | e- |s- | e—Jst-]}m—-|6-|;8—|42—-|s—]s-]e- |e] I~ ]0 |O 10 [OF [0 Or+ | 000 € ge— | SE~ | Te— ; 8— | H— | IZ— | sI— | SI- ] eI— J OI— 2 —] G— J e- | s-— | et—]u—]}o-|;6—-|8—/9—-|s—)F-]e-7pe-]I-]O |O 10 | set [0 se+ | 008 Z OF— | 9e— | ce— | ez— | S2— | ee— | 1— | 9T— | eI— J OI—] 2—] 8— J e- |s- |a—|u-jor—|s—|z—/9—-|F—]e-]je-]I—}oO |0 |O JO | er JO ge+ | 009 @ we— | se— | be— | og— | 92— | e2— | or— | 9t— | et— ] or— | 8 — | s— | #- |e- ;1-jor-|6é—|s—|/9—-|s—]F—]e-]e—/I-]O [0 |0 [oO |r JO ret | 00F Z H— | OF— | 9e— | ce— | 8a— | Fe— | Os— | AI— ] FI- ] TI-] 8— ] S— | F- J 2- Jor-|6—-—]|8—Jz—j|9-]F—-—]e—]se-]t—/O [I+] I+ |T+/ 1+) cer 10 ze+ | 008 Z S— | a | Be— | Fe— | O€— | 92— | e— | SI— | SI- | I- | 8 —] 9- Je js- |e-—|s—|z—-j9-|F—]eF—-]|e@—]T-J]oO JO [It| T+ [ct] T+/ oF | 0 og+ | 0002 L— | eh— | ee— | se— | Te— | ge— | ec— | o— | 2I- | eI—] 6 —] 9- |#- js- |8—J2—-[9-J]s-—j|rF—-]e-—]e-]o JO [i+ }et)e+ [etl T+ | set | 0 8c+ | 008 T 9»— | er— | 6e— | Se— | ce— | 8e— | — | O2— | ZI— | eI— | 6 — | 9— |e Js-— JL-—]9-[S—]F—- |e —]e-—]/T—]o [I+] t+ et | st [et t+) vt fo 9+ | OUT 9¥— | eb— | OF— | 9e— | ze— | os— | Se— | e— | BI— } FI— J or— J 2- | S— Je— J4—-|s-—]|F—-]e-—j;s-}Ir—fo JO [I+ ) st fet) st fet | t+ | set | 0 Sot+ | 009 T bh— | ch— | eg— | ye— | ee— | eo— | Se— | 12— |] SI— | FI- J Or— J A- | S— fe- J9-|s-—je—-]e-—je—|E—-|oO J|t+ [et jet | et] et ct | it) eet | 0 s+ | 008 T WH | Tr | e€— | 9e— | ce— | es— | se— | te- | GI— | SI— | 1I- ] S— | S— Je- |S-—|F—-je-|s-JT—}oO [oO | t+ set |etler eter [Tt ar | 0 c+ | OOF T OF— | Le— | ce— } ze— | ee— | Ss— | T2— |] 6I— | sI— ] 1I- S— _ }s— fe- JF—-fe-—]e-f;T-]/T—-joO |T+/etlsetjet jer ler ler [it | ost | 0 Oc+ | 008 T ze— | e— | Te— | 82— | G2— | Z- | SI— | sI- | T1I- | 8— | s— fe- |F-je-—]se-|r—-jo jo J|r+jerljerlet)erjer ler | t+ err | 0 6I+ | ose T ge— | es— | te— | se— | se— | te— | st1— | st—| 1—-|8- |s- Je-— Je-Je-|]e-J|r-jo jrt+ |otjerjet [et | ster jer | t+ | sry | 0 sI+ | 008 T ge— | og— | 22— | — | — | 8I— | T- | 1- ; 8- | s-— Je-— fe-—|e-jI—]o |O |T+istletjet|er|rt pet pet lit yur jo Zit | OST T zée— | 6o— | — | H— | — | 8I— ] I- | TI—]8- | s-— Je- fe-|rI-|I—]o j|T+ie+let [et] et | ot let let pet lit) ort | 0 9+ | OOLT 6c— | 9e— | &@— | O— | SI— | FI— | 1I— |] 8- |S— Je- |e-]1I—-jO Jo, |r t+tjetjet ler [er [e+ | et let [et | tt | sit | oO SI+ | 0S0 T se— | so— | ee— | oe— | SI~ | FI— | TI—]} 8- |s- |e- | T-—]O Jo JrtlT+]etjpet ler e+ etl rt let yet | i+ |r | 0 FI+ | 000 T s— | — | et— | zi— | at—for—}z— Je— fe-— |r —-fo Jrtl[r+[otle+l|et st] et let [et | ot] et] t+ | ert [oO I+ | 096 w— | te— | 6r— | zI— | wI— |or—|z- |}s—- Je- Jo Jo JrI+[et|etfet|et | et | Fe [et | et] et pet i+ )art | 0 I+ | 006 w— | st—}9t—]et—jor—Jz- ]s- Je— Jo JI tli +i[ot+i[et et] tlre ot |e | ot [e+ pet i+ |) Tt 10 Ti+ | 098 - os— | st— | 91— | et— Jor— J z— | s— Jem JT + [T+ [eti[etjetljetl| e+ )et [et | ot) e+ Ft | et | t+ [ork | 0 oI+ | 008 4i—|st—}ai—} 6 —j2- | s— fe Jr tl[otletyetl[etlet+ l/r + let yet rt et et let jit je + | 0 6+ | OcL wi~|ot—]et—}6e—|z- |s- Je- Jetfetrietiets/etle+] r+) et) et set e+ ot let jit] s+ | 0 s+ | 002 si—|a—|6—]z—- |s- Je- |atietjerletle+/et+] r+] et [rt et | bt] pt pet i+} et [0 s+ | os9 a—|tm—|6—]2- |s- le- Jatletletjet|r+)et]e +] et jot et et] ot pet i+ et oy L+ | 009~ t—|6—/9- Je- le- Jotletle+letla+ [e+ let] stlrt jot lot | ot let jit jot | 0 9+ | og o—|s—fo- ji— Je-— Jotietl[etset+|r+]e +] e+] ot [et] et let] et [etl t+ ys + jo s+ | oos L—-|s- Je- J[e- Jetletletlet|e rye +] e+ et [or] tt [ot /et [et [i+] r+ | 0 b+ | 09% g—|e- Je- fa- Jetietl{[etset|r+] r+] e+] Ft ]et] ot ppt [et fet lit je+ jo s+ | 006 s— Je- Je- Jetletletfet]rt+ pt] rt+ let let fet [et pet per jit set [0 e+ | ose e— |e- Je- Jetfetl[etl/etl/etrletjer rt jeter jer ser yet [it (et | 0 e+. | 008 o |0 Jo Jo |o |o |o0 |o0 Jo |o }6 }O |0 Jo jo 0 0 0 V2 Z- j@- |e- |e- |’ | s- | s- |e ]%—-]z-/e-]e@—/T-|I-|r—- jo Ba< |e e— |&- [%s- |e—-]|%@-]e-|e@—-]%—-|e—-]1-|F- |0 b— | 00r— z— |%—-|e—-|@—- | s—-|%—-]1I-|2—- |o dee) 0ST —|%-]s-|I-]|m- |o u— | 00¢— SULO"IDT gaag | aap | aaer | gear | a2oy | a9ay | aay | gear | a0ay | aay | gaay | aoap | 4004 | aay | gaan | 9905 | ga0r | a00p | 9005 | aay | g20y | 900y |a00p | 3a0p | 409 | g00x | goer | 920F | eStrod OorE | Oost | Cozt | Cort | oot | 006 | O08 | OOL | 009.| OF | OOF | ODE | O0Z | OOF | OOFT | OOET | OOZT | OOTT | OOOT | 008 | OO | ODL | O09 | OOS | OOF | OOF | OZ | OOT | -m0os | NOM. | Aades : pue ae -sodoL| anew sydea 3 ~yredur0a queuyiedu10) MOTEg quawy1edai0a oA0ogy -80do.1, JO WOT}CA -ofa Uva [‘syuemyredu0o xIg ‘siajeUT NGG ‘SNIPs JeyNO ‘srajyauT Oge*SNIpPwI 1oUTT) ‘q 2u0Z 33 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. ee— | Sc— | 8I— | 9I— | SI— | OI— | 8 — | 9 —|F—)}e—-]IT- 0 eo— | 6I— | FI— | GI— | OI— | B— _| 9- i Aad — ad 0 0 elt+ an LL+ 000 ST 66 || S6—-") OL [20 | He | | 8 pe | eS ea la «10 te— | 6I— | FI— | GI— | OI— | B—_— | 9— i s— I- 0 0 @Lt+ $— OL+ 000 #1 se— | 96— | 6E— | LI— | FI— | TI | 6 — | 9 — | FP - |e] i fo S— | 8I— | 8I— | TI— | 6 — | 8— | 9- t—- Cr ic 0 0 Th+ &— Lt 000 €1 $8— | 42— | OC— | LI— | FI— | T1I- | 6 — | 9 -—|E¢—-—)}E—]e- | 0 &o— | 8I— |-8I-— | T1I—- | 6 — | 2— | S— &- ot i Ss 0 0 69+ e—- GL+ 000 21 9s— | 82— | Te— | 8I— | SI— | 7I— | OT—- |} 2 -— | $ —| § —]fVe- | 0 co— | 4I— | sI— | OI— | 6 — | L—- =| a s— o— = 0 0 89+ &—- be 000 IT 8&— | O&— | G2— | GI— | 9I— | SI-— | OT-— | 8 —| 9 —|; F—]fse- | 0 @o— | LI— | @I— | OT— | 8 — | 9— | F— &— oT t— 0 0 99+ &— 69+ 000 OT OF— | G— | &e— | OC— | LAI— |] FI- | TI- | 8 —| 9 -—|F—-—]s- | 0 Té— | 9I-— | 1I-— | 6 — | 8 — | 9- =| F—- g— G— oe 0 0 go+ os cot 000 6 Gh— | &&— | F— | Te— | SI-— | SI— | GI-— | 6 — | 9 -—| F-—]s- | 0 O@—- | 9I- |} Ti-— | 6 —|L—]9- | F o— [= [= 0 0 zo+ C= pot 00 8 Sh— | b— | SS— | Ge— | SI— | SI— | AI—-— | 6 — | 9 —| F—J]s- | 0 0¢- | SI- |} 11-6 —| 2—] s- | t— co rs 1 0 0 09+ o— Zot 000 8 St— | 98— | 9— | GB | BI— | SI-— | AI— | 6 —| L -— | F-—] S— | 0 6i—' | FI— | 0l— |} 8 -— 19 — ps | 8 o— I- 0 I+ 0 8g+ o— 09+ 00g 2 Le— | L8— | Le— | Se— | 6I— | 9I— | EI— | OI— | 2 — | $ — | S— | I- | 8I-— | FI-}]6-—|s—}9-—]r ]E- [= 0 0 I+ 0 gst oT so+ 000 2 Le—- | L8— | Le— | Fe— | OC— | 9I— | SI— | OI— | 2 — |} $ — |] E— J I- | kI-] &l-—]6é—j}2—-—]9-—]Fr |] o- I- 0 0 I+ 0 got o— zg+ 008 9 8h— | 88— | 82— | He— | OC— | 9I— | SI— | OI— | 8 — | S$ — | O— _ J I- | ZI- |] &I-]6—-—}]2—-—]s—]r—- | e- i 0 0 I+ a G+ o— g¢+ 009-9 Ge [GE 86 | S6= | Te 20 | SS | OR ee Pe eS Ee | ce | Se | ae he SS re Pe t= 0 0 I+ T+ est 6— gc+ 00F 9 Os— | OF— | 62—- | SZ— | To— | LI— | FI— | OI- | 8B — | $— |] E— | I- | o9—-]etI-|;s—-—]9—-—|s—]e- J se- i= 0 0 I+ Le ost o— $+ 002 9 Ts— | Te— ; O&— | 9G— | Ge— | SI— | FI— | TI- | 8 — |; F —| O- | I- | 9I—-]} @-|s—-1|9-—|s—]e— | s- 0 0 I+ Aa I+ Igt+ o— est 000 9 CS | ERS | OS | Ze ee |) ST | SE | ES | 8 ee eh Se a | ee 1 oe eS |e 0 0 I+ It It os+ o— oo+ 008 ¢ FS— | G— | Te— | Le— | &S— | GT— | OT— | @I— | 6 — | 9 — | B— Y= |S BS | he es ee SH eS 0 0 I+ I+ Lae 6r+ I= os+ 009 ¢ gG— | &— | c&— | 8o— | Fe— | OZ— | OI— | VI-— | 6 — } 9— | E- | I- | FI—-];OI-|9—-—;F-—]e-—|Fe jI- T+ T+ or G+ I+ 8h+ I= 6r+ OOF ¢ 9S— | Fr— | E8— | 62— | Fe— | OC— | 9TI— | SI-— | 6 — | 9 —| F— | I- | I-] OI-]|9-]|F—|E—-—]e 10 i I+ or ot+ I+ Le+ Ee 8b+ 00z ¢ £g= | Sh= | SE—" || Go=— | ‘SG— | OS= | 2h | St |: Ol | ee |S | SH 6 eee eS eH Pe 0 i I+ G+ ot I+ cpt I- 9+ 000 $ sc— | 9F— | be— | OF— | 9E— | Te— | LI— | SI— | OI— | 2 — | ¥— | T- | VI-|s8—j|;s—-—|/F&—-—16-—]|I- 10 Le T+ ot ot At tet t= cet 008 + 6S LP SS TE | Oe To ZI SE OFS 2 Se ES EE Se ee ee SS TS «10 ot ot o+ ét+ T+ a+ T= e+ 009 + OO= | SF= | 9S— | C8— | Zo= |e | SES | BIS Ol | 2H EO Ole | te lk eH eS Ee T+ ot ot o+ ot I+ Tet+ i ch+ 00 + O9—- | 8h— | 9E— | GE— | LE— | @— | SI— | F1— | TI-— | 2—]} F—- | I- | Ol-—]9-j;e-J;e-|t-jJo T+ ot G+ ot+ a+ I+ 6g+ t= Ort 002 + O9— | Gh— | LE— | GE— | 8Z— | ES— | BI— T= | 2=(%> | I> | 6— |S —te=—)}1—|90 It | t+ Gar e+ e+ é+ T+ Le+ eo get 000 + I9— | OS— | 8E— | ES— | 8E— | FE— | BI— cL= | S=— | s= | E= | s— | PH | Pe | 0 I+ | 3+ e+ e+ e+ e+ T+ 98+ tT 28+ 008 & 0o9— os— | 68— | F&— | 6B— | Fo— | OZ— éI— | 8 — | $— o— £2 ES EO T+] e+ €+ ot e+ Sar e+ [+ ret = oe+ 009 € 6S— | OS— | 6e— | SE— | 6Z— | SV— | O2— GE GE | Gee Gf Quer g at T+]/e+ ]e+ | + yt e+ e+ e+ E+ ze+ = Se-F OOF € ss— | OS— | 6E— | SE— | O8— | S2— | O6— GI |G — | S— |S | SESS | Lae Pe Se | Se ee e+ a+ oe e+ ot og+ t= Eee 002 € gS— | 6h— | 6E— | SE— | O&— | 9Z— | IZ— I> |} 6—|s=— | oS | RSL | Lae oa ee | ee Pe g+ e+ e+ e+ ot 8o+ = 6+ 000 € Le— | 8&— | FE— | O&— | 9S— | T2—- &I-|6—-|]§- |@- | 8-10 @t+)/e+)]/et] e+ [ot g+ yt a+ e+ e+ 9+ i= l¢@+ 008 z Gr— | LE— | FE— | 63— | S2— | Te— = |e— | 9=— |e | Se— | 0 e+/e+)/e+)e+ | e+ 9+ G+ gt e+ ot + i Fo+ 009 Z 98— | €&— | GE— | SS— | le— sI-—}6—|]9- J@- JT -;t +l] +)/e +] 9+ ]e+ | 9+ 9+ o+ ¢+ $+ e+ 1+ [= oot OOF Z FE— | SE— G— | $S— | Te— sI— | OI— | 9- o—- 0 @+tirt+]e+}] e+ )ot 9+ 9+ g+ g+ e+ a+ 6I+ t= 0+ 002 & cE— | O&— | LO— | EZ— | O3— eI-— |6—}9- |@- | T+/€+}/¢4+/9+)/9+]9+ | 9+ it S+ gt e+ e+ oT+ [= Li+ 000 Z 86— | SG— | Ge— | EI— @I— |6 —|}$—- 1e- J7tlr+]/9t+}9+)]/94+)] 9+ 9+ i+ c+ G+ b+ a+ FI+ 0 FI+ 008 T * Go— | OG— | LI— ti | 8 | s= o- $+ )r+]/9+}9t+]}L2+] lt b+ i+ c+ g+ $+ o+ Ti+ 0 II+ 009 T 8I— | 9I- Ol— |S=— | S=— | o— (Rr SE 2+ 2a) ea] eb | l+ c+ ot t+ a+ 6+ 0 6+ OOF T WM | te (6K | 2H ps (ee Re se Le he ea ce oP L+ ot e+ e+ ot Loe 0 L+ 006 T Ol-18-—|9-|]F—- |e- |G+t}|Stlo+tl}/a+i19t+] 9+ [9+ 9+ g+ e+ e+ ot gt 0 Gb 000 T 9-|o¢-—[e—- |T- |St+t}G¢+{/9+]}/9+]/9+1/9+ | 9+ 9+ c+ e+ or T+ e+ 0 er 008 Be fe P+tig¢til/G¢t+y;o+]}/o4+)] ot c+ e+ e+ e+ ot I+ ot 0 ot oo9 o— I- [e+] +]e +] +]e +] e+ | et e+ e+ ot ot I+ T+ 0 Le 00F 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 pad Leo |e oH Pe | t= [tk t= = I- I- 0 Di 0 t= os — CH SS eS eS (eS te o— o- on I- I= L= 0 b= oor — PRS |e] ee be t &- o— o— Ls @—- 0 6 osI— =F —|F — | P=" | F— r= = o- o- I- tO 0° - 00¢— = Bry ee | r= ¥— = &- oc [= a> 0 0 = ose— B= | = —— §= = o- 1 So 6 0 6 — o0g— — &— &—- o— {= H= 0 i- ose— c= o— I= = 0 eI- 00F— o— ‘= sT— T+ 9I-— Osh— la aA I+ 8I- oos— SULOy DT qoay | Jook | Joog | Joos | Jooy | Joos | Jaoy | Joo | Joos | JooF | Joo} | Joos | Joos | Joos | Jooy | Joo} | Jooy | aay | yooy 902} qo} qaoy 40a} 429} uns 000 | 00S% | 000Z | OOST | ODOT | OOFT | OOZT | OOOT | 008 | 009 | OOF | 002% | 0D0E | ODSZ | 000% | OOST | OOOT | OOFT | OOGT O00T 008 009 00F 002 | _gadiuoo woryes sud T Aer pue Ayd | -usdur0g | -Sodoy, queued queuyredm00 Mojeg quewyieduros saoqy ~eiBodoy, -ui00 JO uoT} BAaT? UBIAL —U01}84S JO WOT}BAI]O JOJ WOT}DILI0D —J0} WOT}daLI0D [-squourqseduroo yy 4t97 ‘s1012U OSZI ‘SNIPBI 19}NO {s1ajaw YEG ‘SNTpPeI JoUUT] “ql au0g 3 15593°—12 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 34 —UO0T}21S JO WOTJBAV]a 1OJ WOTJIeII09 Io} WoOT}oeLI09 Ie— | 6I— | 24I— | SI— | gI— | 11- | 8 —} 9 — | e— &—- |3- I- L= 9- c= | t= o— Ls 0 0 I+ Tt I+ 0 got y- oL+ 000 ST Te— 6I— | ZI— | SI— | &I- | TI—- | 8 —} 9 — | g— 8— | 3- r= §= | $= |= &— o> I= 0 0 I+ I+ I+ 0 got - 69+ 000 #1 &— | O— | 8I— | 9I—- | fI-— | 11- ] 6 — |} 2 — |} s— _ | e- J z- I= |S= |= (8 se fe |-0 I+ Tt T+ I+ It I+ 9+ b—- L9+ ~| 000 &T ve— =| Se— | GI- | I= | SI— | ZI— | OI | 2 | ee Lee | ee eS ea Wig) TS '10 0 T+ I+ I+ It ir I+ 09+ y- ihe 000 I 9— | &— | 12— | 4I— | FI- | I— | OI— | 4 — | ¢— J e— Je- |I-— J e-— Jz—- Je- J1- Jo I+ | 8+ G+ ot T+ I+ T+ 6o+ s— 9+ 00 IT 2e— | — | @W— | SI— | St— | zI— | oI— | 8 — | g— _ | F— J s- | I- |e-— |e- |e- |t— Jo T+ | ot ot ot ot ot+ I+ Lg+ o= 09+ 000 IT 82— | S2— | e%— | GI— | 9I— | SI— | TI-— | 8 — | s— | r— Je- J I- }e- fe- rs 1/0) I+ {@+ | ot ot ot ot e+ I+ oo+ §= sc+ 00S OT 6Z— | 98— | &— | OS— | AI— | FI- | sI— |} 6 — | 9—-—« FH | Z—- I- |@- |@- |I- |0 T+ | 2+ | d+ ot a+ a+ e+ I+ wet s— g+ 000 OT o&— | Zo— | H— | OS— | LZI— | SI— | eI— | OI— | 2— | F— | Z2-— |I- | I-— |1I- Jo Tse: TEE | egcbe | BE t+ a+ a+ ot I+ oot 2 got 00S 6 O&— | 42— | %— | O2— | AI— | SI— | EI— | oOI— | 2— | F- | 2—- | I- |I- Jo 0 I+ | 8+ | ot | &+ a+ e+ t+ a+ I+ 6h+ C= t+ 000 6 Te | 8e— | — | We | 2I— | et | er OI- |4— |%- |@- | I- JO It | T+ | Ut | Ut | ot | + e+ e+ ot+ ot T+ Le+ ci og+ 009 8 ‘TE— | 8— | FB— | I2— | 8I— | 9I— | €I— | OT— | z— | S— | z—- I- |0 I+ | 3+ [t+ | ot | e+ | E+ e+ e+ e+ o+ I+ Spt o— Lb+ 00Z 8 ee— | 6@— | S2— | @— | SI— | 9I— | EI— | OI— | Z—- | S— | s— | I— | it | t+ Jet fat fet jet | rt e+ SF ot ot Tt eet o— Sb+ 008 ZL ee— | 6@— | S2— | @— | BI— | LI— | FI- | TI— J z— | e— Je— J i— Jit fet fot fat fet fat fet e+ e+ ot ot T+ Tet o- ee+ 00F Z Fe— | O&— | 98— | €S— | O— | ZI— | FI- | TI— | 8— | s— J e— | I— Jot fet fat jet |e+ | et | et o+ e+ e+ e+ I+ 6E+ o— Tr+ 000 2 ve— | TE— | 9B— | Es— | OS— | AI- | SI— | gI— | S—_ J 9-— | e— | I— | st fet fet Jet [ot | e+ [ot o+ e+ e+ or T+ gé+ o— or+ 008 9 gg— Té— | 246— | be— | OZ— | SI— | SI— | 7aI— | 6— | 9-— | e— | I— Jet fet Jet | e+ | ot Jot fot c+ e+ e+ or I+ Le+ o— 6g+ 009 9 gg— Té— | 22— | Fe— | IZ— | SI— | SI— | maI— | 6—_ | 9— J e— | I— Jet | e+ [H+ | o+ | Ft [ot | ot c+ ot s+ e+ I+ 96+ a 8e+ OOF 9 gs— Té— | 240— | be— | TZ— | 8I— | SI— | Z7I— | 6—_- | 9— ~ | e— | I-— Jet | e+ Jot | ot [ot | ot Jot c+ 7+ e+ e+ T+ e+ C= 98+ 002 9 se— | T€— | 246—- | e— | T2— | SI— | SI— | I— | 6— -9— fe-— | I— | ot pot Jet fot | ot jot fet G+ e+ e+ é+ Tt eet o— sgt 000 9 se— | 1€— | 4e— | — | IZ— | 8I-— | sI— | gI— | 6—_ | 9— J e— | I— | et | e+ [ot Jot |ot+ fot fot c+ au $s e+ T+ eet o—- bet 008 ¢ se— | Té— | 2e— | — | IZ— | 8I-— | SI— | zI— | 6—_ | 9- | e- | I— | et | et [9+ | ot |ot+ Jot Jot o+ ot b+ e+ I+ og+ oF cet 009 ¢ se— Té— | L6— | ¥o— | — | SI— | 9I— | ZSI— | 6B—_—-« | 9— S| E-— «|S CL GH+ 1 GH+ [St CJ ot J OF Jot | o+ g+ ot e+ e+ I+ 6+ @- Té+ 00 ¢ ge Té— | 20— | H— | I2— | SI— | SI— | ZI— | 6B—_- | 9— | E-— | s— | 9+ | 9+ | Ot | Ot JOH | OF | 9+ c+ ot b+ e+ I+ 8o+ e- og+ 006 ¢ be— Té— | L0— | #— | — | SI— | SI-— | AI— | 6—_- | 9— S| -— «| S— | OF Lot J Ot Jot Jot [ot | 9+ 9+ c+ tt e+ I+ 9¢+ oT 82+ 000 ¢ Fe— T&— | 26— | %— | l— | SI— | SI— | AI— | 6—_- || 9— «| E— | s— | 9+ | 9+ J Ot Jot Jct fst | at 9+ c+ ot SF I+ Sor a 98+ 008 + yE— | €— | 8B— | H— | Z— | 8I- | STI— | SI-— | 6@— _- | 9- J e— fo- Jzt fet fzt [Lt Jat fat fat 9+ o+ ot e+ ot Ft iL Sot 009 > g&— | O&— | Zo— | — | T7— | 8I— | SI—-} @I— | 6—_- || 9— FW LB Cf At fet fet fet fet fat fat 9+ o+ e+ e+ ot c+ t= e+ OOF * s&— | O&— | La— | H— | IZ— | 8I-— | STI— | ZI-— | 6— | 9— «| B— YU lt fet fet fet fat fat fat 9+ gt+ ot e+ o+ Tot [= cot 006 + c&— | O&— | Z2— | €Z— | OZ— | ZAI— | FI— | ZI-— | 6— | 9- J e-— | s— | st | st | st fst fst fat fat 9+ ¢+ ot e+ ot 61+ | 0+ 000 + Té— | 6Z— | Ze— | €@— | OZ— | ZI- | FI— | TI— | 6—_ | 9— | e— | s— {st | st |st [st fst fst fat 9+ G+ bt e+ ot git I- 6I+ 008 € O&— | 82— | 9Z— | SZ— | BI— | 9I— | FI— ] TI— | 8B— |} 9- J e— | I— | st | st | st | st | st Jet fat 9+ c+ $+ e+ or 9I+ I- Zit 009 € 62— | 4o— | SZ— | e2— | 6I— | 9I— | EI- ] TI— | 8B—_ | 9— | e— J I— | 6+ | 8+ | st | st | st | st fat 9+ c+ e+ e+ ot Sit I- 9I+ O0F € 83— 93— | F— | I2— | 6I— | 9I— | EI— | OT— | 8B— _ | 2z—- c= I- |6+ | 8+ 8+ | 8+ |8t+ | 8+ | 2+ 9+ G+ $+ e+ ot FI+ T= ST+ 00Z € 9— | 86— | $— | OS | SI— | ST = | BI | Ol | a= | 9m |e | I 16+ Le | ee et let ol et | et 9+ c+ tt e+ a+ Zi+ fC eit 000 € we | Ge | GI | LI | FI | ZI—- | Bm 1 em Le Le Le 1) | ot pet lst ee pee | et 9+ c+ e+ e+ ot Ti+ T= ZIt+ 008 Z T@— | 8I— | 9I— | FI— | BI— | 6 — | L— | S— «YW LL T— | 6+ [8+ [8+ [8+ | 8t | st | 2+ 9+ G+ ot e+ ot OI+ t= Ti+ 009 @ Ob | SE | 8S | HS | 8 = ee SS = PS Per 1S PS |) B+ SAR cede tzoe 9+ c+ e+ SH ot+ 8+ load 6+ OOF @ P= "| SI | OLS 18 — |e GS Em | TO 1 6 OL BH PSH OL 8H Of ete tks | 9+ ot+ F+ im I+ Boe T= St 00% @ I= | t= 1 Ol FS — 19 |G | SS | b= es | Bae PSHE | ete ese ae ee 9+ c+ a+ e+ I+ 9+ Ls bec 000 & S— I= Ee | F— be= | T— LB | ab | 2 | 2+ Lor [ot | o+ G+ b+ S$ ot I+ y+ 0 Pate 009 T y— |8— | o- | I-— | 9+ | 94+ | 9+ | G+ | St | Gt | e+ e+ e+ ot ot+ I+ Cae 0 e+ 00¢ T Ge hi TS 20 G+ | b+ | b+ | b+ | F4+ J O+ | O+ e+ e+ e+ T+ I+ Et 0 i 008 0 €+ | o+ | o+ | ot | st | St | Ut ot I+ I+ I+ 0 0 0 0 OOF 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 WOT @—- | I- Eo Re ee I- I- i od I- 0 0 0 0 og — o- | o- o— o—- o— o— I- I- 1S 0 0 0 0 ooT— o- | o— o— | o—- o— I- Le t— te i ad 0 Loa ost— B=. /e> | s= o- o— [= T= i [— 0 Ee 00¢— E> [E> &— o- o— l= i g— 0 o- 0se— e- s= G— o— = .S 0 so o0s— a= 6 o— t= nS 0 r= ose— oT oT I- So T+ 9 — 00F— o— r= = I+ = Osk— I- L= T+ 8 00g — SuLoyIDT 4oay | Jooy | yeas | yoos | yooy | qooy | qooy | yoay | yoy | yo0y | y00y | goay | goay | qooy | aay | yoay | yoay | yoay | yooy | yooy | yooy | yooy | yooy | yooy | ures OOOE | OSZZ | 0OS% | OSZZ | 000% | OSZT | OOST | OSZT | OOOT | OFZ | OOF | OSZ | OOOE | OS2z | CDSZ | OSzz | 000% | OSZT | OOST OSZT O00T OSL 00g OSs -uedui0o woryes Aqder pue | -uad: -Sod yuour kyder | ~uedur0g jodoL, Ys ~redm00 quoulyieduios Mofag quourjieduros ea0qy -dodol, JO WOT} 3 -BAg[a URE_L [‘s}asmyredur00 uaz, ‘Wf au0g “S19JOU 06ZZ ‘SNIPCI Jo{NO ‘s1oJoU ORZT ‘sniper J9UUT] 35 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. ae = a = = a = ice = io e- l= 0 T+ re ot e+ e+ €+ ot+ I+ 6r+ c= e+ 000 ST ee os 2 or or a = 6—- L- c= §= ik I+ ot ot+ e4- e+ e+ Ser e+ T+ 8h+ LE aot 00S #1 I€— | La— | &— | OZ— | 9I— | FI— eI— G= = c= io t= ie ot o+ Soe e+ St e+ ot+ I+ 9F+ F- og+ 000 #1 = = = = = = L— 6 — i= s= o> o- ot s+ Se b+ et e+ §+ e+ ot+ ce+ t= 6F+ oos et * a os iz ie ot rH ot oI— l= c= g= co o+ SHE e+ ot e+ e+ e+ a e+ eet i= Let 000 &T Z&— | 8Z— | S2— | Te— | ZI— | STI— GI- oI— wire C= s= o— e+ s+ e+ bt e+ e+ $+ e+ ot oh+ = 9b+ 00 ZT ZE— | 6— | SZ— | TZ— | 8I— | SI— sI— oI— L= c= 8— o— e+ s+ ot e+ e+ e+ t+ e+ ot OFT i a ppt 000 @T Z&— | 6Z— | 9Z— | G— | BI— | SI- gI— or— L= 9- F- o- e+ e+ g+ g+ cr t+ tt e+ a+ set 8= Te+ 00g IT = — | 9—- — ae — Si oI— 8- 9- = a b+ e+ g+ c+ o+ e+ a+ €+ o+ LE+ = 0F+ 000 IT ie Oe - ee co or sI— OT 8—- 9—- y— o~ o+ o+ 9+ 9+ G+ gt+ t+ §+ ot set e-— ge+ 00S OT $e— Tgé— | 42— | €2— | 6I— | 9I— FI— TI- 6—- 9- —— o— o+ St+ 9+ 9+ 9+ gt+ gt e+ ot e+ $= | oer 000 OT Fe— Té— | ZE— | &]— | OS— | LZI— FI— TI- 6— 9- a o- 9+ 9+ 9+ 9+ 9+ 9+ c+ §+ ot Té+ &— bet 00¢ 6 ¥e— Te— | La— | Fe— | OZ— | LZT— FI— II- 6— 9— t= o— 9+ 9+ Lt+ L+ 9+ 9+ c+ §+ a+ 6o+ 8 eet 000 6 - = = = = A bI— TI- 6— 9- = c= Lt Loe Lt Lt 9+ 9+ c+ e+ ot Lo+ c= og+ 009 8 i i ie ie oe i bI- II- | 6— 9- 7 Z— L+ L+ 2+ 2+ 9+ 9+ G+ e+ ot 9e+ a 8o+ 002 8 ee— | og— | 2e—| ee— | 0¢— | 2zI-— | FI- | aI- | 6— 9- — o- 2+ L+ 2+ L+ 9+ 9+ St e+ ot P+ o 9¢+ 008 2 ge— | o&— | 92— | &@— | OZ— | ZI— FI— oI— 6—- 9= t= o— s+ 8+ 8+ 8+ oe 9+ gt e+ ot oot c= tot O0F Z ZE— | 6Z— | 9Z— | EV— | OS— | 2I— sI- 6I— 6—- 9- F— o—- 8+r 8+ 8+ s+ i+ 9+ g+ e+ ot Te+ eo e+ 000 2 — < = = =| i= FI— gI— 6- 9- F—- o- 8+ st 8+ 8+ Lt+ 9+ gt a+ ot 0+ oT cot 008 9 ie = oe eo a it FI— si= 6= = : oo st gt oe st i+ 9+ + e+ of 6t+ a lotr 009 9 (— | te— | Se— | we | GI | BI i See aI— 6— 9- r= o— 6+ 6+ st st it 9+ G+ e+ ot+ sIt+ ae Go+ 00F 9 6z— | 92- | #— | 12— | st—|9t- |FI- |ezi- | 6- 9- — o= 6+ 6+ s+ s+ L+ 9+ c+ $+ et 21+ o— 61+ 002 9 6s— | 9e— | &—- | 12— |} sI—|9I- |rI- | II- | 6- 9- a a 6+ 6+ 8+ s+ L+ 9+ c+ t+ e+ 9t+ a en 000 9 —_ ss = — Fa oe ra 1 Sd 6—- 9—- i oa co 6+ 6+ st gt L+ 9+ c+ $+ e+ cT+ o— 2T+ 008 ¢ se ao = oe ar or i i 6— 9- ¥—- o— 6+ 6+ 8+ st Lt 9+ c+ a+ e+ ci+ o— Ze 009 ¢ L@— | HE— | G— | OS— | BI— | 9I—- FI— cL= 6— 9- a o— 6+ 6+ 8+ 8+ Le 9+ G+ e+ e+ FI+ C= ot OOF ¢ 9%— | — | G— | 6T— | LI— | SI—- &I— TI- 6—- o= t—- o= 6+ 6+ 8+ 8+ bb 9+ ¢t+ ot ot e+ o- GI+ 00¢ ¢ Sc— | 86— | I6— | ST— | 9T— | FI éI— oI- 8- 9—- t= a- 6+ 6+ 8+ st 2+ 9+ g+ e+ e+ aI+ o- I+ 000 ¢ ze— | et— | 21— | gt— | et- - - - |e- Je- |e- fer fot fet fst fat fot [ot fet fat fort I- ut 00s + e i ot i a1 _ {TH b a g= .