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\ 
 
 AN ATTEMPT 
 TO TEST THE THEOKIES OF CAPILLARY ACTION. 
 
ionlinn: C. J. CLAY, M.A. & SON, 
 
 CAMBRIDGE UNIVEESITY PBESS WAREHOUSE, 
 
 17, Pateknostee Kow. 
 
 CAMBRIDGE: DEIGHTON, BELL, AND CO. 
 LEIPZIG: F. A. BROCKHAUS. 
 
AN ATTEMPT 
 
 TO TEST 
 
 THE THEORIES OF CAPILLAEY ACTION 
 
 BY COMPAEING 
 
 THE THEORETICAL AND MEASURED FORMS 
 OF DROPS OF FLUID, 
 
 FEANCIS BASHFORTH, B.D. 
 
 LATE PEOPESSOK OP APPLIED MATHEMATICS TO THE ADVANCED CLASS 
 
 OP KOTAL ARTILLERY OFFICERS, WOOLWICH, 
 
 AND FORMERLY FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE. 
 
 WITH 
 
 AN EXPLANATION OF THE METHOD OF INTEGRATION 
 EMPLOYED IN CONSTRUCTING THE TABLES WHICH GIVE THE THEORETICAL 
 
 FORMS OF SUCH DROPS, 
 
 BY 
 
 J. C. ADAMS, M.A, F.RS. 
 
 FELLOW OF PEMBROKE COLLEGE, AND LOWNDEAN PROFESSOR OF ASTRONOMY AND GEOMETRY 
 
 IN THE UNIVERSITY OF CAMBRIDGE. 
 
 AT THE UNIVERSITY PRESS. 
 1883 
 

 A. 'S^joQ 
 
 UNIVERSITY 
 LjBRARY,/' 
 
 TAe writers take this opportunity of returning their thanks to the Syndics of the 
 University Press for undertaking the expense of printing this work. 
 
INTKODUCTION. 
 
 Many years have elapsed since this work was commenced, and it is even now only 
 partially completed. My object was to test the received theories of Capillary Action, 
 and through them the assumed laws of molecular attraction, on which they are 
 founded. To this end it was proposed to compare the actual forms of drops of 
 fluid resting on horizontal planes they do not wet, with their theoretical forms. 
 
 After some trials a satisfactory micrometrical instrument was constructed for the 
 measurement of the forms of drops of fluid, but my attempts to calculate their 
 forms as surfaces of double curvature failed entirely, and my undertaking must have 
 ended here, if I had depended upon my own resources. But at this point Professor 
 J. C. Adams furnished me with a perfectly satisfactory method of calculating by 
 quadratures the exact theoretical forms of drops of fluids from the Differential 
 Equation of Laplace, an account of which he has now had the kindness to prepare 
 for publication. After the calculation of a few forms, application was made to the 
 Royal Society for assistance from the Government grant in making the needful 
 calculations. The following extracts from the application (Oct. 27, 1855) will explain 
 the objects of the undertaking. " I have carefully examined all the published 
 " experiments that I could meet with, but these have been generally made with 
 "capiUary tubes, and in consequence of the diflSculties inherent in this mode of 
 "observation they have not led to consistent and satisfactory results. 
 
 " It appeared to me that the best test of theory would be obtained by making 
 "careful measures of the forms assumed by drops of fluid resting on horizontal 
 "planes of various solids " 
 
 "At first I knew of no better mode of arriving at the theoretical forms than 
 " that given by geometrical construction, but I am indebted to Mr Adams for a 
 "method of treating the differential equation 
 
 ddz \ dz 
 du" , u du „ _2 
 ■ + '■ d^ " ^''^ - b ' 
 
 hsr (^-£/ 
 
 ' when put under the form - + - sin ^ = 2 + 2ab' 7 = 2 + /8 r , 
 
2 INTRODUCTION. 
 
 "which gives the theoretical form of tha drop with an accuracy exceeding that of 
 
 "the most refined measurements. Values of r, y and ^ have been calculated by this 
 
 boo 
 
 "method for values of <f> at intervals of 2|° to 5°, from tj} = to ^ = 145°, for 
 
 "values of j8 equal to ^, ^, 1, 3, 6, 10 and 16. It is however very desirable that 
 
 "calculations should be made for more numerous as well as for larger values of /8. 
 
 "I also propose to make accurate measurements of the forms of the common 
 "surfaces of two fluids that do not mix. The form of a drop of fluid (A) will 
 "be taken when immersed in a fluid (B), and also the form of a drop of the fluid 
 " {B) when immersed in fluid (A), and for this purpose a plate-glass cell has been 
 "constructed, so that the observations can be made whether the drops rest on the 
 "bottom, or float in contact with the upper surface. The forms of drops of fluids 
 "(A) and (B) will also be taken when resting on horizontal planes surrounded by the 
 " atmosphere." 
 
 "The objects of the experiments are 
 
 "I. To compare the actual forms assumed by drops of fluid when resting on 
 "horizontal planes composed of substances which they do not wet, with their theo- 
 "retical forms. 
 
 "IL To dotermine the effects of supporting planes composed of various sub- 
 " stances. 
 
 "III. To examine the effects of different degrees of roughness of the supporting 
 "planes composed of various substances. 
 