= o— 6+ 6+ 8+ 8+ lt 9+ c+ e+ ot 8+ I- 6+ 000 aiI-|1- |6- |s— |9- - |e |t- Jé6t jet jst jst Jet jot fst Jet fet fot I- L+ 008 € oI- 8—- LCS o= p= o- 1 8+ 8+ tr a 9+ c+ e+ e+ et c+ I- 9+ 000 € 1 Go c= c= o— | 8+ + as 9+ o+ b+ ot ot+ T+ Sar I- e+ 00S Z GI t= c— 67 a L+ 2h 9+ 9+ c+ t+ a ot I+ ot+ i © ab 000 @ I- I- I- c+ ot+ pt t+ t+ or e+ e+ it EF 0 Te 00¢ T 0 0 F+ Pt e+ Cr e+ e+ ot T+ I+ 0 0 0 000 T 0 e+ e+ ot a+ ot Bate I+ 0 0 0 0 0 00s 0 0 0 0 0 0 0 0 0 0 0 0 0 * 209 t= i I= T= I= 0 0 0 0 0 0 0. os — be LS a = iL [= 0 0 0 0 0 00I— om om o— o— i= i I= [= 0 0 0 osT— o- oT o— = l= i c 0 0 0 00— o- o— o— oC [= i= T= 0 Eo ose— C= S= a oT i Same I- be 0 t= 00e— e= o- o- I- I- La 0 In- ose— c= go io [= T= (on 0 6 00%— &— = = i [= oT I+ GS Ost— o— 6 i= o= I+ oo 00g — = ‘Ge? I- = I+ Ga os¢— oT I- b- T+ g— 009— oT I- C= I+ = os9— i= 9 = I+ eS 00ZL— L= 9- I+ i= osL— C= i I+ 8 008— SwmoyyDT aay | qaey | qaay | 499, yoo} oa} qyooy qooy yooy | qaoy | yoy qeoy qyooy qeoy qooy 4oay 490} qaay qeay 4aay uorjes cosy OorF ooo ove ode 0086 00%Z 000e o09T 00eT 008 00% Oost o00F oosé 0008 00Sz 0006 OOST 000T 00s -uadui0 nores sydes : uae -uedmo)| -Sodoy, yueur —— det -j1edureo jueujreduroo Mojeg jueurjiedui0o aaoqy -dodoy, yo uot —UoTjJR]s JO MOJVA|O 10} UOFIOELIO Jo} WOT}0e1I09, -CAI]o UBT [-syueursedu0o SATOMI ‘SIOJOU OZGE ‘SUIPeI JojNO ‘siajeuT O6zZz ‘sn per JauMUy] "9 au0g 36 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. — W01}¥}S JO MO}VAV[d JO} MOT}0eII09 —J0} WOTVaII0N “QE= | BET | BE— | S— | Te— | BI | st | a— J or—]4— Je— ]I- |e t]et]et Jor Jot Jot Jot Jot Jot fat Jet fart | eet c= set 000 st 9e— | GE- | 8o— | SZ— | Te— | sI— | st— | et— | or—] z— | ¥—- |I- ]o+]}o+# ]o+ Jot [ot Jot | ot s+ g+ b+ gt I+ cet s— “et 008 FI 98— | BET | 8B— | Go— | Te— | 8I- | sI— | GI— | or— | z— J ¥— |I- |o+]o+}o+r Jot fat fat fat 9+ gt a+ et I+ ost t= ret 000 #1 9E~ | Be— | 8o— | Go~ | Te— | SI- | sT— | eI— | or— J 2- FF JI- Jo+]o+fot fat [zt fat fat fot fot Jet fer frt | ect tS get 00g &T 9— | BE | BS— | Go— | Te— | BI— | SI— | T— J or— J 2— | H— JT— Jatfatfat fet fae fat fat fot fot [ae fet fre | eet ae et 000 &T ge— | GE— | SZ~ | So— | Te— | BI~ | sT— | eI— Jor— J z— Je— J I— Jet fatyet fet fot fat fet fot fot fat fet frt+ | ot rs ost One et e~ | Te | 8E— | So | Te— | SI- | SI- | I— J OTI— J 2— | F— J t- | St [et]st fst fet fat fst fot Jot fat fet frit [eat a 66+ 000 2t E- | Te— | 8e— | Fe— | Te— | SI— | sI— | I— | or— | 2— | F— Jt- |st+][et ist fat fat [at [r+ fot for fat fet frit | ect a Le+ 00¢ II Fer | We~ | 82— | Fo= | Te— | SI | St— | eI— } or— | z— J e- Jt~ Js+]st |st fst fat fz+ |zt fot fot fot fer fat | act f= G+ 000 TI 8e— | O&— | 2e— | H— | TE— | SI— | SI— | sI— J or— | Z— J F— JI- Jot+let lst jst (st fzt+ fzt fot fot fot fet fat | ost t= e+ 008 OL Be | O— | 2e~ | FE— | OS | SI~ | SI | GI- J or— | A~ | F— | I- Jo+]ot]st |st jst jst fet fot fot fot fet Jet fort s— aot 000 OL Te | 6— | 92— | Se— | OS | BI— | SI— | GI— | or— | z— | F- | I- |6+]oet]ot |st |st+ |st+ |st fot fot fet fet fat fart Sa 06+ 00S 6 OF | 8B— | Se— | EZ— | OS | SI— | SI~ | @I— J or— | z— JF ]I- |o+]ot et Jot 8+ |st [st fst Jot [ot fet fet fort c= 6I+ 000 6 O&= | 8B— | Se | EE— | OC | SI- | SI | eI- | 6 — | 2— | F- | e— J ort }ort}et Jot jot {st [st fut fot fot Jet fet fort e— st+ oo 8 Go~ | LO | HE | GE | GI— | AI— | FI | I- ] 6 — | 2~ FH | e— fort fort |et jot jet [st jst [zt Jot fot fet fat | ert &— 9+ 000 8 E> | 9~ | HE | G— | GI | AT | FIT | I- | 6 —]2— | F- | s- Jort+]or+]et fot jet jst jst fut fot fot fet fat | ert g- s+ 008 2 8o7 | OT | FET | Te— | BI | Ot | HIT | I~] 6 —] 2— J o~ | e— J ort fort }et jot yet [st jst jst fot jot fet fat | ert o— w+ 009 2 ke~ | So— | &S— | Te~ | SI— | OT | HI | I~ | 6 —|2— Je— | e— jort|ort+}et+ jer Jet Jet fst fet fot fot fer fat jit a5 s+ 00% 2 9— | FE— | G— | OS | SI— | MI | &I- | TI—- | 6 —] 2— | F— | e— fort ]or+]et jet Jot jst fst fet fot fet fet fet fart a= s+ 008 £ Se | Ge | OS | SI— | SI— | EI— | TI— | 6 — | z2— |e | S— fj ort | ort jer jot fot fst fst fut fot fet fet fat fort e- a+ 000 2 2 | Ge— | OS | LI- | SI~ | EI— | TI—- | 6 —] z— | e— | e— Jort+}ort+}et jet jet [s+ jst fst fot [ot fet fat ort oa a+ 008 9 eo— | Te~ | I~ | AT— | SI— | SI— | TI- | 6 —] 9— [ F— | e— fort sort ]6+ Jet jot jst Jst fst fot fot fer Jat for o— T+ 009 9 Te | 6r— | At— | st~ | &I~ | 11— | 8 ~ | 9—- |e |Z Jort+ }ort+|6t+ |6+ Jot | s+ st fst fot fot fet fer Jot o— a+ 00F 9 OG~ | SI~ | OT~ | FI— | GI~ | OI— | 8 — | 9— | F— [I- | Ort] ort) 6+ Jot fot | 8+ st fat fot for Jet Jat fst o— or+ 00% 9 6I— | I~ | 9I— | FI— | GI— | OI— | 8 — | 9— | e— J I— J or+|or+]} 6+ jst jst fat fzt fot Jot fot fet fet fet o- or+ 000 9 SI~ | sI— | &I— | 1I- | 6 ~ | 2—]/o- |e J I- jort+jort+ ot |st |st fat fet fot fot [ot fet far fat o- 6+ 008 ¢ 9t~ | t— | &I— | 1 | 6 — | 2—]s- |e J i— fort jort+]} 6+ fst fst fat jet fot fot fet fet fre fat o— 6+ 009 ¢ SI— ) FI— | I~ | OL— | 6 — | 2—]e- |e |I- [6+]eot+]st+ [st [st fzt+ zt fot fot fet fet ftt fot a gt oor ¢ BI— | eI— ] OI | 6 | 2—/S—- |e JI- |6+]}6+/st [st fst [zt J zt fot fot fet Jet ftt fot z- 8 +. 002 ¢ sI— | 1I— | OI— | 8 —] 9 —]e- [e— JI- |6+]6+]st+ [st fst jet fzt fot fot Jet fet Ti+ Jet o— a+ 000 ¢ oI— | 6—|8—|9—]Fr— | [I- |6+]6+] st fst fst fet [zt fot fot [rt fet [rt Jet i= g + 00g & L—|4—|¢-)e— |e JI- |6+]o+]8t |st fzt |ot Jot fot fot fet fet rt Jet I- be 000 + 9-|S$—]e- |e JI- |St]st]et fst fat |ot fot fst fet fer fat It a+ I= e+ 00s € G$—|r-]e- |e |I- JL+]}L+]/9+ [ot Jot [ot fst fet fet fet far fit fer T= e+ 000 £ €—|%@- |I- }I- }o+]o+ ]ot fot fot [et [e+ fet fet [e+ Jit jo T+ t= a+ 00s @ o— |I- |t- |gt]o+]ot jot [ot fet fer fet fat fot fit fo 0 I= T+ 000 z O Jo \e+le+)et fet fet fet fet fet f2t f+ Jit jo 0 1= r+ 00S T 0 e+tletjet+ jet zt |zt j2t fet fit Jit fit fo 0 0 0 000 T @+]ot+)et fot [rt fit | t+ ft fit Jitt Jit Jo 0 0 0 00s 0 |0 /@ |0 Jo jo jo 0 0 0 0 0 0 0 0 0 “ya 0 PPA t= [t= te yr rete te = 0 0 0 0 cor — P= [tee ee ie be T= 0 0 0 0 002 — Z— |@- |%@ |e- J@- |I- |{I- Jt- Jo 0 0 0 00g — €— }%@- [Z- |e- |e- |I- |I- Jo i 0 i 00% — t= (e=> fee |e. fae pita fre: tre T+ e- 00s — rm [t= |g= |i |e t= lee t+ z— 009 — e- js J@- Je- fI- Ja- I+ eS 00h — s- |s- |z- |I- |e- I+ we 008 — Gee ee eee I+ gh 006 _— a (t= es t+ 9 — 900 T— tee es t+ 9= oor 1— TS Wes Te al 008 T— SULOYID 409} | Joo | Joos | Joos | qooy | qoay | Joos | Qaoy | yooy | qoos | qaox | Joos | gaa | qooy | yooy | yoog | yooy | qaoy 4909} qooy {oo} yooy qooy qooy | ~=uolts 0022 | 0099 | 0009 | OOFg | COSr | OUZF | OO9E | OUDE | GOFS | GOST | GOZT | 009 | OOzL | 0099 | C009 | OOFS | Cor | doz | OOse | Ooos | OFZ -| Oost | Cozt | O09 “ed 09 | owes | sydus P -uadt “id Wout - ude. uadurog | -odoy, aie ree quouljiedui0s Moleg quouljziedui0s saoqy -sodoy, Jo HOWeA : -9]9 UE ['squeujiedur0s u997x1g “S19}9UL OFSE ‘SNIPS Jo}NO ‘s19}eUN OZEE ‘SNIPB1 JaUuU]] "H auoz 37 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. og— 9F— | GF— | 8E— | EE— | 8S— | FE— | OS— | ST— | TI— | 2—- = sit | ett | et+ | e+ | st+ | att | I+ Ti+ 6+ gt 9+ e+ wot L= Teé+ 000 ST 6F— ch— | Ih— | LE— | GE— | 8Z— | F— | OS— | ST— | TI— | 2—- e- FIt | FL+ | E+ | PEt |-FT+ | SI+ | ott Ti+ oT+ 8+ 9+ e+ got Li og+ 00¢ #1 8h— vr— | Te— | 9E— | GE— | 8Z— | FO— | GI— | ST— | IT— | 2—- o= FI+ | FL+ | PI+ | FT+ | PI+ | €T+ | 2t+ TI+ oI+ gt 9+ €+ oot 9—- 8+ 000 #1 Le sh— | OF— | 9E— | TE— | 4O— | SS— | GI— } ST— | TT— | £—- s= I+ | H1+ | FI+ | FE+ | FTF | SIt+ | a+ Ti+ oI+ st 9+ e+ 0¢+ 9- 9o+ 00¢ &T 9F— Gh— | 6E— | SE— | TE— | 4e— | ES— | BI— | ST— | TI— | £Z- = Pit | I+ | PL+ | PT+ | PTF |-8E+ | I+ Ti+ oI+ st 9+ e+ 6r+ 9—- Sot 000 &L Sr TF— | 88— | FE— | OF— | 9Z— | GS— | BT— | SI— | TI— | L—- e— | e+ | e+ | tt | Ft | T+ | Ott | att Ti+ oI+ 8+ 9+ e+ Sit 9—- oot 00¢ GT oe OF— | LE— | EE— | OF— | 9Z— |} GB— | BI— | ST— | TI— + 2— &— Git | bI+ | FI+ | FT+ | FEF | SI+ | att Ti+ oI+ gt 9+ e+ 2T+ g— oot 000 oI 6E— | SE— | G&— | 62— | GS— | To— | 8I— | FI— | TI— | Z— &- GI+ | FI+ | FI+ | FT+ | FT+ | SIt+ | at Ti+ 6+ Lt+ c+ e+ sIt+ ¢— 0¢+ 00S TT LE— | FE— | TE— | 8S— | GZ— | Te— | SI— | FI— | OT— | 2—- 8 Sit | FI+ | FI+ | HI+ | FT+ | ST+ | ait Ti+ 6+ Le G+ e+ FI+ G— 6I+ 000 TL FE— | TE— | 83— | FE— | T]— | LT— | FI— | OT— | LZ—- = Git | I+ | PI+ | HE+ | FE+ | ST+ | att Ti+ 6+ Lt G+ €+ vit c= 61+ 008 OT €E— | O€&— | LE— | FB— | OS— | LT— | ET— | OT— | Z— o> St+ | FL+ | bt+ | F1+ | FI+ | ett | ZI+° Ti+ 6+ L+ s+ e+ eit+ g— sI+ 009 OT €&— | O€— | L2— | &U— | OS— | 9T— | ET— | OT— | 2Z— c= SI+ | PI+ | 1+ | T+ | FT+ | Ott | 1+ oI+ 6+ Lt G+ e+ sI+ Gc 8I+ OOF OT GE— | 6G— | 9Z— | EV— | OS— | 9T— | ET— | OT— | 2Z— ¢= GI+ | PL+ | FE+ | PL+ | I+ | OEt | IT+ oI+ 6+ Lt+ G+ e+ a+ c= 1+ 002 OT ZE— | 6G— | 9V— | G— | 6I— | 9T— | SI— | OT— | L— c~ Git | PE+ | PT+ | FT+ | I+ | VE+ | 1+ oI+ 6+ Lt ct ot+ a+ ; om 9T+ 000 OT 8Z— | SS— | G— | GI— | 9I— | SI— | OT— | L— &- GI+ | FI+ | PE+ | FI | PTF | Ut+ | TT+ oI+ 6+ Lt G+ ot Ti+ - gt+ 008 6 8G— | SZ— | G— | 6I— | 9T— | ST— | OT— | L— 2 SI+ | HI+ | bI+ | FT+ | SI+ | B+ | I+ or+ 6+ l+ S+ ot Ti+ y= 7 009 6 L6— | ¥O— | GB— | 6I— | 9T— | ET— | OT— | L— e= PIt | FL+ | PI+ | FT+ | St | r+ | Ti+ oI+ 6+ Lt c+ ot or+ ~— I+ 00F 6 L6— | ¥6— | T@— | 6I— | 9T— | SI— | OLT— | LZ— S= Pit | PT+ | I+ | E+ | Ett | 2+ | 11+ oI+ 6+ Lt G+ ot oT+ p— PI+ 00é 6 9%— | ES— | TZ— | 8SI— | SI— | GI— | 6 — | 9— C= I+ | I+ | FI+ | ST+ | Ett | Or+ | I+ or+ 6+ L+ ct ot 6+ $- gI+ 000 6 €o— | O@— | 8SI— | SI— | @I— | 6 — | 9— = It | PI+ | PI+ | St | I+ | a+ | t+ or+ 6+ L+ G+ a+ 6+ ¥- ei+ 008 8 €c— | OC— | 8I— | SI— | @I— | 6 — | 9—- e— | bi+ | w+ | br+ | ett | Sit | a+ | 11+ oI+ 8+ l+ c+ ot 6+ ~—- ett+ 009 8 co— | 6I— |.4I— | FI= | 1l— | 8 — | se c= PI+ | I+ | St+ | &T+ | I+ | at+ | t+ 6+ 8+ 9+ gt G+ 8st r= ort 007 8 eo—.| 6I— | ZI— | FI— | TI-— |] 8 — | S— €— | bit | bI+ | et+ | et+ | Ett | ott | oft 6+ 8+ 9+ G+ ot gst F= ait 002 8 T@—- | 8I— | 9I— | €I— | II-— |} 8 — | s— = I+ | I+ | et+ | t+ | eit | I1+ | of+ 6+ 8+ 9+ e+ o+ L+ = Ti+ 000 8 A= | A= | l= | Ok— | 8S— | = o- PI+ | I+ | ert | a+ | att | Ti+ | or+ 6+ 8+ 9+ a+ e+ 9+ = 64+ 00S 2 9I— | FI— | at— | OI-— | 8 — | S— o— ert | st+ | a+ | at+ | at+ | 1i+ | or+ 6+ 9) ae 9+ y+ o+ G+ a= 8+ 000 2 FI— | ZI— | oI— | 2 — | S—- o— ett | ot+ | a+ | 11+ | 11+ | Or+ | 6 + gst Lee 9+ ian e+ q+ s- 8+ 00g 9 GH T= | 6S | 2— 7] se oc it | 2+ | 1+ | 11+ | 11+ | O1+ | 6 + 8+ Lt+ G+ e+ o+ y+ c= 9+ 000 9 6-— (219 — 1 2= o- @i+ | Ti+ | 11+ | Or+ | Or+ |} 64+) 8+ Lt+ Oot G+ e+ o+ e+ o— c+ 00s ¢ Ree | ores Qa o— Ti+ | 11+ | or+ | OT+ | OT+ |6+)8 + Lt 9+ e+ e+ ot ot+ os e+ 000 ¢ 2S | eS o— OI+ | Olt }oT+ | 6+; ;6e4+);8t 2+ 9+ 9+ e+ e+ ot a+ o> y+ 00S F QS eS eS I- Ort | ort }e+l]oe+/8t{/styat 9+ gt+ e+ €+ o+ toe Z- e+ 000 + Ber ee; i 6+/6t+) Sti stlz+tl{[z2+)/9+ g¢+ G+ e+ e+ T+ Tt oT e+ 00¢ & P| oF dos St stlzaty~2ti ~atloa+ty et ge y+ g+ a+ I+ 0 = T+ 000 € oT pie L+/L2+,}/9+/9+/9+]/¢+/¢4+ y+ et €+ ot I+ 0 ie T+ 00S Z o- | oe 9+/9+]/o+]/o+);o+}]ot+]/Ft+ e+ et+ G+ ot I+ 0 LE T+ 000 3 iL P+l[rtl/r+i r+ {rt+jetrlyet e+ ot+ a+ T+ I+ 0 i= Le 00s T er etietiletiletj et etlor ot ot+ It T+ 0 0 0 0 000 T GHC lS ITE | Le | te I+ Ee 0 0 0 0 0 0 oo¢ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0d Go| Se — | oH | oe | r= L= Le l= [= 0 0 0 0 002 — Pe) | eS a [8 eo G6 67 o— i i 0 I+ LS oor — GS | RS e=— |e = f= == o— i = £= I+ (a 009 — . 9—-—|]9-|\S- P= ¥- s- oT i oe L= It C= 008 — 99> c= Ea c= o—- = g7 ot = 000 T— 9- Ques = o—- = oo ot GS 006 T— = r—- = tT a ot LA OOF T— = ¢= = f= a+ 6—- 009 IT— &—- oc 6 — a+ II—- 008 T— o— I e+ FI 000 2— SUOUDT 4a0y | rey | yaex | qo0y | goog | aoaz | goax | ao0z | qo0y | ye0y | goog | yey | goog | y003 | yo0p | a00p | yoy | yoay | aooy | soar | gooy | gooy | gaoy | yoop | uoMes 000ZT |O00TT | COoOT | 0006 | 0008 | 0002 | C009 | CODE | 000% | CODE | 000% | OOOT | OONZT | OOOTT | OODOT | 0006 | 0008 | 0004 | 0009 000¢ 0007 0008 0002 O00T — mores Ayer -ued m0: -sodo, \ueul Ayder 9 ue ~yxed 0100 quaurjredur0s Mopoyt quourjredui00 eaoqy “Bodo, JO NONBA -919 Weep, —U01}8]s Jo WOTLBANTA Joy UOT}DEII0;) —JoJ WOT}Oe1I09 [‘s}aeujiedui00 £400M LT, "SIO{OUL OFFS ‘SNIPVI Jono ‘sIo}our OFZG ‘sNIpeI souUT] "I 90g EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 38 6e— 93— | Z— | 6I— | ST— | TI—- 9- o— Te+ | 0+ | 6T+ | SI+ | ZT+ | ¢T+ I+ oI+ 8+ 9+ e+ FIt OI— Ft 000 ST 88— 9B— | Z@— | SI— | SI— | II—- 9- | 3- Tot | OS+ | GI+ | ZI+ | OT+ | FI+ TI+ OI+ st gt+ e+ sI+ oI— e+ 00¢ FT Le— St— | T— | 8SI— | FI— | TI—- 9- | 3- To+ | OS+ | 6T+ | LI+ | 9T+ | FI+ Ti+ 6+ Lt Gt et ait G= Tot 000 #1 9g— Fo— | T2— | LI— | FI- | TI 9— | 2— | 02+ | 6I+ | SI+ | ZI+ | OT+ | FI+ Ti+ 6+ Lt G+ et Ti+ 6 — oo+ 00S eT se— €6— | OG— | LI— | FI— | IT— G— |e— | Oct | 6T+ | SIt+ | OT+ | ST+ | FI+ Ti+ 6+ L+ q+ a+ oI+ 6—- 61+ 000 &T ee— @3— | OC— | LI— | FI— | OT— G-— | e-— | oo+ | 6T+ | ZT+ | 9T+ | ST+ | I+ Ti+ 6+ 1+ o+ c+ 6+ 3 = LIF 00S 21 se— @o— | 6I— | 9I— | I— | OT— G— |%—- | 03+ | 8T+ | ZI+ ] 9T+ | ST+ | F1+ Ti+ 6+ L+ e+ ot 8+ = 9I+ 000 ZT T@— | 8I— | SI— | €I— | OI- S— |%— | 61+ | 8tt+ | 2ZI+ | 9I+ | I+ | eT+ oI+ s+ L+ e+ ot L+ 8 sit 00S TT Ié@— | 8I— | SI— | @I— | OI— G— | o— | 61+ | 8It+ | 9I+ | ST+ | FI+ | sI+ oI+ st 9+ ot o+ be b= I+ 000 IT O@— | LI— | #I— | @I— | OT— g— |e— | 81+ | 2zt+ | 91+ | Stt+ | 1+ | ett or+ gt 9+ e+ ot 9+ Lo sit 00¢ OT 6I— | 9I— | FI— | TI— | 6 — g— ot SI+ | LI+ | ST+ | FI+ | ST+ | I+ 6+ st 9+ e+ a+ gt+ (ed git 000 OT 8I— | 9I— | €I-— | TI-— | 8 — $— | 2— | 241+ | 91+ | ott | bI+ | STt+ | ZI+ 6+ Lt+ 9+ pt ot+ a+ g9- OI+ 00¢ 6 LI— | SI— | 8I— | OI- 18 — F-— |t- 20+ | OT+ | GSI+ | I+ | I+ | I+ 6+ Lok: 9+ p+ ot e+ 9 64+ 000 6 9I— | FI- | aI- | OI-— J 2 — b— | @— | Ott | STt+ | HI+ | Ott | att | 11+ 6+ J2+ [ot b+ ot e+ o= 8+ 00g 8 SL" | STS 6: = 2 = bcs o- QT+ | SI+ | FI+ | I+ | att | Ti+ 8+ bok G+ e+ ot ot oo sob 000 8 TI-— | OI-— | 8 —|9—- &— oe I+ | bI+ | Sit | SI+ | 11+ | OT+ gt+ Dee c+ e+ ot Ls oe 9+ 000 2 eh Sh = i eI+ | et+ | ci+ | 01+ |}6+)8 4+ Lor gt e+ e+ I+ 0 P= r+ 000 9 == pS i= a Ti+ | Ot+ | oT+ |]6+/}/8+)2+ 9+ G+ e+ et T+ 0 oo Ce 000 ¢ g= I= |0 6 FE Se Se Be Oe bp oe G+ pt e+ ot Tt y= or o+ 000 F I- |0 L+]/9+}/9+]9+/¢9+ )/¢9+ r+ e+ a+ ot+ It 1 Si o- T+ 000 € I> 10 G+t)/p+l]p+ipti p+ryet+ e+ e+ a+ ot I+ t= LS 0 000 & It [@+]e+r [et ;otlyeti tt I+ Tt T+ I+ 0 LS PS 0 000 T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 198 €-|}@-|/¢-|¢- > LS I= t= Tt Esk T+ 0 00F — y- |b ‘aa c= o—- I- t= 0 a+ og 008 r- S= o—- I- bate 6: E R= 006 T— smoyyog aay yooy | ya0q | gaaq | yooy | yooy aay | yaay | ya0z | qooy | yoy | goog | yooy | yooy qa0y 4oay qoay 40a} 400} woryes o00zT 0008 | 0002 | ooo9 | ooo | coor 0002 | ooot | co0zt | ooott | cooot! o006 | 0008 | o002 coos | coor | aooe | ooo | oot | -wodtt0>| goes | xqdter soae -uedwm0g | -80doz, jueul Yer 4redui0o queurredui0s Mofeg yuewyredu100 eaoqy -dodoL Jo WOTyeA -9[9 URAyy —U01}81S JO WOLYBANTA JOJ WOTJDeLION —J0} WOTJOeIIOD [syueujsedu0s ueeyxIg *s10}9TT OOF ZI ‘SNIPeI J0jNO ‘s19J9UI OFFS ‘SNIP JouNT] fee 39 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. tI— | F1—|et— | 2t—|or—|6— |2— | 9- | ¢—- | e—- | s- | I- | ett | 1+ | ort | 6+ st | 2+ | 9+ g+ pt e+ ot T+ Sok @I— SI+ 000 ST tI— | et—|zi— | 1—|or—|]s— | 2—- |9- | s-— | e- | e- | I- | ett | T+ | ort | 6+ 8+ | 2+ | 9+ gt ot e+ ot Ta ot sI— PI+ 00S FT t= eI— | 2I— | T1— | O1— | 8— L= 9- c= ; oe o— i It+ | O1+ | OT+ | 6+ 8+ Lt+ 9+ gt e+ e+ a+ I+ e+ i= sit 000 FT ei— |zti-|}t1—]}]or-}6 —|s— |2— |9- | ¢- | #- | @- | I- [1+ |] ott oIt |} 6+ | Bt | 2+ | 9+ g+ b+ e+ ot Tt I+ Ii- It 00¢ &T eI- Cl} TE= | 01> | 6 =| 8— L a= c= c= o— i= Ti+ | oOl+ | 6+ | 8+ 8+ Lt+ 9+ ct a+ e+ a+ It I+ T= Zit 000 €T oI— II-— | 0I- }6 — | 8 —]2- 9= 9—- c= 2 o— 0 Ti+ |ot+ |6 +] 8+ i+ 9+ 9+ c+ e+ e+ ot T+ 0 oI— OIt+ 00¢ ZT ZI— | t— 1 0t- |e — fem | 2— | 9 |e [8 |e Te ot+}/6et+ioa+]st |zt | ot | ot e+ e+ e+ a+ T+ 0 oI— oI+ 000 21 6I— l—|0i— |@ — 1s —|i- | 9- |S Fee |e Pe 10 ort |/6+]/6+]st | zt | ot | ot a+ P+ e+ o+ I+ 0 6- 6+ 00S IT i- |oi-}6—]8s—|24—-—]9- |s—- |¢s- | F#- | &- | I- 10 ort |e+{st}]zt jzt | 9t | ot e+ P+ e+ e+ T+ t 6— gt 000 TT i |e 14-1 S— Lee fs |e i= fem tie ie 6t+}/sti stl] z+ |9t jot | ot e+ e+ ot I+ I+ tS 6= gt 00S OT Gi 1c |6—1e—|2—19- |. |r pe | em I-18 6+ti/stist}]z+ jot [ot | St ot e+ ot I+ t+ T= = Le 000 OT oI- 6-18 — | L=|9—|e= | eo leo [eo |e LT 10 6+ istist]zt+ [ot [9+ | Gt au e+ ot T+ It o— 8 — 9+ 00S 6 = 6—le-lb—|a—-|@—- |e le | t— 1e- TT | 0 gst{/L2+]L2+]9+ [ot | ot | o+ e+ e+ a+ T+ T+ o— ia g¢+ 000 6 G— 1F=—(R=—|2=—10—|S— | F—- |e 1a | a= TT 18 Stlz+]2+}]9o+ [ot [St | t+ et e+ ot Tt I+ o— £5 G+ 00¢ 8 gs—- |8—l|/2—/9=—19=—/|8=— | F- | 8- Le | oo I- |0 L+i/L+]9+]9+ | G+ [St | + e+ e+ e+ T+ I+ o— Le Ge Hk 000 8 L—-{]9—-/]9-—|]¢¢-—]*%- |#— | €- | e- | s- I- |0 L+i/9+/9+]¢+ |o+ |F+ | Ot ct o+ et T+ It o— 9- a+ 000 2 QS = S| RS e o. €— o— T- |.0 9+/G$+/E+]%4+ $+ e+ e+ e+ ot+ a+ T+ I+ o— go iS) te 000 9 = | ES e- oe eS oT b= 0 G¢+/G+]e +) Ft e+ e+ a+ ot T+ T+ I+ 0 o— i= c+ 000 ¢ a o— o— a Lt 0 Pr+yjrt+]e+)et+ e+ ot @t+ a+ T+ I+ I+ 0 o— aa L+ 000 F i Se = LL I- 0 et+ijetryaor|ot ot ot e+ a+ I+ I+ T+ 0 o— G— 0 000 € 1 I-— |0 ZeiS+iscw ler | et | ee Pre T+ I+ T+ I+ 0 i Le 0 000 @ 0 TEP eh tee Pe Lie I+ 0 0 0 0 LS Le 0 000 T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 at PH |b tH t= (he ie t= [= tT L I- 0 I+ tt+ 0 oor — ee i I- i E= 1I- 0 T+ et i= 008 — t= = i I+ Pa oe 006 T— SULOYIOT yooy | yo0y | y00y | yo0y | yooy | 4003 | Joos | Joor | ooy | Joay | JOOE | JOOF | JooT | Jooy | JOO] | 400} |, 790y | Jooy | Joos 409} 00} 402} 490} Joo} wOT}es 00zz | 0099 | G009 | OOFS | OOSF | OOS | CODE | ODE | OOFS | OORT | OOST | 009 | 00ZL | 0099 0009 | OOPS | OOSF | OOZF | 0098 0008 00S 008T 008T 009 -uedu0o mores Ayder Codex [wedurog | Bodog, qyueur eed ~yredu100 qyuewji1edur0s MoTeg queuyiedui0s sa0qgy -sodoy, Jo WOTywA -9[9 Wee, —U01}246 JO WOLZVASTS IO} WOTJOeI1I09, [sqyueu7zredui0s A,U0MT, “SIOJOTI QOS ST ‘SNIPeI JoyNO ‘sI0JeU OOF ZT ‘snipes Jouuy] BA 10} WOTJD91I0Q EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 40. OI— | 6— |8— J4~ | 9— | 9~ |s— |¥- |e- |e- Js- | t- Jot |et [et Jat Jor fot Jot Jot Jot fet fat fit for sI— 6+ 000 $T 67 |8— |4= |9~ [s= [s— ]F— |e | e- Js- JH JIi- [st fet [et Jot fot fot Jat fet fet Jet [re fre foe I- st 000 1 6 |8~ |4= Jom [e— | s~ ]F— Je— | e- [s- |r JI- [st fet fat jot fot [ot fat fet fet fat fit jt foe eI— Lt 000 81 6— |8— Je4— | 9- | S— | S—- J- [e- | e-— |e- | I— |i [st fst fet Jot fet Jot Jat e+ e+ e+ Tt te 9- gI—- 9+ 00$ eT 8— |LZ- |]9- ]9- [G- |e- |F— Je- Je-— Je- |I- ]I— Jat |zt+ fot Jot Jot | ot | ete e+ e+ e+ I+ It 9- eI— or 000 @t B— [4~ | 9- | S- |e |e | F- [e- Je- |e |I- | t— |2t+ fot Jot [ot fot fat |at e+ e+ + I+ I+ 9- I— Sr 00¢ TT L- jL~ |9- |G- |%- ]F- J e- Je- [z- |e- JI- JiI— [z+ }ot Jot Jot [et | ot e+ e+ a+ ot iF T+ 9- II- ot 000 IT L- |t- [9- |G—- |$- |e- Je- Je- [e- |e- |I— | i— Jzt ot Jot [ot Jet b+ | e+ et ot o+ I+ I+ 9- oI— e+ 00S OT 4- |9- 19- |S- |%—- |#- Je- [e- |e- Je- | I— | I— Jot Jot [ot [ot fot ot | et e+ o+ o+ It It 9- oI— b+ 000 OT 9— |9- |S—- |G— |%— |F- |e- J e- | e- |e- |I- Io 9+ | 9+ | G+ | Gt | e+ | et Jet e+ ot+ ot I+ 0 o> 6 — e+ 00S 6 9— |s- |S- |F- | F- J e- | e- Je- |e- |I- |I- Jo 9+ | St | ot | e+ | o+ | et | e+ o+ o+ T+ I+ 0 = 6 — e+ 000 6 Ss |e | e= Les (ea [ee ie | |e. Io Gt | G+ (G+ | b+ |F+ | et | e+ o+ ot T+ I+ 0 = 6e== e+ 00S 8 ge (b= |e | 8= [ee 1 e= bee | re it [0 Gt |} c+ [e+ | b+ | P+ | et Jet e+ ot T+ It 0 c= 8 —- et 000 8 P= | F=— 1S PS [oS Pee be I- |T- |0 Gt |}G+ [$+ | b+ Jet | et Jat ot Tt I+ T+ 0 cf 1 a+ 000 2 e-— |e- |@- | e- i = I- |0 + | Ft | F+ J Ot fet | ot [z+ ot+ T+ I+ T+ 0 i an os I+ 000 9 o-— |%@- | B- i to I-— |1- |0 et |e+ fet | et fet | z+ {ot a+ T+ I+ I+ 0 Ee. C= T+ 000 ¢ I- i oe I- |1T- |0 St [et 1 e+ (ee | ee 1 i+ I+ I+ I+ T+ Tet 0 ¢= = I+ 000 * ke [0 0 0 c+ | 2+ [ot | 3+ [2+ | t+ I+ T+ 0 0 0 0 s— 2 0 000 € 0 0 T+ Te |e re T+ T+ T+ 0 0 0 0 o— Oo 0 000 @ 0 iP Pie | re [ede ie ie 0 0 0 0 0 0 L= i= 0 000 T- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19g = {t= re IS = Te T= I- 0 0 0 0 I+ Lt 0 o0g — I- |T- | I- I- Li I- f— 0 at ct 0 009 — I- I- LS T= os i 0 ot ag i 006 — t= te 0 e+ y+ i 00 I— 0 e+ 9+ o— 00S I— SULOYIOT yoo | 409 | ya0q | eax | go0q | Joos | gooq | 00x | 400g | 4095 | goa | goog | ga0q | goaz | go0q | goa.) yoo | yoay | gooy | oor | yoor | yooy | yooy | aoa | tones OOFS8 | OOLL | OOOL | OOED | 0O9S | DOG | OOS | OOSE | 008% | OOTZ | OOFT | OOL | 00FB | OOLL | COOL | ODES | ODDS | OD6F | OOF oose 008¢ 001% OOFT 00L -uedui0. mores Aqder pue i Tau -uedurog | -Zodoy, z Aydei 4redui0o queuyreduioo Mopeg yuounredur0s evoqy -Bodoy, Jo mora -9[9 Ueayl —01} 81S JO MOLVA IO} WOT}IILION —I0} W0{}001109 [‘syueurjzedur0o inoj-AyUOM J, *SI9}OUL ON 8z ‘SNIPeI 12]NO ‘s19}9UT OO ST ‘sNIpeI JaUUy] "T au0g 41 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. “190] G6P ST OINAVAIND OY} PUB (SOT Z'LZ—) WY S'Eh Sl OUOZ SII} JO O[PPIUL OY} 0} WOTZS4IS OY} UOT OOURISIP OY, ct— |#1—|2tI-| m—]|ot—|6— |8s— | 2—- | s-— J e— | e— | s— | OTF | Stt+ | ett ait | m+ ]6+t |} st J lt | st jet | et a+ I+ Le— eo SIt+ 000 ST FI- «| et— | m—|or—|6—|s— |8s— |24- |s— |e | e- | o- | Stt+ | oIt+ | ett | 1+ OIt | 6+ | 8t | Zt | ot | e+ | et ot It ch— 8g— sit 000 #1 ei— |zi—|m—|or—|6—|s— |4- |9- |S-— |e | &— | @— | I+ | ett Zit] 1+] ott] st | sat [ot [ot | e+ | et ot T+ er— 2 Ti+ 000 €T oI— ti— | oi-|6 —|8—]8s— |4—- |9- |S—- |#—- | e- | e- | ett | sit] 1+] ott] 6 + gst | 2+ | 9+ | S+ | e+ | O+ a+ T+ a a oI+ 00¢ 2T eI— ti—- | o1- |6 —}8—|2- }2- |9- | S— | F—- [| e- | o- | ett | t+ | 11+ | O1+ 16 +/]st+ [2+ | 9+ | Gt | F+ | E+ ot I+ Or— 6h— 6+ 000 ZT ti— |oi-|6—|s—|/s—]4- |9- |S- |#- Je- | e- | e- | att | 1+] Olt] 6+] 8 pit [ok | ge | er Tee | oe ot It 6e— 8h— 6+ 00S TT ll- |oi-|6—|8—|{8—]24— |9- |S {F—- [&— | &- | s- ait |t+|ort}/6e+istl{2zt+ |ot Jot | F+ | et | et oF T+ 8E— oF— Bote 000 TT oI— 6—|6— | 38 — [8 —)12= Lo= i = g- ca o— Ti+ | O1+ | O1+ |]6 +) 8 +4 24+ 9+ G+. | #+ e+ e+ ot T+ 9E— er— Look 00S OT or- |6-—|6—|s—]2—f]4—- ][9- ]S- Jem Fe-— JV J e— | Wt | ort OIt+|/st{2+] 24+ | 9+ | St Jet [et | ot ot It ce or LoP 000 OT 6 — 8S 8 | b= [2a o= g- ¥—- T= ¢= o- oT Oot }6+1/6+/8+}24+) 9+ G+ et et e+ ot ot I+ ee— 6e— 9+ 00S 6 6—- |s—|s—/z—]9—J9- |S-— |#- |*%- ]e- | 2@- | e@- JoIt+|6+/8 +/2+]9+]9+ | ot [e+ | + | Ot [ot a+ T+ se— Le= G+ 000 6 gs—ls—/|z—-/9-—J/9- |¢- |#- ]#- ]e@- J e- |I- Jot+]st]stjezt+yig + | EG Pot [PP Lee | ee It It og— ge— g+ 00¢ 8 g—|/s—/z-/]9-—|9-— |S- |#— |#—- {@- ]2- JI- |6o+]stl[ sty +]}/9+/]9+ | G+ |o+ | F+ [Ot | ot It I+ 6Z— eg— pet 000 8 LZ—-|/|9—-{¢—|¢e—- |#— ]@- Je-— ]e@- Je- JI- |8tl[ez_tlet+rjotis + | ob [er Feb |e | er | ot I+ I+ 93— 66— Gob 000 2 Pa | e— o= oT o- o- |- L+/9+{/9+/;9+])¢b 4+] 6+ e+ e+ ot ot ot I+ I+ &— So— a+ 000 9 e=> pe= (seo eS I- |1- |/otiogtietyr+]pt+i)et fet jet [et fot [tt t+ 0 6I— Te—- + 000 ¢ C= i f Soe I- I- ¢+/o+{r+]pr+{etr]et ot a+ I+ I+ I+ I+ 0 9t— tI—- t+ 000 F l= LS ee 0 p+l[p+ti{erletl]etr]str ot+ a+ tr T+ i 0 0 oI— sI— Le 000 € 0 0 et+tletl|/ze+]et+]/e+]It+ |1t+ [1+ |1I+ | |0 0 0 8 —- s= 0 000 @ 0 etl]erl;rTt+}]t +i] iT+i)] itt T+ |1I+ | 1+ | 1+ |0 0 0 ee = 0 000 T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10T SoS [oe eT a Te c= TS. 30 0 0 0 pt poe 0 ooe — Go = e= i= {T= PES LE 10 0 6+ oI+ Le 009 — 2- | 2= |= I- i T- eit FI+ Le 006 — I e= t= AI+ 61+ g— 00@ I— T- 0¢+ eo+ e- 00s I— smOyIDT yay | yaaz | qooy | yoy | yoy | ya0y | gooy | ye0y | goay | goay | goog | y00F | yoay | Joy | goay | oax | Joos | JOO] | 400} | Joo] | 3095 | Joos | Joay | 3095 ed woryes ors | oozz | cooz | ooe9 | 009% | cos | ooze | Gose | dose | OoTZ | 90FT | 002 | OOFS | COLL | 0OOL | COLD | OOS | COBF | GO | OOSE | COZ | COTS | OOFL | O0L | -tod ~wedtt09 | pores seuoy| Ue | -wedu &qdex queut i * 0D Bada uae Bodo | 4redm09 quemyiedmi00 Molog quaunredui0s sA0q y auLes}V L a pees —T10T} 24S JO WOTJBASTO JOY WOT}DaLI0D — JO} 101}901109 [‘syuourzedui00 u20}.M0g *819}9UI 008 8g ‘SNIPE 10jNO ‘s19}9UI 008 8% ‘snipe 10uuq]) WW au0z EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 42 ‘J90J OOOT Sl OANQBVAIND OY} pUB (sopTUT 0'6P=) UY 6'SL Sl OUOZ SIT} Jo O[PpIUT oY} OF UOTZRYS OY} WOT SOUBISIP OT, — ¥— &— e— oT o— o— G = 0 0 0 c+ G+ e+ e+ e+ e+ e+ e+ ot I+ I+ I+ I+ vr 6F— c+ 000 ST = b— e= s— o—- oT {- [= l= 0 0 0 St ct p+ e+ e+ e+ ot+ a+ ot+ I+ I+ I+ I+ lh Sh— et 000 FT $= > (&- |2= |e |e— | i= (T= 1 | 0 0 0 + | b+ | b+ | Ot [E+ | O+ | ot+ | St | 2+ J I+ J I+ I+ Tt 6g— sh— e+ 000 &I s- S> Po= [o— | oe 1 t= i= ji |i 10 0 0 Fr | b+ | O+ Jet [+ | 2+ | e+ | ot | s+ fF t+ [t+ I+ I+ 98— 68— €+ 000 21 c= = G-— | 3- | 3- I- i I— | T= }:0 0 0 pt | e+ [6+ | St [e+ | s+ | ot | 2+ | I+ | I+ Jit Tee T+ se= 98— Str 000 IT g- o- o— oT E- LS = l= t= 0 0 0 $+ Ca €+ e+ o+ o+ t+ o+ I+ T+ T+ oe I+ og— oe ot 000 OL o—- o> Lee. | LS TS LS | eS 1 0. 0 0 e+ | e+ | et | st | ot | ot | ot | t+ | T+ | I+ | + I+ T+ 8o— og— ot 000 6 o = l= L= i= t= I= We 0 0 0 e+ e+ ot o+ o+ ot I+ T+ I+ T+ I+ I+ 0 So— 96—- I+ 000 8 i c= [eS 110 0 0 0 0 0 0 e+ | ot | ot | ot | ot [i+ Jit |] t+ | T+ [T+ | I+ I+ 0 co— oe I+ 000 2 I- |0 0 0 0 0 0 0 Gt jot | St | st | ot J It Jit | t+ | T+ [It | 1+ 0 0 6I— 0e— I+ 000 9 0 0 0 0 0 0 0 Gt | St [ot | St | ot [It Jit | t+ | T+ [I+ | T+ 0 0 9I— LI— I+ 000 ¢ , 0 0 0 0 0 0 ot Jot jet | ot | t+ | t+ | t+ | t+ J+ Jo 0 0 0 I— st— T+ 00S F 0 0 0 0 0 t+ | t+ | T+ | T+ | T+ | I+ | T+ | T+ | T+ | 0 0 0 0 sI— eI— 0 000 F 0 0 0 0 0 Gt | St | It | I+ | T+ J I+ | I+ | T+ J] 0 0 0 0 0 t= II- 0 00S € 0 0 0 0 T+ I+ I+ T+ I+ I+ T+ T+ 0 0 0 0 0 oI— oI— 0 000 € 0 0 0 TP te [Te pe Pe. | | Ee [T+ 1 0 0 0 0 0 8 —- a= 0 00S & a 0 0 T+ | 1+ | 1+ | 1+ | T+ | 0 0 0 0 0 0 0 0 OS oS 0 000 & 0 0 I+ | 1+ [14+ | 1+ |1I+ | 0 0 0 0 0 0 0 0 CS Ss 0 00S T 0 I+ | 1+ |0 0 0 0 0 0 0 0 0 0 0 o= os 0 000 T I+ |1+ |0 0 0 0 0 0 0 0 0 0 0 Go 6 0 00g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19d Be CS | | ee 0 0 0 0 0 0 0 0 pt e+ 0 00g — Ss Ss r= | E= [8 0 0 0 0 Lt+ Bae 0 009 — ES | eS te 0 0 oI+ or+ 0 006 — I | i= T= 0 bI+ SI+ I- 002 T— = 0 LI+ sit [= 00g T— swoyny y0aJ | JooF | Joay | Joo¥ | Joos | Foot | Jooy | Jaoy | Jooy | Jooy | qoay | Jaoy | Joay | Jao | Jooy | Joay | yoay | Joay | Joay | Jaoy | qoay | qooy Jooy Joo} eed aor} OOFS | OOLL | ODOL | ODED | 009S | OOnF | OOF | OOSE | 008% | DOTS | OOFT | OOL | OOF8 | COLL | ODOL | OEY | 0D9E | OOGF | OOF | CDSE | 008Z | OOTS | OFT O0L -m100 -esuad w0148s syder ‘ se wor | “KP? PUF | -uadurog | -Sodoy, quaut “Bade saint, 4redui0o rf aures * Jo uoreA quewmyieduros mopag quemyiedur0o aaoqy VW “1a Wea —U01}8}8 JO MOTZBAIA 1OJ WOT}DEIION —10} WOT}DaIION [‘squeuny1edurus 0904XI19 “$19]2UI 000 66 ‘SNIPeI J9yNO ‘srajoUL 008 gg ‘snIPeI JouUT] ‘N auog 43 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. “490} OOGF Sl OINZVAIND OY} pUB (SOTLUI G°'Zg=) WY Y'ZET SI 9MOZ STG} Jo o[pprlur oY 07 UOTZRIS oY} WIOIZ 9OUISIP OL, i b= i T- |0 0 0 0 0 0 0 0 e+ | St | ot | St | St | ot J T+ | T+ | T+ | 1+ | 1+ T+ I+ co— Lo—- a+ 000 ST tf eS r= I- |0 0 0 0 0 0 0 0 + | Sot | St | ot I+ | 1+ | 1+ T+ [T+ | 1+ | T+ I+ I+ &o— So— ot 000 #1 ie {= 0 0 0 0 0 0 0 0 0 0 t+ | e+ | It | I+ It | T+ [T+ | t+ | T+ | 1+ | t+ I+ i oo— &o— I+ 000 €1 a 0 0 0 0 0 0 0 0 0 0 0 I+ It | Tt | T+ | I+ | IF Jit | t+ Jit | t+ | 1+ T+ 0 0Z— 1o—- T+ 000 GT = 0 0 0 0 0 0 0 0 0 0 0 It [I+ [TH [th [tbh | t+ | i+ | t+ | t+ J i+ | i+ I+ 0 6I— 0Z— T+ 000 TT T= 0 0 0 0 0 0 0 0 0 0 0 I+ [1+ te TEP LES Ee EE LE Pe Pe 1 te 0 0 LI— 8I- LF 000 OT 0 0 0 0 0 0 0 0 0 0 0 I+ Tt | I+ | I+ | T+ | T+ [I+ | t+ | I+ | 1+ | 0 0 0 ST- I I+ QU0 6 0 0 0 0 0 0 0 0 0 0 I+ TH |b | te PTE | te | tt | ie PE Io 0 0 0 FI— tL— 0 000 8 0 0 0 0 0 0 0 I+ | I+ I+ I+ | 1+ | 1+ | T+ [0 0 0 0 0 0 4 ol= 0 000 2 0 0 0 0 0 0 0 TE | Tab I+ T+ T+ /0 0 0 0 0 0 0 0 oI— oI— 0 000 9 0 0 0 0 0 0 TH | Te DT [0 0 0 0 0 0 0 0 0 0 Ge S603 0 000 ¢ . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 06. |0 0 0 B= b= 0 000 % 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a y= 0 00s € 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gem Gi 0 000 ¢ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 ie p= 0 00S & 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r= ¥- 0 000 @ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g= oe 0 00¢ T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o- GT 0 000 T 0 0 0 0 0 0 0 0 0 0 0 0 0 LS bo 0 00g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 is ag 0 0 0 0 0 0 0 0 0 0 0 0 or o+ 0 00g — 0 0 0 0 0 0 0 0 0 p+ e+ 0 009. — 0 0 0 0 0 0 9+ 9+ 0 006 — 0 0 0 0 0 8+ s+ 0 002 T— 0 0 0 oI+ OI+ 0 00g I— 0 I+ ai+ 0 008 I— SULOYOT gaay | qaay | aaey | gaax | aa0y | aaay | aaay | ga0y | gaay | goog | goar | ga0y | ga0y | a0y | aay | yaay | yay | goog | goag | yaay | aoay | gay | gooy | aeoy | Rieg | von 0096 | 0088 | coos | doz | OOF9 | Og | OS® | door | dE | OFZ | Coot | O08 | C096 | Coss | CODE | OOZL | OOF | OOD | ORF | OOOF | COLE | Cor | OOM | 008 | “tron | -*SU°d | ones | suder s@ wor | ZOOPER’) ordutog | -Borlo, queur eae Salk, 9 UY | aredur0a quewyiedui0s Mojeg queuljiedut0s aaoqy aures yy oa weet —I101}248 JO WOT}BAV]A IO} W0109I109 —J10} WOT}9eII09 [‘squauyzedui09 qyStIe-AjUEM TL, S1049UI OOL 99T ‘SNIPeI 104NO ‘si9}9UI ONO 66 ‘SNIP JaUUT] ‘CQ aU0Z 44 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. REDUCTION TABLES FOR NUMBERED ZONES. Zone 18. [Unit of elevation 100 feet (27.1 fathoms for depths). 9=1° 41’ 13” to 1° 29” 58”. One compartment. ] Correction | Correction for elevation of station at— Correction {| Correction for elevation of station at— ae as Moe departure! ones as Ae departure - ropor- see Somali” 5 000 feet 10 000 feet 15 000 feet siete “tionality 5000 feet ‘| 10000 feet 15 000 feet +150 +1 —2 —5 —7 0 0 0 0 0 +125 +1 —2 —4 —6 — 25 0 0 +1 +1 +100 +1 —2 —3 —5 — 50 0 +1 +2 +2 + 75 0 —1 —2 —3 — 75 0 +1 +2 +3 + 50 0 -1 —2 ~2 —100 0 +2 +3 +5 + 25 0 0 —1 -1 Zone 17. [Unit of elevation 100 feet (27.1 fathoms for depths). 9@=1° 54’ 52” to 1° 41’ 13. One compartment.] +150 +3 —2 —4 —6 0 0 0 0 0 +125 +2 —2 -3 —5 — 25 0 +1 +1 +1 +100 +1 -l —3 —4 — 50 0 +1 +1 +2 + 75 +1 —l —2 -3 — 75 +1 +1 +2 +38 + 50 0 -1 —l —2 —100 +1 +1 +3 +4 + 25 0 -—1 -l -l Zone 16. [Unit of elevation 100 feet (27.1 fathoms for depths). 0@=2° 11’ 53” to 1° 54” 52”. One compartment.] +150 +4 —2 -—3 —5 0 0 0 0 0 +125 +3 -1 —3 —4 — 25 0 0 +1 +1 +100 +42 —1 —2 -—3 — 50 0 +1 +1 +2 + 75 +1 -1 -1 —2 — 75 +1 +1 +1 +2 + 50 0 -l1 -1 —2 —100 +1 +1 +2 +3 + 25 0 0 -1 al Zone 165. [Unit of elevation 100 feet (27.1 fathoms for depths). @=2° 33/ 46” to 2° 11’ 53’. One compartment.] +150 +5 =i = =4 0 0 0 0 0 +125 4d i =2 =3 — 25 0 0 0 bel +100 +2 ==if af i ag — 50 0 0 1 +1 + 75 41 =i -1 2 — 75 Ay ey. +1 +2 + 50 +1 0 -1 -1 —100 +1 +1 +2 +3 + 25 0 0 0 =i Zone 14. [Unit of elevation 100 feet (27.1 fathoms for depths). @=3° 03/ 05’ to 2° 33’ 46”. One compartment.] +150 +6 -1 —2 —3 0 0 0 0 0 +125 +4 —1 2 —3 — 0 0 0 0 +100 +3 —l —l —2 — 50 0 0 +1 +1 + 50 +1 0 —l —1 —100 +2 +1 +1 +2 + 25 0 0 0 0 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 45 Zone 13. {Unit of elevation 1000 feet (271 fathoms for depths). @=4° 19’ 13’ to 3° 03’ 05’. Sixteen compartments. ] No correction for elevation of station. No correction for departure from proportionality. Zone 12. [Unit of elevation 1000 feet (271 fathoms for depths). @=5° 46’ 34” to 4° 19/13’. Ten compartments. ] No correction for elevation of station. No correction for departure from proportionality. Zone 11. [Unit of elevation 1000 feet (271 fathoms for depths). 97° 51’ 30” to 5° 46 34”. Hight compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 10. [Unit of elevation 1000 feet (271 fathoms for depths). 9=10° 44’ to 7° 51’ 30’. Six compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 9. [Unit of elevation 1000 feet (271 fathoms for depths). 0@=14° 09’ to 10° 44’ Four compartments.] i No correction for elevation of station. No correction for departure from proportionality. Zone 8. [Unit of elevation 1000 feet (271 fathoms for depths). 9=20° 41’ to 14° 09’. Four compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 7. [Unit of elevation 1000 feet (271 fathoms for depths). 6=26° 4/’ to 20° 41’. Two compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 6. [Unit of elevation 10 000 feet (2710 fathoms for depths). 6=35° 58 to 26° 41’. Eighteen compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 8. [Unit of elevation 10 000 feet (2710 fathoms for depths). @=51° 04’ to 35° 58’. Sixteen compartments. ] No correction for elevation of station. No correction for departure from proportionality. Zone 4. [Unit of elevation 10 000 feet (2710 fathoms for depths). @=72° 13’ to 51° 04’. Twelve compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 8. [Unit of elevation 10 000 feet (2710 fathoms for depths). 9=105° 48’ to 72° 13’. Ten compartments. ] No correction for elevation of station. No correction for departure from proportionality. 46 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. Zone 2. [Unit of elevation 10 000 feet (2710 fathoms for depths). @=150° 56’ to 105° 48’. Six compartments.] No correction for elevation of station. No correction for departure from proportionality. Zone 1. [Unit of elevation 10 000 feet (2710 fathoms for depths). 6=180° to 150° 56’. One compartment only.] No correction for elevation of station. No correction for departure from proportionality. SPECIAL REDUCTION TABLES FOR SEA STATIONS. [Corrections in dynes in units of the fourth decimal place. Station at sea level.] Depth Zones | Fathoms A B Cc D E F G H I J K L M N Oo } 5 000 —1 -1ll -—27 -39 -—53 ~-55 -—48 -40 -—40 -—34 -18 -3 +441 +51 +32 | 4 800 —1 -1l1 -27 -39 -—53 -55 -—47 -—39 -—38 -~32 -17 -—2 +441 +449 +831 4 600 -1 -ll -27 -~39 -—53 -—54 -—46 -37 -—36 -29 -15 —-1 +440 +47 +429 4 400 —-1 -ll —27 -38 -53 -54 -45 -36 -—34 -27 -—14 -—1 +40 +45 +28 4 200 —1 -ll -—27 -—38 -—53 -53 -—44 -34 -—32 -—25 —12 0 +389 +44 +27 | 4 000 —1 —ll —27 —-38 -52 -—52 -—43 -—33 -30 —23 —11 +1 +39 +42 +26 | I 3 800 —-1 -ll -27 ~—38 ~—51 —51 —42 —31 —28 -—21 —9 +41 +38 +440 +24 | 3 600 -1 -ll -27 -38 -—51 -50 —41 —29 -—26 —19 — 8 +42 +37 +38 +23 | 3 400 —1 —-ll -—26 -37 -50 -49 -—39 -—28 -24 -17 —7 +42 +36 +36 +22 | 3 200 -l -ll -—26 -37 -—49 -48 -—38 -—26 -—22 -15 —5 +43 +35 +34 +421 3 000 —-l1 -ll -—26 -37 -48 -46 -36 -—24 -20 -13 —4 +3 +433 +32 +19 2 800 —l -ll —26 -37 -—47 -44 -34 -—23 -18 -ll —3 +44 +32 +30 +418 | 2 600 -1 -ll -26 —36 -—46 -—43 -—32 -21 -16 -10 — 2 +4 +30 +28 +417 |} 2 400 —1 -ll -26 -36 -—44 -—41 -30 -20 -14 —8 —1 +4 +28 +26 +415 | 2 200 -1 —ll —26 -35 ~—43 -39 -27 -17 -12 —6 —1 44 426 +24 414 |; 2 0N0 —1 -10 -—26 -35 -42 -37 -25 -15 -ll —5 0 +4 +25 +422 +412 1 800 —1 -10 -—26 -—34 -—41 -34 -22 -13 -—9 -—4 0 +4 +23 +20 +412 | 1 600 -1 -—10 —25 -—34 -—40 -32 -—19 —-10 —7 —2 41 44 +420 +418 +10 | 1 400 —1 -10 -25 -34 -—38 -—28 -16 —9 —5 —2 +1 +44 419 416 +9 }: 1 200 -—l1 -10 -25 -33 -35 -25 -13 —7 —3 —1 +1 +483 417 414 48 1 000 —-1 -10 -—25 -—31 -—31 -—20 -10 —5 —- 2 0 +1 +42 +14 412 +6 | 800 —-1 -10 -25 -30 -27 -15 -—7 -—3 1 0 +1 42 411 +9 +5 |: 600 —1 —10 -—23 -—26*-21 -10 —4 -1 1 +141 4249 47 44 400 —1 -—10 -—22 -—21*-13*-5 —2 —-1 0+14+1 41 464542 200 —1*-10 *-17*-l1l*-4*-1 0 0 O +1 41 41 43 42 41 0 0 *0 *0 *0 *0 *0 0 0 0 0 0 0 0 0 0 \ * Use table following for these values on account of large second difference. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 47 Supplementary table for use in connection with gravity stations at sea. [Correction in dynes in units of the fourth decimal place. Station at sea level.] Depth Zone Fathoms B Cc D E F 800 —25 —27 750 —24 —26 700 —24 —24 650 —24 —22 600 —23 -—26 -—21 —10 550 —23 -—26 -19 — 9 500 —23 -25 -17 —7 450 —23 -—23 -15 — 6 400 —22 -21 -13 — 5 350 —21 -19 -ll —4 300 —20 -17 -—9 —38 250 -19 -14 —6 —2 200 —-10 ~—17 -ll —4 —-1 150 -—10 -15 —7 —2 —-1 100 —-9 -ll —4 —-I1 0 75 -—-8 —9 —2 —1 0 50 —-7 —-5 -1 -1 0 25 -5 -1 -1 0 0 10 — 2 0 0 0 0 0 0 0 0 0 0 USE OF TEMPLATES. For each scale of map or chart to be used in the computations there was prepared a sheet of transparent celluloid with the circles and radial lines which define the limits of the zones and compartments drawn to the same scale. Such a template is shown in illustration No. 10a as used for maps on a scale of 1/10000. The zones are marked with their designating letters, and the scale of the template is ordinarily marked on each. No attempt has been made to reproduce the illustration to the proper scale. Each template consists of a sheet similar to that indicated in illustration No. 10a carrying lines bounding the compartments which lie on one side of the reference line. By turning the template 180° in azimuth on a map it serves also to fix the position of the remaining compart- ments. While in use the template is placed on a map with the center of the circles at the station and with the reference line lying in the meridian. As a convenient designation the compartments in any zone are numbered in the clockwise direction commencing with the first which is to the eastward of north from the station. Illustration No. 106 shows a template such as was used on maps on a scale of 1/6013500. This necessarily shows more distant zones than illustration No. 10a. The dotted radial lines in zones 14 to 18 are not compartment boundaries. Each of these zones has one compartment only. They are lines dividing each of the zones into ten equal parts, as it was found convenient in estimating the mean elevation for such large zones to make separate estimates for each part rather than to make an estimate at once for the whole compartment or zone. For the same purpose dotted lines are shown in zone 7 separating each of its two compartments into five equal parts. ; : ates By the use of these transparent (celluloid) templates the many circles and radial lines fixing the limits of the zones and compartments on a given map for any station were super- posed on the map by the mere process of laying the template on the map in the proper position. The use of the templates saved a very large amount of labor which would otherwise have been necessary in drawing the many zones and compartments on several hundred maps. It also left the maps without damage or defacement. 48 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. In computing the correction for topography and isostatic compensation for a given station the computer places the appropriate template in the proper position on the best contour map available. He then estimates the mean elevation of the surface in each compartment from the contour lines on the map, seen through the template, and at once takes out from the reduc- tion tables the two corrections for that compartment and records them in the proper places on the computation forms. As he has the reduction tables constantly before him he is con- tinually guided as to the accuracy with which the estimate of mean elevations must be made in order to secure the corrections with the required degree of accuracy. As a rule this estimate Reference Line Reduced trom template used on maps of 1/60/3500 scale Reauced from template used on maps of Y10000 scole Ls ILLusTRATION No. 10 (a).—Template for maps of ILLUSTRATION No. 10 (b).—Template for maps of scale scale 1/10000 (reduced). 1/6013500 (reduced). may be made very quickly, for as indicated in the reduction tables, an approximate elevation of a compartment is sufficient. This is especially true in the numbered zones 13 to 1, for which the unit elevations are either 1000 or 10 000 feet. EXAMPLES OF COMPUTATIONS OF CORRECTIONS. The following table is a sample of the computations, and in it are given the values (in units of the fourth decimal place in dynes) of the correction for topography and isostatic compensation for each compartment of zones A to 1 at the San Francisco gravity station. This station is near the open coast, is 85 miles from the 1000-fathom line, and is only 375 feet above sea level. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 49 San Francisco, Cal., gravity station No. 54. [p=87° 47’ 22/7. 4=122° 25’ 40’. Elevation=375 feet.] A B c D E F G H I J k L M N Oo +2 0/4+13 —1 +3 0) lL 61 0 0 0 0 00 00 00; 00 0 0 0 0 00;—2 0; -10 +13 -1 +9 0; +1 +1] +1 «0 0 0 0 0 0 0 0 0 0 0 0:0 0 0 0 0 0 0 0 0 +13 -1 49 0] +2 +1 0 0 0 0 0 0 00 0 0 0 0 0 a 0 0/— 4 0 0 0 0 0 tig +9 0/42 +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0/—1 0}/-— 40 0 0 0 0 +2 +1 0 0 0 0 0 0 00 0 0 0 0 0 0/—1 0/— 1 0/-2 0 00 aL bl f 4 : . ; 0 0 0 0 ‘a 10 { 00;/—3 0)/ -—10 00 00 0 0 00;/—3 0} -—10 00 0 0 0 0 0 0 00 0 0 0 Olf—1 O0/— 8 O;f—1 0 00 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 { 00 00 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0;/+2 0) —2 9 0 0 0 0 0 0 0 0] -1 0 0 0 0 0 00;/+2 0} —20 0 0. 0 0 0 0 0 0 0 0 ‘0 0 0 0/ +10 0 00 { 00 00 0 0 0 0 0 0 0 0 0 0; +10 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0;+4 0 00 00 0 0 0 0 0 0 0 0 00 0 0;+1 0 00 0 0 0 0 00 0 O/—1 O/f—2 Olf—-1 0/1 +2 0 00 0 O;/-1 0 00 0 0 0 0 GO| + 8 0 0 0 0 0 0 0 0 0 00 OO) = 2:0) fl 0 0 Oo}; 00 0 0 0 0 0 0 00 +12 0 00 0 0 00 0 0 0 0 +13 0 0 0 00 0 0 0 0 +13 0 0 0 00 0 0 +13 0 0 0 00 0 0 +13 0 0 0 00 0 0 +12 0 9 0 —1 0jf-1 0 +9 0 00 0 0 00 +60 0 0;-—1 0 eG 0 0 00 —20 = 00 00 -—30 0 0 —-20 +2 04/452 -—4}] +35 0/+9 +6] +1 0 0 0 0 0 0 0 0 0; —2 0] —1 0} —7 0] —14 0] +15 0] +99 9 +2 +48 +35 +15 +1 0 0 0 0 —2 -1 -7 — 14 +15 +99 18 17 16 15 14 13 12 Bl 10 9 8 7 6 5 4 3 2 1 —4 0 Te Oo Kaee @aas OSE b —- 56 —6; —5] —5] —7] -2 0 0 ol 0 0 0 0 +28 0 (+28 0 [\+32 0 1\+35 0 [\+38 0 0 =a mes Stheals a <4 . c a e : +1 Sel este es ey ey th a at ae -— 7 +5 +5] +8 = 3 +9] +8 tos 749 0 0 +1 oi +1 { 0 +11] +10 { 0 . 0 ad 0; +1 — 2} +n] +9/\+4 0 0 o} +1] 41 + 5] +7 0 0 oO} +1] 41 { 0 + slp -1N4+4 0 0 +1] +41 0 + 9 { -ol’ Oo} +1 41] 41 + 10 Oo} +1) 41] 41 + 10 0 +1 +1 0 Sg oO} 41 oli 0 + 8 +1 +1 0 4 4 +1] 44 0 = ff +1 ek 0 { 0 +1] 41 +1 0 +1 0 +1 +1 +24 0] +21 0] +20 0} 419 0} +17 Of +25] +23) +21) +14) +10 +15}; +10| +9/ +9 +8) +5] +4] +1! +24 +21 +20 +19 +17 Sum of all zones = +446. At the top of the table are given the latitude, longitude, and elevation of the station. In actual practice the zones may be arranged in any convenient manner on a single sheet. Here they are placed in such a way as to show them in as compact a form as possible. The headings of the several columns indicate the zones by letter or number, it being under- stood that the zones are in the order of their distances from the station, namely, A to O, and 18 to 1, the zone A being at the station with its inner radius zero. In a zone having more than one compartment, the compartments are numbered clock- wise, the first one being to the north of the station and just to the east ot the meridian passing through the station. Having this arrangement of compartments in mind, one can readily see 15593°—12 4 50 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. in the table for any station the effect of the different prominent topographic features. This is noticeable in zone O for both the San Francisco and Pikes Peak gravity stations. At San Francisco for the first 13 compartments of zone O the corrections are all zero or negative. These are all land compartments. In compartment 14, nearly due south from the station, a positive correction, due to the ocean, first appears. In compartments 15 to 25, all to the westward of the station, the corrections are all positive, showing the influence of the deep waters of the Pacific. Compartments 26 to 28 are land compartments, showing the influ- ence of the Coast Range to the northwestward of San Francisco. The similar influence of the part of the Coast Range to the southeastward of San Francisco lying within this zone is shown in compartments 10 and 11. It will be noticed that there is a double column for each of the zones B to O. The first column gives the effect on the intensity of gravity at the station, due to the topography and the isostatic compensation of the several compartments based upon the assumption that the station is, in each case, at the same elevation as the compartment. The mean elevation of the compartment is obtained from the map or maps used. Entering the table for the par- ticular zone with this elevation, this correction is obtained from the fourth column, which is headed ‘‘Correction for topography and compensation.” In the second column for zones B to O is given the effect of the intensity of gravity due to the elevation of the station above or below the average elevation of each compartment. These quantities are given in the tables under the headings ‘‘Correction for elevation of station above compartment” and ‘‘Correction for elevation of station below compartment.” In taking out the second correction it must be kept in mind, as already noted on pages 22 and 29, that it does not become zero in zones M, N, and O when the station is at the same elevation as the compartment, but, instead, has the values shown in the special column in the reduction tables for these zones. For zones B to L the second correction is zero when the station is at the same elevation as the compartment. Two columns are given for each zone 18 to 14, the first one showing the correction as read from the map and given in the first columns of the reduction tables for those zones, while the second column contains the algebraic sum of the corrections for the departure from pro- portionality and for the elevation of the station above sea level. For each of the zones 13 to 1, there is only one column of figures, which are the corrections for the compartments as read from the map, each compartment of zones 13 to 7 having a cor- rection of 0.0001 dyne for each 1000 feet in elevation (271 fathoms for depth), and zones 6 to 1 having a correction of 0.0001 dyne for each 10 000 feet of elevation (2710 fathoms for depth). The algebraic sums for each column is given at the foot of the column and immediately below these separate sums is given the algebraic sum for the zone. The sum-for all zones is +446 in the units used in the computation or +0.0446 dynes. This is the correction at San Francisco for the topography of the entire earth and its compensation. It was found at times to be desirable to treat in two parts the corrections for a compart- ment which contained both land and water areas. The corrections for land and water for the compartments treated in this way are connected in the table by brackets, the first number being for the land portion and the second for the water portion of the compartment in question. In determining the correction for any portion of a compartment the table is entered with the elevation of that portion as the argument as if it were the elevation of the whole compartment, but the correction entered in the computations is only that proportion of the total correction which the area of the portion of the compartment bears to its total area. The elevations close to the gravity station at San Francisco are low and in no case inside of zone I is the height of a single compartment more than 700 feet above sea level. In zone L one compartment to the eastward of San Francisco, in the Coast Range, has an average elevation of about 800 feet. Zone I. is just beyond the change of sign due to distance (see p. 65), and therefore the correction for that compartment is not over 0.0001 dyne. In zone M there are several compartments near the compartments of zone L in the Coast Range, already referred to, EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 51 with elevations of about 1000 feet, each of which causes corrections of — 0.0004 dyne (see third and fourth numbers in the column for zone M). In zone N there are two compartments having depths of about 900 fathoms, which cause corrections of +0.0010 dyne. The land compart- ments in this zone do not have elevations above 1000 feet. A portion of zone O extends well beyond the 1000-fathom line, which causes corrections as large as +0.0013 dyne for several water compartments. The corrections for the land and water portions of each of the zones 18 to 14 are given separately, the correction being minus for the land and plus for the water. Most of the water sections of these zones are far out in the Pacific Ocean. Each of these zones has only one compartment, but for convenience in reading elevations and depths from the maps, each zone is divided into 10 parts and for each part the correction is taken from the reduction tables as one-tenth of the value given for the whole zone for an elevation equal to that of the part in question. The table was entered only once to obtain for the zone the correction for the elevation of the station above sea level. For each of the zones 18 to 14 at San Francisco the algebraic sum of the corrections for departure from proportionality and for elevation of station is zero. Each of the zones 13 to 1 has only one column of figures in the table, as there are no cor- rections for elevation of station nor for departure from proportionality. The total correction for each of these zones is plus, showing that the effect of the water compartments predominates. There was no interpolation of values in any of the zones for the gravity station at San Fran- cisco. All values were computed directly from the maps and charts. The following table gives in detail the computation of the effect of topography and its isostatic compensation at the gravity station Pikes Peak, which is a mountain station far from the ocean. The station is much above the general elevation of the surrounding country. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 52 ‘Ajeatoadsar “ojog ‘uostuuny pure ‘Iesueq ‘ssulidg opes0jog 48 ‘Gp pue ‘FF ‘Zh “SON SUOT}EIS APIAVIS WO UOTejodsajuy Aq PoUre}qo aoa SOl[e}] UT WMoYs san[eA—ALON "TLSI-+ = seuoz [Ye Jo wing I+ + 9+ s+ 6+ 6+ + + - = st— s8— 6o— 19— s9- 3S 3 53 5 - = i It 0o9—- | 2+ 99- | e+ L- | 8+ - {et L- seats ; ze ane ma ee: wee tS ven - . so Tt oo fat go- | et u- Jet uw }et T ra g ¥ ¢ 9 L 8 6 or IL ar &T tI ST oT 21 81 6z8— FeE— o6e—- —«. oe 6L+ Tos+ olb+ Shot cist co9+ ozs + oTe+ LST+ al+ G+ Set Lse— | e+ o9e— "| 06+ Ose— | L6+ G2I— | LIT+ se— | 9I+ GOt | OLT+ BPEt | G+ Tet | 09+ Wt | &— GOO+ | GF—- GOot | FE- E+ | TI SoTt | O e+ 0 Vt 1 We I+ 2I- I+ 8I- a It LI- y+ 9- I+ 6I- b+ 9- te ti rb o~ ee ae t+ 9- Te 2 F+ 9- 9+ B- 6+ I+ I 2I= b+ 9— 9+ B-— 6+ @I+ I~ $I- Br -9=' G+ 3- 8+ Ft I+ FI- t+ 9- 9+ b— 8+ eit IF $I- St S— rm. G= Ge ge 8+ Ft 6+ oIt 9+ Tot I+ FI= gt 8— tr g= 9+ 3- 8+ + 8+ eit G+ e+ I+ @I- Gt 0&— 9+ 08— gs 9= 9+ 3- 8+ t+ St Sit §+ 96+ I+ 0OIT— e+ 08— L+ $8 $+ 9- 9+ o- 6+ €+ 8+ e+ Ft 96+ I+ 6—- G+ Lo— Sa $8 Beg ot 1- St Ft Lt I+ Gt Set G+ get e Te 3 = gt &e~- 9+ ge— $+ 9- 9+ B- St St Loe Stet Gt &%+ e+ Trt It g-= e+ ig- o> t= e+ G— oF 2= s+ 9+ 9+ SIt+ Gt + F+ OF+ 0 T9t+ Tse 80 St 9B— 9+ 22— Tr G= 9+ 3g- 8+ St 8+ FIt+ b+ GB+ e+ 66+ T+ 9+ re 6-= G+ 8I- L+ €4 b+ F-— Oa SE L+ 9+ 9+ ST+ 9+ Gt G+ set I+ 29+ 8 = 61+ I+ 6- G+ 9I- 9+ Te- BP it 9+ L+ 6+ @It+ 9+ f+ y+ OF+ 0 19+ 0 oL+ I+ 0I- G+ 8I- L+ To- St b- 9+ B- 6+ F+ OI+ OT+ L+ 0¢+ G+ e+ I+ 19+ eS Tet P= 19+ Te Or c+ 8I- 9+ 0¢- G+ 9+ @- Go OIt 6+ L+ 61+ Gt e+ It 19+ - t+ g— sot IF 6—- G+ 0@- Lt+ t-— ce 3 9+ 3- OI+ 3+ OI+ 6+ L+ 61+ 9+ set I- 09+ PS TE €— got — r+ 10 8It Ts 6° E+ 02— be Lo e+ G— 9+ 6- OI+ 3+ Olt 6+ Z+ 61+ g+ get Z— 8o+ II- 0L+ Bo it 0 a+ | 0 Sit Tt 6-— t+ 0— L+ lg- J+ 9- ch 3= OIt &+ TIt 64+ 8t sit g+ get e— 6o+ 6— OL+ i 294 f&- G+ |0 8It te a> Gt 8I- Lt 8%— &+ 9—- oF 2= oI+ €&+ 6+ I+ L+ 61+ 9+ 98+ Z— got 2° 0S GIS $- Gt |0 8It+ 0 e+ oO N n 1 x f I H 9 az a a 0 a v ("30037 ¢80 PI=UOTCACT A ‘00 460 oSOI=Y “st oN uoung hnavipn “ojo ‘yDrq $2%%d ‘BI AG 88=F] EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 538 The arrangement of this table is the same as that for the station at San Francisco, which was discussed in detail. The corrections for the zones A to 14, at Pikes Peak, were computed from the elevations read from maps. For zones 13 to 1 they were interpolated (in the manner explained later under the heading ‘‘Saving of time by interpolation’’). from stations Nos. 42, 44, and 45, which are at Colorado Springs, Denver, and Gunnison, Colo., respectively. The total value of the effect of the topography and its isostatic compensation as obtained by the inter- polation is given in the table for each of the zones 13 to 1. The leaders shown in the columns for these zones indicate the number of compartments in each zone. As Pikes Peak is an inland station, there are no water compartments within the computed ' zones A to 14, As was the case in the table showing the corrections for the different zones at the San Francisco gravity station, zones B to 14 at Pikes Peak have two columns of figures each. In each zone the first column shows the effect of the topography and compensation with the station at the same elevation as the several compartments, while the second column of figures shows the corrections due to the elevation of the station above or below the compartment. It is interesting to notice the change of sign at zone F of the correction for elevation of station (see p. 52), the change of sign due to distance between zones J and K, in the first column for these zones, also the change in the sign of the total correction between zones K and L. Pikes Peak is a conical-shaped mountain, which accounts for the corrections for the several compartments of each of the near-by zones being of about the same size. The effect of the mountains to the westward is clearly shown in zones M, N, and O, but especially in zone O, the corrections being larger in the lower half of each column corresponding to compartments west of the station than in the upper half of the column in each of these zones. CORRECTIONS FOR TOPOGRAPHY AND ISOSTATIC COMPENSATION, SEPARATE ZONES. In the following table are given the total corrections for each zone, for topography and its isostatic compensation, to the intensity of gravity at each of the 89 gravity stations used in this investigation. There is also given the total correction for each station, this necessarily being the sum of the corrections for the separate zones. The values are given in units of the fourth decimal place in dynes. The names and numbers of the stations are given in the headings of the table, while the letters or numbers of the zones are shown in the first column. The value for each zone at a station was obtained from the computations of the corrections for the separate compartments of the zones. Samples of such computations made at a station are given in tables on pages 49 and 52 for the gravity stations at San Francisco and on Pikes Peak. The figures in italics represent the accepted interpolated values for the correction for topography and its compensation as explained on pages 58-60. The other figures are the values for the zones for which the corrections were obtained directly from maps and the reduction tables. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 54 Correction for topography and isostatic compensation, separate zones sig!| MRESESOS ORR SE RSNA Hoh RsREHMessaehy Ss) i) | CBee eRaa Reise sasha Sees h ema geo | reer ETE) Pee ee ee PE beet es | F 282 AR S$ NEBR SSSSS MS ASD M SM SMAI SAMHOASCOAORN |S oe AQABSIMOCOMNAAAENHOAON MOMMA WINS OMoon | 1G ee || See TUTTE E ITT TT 1+ $4+FF4444 [$f eee | 444444 DTU TTT TEESE F 4444444 | + 4 OB4 5° S38 | ++ ftp eek i Recess @ as | + FAEEEEEELE HEHEHE H+ | F ae a Ee | Sap ee a eee a a a baa Sete LEFEEEEEEEEE ETE ETS | F a S 5 oS) (ee apm esiete Seater te Ml eget) eeeted te Phi PIVOT) ) biteee eee |e S. ° % 3 " = qi 4 £24 oq gis | + ieee ga) Saha PEEP bbb tb dL bbb HHH | S : = g23 | + fide gate | Es a 3 < 5 ge a 2° PA Ee De Diseases lt ef Bes | + Petites paeeania ey | eee As 3 wa iS | + Sea ie ee iat | £5 5 & 2 . Bs || ep Lieb later eel os x “ g ° g e 8 a EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 55 Correction for topography and isostatic compensation, separate zones—Continued. ee sagitngton, i i ; Cam- Zone IS 8. thsonian| Baltimore, | Philadelphia, Princeton, Hoboken, New York,| Worcester, | Boston, bridge Office, Institution, No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. ae No. 21 No. 22 ; A +2 +2 +2 +2 + 3 +2] + 2) + 2) 4+ 2 Ae B +12 + 8 +24 +12 + 40 + 8 + 27 + 56 + 16 + 12 C + 2 0 + 4 +4 + 16 0 + 7 + 64 + 4 0 D 0 0 + 6 0 + 6 0 + 2 + 31 + 1 0 E 0 0 0 0 0 0 0 + 11 0 0 F 0 0 0 0 0 0 0 + 7 0 0 G 0 0 0 0 0 0 0 0 0 0 H 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 J 0 0 -1 0 0 0 0 — 10 0 = ¢ ee ee es ee er 0 0 — 3 0 0 - - M —12 —12 —20 — 6 —- ll —12 — 12. — 27 —- 4 — 9 N -17 —17 —16 —10 — 16 —18 — 18 — 25 — 12 — 15 O —23 —23 —20 —19 — 22 —25 — 26 — 28 — 10 — 14 18 =5 — 65 — 6 -3 — 4 = 6) == 6) = BP 278 — 2 17 —8 — 8 Sait — 6 -— 6 -§) — 5] — 8] - 1 anf 16 -—9 -—9 a) —6 -— 6 —s| -— 8] - 4] -1 — 2 15 — 8 —8 — 8 — 3 ae +i} +°1/ -— 2g] - 2 — 2 14 —4 — 4 = 0 + 1 +3/ + 3} -— 2] ~ 2 -— 2 12 +38 +38 +7 +12 + 18 +4) +44) 4+ 9) 411 + 10 12 +13 +13 +14 +19 + 20 +22} +22} + 24] + 26 + 25 1 +18 +18 +19 +21 + 21 +21{ +21] +23} + 25 + 25 10 417 +17 +17 +16 + 16 4+16| +176) +17]; +18 + 18 9 +11 +11 +10 +11 + 11 +11 + 11 + 12 + 12 + 12 8 +12 +12 +13 +14 + 14 +15) +15) +127) +17 + 17 7 +6 +6 +6 +6 + 6 +6|/ + 6} + 6) +6 + 6 6 +6 +6 +6 +6 + 6 +6| + 6| + 6) +6 + 6 5 +7 +7 +7 +6! + 6 +6| + 6} + 6] + 6) 4+ 6 4 +6 +6 +6 +6 + 6 +6) + °6) + 6] +6 + 6 3 +6 +6 +6 +6) + 6 +6) + 6] + 6] + 6| +6 2 +4 +4 +4 +4| + 4 +4] + 4/ + 4) + 4] + 4 1 +1 ed by +1 + 7 +1 + 7 eed |! Pepe + 1 Total +40 +34 +57 +93 +130 +79 +106 +178 +133 +101 i Ithae: Cleveland, | Cincinnati, | Terre Haute,| Chicago Madison, | St. Louis Kansas Ellsworth, Zone No. ai No. 33 No.33 | No.34’| No.85 || No.36) | No.37' | No.38 | %, | “No. 40 2 2 2 + 2 + 2 + 2 + 2 + 2 7 i a a pe Lr ‘ee +56 +62 +56 +64 + 68 C + 4 +88 +78 +84 +60 +72 +95 nee Be oc D + 4 +59 +48 +57 +28 +42 +70 + + + a E 0 +27 +20 +22 +12 +16 +30 +13 +3 ee F 0 + 6 +10 0 +2 +4 +10 0 oe + a G 0 0 0 0 0 0 ‘ ; a + ; = i a ; ‘ 0 0 0 0 } 5 —16 —16 —11 — 8 —7 —16 — 3 —16 — 16 K ) —20 —20 -17 —10 -— 9 a — ae - rs Ll 0 —32 —24 —20 —12 -l1 - _ ; = _ a M — 6 —50 —42 —42 —30 —22 —57 = ae _ a N — 4 —56 —41 —50 —37 —26 —48 = =a = O — 15 —58 —45 —48 —35 Be = = : =e ca ve ay SS =e Zi = as Gil vee)! asad) ete) ae oh SS = ai ae a4 Fl «fel Set Sto] sag ae =4 2 ae —6 oe| =@| 271 em| = 46 ee =2 =H af a S6|| sol soe) sie) ip te Ee a ae = ~16 =i¢| Ie) =] + 8 6 +/e +/#1- |@ + |e +] 22 +]ot+ |es +)o9e +)]H~t+ fort eg + es + oe + oe + gg + 6 TC —~}l —j);86— |6 + | 9F +]99 +) 66+ }29 +) 49 +) 09+ | 09 + 6h + 0g + ig + IS 69 + OL 6 —/|& +)¢8- |et+ |¢9 +]98 +])9r + | 28 +/118 +]/I8+ | se 4+ go + 99 + 89 + 69 + 8 + Il 4 -!}9 —)|]T14- |FE+ |} 82 +] 80 +) 0¢+ | ZO +} 96 +]26+ | sor+ 68 + 94 + e+ gg + gott+ ZL 6 —]1IL — | 240I- | 8T + | 981 + | 891 + | G4 + | 9ST + | BET + ] SETt+ | Fart eli+ €gI+ osi+ S3I+ 991+ 81 st -—"}1l —|]42—- /8 + | 48 +)]T0L +] 2+ 178 +) 82 +) 22+ | 06 + 68 + gz + GL + GL + 201+ bL 46—-—|140 —|G2- 1/8 + |18 + ]TOL +] 68+ | 68 +1/F2 4+/28+ | 88+ 06 + 08 + 04 + 69 + 901+ SI 86 — |}8& —]/0L- [4 + | 6L +] TOT +] 2864+ | 2 +/08 +)98+ | 684+ 00T+ gg + go + gg + 901+ 9T Gg —)T —/G2—-— 1}9 + | 42 +5246 +/60+ 162 4+/868 +118 + | 98+ TOoI+ 68 + 8g + Tg + ¢cor+ LI 8 — | 2 —/G2- 19 + | FL +1246 +/00+ |] 34 +] 22 +] 62+ | ¢¢+ 901+ 16 + gg + Tg + Gott 81 TST— | O9T — | T6E— | FL + | 9E + | Toh + | GE + | OE + | HE + | zzst+ | OZEt+ Feet sort 9bo+ PSo+ 0oSt oO 966— | LT — | e8E-— | & + | 828 + | 466 + | 23 + | Set + | Sez + | OLZ+ | 628+ 399+ 9¢g+ ohot sié+ 8Lb+ N L08— | 966 — | P8F- | T + | 093 +] 22g+)]8 — | 64 — | TOL + | 249I4+ | ¥88+ o9g+ 067+ 996+ 993+ Shrt+ WwW 6sI— | 46 — | s8I-— | 0 4¢o+/¢¢ +/8 — |13 -—|9 +)0¢+ | 96+ eo cr + 96 + 96 + 96 + T goI- |6 — | gG0I- | 0 G¢ +/8 +/¢ — |e. +/¢ +/9t+ |02- 00g— ost— oe + 06 + 09 — M CL = | SED 0 6 —;G +]/T — |O9T+);8 +1/9 — |! 96- POF 966— ae oe — 9L1— £ 88 — | sé +} or + |0 0 0 0 918 + | 0 &L — | OFB— 0¢L— gos— 0¢I- 0ZT— o9g— I og — | 682 + | #9 + | 0 0 0 0 6crF +} T +/08 — | 328- Z6g— FOP PPI PRI 39e— H 12 — | 126 + | ozI+ 10 0 0 0 ecg + | 0 SII— | $28— oag— c6h— 0G— #0G— 80%— 9 91 + | 806 + | 022+ | 0 L —1([0 0 ceo + 10 O8I— | 06g— Ors— 00s— 00g— 00s— OFF a s¢ + | 486 + |] sze+ | 0 S150 0 g9¢ + 10 ceo— | RE Fob 80%— oIs— e1s— g1e— - a FIT+ | TIS + | 28st | 0 iT — 10 0 ars + | 0 O8I— | O1Z— FES 86o— ¥0Z— $0Z— ZES— a OZI+ | O9T + | 9GT+ | F + | —]0 br + | 39T +] 0 OOT— | FOT— s0l— s0l— 00T— 00T— FOT— 3 m9 +) 9 +/89 + /9T+ 18 +12 —]9t+ |% +)0 OF — | br — br — OF — OF — PP — a G@ chee HR Se OE ee 1G See ye See SB ee eee Se aS | OS ES Lo = L = L= t= Vv 0 0 "O “ ZI "0 ‘ 2 i T ‘oO. Yaa | “pu | oes |, ee Duets ‘eat wee | Ce | SN | eo | ae a ON FON £°ON 2 ON ‘onsite 7TOZHIMS ~TOZIIMS OUnL ‘qaseea enspoH IS} ‘sagz00. 2) “aqou. VI ‘nin : ‘nyqeo ueireaey : deact daa. ‘neeye[q nB3}81q ueg pue eu0z, DUNT yeas quinyg re UM0} ‘i 3s vuneyl -om0 IvoN B3U0L BBU0T, e3U0], Bsu0.L nyNjouox{ 49 ~19UIOL) -Fe]e3 198 -soulvs 48 TMC 1vON. W88M Jo ['sey21g poyag oy} Ur you suoKeyg] *sauoz aynsndas ‘uoynsuad mos asymjsost pun fydvibodo; wof woryoatLo9 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 85 Note that for the ocean stations, Nos. 1 to 7, there is in each case but one change of sign as successive zones are considered, namely, the change of sign due to distance. The water compartments predominate in their effects in every zone. At station No. 1, over a deep part of the ocean where the bottom is nearly level to a great distance from the station, the positive corrections beyond the change of sign due to distance have nearly the same aggregate as the negative corrections before the change of sign, and, therefore, the total correction for topography and compensation is small. At stations Nos. 2 and 3, near the Tonga Deep, this balance of positive and negative corrections is slightly dis- turbed in one sense—the positive correction predominates; and at stations Nos. 4 and 5, over the Tonga Deep, the balance is greatly disturbed in the opposite sense, the negative corrections being largely in excess: These are probably typical cases. Note that stations Nos. 6 to 9 constitute a progressive series of four in relation to topogra- phy. No. 6 is over deep water near an oceanic island, No. 7 over water of moderate depth nearer to the oceanic island, No. 8 near sea level on the coast of a high oceanic island, and No. 9 on a high summit of such an island. Note that the corrections for topography and compensa- tion stand in order, namely, +0.019, +0.078, +0.162, and +0.469. A comparison of values for corresponding zones in the preceding table for these four stations will indicate the manner in which the positive corrections gradually gain predominance as the station is made-to approach from deep water to the summit of an oceanic island. While making this comparison it will be well to consult pages 65-71 in regard to the change of sign due to distance. Stations Nos. 11 and 12 are like station No. 8 in being near sea level on the shore of an oceanic island surrounded by deep water. Note the resemblance between these three stations as to the correction for separate zones. In each case the sum of the corrections out to zone L is small, but beyond that large positive corrections appear and the total correction for each station is positive and large, corresponding to the known fact that large values of gravity are ordinarily observed in such a location. . Station No. 13 is remarkable for having unusually small corrections in every zone—all positive. Station No. 14 shows a succession of values characteristic of stations on a high plateau far from any ocean. The large positive corrections for near zones are more than offset by still larger and more numerous negative corrections beyond the change of sign due to distance, which occurs at zone J, and the total correction is, therefore, large and negative. The very large negative values in zones K to O are due to the fact that the high plateau extends far enough from the station to fill these zones. The negative corrections are numerous because, the station being far from the nearest ocean, the water effects do not predominate and positive corrections do not appear again until a very large zone is reached, namely, No. 6, of which the inner radius is 2900 kilometers. A comparison in detail of the corrections for separate zones at stations Nos. 15 and 16. will show why the corrections for topography and compensation tend to be large and positive for a station above the general level in a mountainous country and negative for a station far below the general level in the same region. Note that the positive corrections for small zones are much smaller at station No. 16 at the bottom of one of the deep valleys than at station No. 15 on a high summit of the Alps, and that the change of sign due to distance occurs before zone G at No. 16 and after zone J at No. 15. These two differences between the two stations are due largely to the effect of corrections due to the differences of elevation of the station and the zone (‘station below compartment” and ‘‘station above compartment”) shown in the reduction tables on pages 30-43. Consult especially the reduction table for zone G on page 35 in connection with the correction for zone G at these two stations. It will also be noted that for the same reason the negative corrections, beyond the change of sign due to distance and before the water effects begin to predominate; are larger for corresponding zones at station No. 16 as arule. This is especially noticeable for zones K, L, M, and N. 86 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. DISCUSSION OF ERRORS. As the methods of computation used in this investigation are novel in many respects, it is important to consider the accuracy of each part of the process. As it has been stated that the desirability of selecting such methods as would give the required results with the minimum expenditure of time has been continually kept in mind, it may seem probable that this close attention to the economics of the problem has diverted attention from the requirements of the problem as to accuracy. Throughout the investigation very close attention has been paid at every step to insuring the maintenance of the required degree of accuracy. It is not feasible within the allowed limit of length of this publication, and without printing all the details of the computation, to put before the reader all the evidence which has been considered by the writers in estimatmg the magnitude of errors from various sources. The discussion of errors which follows serves, however, to show in a general way the methods by which the estimates of error were made and to put the estimates on record for future reference and for reexamination by others. Let it be assumed for the moment that the purpose of the present investigation is to compute the value of gravity at each observation station by taking adequately into account the effect of every portion of the earth’s mass in producing an attraction at the station. In order to accomplish this the computation must take into account adequately all the facts as to the shape of the earth’s surface (its topography) and all the facts as to density at all points within the earth. These two sets of facts serve to locate with reference to the station every portion of the attracting mass. If this be considered the true purpose of the investigation, the real measures of the total errors made in the attempt are the residuals of the attempt, namely, the apparent anomalies by the new method shown in the table on page 74. Each anomaly is the difference between the computed value of the attraction upon a unit mass (1 gram) at the station and the directly observed value of that attraction. The degree of accuracy attained may be expressed by saying that the largest anomaly is — 0.095 dyne (at stations Nos. 53 and 56, Seattle, Wash.), that the mean anomaly without regard to sign is 0.017 (p. 76), and that as computed from these anomalies considered as errors the probable error of the result at a single station is +0.014 (p. 75).* The total error, as defined above, the apparent anomaly at each station, is the aggregate of errors of three different classes. The first class comprises the errors in the observed value of the attraction at the station. The second class includes all errors in the computed values of the attraction at the station. Among these are errors due to numerical inaccuracy in the computations, due to errors of approximation in the formule used, and errors due to the faults and incompleteness of the maps which were used. The third class includes such errors as are due to the difference between the actual arrangement of density in the earth and the arrange- ment which has been assumed. The assumed distribution of densities is that fixed by the statement that under every part of the earth’s surface the isostatic compensation is complete and uniformly distributed with respect to depth down to a limiting depth of 113.7 kilometers (p. 10). The purpose of this discussion is to give the reader an estimate of the probable average magnitude of the errors of the first and second classes and to compare this with the total error as expressed by the anomalies, thereby securing an estimate of the magnitude of the errors of the third class. From this point of view the errors of the third class are the portions of the apparent anomalies which may not be accounted for as due to errors of the first or second class. The smaller the errors of this third class are found to be the more nearly the assumed distribution of densities agrees with the actual. The errors of this class furnish a good basis for further investigation as to the actual distribution of densities in the earth. * This mean and probable error are based upon the anomalies at 87 stations in the United States, the two stations Nos. 53 and 56, at Seattle Wash., being rejected. 7 : 1 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 87 ERRORS OF OBSERVATION. The half-second pendulums, described in Appendix 15, Coast and Geodetic Survey Report for 1891, were used in the relative determination of gravity at each of the 89 stations in the United States used in this investigation. The observations were made during seasons of less than 6 months each, and the pendulums were standardized at the base station (in the basement of the office of the Coast and Geodetic Survey at Washington) both before and after each season. Three pendulums constituted a set, each pendulum being swung through at least two periods of approximately eight hours each in determining the intensity of gravity at a station or while obtaining the periods of the pendulums at the base stations. The necessary time observa- tions were made with a portable astronomical transit set up in the vicinity of the gravity station. The apparatus was used during standardizations in the same manner as in the field. The following table shows the magnitude of the probable errors of the relative intensity of gravity at 85 of the stations in the United States used in this investigation. The stations for which no probable errors were computed are the base stations, the Smithsonian Institution, Washington, D. C., Baltimore, and Seattle University. Stations Probable error, in dynes 8 +0. 003 14 + .002 58 + .001 5 . 000 Average + .0013 The probable errors shown above are those due to the accidental errors made at the stations in the field. Let it be assumed that the accidental errors in obtaining the mean periods at Wash- ington from the standardizations of the pendulums are approximately equal to the probable errors in the field means. Then the total probable error for a station may be considered as a combination of the probable error of the standardization and the probable error of the field station. On this assumption the maximum probable error is +0.004, and the average probable. error is +0.0018 for the mean result at any station. The actual error is probably at no station more than four times the average probable error, or 0.0072 dyne, and the average actual error is much lower than that. It is believed that the assumption stated above tends to give estimates which are too large rather than too small. . The following special statement is necessary for the seven stations, Ely, Pembina, Mitchell Lake Placid, Potsdam, Wilson, and Alpena. Upon the return of the gravity party to the base station, in November, 1909, after having observed at these stations, it was found that the period of each of the three pendulums used during the season had considerably shortened. After having made two complete determinations of the periods a very thin film of foreign substance was dis- covered on the supporting plane of each of the three pendulums. Upon the removal of this substance the pendulums resumed their former periods. In addition to the stations mentioned above, North Hero and Iron River were occupied while the pendulums were probably affected by the foreign substance on the planes. These two stations were reoccupied during a subsequent season, and the values obtained for the intensity of gravity agreed closely with those obtained during the first occupation of those stations, provided it was assumed. that the foreign substance affected the periods of the pendulums to the same extent at those stations as during the first determination of the periods at Washington in N ovember. North Hero and Tron River were considered as base stations in determining the value of the intensity of gravity at Lake Placid, Potsdam, Wilson, and Alpena, which stations had been occupied after North Hero and before Iron River. Iron River and Washington were considered as base stations for Ely, Pembina, and Mitchell, these three stations having been occupied after Tron River and just before the return to Washington after the close of the season. The intensity of gravity used for North Hero and Iron 88 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. River was that determined during the reoccupation of those stations in 1910, and the value of the period at the base station was that determined by the first standardization after the close of the season in 1909. The periods for the first and second occupation of North Hero differed by 0.0000057 second, while at Iron River they differed by 0.0000042 second, and at the base station the difference between the period given by the first standardization in November, 1909, and the mean period of the two standardizations in May and October, 1910, was 0.0000033 second. This seems to indicate that the effect of the foreign substance on the periods at the stations between North Hero and the base station gradually decreased during the season between July and November, 1909. If the error in the adopted mean period at any of these stations is as much as 0.000 002 5 second, then the error in the value of the intensity of gravity at the stations from this cause is 0.010 dyne. If a similar error was made at one of the base stations (North Hero, Iron River, or Washington), the error due to this cause is 0.005 dyne. Hence, it is possible that there may be errors as great as 0.015 dyne in the adopted values of the intensity of gravity at the stations Lake Placid, Potsdam, Wilson, Alpena, Ely, Pembina, and Mitchell. It is believed, however, that the actual error for each of those stations from all causes is less than 0.010 dyne. In general the pendulums show approximately the same period at the base station in Wash- ington during successive standardizations. There is given below a table showing the mean period of the three pendulums forming the “A” set for the base station: Date of stand- Period in ardization seconds Jan., 1909 0.500 707 5 June, 1909 .500 707 7 Dec., 1909 - 500 706 4 May, 1910 .500 705 7 Oct., 1910 .500 707 0 Mean . 500 706 9 It was assumed in each case that the pendulums were in normal condition. The values obtained at the base stations in November, 1909, were not included in this table, on account of the presence of foreign substance on the planes in the heads of the pendulums during those standardizations. For the gravity work done during the years 1909 and 1910, the period adopted for the base station in reducing a season’s work (except the season between July and November, 1909) was the mean of the periods obtained at the beginning and at the end of the season. ERRORS OF COMPUTATION. The first step in computing the attraction at a station was to compute by the Helmert formula of 1901 the attraction 7., at a point on an ideal earth at sea level in the same latitude as the actual station. The ideal earth referred to is one having the same size and shape as the ellipsoid of revolution which most nearly coincides with the sea-level surface of the real earth and having no topography and no variations in density at any given depth below the suresee, (See p. 12.) The Helmert formula of 1901 is based upon many gravity determinations widely distributed over the earth’s surface, and in consequence probably gives a close approximation to the desired values. The available indirect evidence gives strong support to the belief that this formula, in which the constants are computed from gravity observations, is of a very high degree of accuracy: For example, the values of the flattening of the earth, as computed by this formula and as com- puted from geodetic observations in the United States, are of about the same degree of accuracy and agree closely. The value of the reciprocal of the flattening derived from the Helmert formula of 1901 is 298.3 +0.7, and from geodetic observations in the United States is 297.0 0.5.* * Supplementary Investigation in 1909 of the Figure of the Earth and Isostasy, p. 60 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION-ON GRAVITY. 89 This is a confirmation, by independent observations of a different kind from those on which the formula is based, of the accuracy of the second constant in the Helmert formula. But, on the other hand, the Helmert formula of 1901 is based upon selected coast and inland stations. The present investigation indicates that even at these carefully selected stations there is probably small systematic error due to the failure, by the methods of reduction used in connection with the derivation of the Helmert formula, to take account properly of the effects of the topography and its isostatic compensation. A correction (0.007) serving to eliminate this systematic error as completely as is possible at present, has been derived from the observations in the United States and applied to the first constant in the Helmert formula of 1901. (See p. 75.) It is believed that the Helmert formula of 1901 so corrected is a true representation within less than 0.003 dyne on an average of the attraction at sea level on the ideal earth, if the formula is limited in application to the range of latitudes occurring in the United States. . The correction for elevation (p. 13), the next step in the computation, is of such a nature that it is reasonably certain that the errors made in computing it are very small, usually not more than 0.001 dyne. An error of 3 meters in the elevation makes but 0.001 dyne error in the computed correction. For the gravity stations in the United States the elevations are known as a rule within 3 meters and at very few if any of the stations is the error in elevation more than 15 meters. . The value of the gravitation constant (k) adopted in this investigation is 6673 (10-4), and it is estimated that the probable error of this adopted value is one part in 1330. (See p. 14.) This constant enters directly as a factor into each formula for computing the correction for topography and isostatic compensation. (See formule (10), (15), (16), (17), and (18), pp- 15-17.) Hence, the probable error of one part in 1330 in the gravitation constant produces an error of the same proportional part in each computed correction for topography and compensation. The largest of these corrections (see p. 74) is only 0.187 for station No. 43, Pikes Peak. Even for this case the probable error in the correction due to error in the gravitation constant is only 0.0001 dyne (0.187/1330), and is therefore negligible in connection with the present investigation. Similarly, any error in the assumed mean surface density of the earth will produce an error of the same proportional part in the computed correction for topography and compensa- tion corresponding to each land compartment. The mean surface density has been assumed to be 2.67 in this investigation. It is reasonably certain that the mean density of the whole of that portion of the earth which lies above sea level does not differ from this by as much as one-twentieth part.* At Pikes Peak, station No. 43, the sum of the corrections for all land compartments is probably greater than for any other one of the 89 stations in the United States used in this investigation. At this station this sum is about +0.180 dyne.+ An error of one- twentieth part in this would be only 0.009 dyne. An inspection of the tables on pages 54-58 indicates that as a rule the sum of the corrections for land compartments for stations in the United States is less than 0.020 and an error of one-twentieth part would, therefore, ordinarily be less than 0.001 dyne. . In general the density of sedimentary rocks tends to be less than 2.67, not unfrequently as much as one-tenth part less. On the other hand, igneous rocks and rocks which have been buried to a great depth tend to be of density greater than 2.67. These local departures of the densities from the assumed mean, 2.67, produce errors of the third class, which have been defined as errors due to the difference between the actual arrangement of densities in the earth and the assumed arrangement. These effects of local departures of density from the mean are a part of the anomaly at the station rather than errors in determining the anomaly, Hence, the discussion of them will be taken up later as a part of the discussion of the meaning of the anomalies. . sity of the earth, 2.67, and this estimate of its uncertainty are based largely upon the information : : oe Ee Sent, fae ue, by William Harkness, Washington, Government Printing Office, 1891, pp. 91-92, Be te ane a ones A to 10 at this station. (See p. 56.) Zone 9 is the nearest zone containing any oceanic compartments. eee ait the estimates of density of rocks in the vicinity of 10 of the gravity stations here treated as given on p. 530f Appendix T se hs on ouctettn Survey Report for 1894, ‘Relative determinations of gravity with half-second pendulums and other pendulum observa- of the Co: tions,” by G. R. Putnam and G. K. Gilbert. 90 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. On page 15 attention is called to the fact that in deriving the formule by which all compu- tations for distant zones have been made the earth is treated as a sphere with a radius of 637 000 000 centimeters, although it is actually a spheroid. In zones M, N, and O errors due to this approximation are evidently negligible for, as shown on page 22 and in the reduction tables for those zones, pages 41-43 (consult the column headed ‘‘Station at same elevation as compartment”), if all of the curvature were neglected and the earth’s surface treated as a plane, the error introduced would be only 0.0001 dyne in any one compartment of these zones. The curvature of the actual spheroid in any azimuth within the limits of the United States differs by less than one five-hundredth part from that of the assumed sphere and, therefore, the error for any compartment due to the cause under discussion must necessarily be much less than 1/500 of 0.0001 dyne in zones M, N, and O. The errors must be still smaller for near zones. For more distant zones a general consideration of the geometric relations, shown in illus- tration No. 15, page 67, indicates that the error is probably considerably greater. Without a detailed investigation the three following considerations seem to the writers sufficient to assure one that the total error due to this cause is probably less than 0.001 dyne at every station. First, the total correction for topography and isostatic compensation beyond zone O is less than —0.060 dyne at every one of the 89 stations. Second, the actual radius of the earth varies from 6357 kilometers at the pole to 6378 kilometers at the equator; that is, from 13 kilo- meters less (1/490 part) to 8 kilometers greater (1/800 part) than the assumed radius. These differences may be considered as maximum vertical displacements of material in very distant zones from its assumed position. The displacements are small in comparison with the distance to the zone in these cases. Third, on the actual spheroid the radii in various azimuths from the station are different. For example, for a station in the central portion of the United States in latitude 39° the radius of curvature in the meridian is 6361 kilometers, 9 kilometers less (one part in 710) than the assumed value, 6370 kilometers, and in the prime vertical at this same station the radius of curvature is 6387 kilometers, 17 kilometers greater (one part in 370) than the assumed value. Hence, in each zone the errors of the kind under consideration tend to be compensating to a considerable extent, some parts of the zone lying farther from the center of the earth than the assumed curvature places them and other parts of the same zone, lying in different azimuths, being nearer to the center than the assumed curvature would place them. Assuming for the moment that the elevations and depths shown on the maps and charts used are correct, the errors made by the computer in estimating the mean elevation or mean depth within each compartment did not, as a rule, produce any error even in the fourth decimal place in dynes. In zone A an error of at least 5 feet in estimated elevation is necessary in order to make an error of 0.0001 dyne in the computed correction even if the elevation of the station is less than 10 feet. In this zone if the station has an elevation greater than 10 feet, the cor- rection is 0.0002 dyne in every case. In zone F it takes an error of 200 feet or more in the estimated elevation to produce an error of 0.0001 dyne in the computed correction; in zone M 500 feet or more; in zones 18 to 14, 100 feet; and in zones beyond 14, 1000 feet or more. (Consult the reduction tables, pp. 30-47.) In many cases the total range of elevation within a compart- ment, as shown by the map, is less than that necessary to produce a change of 0.0001 dyne in the correction taken from the reduction table. In these cases no error in the correction arose from the estimation of the mean elevation. Still more frequently the range of elevation within the compartment is not more than three or four times that necessary to produce a change of 0.0001 dyne. It is probable that in such cases the estimate of mean elevation was rarely in error by more than the quantity corresponding to 0.0001 dyne. For perhaps one-tenth of all the com- partments the computer found so large a range of elevations shown on the map that his estimate of mean elevations was necessarily made with considerable care and attention to the details of the contour lines, and even then the correction taken from the reduction table may be in error by two or more units in the fourth decimal place. It is believed that the aggregate of such errors for a station is seldom greater than 0.001 dyne. For, as indicated above, difficulties were encountered in making the estimate of the mean elevation with sufficient accuracy at only a EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 91 small percentage of the compartments; the errors so made tend to be in the accidental class; the difficulties were obvious to the computer and he, therefore, exercised unusual care in the extraordinary cases, even to the extent of subdividing the compartments and making a separate estimate for each subcompartment; and finally for each station a second computer inspected the computation made by the first and made a second estimate covering some of the compart- ments in which there were obvious difficulties in making a sufficiently accurate estimate. For compartments for which two estimates were made the mean of the two was used, unless they differed so much as to lead to detection of an error in one or the other. In making the computations for topography and isostatic compensation, the elevations of compartments were read from the maps without making any allowance for the fact that glaciers have a much lower density than the land. A computation was made to show the effect upon the intensity of gravity at station Gornergrat of the defect of density of the glacial ice in its vicinity in comparison with solid earth of the assumed density, 2.67. An inspection of the maps of this region showed that 37 of the 102 compartments in the zones E to K were over ice, and the shapes of the clear portions of the valleys indicated that the average thickness of this ice in the several compartments varies from a few feet to more than 600 feet. The presence of ice in the zones closer to the station than zone E and farther from the station than zone K was believed not to affect the intensity of gravity at the station. An average density of unity was assumed for the glacial material in making this computa- tion. This is believed to be near the truth, for the heavy material carried by the glacier (sand and gravel) is probably approximately balanced by cavities and the lightness of the clear ice in comparison with water. This makes a defect of density of approximately 1.67 in portions of the topography of certain compartments. This should make a minus correction to the computed effect of the topography and a plus correction to the effect of the isostatic compen- sation. The largest correction found for any one compartment due to this lack of density was 0.0004 dyne, while the average correction for a compartment was less than 0.0001 dyne. In the near zones the effect of ignoring the lack of density in the glacier made the computed value of gravity too great, while, owing to the change of sign with distance from the station (see pp. 65-70), the effect of such neglect in the more distant zones was to decrease the computed value of gravity. The total result for station Gornergrat was to make the computed value of gravity too great by 0.0006 dyne, a negligible quantity. It is probable that the effect on the intensity of gravity of assuming glacial ice to have a density of 2.67 in the computations of the effect of topography and isostatic compensation upon the intensity of gravity has not caused an error of more than 0.0010 dyne at any one of the stations treated in this investigation. In using the mean elevation within a compartment as the argument in entering the reduc- tion tables on pages 30-47, it is tacitly assumed that the influence of a unit of area of a given elevation is the same wherever it is located in the compartment. This is only approximately true. For example, in zone 13 (limiting radii 3° 03’ 05’’ and 4° 19’ 13’ ’) Ep is 5000 at the outer edge of the zone and 13 600 at the inner edge. (See p- 25.) The influence of a unit of area of a given elevation on the outer edge of the zone is, therefore, 5000/ 13600 = 0.37 as great as on the inner edge. If, therefore, in this zone the elevations nearer the outer edge in one com- partment happen to be much greater than elevations nearer the inner edge, the correction taken from the table by using the mean elevation as an argument will be too large. Similarly, if the slope in the compartment happens to be downward from the inner edge toward the outer edge the correction taken from the table will be too small. a When the arbitrary selection of radii of zones and of number of compartments in each zone was being made the danger of errors from this source was kept constantly in mind (see Pp. 18), and each compartment was made so small that the estimated errors due to this cause in any compartment would ordinarily be less than 0.0002 dyne. The details of the manner in which this estimate was made can not be conveniently shown here. Evidently the narrower the zone is made the smaller the error from this cause, both because Ep will be more nearly the same on the two edges of the zone and because the difference between the average elevation of the near topography and of the distant topography in each of the compartments of a zone 92 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. tends to be small. Similarly, after the width of the zone is fixed, the smaller the compartments are made within the zone the smaller will be the error, for the less will be the range of elevations included within the compartment and the larger will be the change of elevation corresponding to an effect of 0.0001 dyne. The values of E, were before the investigator for all outer zones at the time each decision was made. For inner zones for which formula 18, page 17, was used, an indirect method of obtaining the equivalent of the change in Ep was utilized. The investi gator also had before him the experience and data obtained in connection with the previous investigation of the figure of the earth * which enabled him to estimate the maximum difference of elevation between the inner and outer edges of any given compartment which would probably be found at any station. Errors due to this cause will evidently be of the accidental class, since in some, zones a downward slope toward the station will produce an error of one sign and in others the reverse slope will produce an error of the opposite sign. In the 317 compartments concerned in the compu- tation at a given station there will be but few compartments, sometimes none, in which this error is as great as 0.0001 dyne, and errors of both signs will probably occur among these few. It is believed that the aggregate error due to this cause at a station seldom exceeds 0.0005 dyne. The errors due to the faults and incompleteness of the maps and charts used are believed to be very small as a rule. The aggregate error for all numbered zones is probably seldom, if ever, greater than 0.002 dyne. For the lettered zones, zones which lie near the station, the aggregate error in some cases may be two or three times this limit. The reduction tables (pp. 30-47) show that for the nearer lettered zones the elevations must be known with greater accuracy in.general than for the more distant numbered zones, and since the compartments are small in the lettered zones it is necessary to know the details of the topography. The magnitude of the aggregate error at a given station, due to faults and incompleteness of maps and charts, therefore, depends principally upon the accuracy of the maps and charts covering the region close to the station rather than that of those covering distant regions. f Some errors are made in locating the compartment boundaries on the maps, due to unavoidable inaccuracy in constructing the templates, to inaccuracy in placing the templates on the maps, to special difficulties encountered in connection with the distortion of scale on Mercator ‘charts, and to shrinkage and, therefore, error of scale of the maps and charts. With the templates and maps before one it is evident that the aggregate effect of these errors at a station is ordinarily negligible. In general the effect of an error in locating a compartment boundary is simply to throw a small part of the area which belongs in one compartment into an adjoining compartment, where its influence on the computed correction is nearly the same as if it had been placed in its proper compartment.. The methods followed in computing the reduction tables have been stated on pages 19-28. The precautions taken were such as to insure that no tabular value is in error by more than 0.0002 dyne, and that in general the tabular values are correct to within 0.0001 dyne. The intervals between tabulated values have been so selected, with due regard to second differences, as to insure that the errors made in interpolating between them, using first differences only, shall ordinarily be less than 0.0001 dyne. How large are the errors introduced into the computed topographic effect on: the intensity of gravity by the interpolation of values corresponding to outer zones? The complete com- putation was made for only six stations. Each new station to be computed was so chosen, if possible, as to lie within the triangle defined by the nearest three stations for which the computation had already been made, and near the center of said triangle. From these three surrounding stations the interpolation, if any, was made. The computation was commenced with the inner smaller zones and proceeded outward. The two rules used by the computers in deciding at what zone it was allowable to begin to accept the interpolated values and to accept them for all larger zones were, as stated on page 63, as follows: * The Figure of the Earth and Isostasy, etc., pp. 125-127. + For a more detailed statement of the considerations upon which the judgment expressed in this paragraph is founded, see The Figure of the Earth and Isostasy, etc., p. 124. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 93 Rule 1—Commence to accept the interpolated values as final with the first zone for which such interpolation is allowable under rule 2, provided that it is beyond the zone containing the nearest of the three stations from which the interpolation is made. Rule 2.—Let 0.0005 dyne be the interpolation limit for any zone. Subject to rule 1, . acceptance of the interpolation may begin with a given zone if each of the three zones next within it shows an agreement between the interpolated and computed values which is within the interpolation limit. _ Under rule 2, at any station the maximum error made by accepting interpolated values would be, in dynes, 0.0005 times the number of zones interpolated, if the error of interpolation I—C (interpolated minus computed) always had the same sign. It was believed, however, that the agreement between the interpolated and computed values (commencing with zones not smaller than those contemplated under rule 1) would tend strongly to be closer and closer for successive zones proceeding outward. It was also believed that there would be a strong tendency for the various differences between interpolated and computed values for several zones such as are interpolated under the rules to include values having both the plus and minus signs, and, therefore, for the errors in the accepted interpolation to tend to be eliminated from the final result for the station. The correctness of these beliefs is established by the results secured during the progress of the computations. From the results of the computations of 48 stations a comparison between the computed and interpolated values was secured at each station on from 2 to 10 zones. In 81 per cent of the cases the average value, without regard to the sign of I—C (nterpolated minus computed) was less for the outer one-half of the zones on which both interpolation and computation was made at that station than for the inner half of such zones. Also in 56 per cent of the cases there were found to be both plus and minus signs of the values of I—C at the station. These tests confirm the theory to such an extent that it is believed that the total error introduced into the computed effect of topography and compensation at a station by the acceptance of interpolated values is seldom greater than 0.0022 dyne and is, as a rule, not more than one-half that amount. In addition to the evidence stated in the paragraph above, this estimate of 0.0022 dyne is based upon the fact that the average difference between the ‘computed and the interpolated values for the three zones (see rule 2) next within the one for which the interpolation is accepted, at any station, is in general 0.0002 dyne or less. The average number of zones per station for which interpolated values were accepted is 11. If the error for each interpolated zone were 0.0002 and all were of the same sign, the error would be 0.0022 on an average. However, as the outer zones have more overlapping of areas, the interpolated and computed values for those zones should agree on an average more closely than these values for the three zones next preceding the zone at which interpolation begins, and as these errors are of the accidental class and not all of the same sign, there is a tendency for the errors of interpolation to be eliminated from the final result for the station. One may, therefore, conclude that the total error caused by accepting the interpolated values is so small negligible. 7 sate oh to which the isostatic compensation extends has been assumed to be fixed by a surface which lies 113.7 kilometers below sea level, but, as noted on page 10, in order to sim- plify and to facilitate the computations the depth of compensation has in the computations really been reckoned from the solid surface of the earth, not from sea level. This computing device has, therefore, virtually displaced the isostatic compensation upward on land areas by a distance equal to the elevation of the surface of the area above sea level, and downward for ocean areas by a distance equal to the depth of the particular part of the ocean considered. For near zones this displacement of the compensation produces negligible effects because the total effect of the compensation is small (consult the reduction tables for zones A to I, pp. 30-37). For the very distant zones, 13 to 1, this displacement of the compensation produces effects which are certainly negligible, since the reduction tables, pages 45 a 46, sea ba is no appreciable correction for elevation in these zones. For intermediate zones J to 14 small appre- 94 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. ciable effects are probably produced in some cases by the virtual displacement of the isostatic compensation introduced as a computing device. Though no special investigation of the aggregate of effects has been made it is believed to be small. In other words, the actual computation made on the supposition that the depth of compensation is 113.7 kilometers measured from the solid surface, is believed to be practically in agreement as to numerical results with the computation which theoretically should have been made on the supposition that the depth of compensation is 113.7 kilometers measured from sea level. Within the great depth, 113.7 kilometers, to which isostatic compensation extends there is probably a slight increase of density with increase of depth, due to increased pressure. No account has been taken of this in the process of computation, as already noted on page 7. It may appear at first sight that this neglect introduces some error into the computed results, but it does not. The isostatic compensation as used in the computation is essentially an excess or defect of density referred to the normal density for each level concerned within the depth of compensation. It matters not in the computation of the effects of topography and iso- static compensation whether the normal relation of density to depth is such that there is no appreciable increase of density within the depth of compensation or whether there is consid- erable increase within that depth, for the excesses and defects of density constituting the isostatic compensation are referred to this normal law, not to a constant density for all depths. The point at which the relation of density to depth enters this investigation, though not explic- itly, is in the derivation of the Helmert formula of 1901. Any actual change in the distribu- tion of density with respect to depth would in general change the observed value of the intensity of gravity and would cause one or more of the constants of this formula to change. There- fore, the constants in this formula as derived from observations correspond to the actual relation’ between depth and density, though that relation is not known. NATURE OF APPARENT ANOMALIES. There have been discussed on the preceding pages the principal possible sources of error of the first and second classes, defined on page 86. Among these sources are the errors in the in- strumental determinations of gravity at each station, errors in the corrected Helmert formula of 1901, errors in the corrections for elevation, errors in the adopted values of the gravitation constant and the mean surface density, the erroneous assumption in certain parts of the compu- tation that the sea-level surface is a sphere rather than a spheroid, errors in the estimated mean elevations in the different compartments, errors due to variations of elevation within each com- partment, errors in the maps and charts used, errors in locating compartment boundaries, errors of interpolation for outer zones, and errors in computing the reduction tables. The errors of each of these kinds are nearly or quite independent of the others, and follow different laws of distribu- tion. In estimating the effects of all these errors at a station one must therefore consider them as accidental errors and that their combined effect is the square root of the sum of their squares rather than merely their sum. On this basis the writers estimate that the probable error of the computed anomaly at a station by the new method is about +0.003 dyne on an average. In other words, the chances are even for and against the proposition that the actual error in the computed anomaly at a station is greater than 0.003. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 95 The basis for this estimate is in part indicated in the following table: Estimate of errors of the first and second classes. : Maximum Average prob- Source of error probable error | able error of ofany station | any station Observations of gravity +0. 004 +0. 002 Helmert formula of 1901, corrected +0. 003 +0. 002 Correction for elevation +0. 003 +0. 001 Gravitation constant _ +0. 000 +0. 000 Mean surface density +0. 005 +0. 001 Defects and incompleteness of maps +0. 004 +0. 001 Acceptance of interpolation +0. 001 +0. 000 From all other causes +0. 001 +0. 001 The square root of the sum of the +0. 009 +0. 003 squares, or the probable error of the the final result subject to all the sepa- rate errors enumerated If the whole anomaly be considered as an error, then the probable error for all stations due to all causes is +0.014 (see p. 75), this probable error being computed from the 87 apparent anomalies available in the United States after rejecting the two Seattle stations. It should be noted that this computation includes the third class of errors defined on page 86, those due to the departures of the actual arrangement of densities beneath the surface from the arrangement which has been assumed. The magnitude of the errors of this third class, the real anomalies sought, may be estimated as that part of the total error computed, as indicated above, from the apparent anomalies, which is not accounted for by errors of the first and second classes, namely, + (0.014)? — (0.003)? = +0.0137. These two values, +0.003 and +0.0137, may be interpreted as follows: The second being about five times the first, the apparent anomalies shown on page 76 under the designation ‘‘ Anomalies, new method,” are upon an average composed of one part errors of observation and computation to five parts actual anomaly at the station, due to the departure of the actual arrangement of densities from the assumed arrangement. The quantities labeled ‘‘ Anomalies, new method,’’ are therefore a close approximation to the real anomalies sought. They are a possible basis for further investigation as to the actual distribution of density within the earth. — THE METHOD NOT SUBJECT TO HIDDEN ERRORS. This discussion of errors would be seriously incomplete if it were closed without calling at- tention to certain characteristics of the computations on which this investigation is based which insure safety against certain classes of obscure but serious errors. The process of integration by the method of computing a large number of separate values of the function (see pp. 23-27), which has been used in this investigation, is very clumsy and inele- gant, as seen from the mathematical point of view, but from the practical point of view of one who desires to solve the problem of computing the effect of all the topography ‘of the world and of its isostatic compensation upon the intensity of gravity at a given station, it has a very differ- ent aspect. From the latter point of view it appears that the method is sufficiently rapid to make its use permissible and that it is clearly safe against errors, whereas the alternative mathe- matically elegant method is unsafe. As to the rapidity of the method, it was found in practice that the necessary reduction tables for zones covering the whole earth were computed in the equivalent of about 800 hours of time for one computer. This seems to be a reasonable time when one considers the importance and difficulty of the problem solved. Moreover, these tables made it possible to make the remaining portions of the computation very rapidly. They enabled the computer in 17 hours to compute the effect of all the topography of the world and its isostatic compensation upon the intensity of gravity at any given station on the earth’s surface, and to be certain that the 96 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. errors of the computed result are confined within the very narrow limits indicated by the pre- ceding discussion of errors. This in turn furnishes a safe basis, and in the opinion of the writers the only safe basis yet available, for an accurate determination of the flattening of the earth from gravity observations; for any effective investigation of the theory of isostasy by means of gravity observations; for any investigation of the real meaning of the apparent anomalies of gravity, such, for example, as those on small oceanic islands; and in fact for any safe general conclusions from observations of the intensity of gravity on the earth’s surface. The method used in this investigation of obtaining the integrals of the expressions (9), (15), and (16), pages 14-16, by computing many numerical values, is safe against excessive or unseen errors because of the fact that the computer has before him in these many numerical values a clear and definite means of knowing how large are his errors of approximation. For example, when facing the actual problem of determining the mean value £, (see p. 23) with various com- puted values of E before him, there is little difficulty in deciding safely how many values of E to compute in order to be certain of a given degree of accuracy in the mean value. Various similar examples from this investigation might be cited. On the other hand, if the computer resorts to the more elegant method, from the mathe- matical point of view, and first transforms formule (9), (15), and (16) by simplification into forms which can be integrated by calculus, he is, while making the simplification, in grave danger of introducing errors of approximation which he believes to be small, but which are in reality large. The writers believe that in this particular problem this danger has not been escaped in the past. For example, the conclusion that it is not necessary to take distant topography into account, a conclusion which has been acted upon in many previous investigations, and which this investigation shows to be erroneous, has apparently been reached in the past by dealing with unsafe approximations in the literal or symbolic form. So, too, it seems to the writers that one can not overlook the necessity of taking curvature very fully into account if one has the numerical values before him, but may easily overlook it if he is dealing with symbols and formule only. Another characteristic of the method of computation used in this investigation, which is very important as a means of securing safety against unseen errors, is the fact that it deals with the actual irregular surface of the earth rather than with a geometrical surface which is assumed to fit the earth’s surface in the vicinity of the station. It is true that the irregular surtace actually used in the computation is made up of 317 level surfaces, one for each compartment of each zone, the mean elevation in each compartment being the argument with which the reduction tables are entered. But the compartments near the station are so small that the surface upon which the computation is based is, in these zones, a very close approximation to the actual irregular surface. The one compartment of zone A is a circle with a 2-meter radius. Each of the four compartments of zone B has an area of less than 4000 square meters. The agreement between the assumed surface and the actual irregular surface of the earth is less close for the more distant topography, but there is still, even for the most distant zones, an approximation to the actual irregular surface. The precautions taken in fixing the size and shape of the separate compartments insure, in fact, that even for these distant zones the approx- imation to the actual irregular surface is sufficiently exact to keep the errors in the computed, effects of topography and compensation well within the allowable limits. In any computation of the effects of topography and compensation in which any part of the earth’s surface is assumed to conform to the geometrical surface, in which, for example, a mountain or an oceanic island is assumed to have a conical shape, or the distant topography is assumed to ke a plain of indefinite extent, it is desirable to consider with extreme care how much error may be introduced into the computations by such assumptions, to consider care- fully what evidence the computer has that these errors are small in each separate case. Such errors once introduced into an investigation remain there regardless ot the degree of mathe- matical elegance and precision which may be maintained thereafter. The writers believe that the more carefully this point is examined the more fully the advantages of the methods of com- putation used in the present investigation will be appreciated. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 97 EFFECTS OF TOPOGRAPHY AND COMPENSATION—WHY COMBINED. In the investigation of the figure of the earth and isostasy by means of observed deflections of the vertical, the whole effect of the topography was first computed and later the effect of the isostatic compensation was combined with it.* In the present investigation, based on gravity determinations, the effects of topography and of compensation have been combined as early as was feasible in the processes of deriving formule and of computing. Thus, as indicated on pages 23 and 24, instead of computing the two effects separately for distant zones they were combined in formula (20) and the resultant effect computed at once and tabulated in the reduc- tion tables on pages 44-47. So, too, for near zones the principal part of the reduction tables (pp. 30-43) refers to resultant effects, not to separate effects. The only columns in these tables showing separate effects are columns 2 and 3, and these were not used in the regular com- putations. Why was this departure made from the methods of the earlier investigation ? This departure was decided upon immediately after a preliminary reconnoissance of the problem. It then appeared probable that, for all zones except for those very near the station, the two opposing effects of topography and compensation would be nearly equal, and their dif- ference, therefore, much smaller than either one. Under these circumstances it appeared that to compute each of the opposing effects with sufficient accuracy to secure the required degree of accuracy in their difference it would be necessary to secure several significant figures in the computation. If this supposition were true, it would be necessary in making the separate computations, either to make the compartments of the separate zones very small and numerous, and hence the computation very slow, or, otherwise, if large compartments were used, it would be necessary to make the estimate of mean elevation in each compartment with such a high degree of accuracy as to be both slow and difficult. On the other hand, it appeared that in the direct computation of the resultant difference of effects, it would be necessary to use but two or three significant figures in the computation, that the compartments could be made large and therefore not very numerous, and that only an approximate estimate of the mean elevation in each compartment would be required and could, therefore, be made quickly and easily. It seemed, therefore, that so much would be gained in rapidity and ease of computation by the proposed departure from the earlier practice that these gains should outweigh all other considerations. Now, this investigation being complete, the writers have an opportunity to review the deci- sion in the light of accumulated facts and greater experience. In that light it appears that the decision was wise for zones which are more than 26° from the station—zones 6 to 1 of the present investigation. For these zones the difference of the effect of the topography and the effect of the compensation is less than one-tenth of either; that is, Zz is less than one-tenth of either Ep or Ec (p. 25). For nearer zones the difference, as a rule, is a much greater propor- tional part. Hence, for these nearer zones the gain in rapidity and ease made by dealing directly with the difference of effects rather than with the separate effects was not great, and therefore the decision was not wise. Moreover, it appears now that if the separate effects had been computed for these nearer zones it would have given the investigator a clearer and more precise insight into the problems involved. It would also have facilitated studies of the rela- tion of the computed results to the assumption as to the depth of compensation and possibly to some other assumptions. If, therefore, an entire new investigation were being made the writers believe it would be wise to compute the two effects separately for zones A to O and 18 to 7, but the gain to be secured does not seem to be sufficiently great to warrant the revision of the present investiga- tion and the remodeling of the reduction tables here printed. * The Figure of the Earth and Isostasy from Measurements in the United States, pp. 68-73. 15593°—12——7 98 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. REGIONAL VERSUS LOCAL DISTRIBUTION OF COMPENSATION.* The question whether each topographic feature is completely compensated for by a defect or excess of mass exactly equal in amount directly under it, or whether the topographic feature is compensated for by a defect or excess of mass distributed through a more extensive portion of the earth’s crust than that which lies directly beneath it, is a very important one. The theory of local compensation postulates that the defect or excess of mass under any topographic feature is uniformly distributed in a column extending from the topographic feature to a depth of 113.7 kilometers below sea level. The theory of regional compensation postulates, on the other hand, that the individual topographic features are not compensated for locally, but that compensation does exist for regions of considerable area considered as a whole. In order to have local compensation there must be a lower effective rigidity in the earth’s crust than under the theory of regional compensation only. In the latter case there must be sufficient rigidity in the earth’s crust to support individual features, such as Pikes Peak, for instance, but not rigidity enough to support the topography covering large areas. Certain computations have been made to ascertain which is more nearly correct, the assumption of local compensation or the assumption of regional compensation only. In making such computations it is necessary to adopt limits for the areas within which compensation is to be considered complete. A reconnoissance showed that the distant topography and com- pensation need not be considered, for their effect would be practically the same for both kinds of distribution. As a result of this reconnoissance it was decided to make the test for three areas, the first extending from the station to the outer limit of zone K (18.8 kilometers), the second from the station to the outer limit of zone M (58.8 kilometers), and the third, to the outer limit of zone O (166.7 kilometers). The computed effect of the topography in each compartment and zone is the same under the two methods. The effect of compensation is assumed to be the same for each compartment and zone which is beyond the limit of the area adopted for the test. The effect of compensation within that limit is computed for each compartment in the case of the theory of complete local compensation, while in the case of regional compensation only, it is obtained from one operation after the average elevation within the area considered is known. The regular computations of the effect of topography and compensation had been completed at 56 stations in'the United States, Nos. 1 to 56, inclusive, and at all of the stations not in the United States, used in this investigation, before it was planned to make computations based on the theory of regional compensation within limited areas. In the regular computations for these stations the effect of topography and compensation for zones A to O was taken from the fourth column of the reduction tables (see pp. 30-43), and no record was made of the elevations of the several compartments as read from the maps. In making the supplemental computations these tables were entered with the previously computed values of the combined effect of the topography and compensation as arguments, and the approximate values of the elevations of the several compartments of zones A to O were taken from column 1 of the reduction tables, and the values of the effect of compensation taken from column 3. The supplementary com- putations were not made for all of the stations between Nos. 1 and 56 on account of the large amount of work involved. While making the computations of the effect of tépography and compensation for stations Nos. 57 to 89 (except station No. 84), a table was made for each station, giving the elevation of each compartment out to zone O as read from the map. With these elevations the reduction tables were entered and the effect of compensation was taken out separately from column 3. The total effect of compensation under the theory of local distribution was obtained for each of the areas considered by adding the values of the effect of compensation for the several com- partments of each of the zones. The mean value of the elevation of each zone was obtained by taking the mean of the elevations of its several compartments, and the mean elevation of * The investigation under this heading was made at the suggestion of Mr. G. R. Putnam, of the Coast and Geodetic Survey EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 99 each of the three areas considered (limited by zones K, M, and O) was obtained by combining the elevations of the various zones, the elevation of each zone being given a weight equal to its percentage of the total area under consideration. The regional compensation for the total amount of topography in the area considered was assumed to be uniformly distributed both vertically and horizontally throughout a column of depth 113.7 kilometers and of a cross section equal to the area of the topography—that is, successively from the station to the outer limits of zones K, M, and O. The effect of the com- pensation upon the intensity of gravity at the station was computed by formula (17), in which the several terms have the same significance as stated on page 17. The table following shows the comparison of the effects of local compensation and regional compensation for 41 stations in the United States and 4 stations not in the United States. It also shows the anomalies by the first method and for 3 cases at each station by the second method. The first column gives the number and name of the station. The second column gives the total correction for topography and compensation by the method of local compensa- tion. In the third column.are shown the values of the compensation for the topography included in the area extending from the station out to zone K, the compensation being assumed to be complete and local, In the fourth column are given the values of the compensation for the topography within the same area, but with regional compensation only, which is assumed to be uniformly distributed and complete, within the area limited by the outer circumference of zone K. Columns 5 and 6 are similar to 3 and 4, except that the area considered extends from the station to the outer limit of zone M. The same statement applies to columns 7 and 8, except that the area considered extends from the station to the outer limits of zone O. The ninth column contains the new-method anomalies, based upon complete local compensation, and the last three columns show the anomalies for the three cases under the theory of regional compen- sation only. 100 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. The mean, without. regard to sign, of the anomalies by the new method for the 41 stations in the United States shown in the above table is 0.017 dyne. For the regional compensation Effect of compensation within outer limit of— Anomaly, Anomaly with regional com- Effect of eh ensation within outer topog- method 1 imit of — Number and name of raphy Zone K Zone M Zone O = station and +0.007) compen- (local sai ‘ c en- ac Local | Regional} Local | Regional} Local | Regional entioa) Zone K | Zone M | Zone O Stations in United States 42. Colorado Springs, : Colo. —0. 007 —0: 036 —0. 036 —0. 094 —0. 093 —0. 165 —0. 164 —0. 009 —0. 009 —0.010 —0.010 43. Pikes Peak, Colo. + .187 | — .052 | — .044 | — .118 | — .100 | — .189 | — .172 | + .019 | + .011 | + .006 | + .002 44, Denver, Colo. — .015 | — .026 | — .028 | — .076 | — .085 | — .152 | — .169 | — .018 | — .016 | — .009 | — .001 45. Gunnison, Colo. — .001 | — .041 | — .044 | — .120 | — .128 | —.212 | —.210 | + .018 | + .021 | + .026 | + .016 46. Grand Junction, Colo. — .051 | — .026 | — .028 | — .082 | — .089 | — .156 | — .170 | + .022 | + .024 | + .029 | + .036 48, Pleasant Valley . Junction, Utah + .024 | — .040 | — .041 | — .103 | — .100 | — .171 | — .159 | + .002 | + .003 | — .001 — .010 49. Salt Lake City, Utah — .041 | — .026 | — .028 | — .075 | — .078 | — .137 | — .143 | + .008 | + .010 | + .o11 | + .014 54, San Francisco, Cal. | + .045 - 000 +000 | — .002 | — .003 | + .009 | + .033 | — .025 | — .025 | — .024 | — .049 55. Mt. Hamilton, Cal. | + .120 — .012 — .012 | — .017 — .009 — .018 — .003 — .005 — .005 — .013 — .020 57. Iron River, Mich. + .014 | — .007 | — .008 | — .020 | — .020 | — .031 | — .024 | + .036 | + .037 | + .036 | + .029 58. Ely, Minn. + .008 | — .006 | — .008 | — .018 | — .021 | — .031 | — .029 | + .021 | + .023 | + .024 | + .019 59. Pembina, N. Dak. | — .009 | — .004 | — .004 | — .o11 | —~.012 | — .023 | — .025 + .017 |} + .017 | 4+ .018 | + .019 60. Mitchell, S. Dak. — .006 | — .006 | — .007 | — .016 | — .019 | — .033 | — .037 | — .001 -000 | + .002 | + .003 61. Sweetwater, Tex. + .009 | — .O11 | — .012 | — .028 | — .029 | — .049 | — .049 | — .031 | — .030 | — .030 | — .031 62. Kerrville, Tex. + .013 — .009 | — .010 | — .024 — .025 — .038 — .0382 | + .029 + .030 + .030 + .023 63. El Paso, Tex. + .001 — .020 | — .021 — .054 — .055 — .098 — .104 + .005 + .006 + .006 + .011 64. Nogales, Ariz. + .038 | — .020 | — .020 | — .046 | — .041 | — .076 | — .069 | — .052 | — .052 | — .057 | — .059 65. Yuma, Ariz. — -010 | — .001 | — .001 | — .004 | — .006 | — .012 | — .018 | + .007 | + .007 | + .009 | + .013 66. oes Cal. - 000 +000 | — .00L | — .002 | — .004 | — .011 | — .024 | — .052 | — .051 | — .050 | — .039 67. Goldfield, Nev. + .027 -030 | — .030 | — .077 | — .078 | — .137 | — .141 | — .015 | — .015 | — .014 | — .O11 68. Yavapai, Ariz. + .034 - 030 — .030 — .080 — .080 — .137 — .129 — .001 — .001 — .001 — .009 69. Grand Canyon, Ariz. — .096 | — .028 | — .029 | — .079 | — .080 | — .136 | — .127 | — .o12 | —.o11 | — 011 | — .02 70. pallup N. Mex. + .014 | — .036 | — .036 | — .095 | — .095 | — .163 | — .156 | — .015 | — .915 | — .015 | — :022 71. Las Vegas, N. Mex.; + .017 | — .036 | — .035 | — .094 | — .094 | — 160 | — .150 + .001 -000 | + .001 | — .009 72, Shamrock, Tex. + .007 — .013 — .012 | — .031 — .031 — .055 — .056 + .030 + .029 + .030 + .081 73. Denison, Tex. _ — .001 | — .004 | — .004 | — 010 | — .009 | — .018 | — .017 | + .003 | + .003 + .002 | + .002 74, Minneapolis, Minn. | — .005 | — .004 | — .005 | — .012 | — .013 | — .022 | — . 024 + .057 | + .058 | + .058 | + .059 75, Lead, 8. Dak. + .044 | — .026 | — .027 | — .064 | — .061 | — .102 | — .089 | + .050 | + .051 | + .047 + .037 76. Bismarck, N. Dak. | — .005 — .008 — .009 — .024 — .026 — .044 — .047 - 000 + .001 + .002 | + .003 77. Hinsdale, Mont. — .017 | — .010 | — .012 | — .030 | — .034 | — .058 | — .067 | + .027 | + .029 + .0381 | + .036 78. Sandpoint, Idaho — .044 | — .014 | — .014 | — .045 | — .049 | — .086 | — .095 - 000 -000 | + .004 | + .009 79. Boise, Idaho — .042 | — .016 | — .018 | — .047 | — .051 | — .094 | — .108 | + .006 | + .008 + .010 | + .020 80. Astoria, Oreg. + .008 000 000 | — .002 | — .005 000 | + .008 | — .015 | — .015 | — .012 | — .023 81. Sisson, Cal. + .015 | — .022 | — .026 | — .058 | — .059 | — .096 | — .0s8 | — .012 | ~-008 | — lon | — - 020 82. Rock Springs, Wyo.| — .001 | — .036 | — .034 | — .093 | — .093 | — 169 | — .177 + .011 | + .009 | + .011 | + .019 83. Paxton, Nebr. + .002 014 | — .016 | — .041 — -043 | — .073 | — .077 | — .008 | — .006 | — .006 | — .004 85. North Hero, Vt. — .009 000 | — .001 | — .003 | — .007 | — .012 | — .016 | — .001 -000 | + .003 | + .003 86. Lake Placid, N. Y. | + .032 | — .011 | — .012 | — .024 | — .021 | — .033 | — .020 + .004 | + .005 ! + .001 | — .009 87. Potsdam, N. Y. — .004 | — .002 | — .003 | — .008 | — .010 | — .017 | — .017 | 4 -o19 + .020 | + .021 | + .019 88. Wilson, N. Y — .002 -000 | — .002 | — .003 | — .004 | — .011 — .017 | — .012 | — .010 | — .011 | — .008 89. Alpena, Mich. -000 | — .004 | — .003 | — .010 | — .008 | — .016 | — .016 | — .022 | — 2093 | — -024 | — .022 Mean with regard to sign - 002 < z Mean without re- af + .003 | + .003 | + .001 gard to sign -017 017 -017 -019 Stations not in United States 15. Gormarsrat, Switz- ‘ee sis i erlan +. — .04 — .04 — .099 — .081 — .140 — .0938 |24. @ 16. St, Maurice, Swite- | : se eee | ae ake erlan . = -. — .024 | — .064 | — .069 | — .103 | — .086 e - 005 _ 6 Honolua, Hawai ne - ne + .003 | + .005 | + .008 -014 ian Islands +. -. -. + .011 }/ + 019 | + .072 137 - 0! . = 9. Menne es ae = PME oe Oe ot Set ats ian Islands + -469 | — .089 | — 036 | — .070 | — .036 | -- .020 | + .108 | + .183 | + .170 | + 149 | + .055 Mean with regard to sign 5 Mean without re- ; + ora + .069 | + .058 | + .007 gard to sign .072 - 069 - 058 021 1 See p. 74. 2 See p. 81. the means, without regard to sign, for the anomalies of the same stations are 0.017 dyne, 0.017 dyne, and 0.019 dyne, respectively, for the three cases of areas limited by zones K, M, and O. The mean anomaly, without regard to sign, for these 41 stations in the United States is practically the same for the two methods of distribution of compensation. regard to sign, for the regional compensation only, with zones the same as for the local compensation—that is, 0.017 d sign, for the regional compensation is 0.019 dyne for zone O. The mean, without K and M limiting the area, is yne—while the mean, without regard to EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 101 The means, without regard to sign, of the anomalies for the six stations, Nos. 54, 62, 65, 66, 80, 81, on or near the coast, are as follows: Local compensation, 0.023 dyne; regional com- pensation to zones K, M, and O, 0.023, 0.023, and 0.028 dyne, respectively. The means, without regard to sign, of the anomalies for the 14 stations, Nos. 57, 58, 59, 60, 61, 72, 73, 74, 76, 77, 83, 87, 88, 89, which are in the interior of the United States and not in mountainous regions, are: Local compensation, 0.020 dyne; regional compensation to zones K, M, and O, 0.020, 0.021, and 0.020 dyne, respectively. The means, without regard to sign, of the anomalies for the 21 stations, Nos. 42, 43, 44, 45, 46, 48, 49, 55, 63, 64, 67, 68, 69, 70, 71, 75, 78, 79, 82, 85, 86, in the above table, which are in the mountainous regions, are: Local compensation, 0.013 dyne; regional compensation to zones K, M, and O, 0.013, 0.014, and 0.017 dyne, respectively. The means for the stations in the interior not in mountainous regions show that there are no differences of importance in the four mean anomalies. This is what one would expect with no prominent topographic features near a station, the effect of the compensation being prac- tically the same whether the compensation is local or distributed uniformly over an area of greater extent. The results for the stations at or near the coast and those in mountainous regions show that the mean, without regard to sign, is practically the same for the method of local distribu- tion and for regional distribution with zones K and M limiting the area considered. The mean anomaly for the method of regional distribution, with zone O limiting the area in the case of stations on or near the coast, is 22 per cent larger than the anomaly of the method of local com- pensation. The mean anomaly for the mountain stations in the case of regional distribution to zone O is 31 per cent greater than the anomaly for the local compensation. If the separate anomalies in the United States be compared, it is found that in 16 cases out of 41 the anomaly with local compensation assumed is smaller than with regional compen- sation assumed uniformly distributed to zone K (18.8 kilometers), and only 13 cases in which it is larger. Similarly, there are 20 cases out of 41 in which the anomaly with local compensa- tion is smaller than with regional compensation extending to zone M (58.8 kilometers), and only 15 cases in which it is larger. There are 26 cases out of 41 in which the anomaly with local compensation assumed is smaller than with regional compensation assumed to extend to zone O (166.7 kilometers), and only 12 cases in which it is larger. In all other cases the two anomalies compared are identical to the last decimal place used, the third. The evidence either for or against local compensation in comparison with such regional compensation distributed uniformly over these moderate distances is necessarily slight and possibly inconclusive. For, as shown in the table, the difference between computed effects of compensation in the two cases compared is very small upon an average. The whole evidence is furnished by these very small differences, which are frequently less than the errors of obser- vation and computation. As shown by the table, there is but one station among the 41— namely, No. 43, Pikes Peak—at which the difference between the computed effect of local com- pensation and the computed effect of regional compensation uniformly distributed to zone K exceeds 0.004. Such a difference tends to become greater as the distance over which the regional compensation is supposed to be uniformly distributed is increased, but columns 7 and 8 of the table show that even when the regional compensation 1s assumed to extend to zone O, a distance of 166.7 kilometers from the station, there is only one station among the 41—namely, station No. 54, San Francisco—at which the difference between the computed effect of local compensation and the computed effect of regional compensation exceeds 0.017 dyne. Nevertheless the evidence, slight as it necessarily is, indicates that the assumption of local compensation is nearer the truth than the assumption of regional compensation uniformly distributed to zone K (18.8 kilometers). The evidence is still stronger in the same direction when the comparison is made between local compensation and regional compensation extending uniformly to the greater distances, 58.8 and 1 66.7 kilometers, represented by zones M and O, It is possible that the assumption of regional compensation only, extending uniformly to some distance from the station less than 18.8 kilometers, may be nearer the truth than the 102 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. assumption of local compensation. But it is evident that it would be exceedingly difficult to test this supposition effectively by gravity observations, for the evidence available would neces- sarily consist in general of still smaller differences than the very small differences dealt with above in connection with the comparison of local compensation and regional compensation extending to zone K. It appears to the writers, therefore, that the large amount of labor necessary to extend this investigation to the remaining 48 stations in the United States, or to smaller assumed distances as limits for the assumed regional compensation, would not be justi- fied at this time by the results, as the evidence secured would probably be inconclusive. At some future time, when more evidence is available from additional gravity stations, an exten- sion of the investigation may be advisable. The evidence shown at the bottom of the table from four stations not in the United States is conflicting and inconclusive. In this connection one should consider the peculiar conditions at the two stations on the Hawaiian Islands. These are islands which are evidently of volcanic origin and where the processes of vulcanism are still apparently active. It is stated above, in substance, to be the belief of the writers that the evidence indicates, though it does not prove, that the assumption of local compensation is nearer the truth than the assumption of regional compensation only, distributed uniformly to a distance of 166.7 kilo- meters, or 58.8 kilometers, or even to the small distance 18.8 kilometers from the station. It is also admitted as a possibility that an assumption of regional compensation only, distributed to some still smaller distance from the station, may be nearer the truth than the assumption of local compensation. If the writers stopped their statement of the case here their real views might be misunderstood. It is hoped, therefore, that the following quotations from page 11 of this publication will prevent misunderstanding: “The authors do not believe that any one of these assumptions upon which the computations are based is absolutely accurate.’’ “It is especially improbable that the compensation is complete under each separate small area, under each hill p D , each narrow valley, and each little depression in the sea bottom. It is exceedingly improbable, for example, that as each ton of material is eroded from a land area, carried out of a river mouth, and deposited on the ocean bottom, the corresponding changes of isostatic compensation occur at the same time under the eroded area and under the area of deposition at just such a rate as to keep the compensation complete under each. The authors believe that the assumptions upon which the computations are based are a close approximation to the truth.” The following paragraph,* written before the investigation of this particular question by means of gravity observations was commenced, expresses the belief of the writers of the present publication: “In the above statement that the separate topographic features of the continent are compensated, it is not intended to assert that every minute topographic feature, such, for example, as a hill covering a single square mile, is separately compensated. It is believed that the larger topographic features are compensated. It is an interesting and impor- tant problem for future study to determine the maximum size, in the horizontal sense, which a topographic feature may have and still not have beneath it an approximation to complete isostatic compensation. It is certain from the results of this investigation that the continent as a whole is closely compensated and that areas as large as States are also closely compensated. It is the writer’s belief that each area as large as one degree square is generally largely compensated. The writer predicts that future investigations will show that the maximum horizontal extent which a topographic feature may have and still escape compensation is between one square mile and one square, degree. This prediction is based, in part, upon a consideration of the mechanics of the problem.” It seems’ clear to the writers that if the area taken be sufficiently small immediately sur- rounding a station, the assumption of regional compensation only, uniformly distributed over this area will be nearer the truth than local compensation distributed strictly in accordance with the elevations within an area. It appears, however, from the inconclusive evidence fur- nished by the gravity observations that the radius of this area is probably less than 18.8 kilo- meters, which radius is within the outer limit indicated in the preceding paragraph. It also appears that the gravity observations will probably not yield conclusive evidence as to which hypothesis is nearer the truth for still smaller areas since the differences between the effects according to the two hypotheses applied to these very small areas are so minute as to be very difficult to observe. * From p. 169 of The Figure of the Earth and Isostasy from Measurements in the United States. EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. 103 TEST OF DEPTH OF COMPENSATION. In this investigation, as stated on page 10, the isostatic compensation has been assumed. to be complete and uniformly distributed to the depth of 113.7 kilometers. This was the most probable value of the depth of compensation available at the time the investigation was com- menced. This depth had been obtained from investigations based entirely upon observed deflections of the vertical in the United States. Later portions of those investigations have shown that the most probable value now available for the depth of compensation is 122 kilo- meters.* It is evidently desirable, before concluding the present investigation, to ascertain whether it is possible to determine the depth of compensation from the gravity observations with as great accuracy as it has already been determined from the observed deflections of the vertical, and whether numerical corrections of importance would result from changing the assumed depth. from 113.7 to 122 kilometers. Accordingly, the approximate test here reported upon was made to settle these two questions. For the assumed depth of compensation, 85.3 kilometers, the values of Ez were computed for a few values of 6 (0 being the distance from the station expressed in angular measure) by the methods and formule set forth on pages 23 and 24. Each of these values was compared with the corresponding values, as shown on page 25, computed for the assumed depth of compensation, 113.7 kilometers. The comparisons indicated that the reduction in Ey caused by changing the assumed depth from 113.7 to 85.3 kilometers, if expressed as a percentage, varied but little from zone to zone among the numbered zones. Accordingly, a few computations only, made it possible to construct the part of the table shown below which refers to numbered zones. Similarly, the effect of compensation alone was computed for some of the lettered zones on the assumption that the depth of compensation is 85.3. It appeared that the change of the assumed depth from 113.7 to 85.3 reduced the computed effect of compensation by amounts which, expressed as a percentage, were practically constant (at 33 per cent) from zones A to zone F, and beyond that point changed in a regular manner, as shown in the first part of the table printed below. Percentage of change in compensation and in Ep when the assumed depth of compensation is : dane from 118.7 to 85.3 kilometers. Zone i ag Zone E, A +33 18 —-17 B +33 17 —18 Cc +33 16 —-19 D +33 15 —21 E +33 14 —22 F +33 13 —23 G +32 12 —24 H +32 11 —24 I +31 10 —24 J +29 9 —24 K 427 8 —25 G 123 7 —25 M +14 6 —25 N +03 5 —25 O -ll 4 —25 3 —25 2 —25 1 —25 By use of this table the changes shown in the following table for 10 stations in the United States and 1 in the Hawaiian Islands were computed. In making the special investigations stated under the heading, “Regional versus local distribution of compensation,” the effect of * Supplementary Investigation of the Figure of the Earth, p. 77. 104 EFFECT OF TOPOGRAPHY AND ISOSTATIC COMPENSATION ON GRAVITY. compensation alone for each lettered zone had already been computed for certain stations, including the 11 used in the present test. Hence, for these zones the required change, as shown below, was obtained at once by multiplying the effect of compensation for a given zone by the percentage shown in the preceding table for that zone. For each numbered zone at a station the total correction for that zone, as shown in the tables on pages 54-58 and 84, was multiplied by the percentage of reduction in Ez for the zone, as shown in the above table, the total correction for the zone being sensibly proportional to Ep. Changes in computed correction for topography and compensation produced by changing the assumed depth of compensation from 113.7 to 85.3 kilometers. [All tabular values are in units of the fourth decimal place in dynes.] é i Mauna ount |Salt Lake} Lake Tron Zone | Fembina, Caren, | ‘Peak himseken, Be eek ee Junction, [Hamilton,| City, | Placid, | River, : No. 69 | No. 43 | No. 54 Tehanas.. o: 0. No. 55 | No. 49 | No.86 | No. 57 A 0 0 0 0 0 0 0 0 0 0 0 B 0 0 — 2 0 -—1 0 0 0 0 0 0 Cc 0 0 — 2 0 — 3 -— 1 - 1 0 —1 0 0 D 0 —2 -— 5 0 — 6 — 2 — 2 —2 — 2 0 0 E 0 — 3 — 8 0 — 8 — 5 — 3 -3 -— 3 —1 0 F 0 — 3 —12 0 —13 —7 — 3 —3 — 3 — 2 0 G 0 — 5 —13 0 —16 — 8 — 5 —4 — 5 —4 0 H 0 — 8 —18 0 —18 —10 -—7 —5 —7 —2 0 I 0 —15 —28 0 —30 —19 -13 —6 —13 — 6 — 6 J =p —18 —28 0 —23 —23 -15 —6 -17 —7 —5 K ay —29 —38 0 —30 —32 —26 —8 —25 -11 —10. L —5 —34 —42 —-2 —26 —39 —36 —4 —32 —12 —10 M -—7 —50 —60 3 -—15 —58 —56 —3 —49 —10 —12 N 0 0 -1 0 0 — 1 -—i] 0 -—i1 0 0 oO +7 +29 +40 —il —37 +35 +38 0 +33 +5 + 6 18 +2 +10 +12 -—4 -—12 +11 +13 0 +11 + 2 +1 17 +2 +10 +12 —4 —14 +11 +14 0 +12 + 2 +1 16 +2 +9 +13 —4 —16 +12 +14 0 +12 + 2 + 1 15 +3 +10 +13 -—4 -17 +14 +14 0 +14 +2. +1 14 +3 +11 +13 —4 —-18 +14 +14 —2 +14 “+ 2 + 2 13 +6 +19 +19 — 6 —36 +21 +23 —5 +25 + 3 + 4 12 +3 +13 +12 — 6 —24 +11 +13 —5 +15 +1 + 3 11 +3 + 6 +7 —5 —20 + 8 + 8 —5 +9 -— 1 + 2 10 +3 +1 + 4 - 3 -—15 +2 + 4 —4 + 3 — 3 + 2 9 +2 — i] 0 — 2 -—9 -—1 -—1 —2 -— 1 —2 0 8 +1 — 3 — 2 —4 —10 — 3 — 3 —4 — 3 — 3 0 7 -l1 — 2 — 2 —2 — 5 — 2 — 2 —2 — 2 —2 -i1 6 —2 — 2 — 2 — 2 -— 5 — 2 —2 -2 — 2 — 2 —2 5 -3 — 2 —2 — 2 —2 —2 — 2 —2 — 2 — 2 —2 4 —2 — 2 — 2 — 2 -1 — 2 —2 —2 — 2 — 2 —il 3 —-l1 — 2 -1 -1 -i1 -— 1 —]) —-1 -1 -1 - 1 2 —l -—i1 -1 -1 -—1 -i1 -1 —1 -— 1 -—1 -i1 1 0 0 0 0 0 0 0 0 0 0 0 Total.| -5 —§4 | —124 | —72 | —432 | =80 | —a7 | 2s poe a 4 \ ~ 77 ; 59, vy mtg SE : ts 7 \ 3 | ¢ pia ~Pentbina Ss Syst \ } i : 1 u- ey e ie ‘ : 45° i nee ee : \ * : M 7 at : Porgy ‘ e ; O N 4 58 Ae 4 e c ad K hi oO Nw rt s T A nN 6 es LA : { pean th ‘ / seit \ elena N A eH DAK p i 4 ' aes Mw se \" / { Butte a is \ / ‘ el : Bism | ee A Se 3 ‘oe Sy i e & AGEN | a O ie o ; | , Hero i Ye eee | hs = pot & G / ‘ Hunt] iy .. | Mais “Jiron River 4 ais fl i 8 ; eee O 4 \ | oe : 4 sara \hee N porsdamt z < / eee fs he rete Pee a RRS Oke gL eaT CNG Oh erm tpn ee ed! a \ Be Ee 86 Aier ; 0 SS =S SS tee es Pashia | itn fe a” . oe as a : ; 89 ee ee Lave Pie : ees rae errs er Basi oO 4 m \ 79 ; Rae iS $9 ae | QO iv ” ‘ ee Sige Cit / ae ‘Ser B f or My ei vs 7s a ty \ \w * Sones | SuPhil fo) ¥ * ay? \ oy : i SPuTIH Dato : : | ae l= VT softens e Lead oe Fe | Lz e L K | spon! 28. 3 — ison Cee Ags? ” / | ‘ s | are” Ss Sissoy, ° Ww a ‘ ! ‘ si eT 2 - oo \ TQ Ocatelly ire M itehell | Be ae ah) > | “atthe wits “LP N C mula Sees A Ee ct Vc | Madison Z eLansin€é \ \ \ o 8 ont = { x - \ . me Crawford I ee ‘ € a: Ye ie | i. : Loe tL WGrercanal A ee , Roto ck Springs 1 | 0 LL Kune aT ae : \ yia N By wc | SaldLake oy c I Chicago % i \ nS ie a ie > 3 4 - , E hae f | , N E R A ie K A q Des Moinese \ es \ \ : ey Ps 83 \ . Hi arrisbU aor, Cars N “Che = $Paxton ve \ a > ] ‘ \ “Wentge ( —y City vi A i \ 0 \ Ue Meee 4 ty \, A | i Linchlne ne Oo i Hl \ aoe ae 23 \ catia, shes tak na aes sek | Pe ee ; . l— . a easant alley- Junetion - \ cy \ oiumbus \2 Ty ae D Tancis - A : SS : fess vey Ny \ e: ae a 2 ‘ c a H nver, Bel aa % e Wi ming” ures BS y E v \ Lh s x 3 een River} go ‘B ee C . ; , 5 7 Ay © q os oO terre Haute \ Gneimai ~ \ / =p ° Golderg ee Te RISA 0 aa] a ee i ; ic i Pikes Peak ¢ ce ie Topeka, ‘ansas Ci ng i om x / Chaxioresvile ‘ unnison i$ 4, & en 5 . S o eee - 9 if | swor | —o ‘ St, Louis oFrankfo : ‘ 3a . ‘, RON Wig A eg oe : aes ‘ | oO . - G me : \ \ t ’ vf CAN ES 38° ; : c Cc Ma age Lo ate i | s EN ON dee : Ppt Ne) ON A aia tae s 2 ae | OAS OE LD manent Se * Ye by eens a / Nah Savoury ( 9 \ e [ Can on T pie a Rele he \ N A J ) De ae ii ree ees VBE) Scien ee ie ca ae Beets UP MRR eC GR AS TAN E DPntn VIL ce AAs IRR Spe lee eI ae Pet ieee i cee ee mae TARR oc Oh) bi aed en Medea ore R G i 70 hea a ate eNashvill ; N Ones “ ae of; $Galhup Santa F, my 71 \ : ‘3 N e Ss S E: c Ue St : \ Bhs ce : Las Vegas ele 7 M4 Columba : ae = sak 7 “pease Ny oP rege, o El Reno : Meee ae! ‘ y A Lg Albujruer 72 © K je \ ; S »~ dQ Rs que Le A Ie uO Lac y : < N A N exist ’ oe Ee 2 Pr \ O F BW M/E x tel y—7 $i Aledter AeoR A Sh As \ ; oO ciysbie ‘ Little Roc . : \ : * | ee Dee er a hit] oe aetna eas 65 x | = h ta N A oe aoe \ Sa! Pe | ees m5 ' is iP ston '~ Denison Pee \ Oo Cusrle cy . i | : ie ie : ae: 6I é . | oa gomey © a -- 64. i Sweetwaterg Fott Worth 6 —. bray ile ; hm 4 ft yee . El Paso ts Jackson | aces (oH ! ra | : ° oeles Rae inched 1k | o : , i “ Md 2 — s \ F [ ereltaneseee ? n Rouge i te A Austin!0 Vv Mine 4 T Y @ a re Trville N A 5 7 ee Kealashiesle a ih 7 a KI | 4 New Orle: een © / Ve : , us yp ee oO \ Galveston i Jf Bets , Renae a i o z wo M y : B e s \ * A aN io FG XS Xe & signs 3 ~. el /o _ 2 4 4 ta. So pak ¥ “Wes palm Beach + L . as | | 3 “ Bs e . | Weer eee i : 2M i : eee 4 \ * ; ass i. < a ‘ ie ei - y f ~ Point Isabel - oc i pee: > | | a we . - : me Oo ee Sa a % Q 0 \ | ae qT. Cc O > - : ao oe 7 GC ir L fF ry ; YY _ *, aan y ipl : : as 7 ; oh | A ae | C U | sf 3 = eae ee . U.S.Coast and Geodetic Survey : ahew® = . 8 O.H.Tittmann, Superintendent % -_——~ ec BASE MAP OF THE UNITED SrArTt> | \ . (Projected on intersecting cone) “ ‘a “ = mn # : Scale |: 7000000 pe X ae =_ \, ta \. a 5 \ \ \ Kingst mais \ , ce ae Kilometers \ 100 100 200 300 400 500 600 700 0 or ae ee! oF , - he lee Bene ee ae Ae | é a 4 xis 0° 105° 100° West from ete a sah = 18° ILLUSTRATION No. 13.—MAp SHOWING LOCATION OF GRAVITY STATIONS USED IN THE INVESTIGATION. PRINTED BY THE U.S.GEGLOGICAL SURVEY a WIRE NAHAS HE PRN AN ce ee ee an en ee eee ae —aaaeenanennnannianetaaat < : 9 aR pte w ong aie cee eeenlatiee Ds Apc yebhaatiak Bae : vein *e whe Rais 20° 9) West from Greenwich - OA 125° 120° ase 80° . 8s 1s 70 \* \ \ 4 | ns | \ \ \ e ‘a ve \ + N 0 . \ 4! | Je m a tg . \ 28 < of” 2 20 = Ps @ Q| b | = Z eLangirs_,, 40° x ; < jaen deerme eer a4 ~ | Pei queeNr es eo eh ee Perr [Saag Boe? ' BiG \ s Moinese | \ me ve . N i \ Ne | | ; : ae ; ° - | ay \ olumbus \ : elndianapol G eee Spfingfiqide \ i 1 pe e ° j Qoi \ son a 3/39 S i Bea ws ag fferson Ci Ey 7 oFrankfo A\ / ° : 00 y 3s er oi eee “ ihe veri me RE MEN c eNashvill \ c ‘S : Ine ee . A ae ot hd Oe oe \ | | +.024 aan ee SNe \ 7240: OOK : Ss x Jerico \ 2 ~ fe ~ a | ; 4 oh ie \ ‘ ~20 \ | > . / N = \ 15 702 ‘ _ \ O29 | ; p 20 | Q i yo eMontgorm 30 Jackson i. i 9 a a \ ot \ icon beagee sioree Nore ee —— \ Ler ‘ ' i O m Ro: a x — “12 & a 2 5 =015 f : 7 ri 20 XN 7 & Kd Rel & | ‘ es q 8 +016 st ‘ 25) We a, gq ¢ gi 4 | py. ; | Ny CS g — “ | Be | \ ee : oy : ~. . ce 0 | 7 {i ‘| 4 4006 ay ‘ne a ———| SOCAN es F A | ‘ ae ae | | i Gree | = | 4 ¥ =», - Ngee eh me “ a - boa te U.S.Coast and Geodetic Survey ‘ eres _ O.H.Tittmann, Superintendent ie -—> . Sate 9 BASE MAP OF THE UNITED: STATES | (Projected on intersecting cone) ao 3 2 Scale |: 7000000 ma 4 — TY Statute Miles 5 Laspehonrmts i ll lll SEES nen 50 ° 100 200 300 0 Gee Kilometers a Pos x —_ SSS do 80 100 200 baa 400 500 600 et —<# \ \ oo ; | ; ke / | OTT has ee A | ee hat | oe i \ us 110° 1055 100 S° 90 8s 80° Ss ILLUSTRATION No. 16.—LINES OF EQUAL ANOMALY FOR NEW METHOD OF REDUCTION. PRINTED BY THE U.S.GEOLOGICAL SURVEY 40° 30° 2s 20° re 125° : 120° us 110° 105° 100° 95° 90° 80° 1S 70° VA T T oP EN. ae / | | \ \ ; ‘ | f Th i ots | ° es ; | | : 2 Oso ” : yi 78-1 : : ( on * I 0 N| ee oxy : } = cf / \ $0 x9 NS iy “4 J / \ A 7014, : a OF L is \ 2 eA os o “\obs / / ; He in ) ~So Si ) -36 i M Ne oe ow: ) a © . . P, \ oO ? t ena ay : N . ‘ Ser ack Re oe \ Ai? : Qs ZO . : N U ~ OM oo | eleng A . ae DjA K be A 4 229 AW ye :\- J ee , » oh pe SS ( site 76 A 7 ee ¥ Noe ae . - & ; 6 uh Oe oO . as : fs = poge® r \ . se ceed ¢ | 00 ~-—- 1 @ 57-008 om \ \ 7-008 rg need : SS e : i A ‘5 mens cam a pS V4 ee io e Y 034 °° ° 4s + ee / 7 a oO 82.¢ \ ° “a oo < o 22 % 4 ee go" “ ; DB oj, 0 \ Cf 2 74 ie < a 7, \ e bom = : i a Ne : St.Paul | oO ws z ral i / \ om x N va 20,40) 007 hy20 75 s OTH AK @°s Z E a « oy 9 *. 89 Pierve® ' uw 2 0 | A PRG 238 | t e-On 200% s a oe 2 Ni L i ol N y : | ae 3 i Pe 2 ! \ i : a ; As BAS *Wocatelia Ql\ uw ‘ Qos7 i | i \ ar a me Z 050 = mm Cos — ays eae : . omen + meee meee cen mee meme ee i.e: ~ = | adisong’s7-pat eLansing: \ y — a 4 ae 8° } x \ t f \ » a. A, Z Be Cr ord “Seo a \ - ae \ a Be fi EO TES s ; 030 So -40 xD 33 ‘ eS xe 160 8 . N be | eae esl a ra i \ V/A ‘ ) Ye 7268 5 a, 36 --02! S AS ° “ Sella t; : ‘7 NX \ \ : N L a3 ' 49-1 So N 3 R IA \ es Moinese \ i \ \ asburg? 4.020 j °Che rc . = 40 } > - "a , : \ | ) nic; V : SG 430 ; \ gi hy tg to) i 17 A nD is £2 c \ \ \ iO fs oa! Tete Bs ay Oo 1 \ Hi r ae _— A 7 ie +8 < ‘ \ eer, L-: a og : 7 " +0050 oe U re — \ slambus 44 \ , , a i . al Comey A 3 550 Yo Springfield e ' : rOlg s -\g0 : M6175 ey Oo | a3. J -50 a r\ Q ‘ oO mk : . : a i ei \ 3 % / \ D : O q a ee 4 ie / ‘cease YW © 8 Chaxlesto™ X Y a 4 wy 6 ee 38.931 ,, eFrankfo ae / Sa ‘id ? 3 ic 7 ‘ So rr N > : j - ¥. fo . Shee 38° be ‘i <<) aa oO 2 Ne ae \ *, Go K FE Ki e Sh) ile ee alae - fs ae 69-19 Dp pores Coa i No) 2 ne te e : . ~ = Ta natn Ps : ) S 3 PE up ee PURE Leg Ap gen: pie ele ashen Mita tt Co tMNC Ls OB Pe tn alaiieab ae gistres daint ed lamem aro ian begat t iS SC Vet ai |e ete et Ke _ ay iy 7\" 320, eNaahvt E N R Ne E 1 ~id 70 iS S$ : eS ‘ Q228 anta Fee ents Zi xy o ‘0 \ ane Le 0 aot: =T0 NS aa iS 4 =O « Albu a iS ‘a S NS y f 3 : : ee N E W. 190 | a okambie, ~ho E | | ¢ a : o -180 | Qo / od : N\ 30 Shae ae ee peeare = tlantak nee 65f.0i = 179 c 15-8 ] 6 om - 160 : \ | | SS 150 Pe LN SB o oM tgomeny 30° ; 9 sl s \ Ae | “Nios = ! fear Sel38 peree ry, eo eto 4g ea piasll ttn IE ; s | 3 a it 7 g i be ee re ae. ome te —— Cae 3 my ; > : LN y L \ oO e smn, Bande ace y * Y orale Lie . ¥F4.006 o i 5-009 i > a Ga \ : 3 \ é ‘ 4 ss ? . 4) ’ x ay Z y A B - : <4 9 4 ‘ e g | 4 . Bf & = x02! 240h0 og 6 a _ 7 “ 25° | a eh a) a Lf ‘ ’ fi . a 4 i asso a a = , . “ ts os 7 a ne "n eo eter att > ‘ Z OS Mee ae Vout nis ees N b C O i bee ay \ M * ‘ oe 5 | 7 Pee er F St = Pita i Me fe = <0 sl i. ; cn i: SINE Se . a loys A ‘ ( U Aes. hi Pat af ay : ae r i Neer U.S.Coast and Geodetic Survey Ps ( ‘ AG Be \ : ° oe pee O.H.Tittmann, Superintendent Oy ts S, ‘7 ee : . od BASE MAPOOFR THE tate D Slates ribuna ’ (Projected on intersecting cone) , .: 4 ee | A: ; 8 oO | Scale | ‘ 7000 000 N | | \ ie \ a ‘ | : ek a : — 2 wan gst \ Sal ; Statute Miles _——— Mexicoe \ \ a a Es a = rent pel ear | Kilometer . ie glare cara i 10 ee a ~ 600 700 Ae ~— 4 ed A Gee ee | ee us’ 110° ‘ 108° 100° ee bg CaN 90° 385° 80° qe° ILLUSTRATION No. 17.—LINES OF EQUAL ANOMALY FOR BOUGUER METHOD OF REDUCTION. PRINTED BY THE U.S.GEOLOGICAL SURVEY eee ae ee \ ES Pee harp tod tack * Wows, foros: 4 1S 80° 8s 90° 95° 160° 105° 120° 125° ——S SS eS ° 100 Pace acer ey S50 ° ; - é ? ape ae a ee i ORR, Goes k eee 3 oes a8 8 Ba ae : q —_ . & 1 a = er s\a\\ : ie ae a Pn ro 7 By ” es > 3 F| = 6°. fe Sag et 5 : 2 S 8 a fie ¥ _ a gi@ 33 =I rd : | aa eae eee ee < Seer nye ° . ° oO —— Nei € bd < : ! wo 9g i e 3 2 ; § | | | ; once Wad | | 4 a | * ® = oO Pa * | ces eae 5 i Bre e et E ‘ O fe = 2 o 2 Soe | & #3 FO O Pp ae ae q 4) a o #5 ss | ome O g ° é ey go oe oo ae ROA vf | a hos ig a ty y g 2 fly { 3 WY —! LJ be =< g 48 | he o > on | y 2 = i) =~ ° | Os 14.02 ere & fe) we, Be ie » e = c Se wm 2 a oO = 4 = CS = m2 © ee = So Be oF eae ol» o > o CS = i O22 62° =} 2 i LJ 2 Ct © = 3 £ = = ss = g 2 cee oe 4s a OP re oH “He Se ee ee ae ee : as a o — ¥ S. = B a 8 See : = =F ‘ LJ WY) ¢ mm ILLUSTRATION No. 18.—LINES OF EQUAL ANOMALY FOR FREE-AIR METHOD OF REDUCTION. ee ee a ere Ae a ne ae Oe RP REED Pe ED hmmm AE Mee iene n No.3. 7 | | we 120° Us" \10° i i‘ i | i | All stations and areas of excessive and defective density. ee SEY a ae maine woyfof O00! a } Oo 3 Scale of Seconds for Residuals > to WS° 0° Pe ee peo oy seers OS See Soa ees ee | 80° ILLUSTRATION No. 19.—ILLUSTRATION FROM SUPPLEMENTARY INVESTIGATION IN 1909 OF THE FIGURE OF THE EARTH AND ISOSTASY, SHOWING RESIDUALS OF SOLUTION H, ALL STATIONS, WITH AREAS OF EXCESSIVE AND DEFECTIVE DENSITY, AND SHOWING ALSO ALL GRAVITY STATIONS WITH NEW-METHOD ANOMALIES. Cornell University Library QB 331.U58 TTT 1924 006 300 820 oi aincves iid ae —— ST ee . fi t cae peer eee te te cote h tte BAG ee es