 " IV. To determine the effects of vaifiations of temperature on the forms of the 
 "drops of fluid from 32" to about 200" F. 
 
 "V. To examine the mutual action of two fluids that do not mix, and the 
 " effects of variation of temperature on them." 
 
 The Eoyal Society voted a grant of £50, the sum applied for. These calcula- 
 tions were completed in 1857. And after the calculation of the theoretical forms 
 and volumes of sessile drops had been carried as far as seemed needful, the money 
 in hand was applied to the calculation of theoretical forms and volumes of pendent 
 drops of fluids. The results of these calculations have been printed in Table IV. 
 
 The delay in the publication of my results has arisen from the interruption of 
 my labours, caused first by my removal in 1857 from College to a country living, 
 and secondly by my appointment in 1864 to the Professorship of Applied Mathematics 
 to the Advanced Class of Royal Artillery Officers, Woolwich. As no systematic ex- 
 periments had then been made since the time of Button to determine the Resistance 
 of the Air to the motion of projectiles, and those for round shot only, I was induced 
 to turn my attention to the subject of Ballistics. The Results of my Experiments have 
 been published under the authority of the Secretary of State for War, as follows— 
 
 I. Reports on Experiments made with the Bashforth Chronograph to determine 
 the Resistance of the Air to the Motion of Projectiles, 1865—1870. London, 
 W. Clowes and Sons, &c. &c. 
 
INTRODUCTION. d 
 
 II. Final Report on Experiments, &c. &c., 1878 — 80. London, W. Clowes and 
 Sons, &c. &c. 
 
 And in connection with these Eeports I published a Mathematical Treatise on 
 the Motion of Projectiles, 1873, and a Supplement to that work, 1881. 
 
 Immediately after the completion of these labours I turned my attention to the 
 preparation for publication of a part of my work on Capillary Action, for I cannot 
 now hope to be able to complete the work originally proposed. The Tables II. and III. 
 appear to give all that is required in order to supply the means for filling up the 
 intervals to five places of decimals for all values of ^8 under 100, and of (f> under 
 180°. The Table IV. for negative values of ^, although not so complete, will afford 
 considerable assistance, and the deficiencies can be easily supplied by original calcu- 
 lation preparatory to interpolation. 
 
 Table V. gives the theoretical forms of free capillary surfaces of revolution about 
 a vertical axis, which was used in calculating the forms of drops of mercury shewn 
 in the diagrams. Deficiencies may be easily suppUed by the help of Table II. by 
 interpolation. 
 
 As a specimen of the work I proposed to do, I have given diagrams and co- 
 ordinates observed and calculated of forms of drops of mercury carefully measured in 
 1863. These shew how correctly the calculated and measured forms of these drops 
 agree, notwithstanding the very considerable variation in their outlines. 
 
 Also, as I found my measuring instrument in good working order in 1882, I 
 have made numerous measurements of drops of the same kind of mercury of 4, 8, 12, 
 16, 20 and 24grs. in order to find the values of a and eo. The values derived 
 from each particular measurement vary considerably — ^but the mean results for each 
 weight of drop are satisfactory and appear to confirm the received theories of Capil- 
 lary Action. Bat as the Theories of Young, Laplace, Gauss and Poisson lead to the 
 same differential equation, and therefore give the same form of drops of fluid, experi- 
 ments of this kind are not capable of deciding whether Poisson is correct in sup- 
 posing that a rapid change of density takes place near the free surfaces of fluids. 
 But more definite information on this head may be expected when the values of 
 a and w at the common surfaces of fluids which do not mix, as well as the efi'ect 
 pf variation of temperature on these quantities, have been determinqd according to 
 ihe original scheme. 
 
 Having given examples of the work I proposed to myself in the first instance, I 
 must leave to others the further examination of this important question, for it still 
 appears to me that this is the only way by which we can arrive at any definite 
 results. 
 
 I take this opportunity to return my best thanks to the Syndics of the 
 University Press for having undertaken the publication of this work. 
 
 1—2 
 
CHAPTER I. 
 
 THEORETICAL EXPLANATIONS OE CAPILLARY ACTION. 
 
 The phenomena which arise from Capillary Action seem to contradict the laws 
 of fluid equilibrium. In consequence, many worthless theories have been proposed 
 with a view to explain apparent anomalies. After long groping in the dark, it was 
 found to be desirable to discover by experiment what were the actual phenomena 
 which required explanation. Hawksbee* found that the height to which a fluid would 
 rise in a capillary tube of given radius was the same for all . thicknesses of the tube. 
 From this it was apparent that the attracting force of the tube was situated at or 
 near the inner surface of the tube. But he does not appear to have taken account 
 of the mutual attractions of the particles of the fluid. Jurin '* also found that the 
 height of the column of fluid supported by capillary action depended solely upon the 
 interior diameter of the tube at the upper surface of the fluid. From this he con- 
 cluded that the column of fluid raised' by Capillary Action was supported by the 
 attraction of the periphery or section of the tube to which the upper surface of the 
 fluid cohered or was contiguous. 
 
 Clairaut" was the first to attempt to explain capillary phenomena on right prin- 
 ciples, by referring them to the mutual attraction of the particles of the fluid, and to 
 the attraction of the particles of the solid on the particles of the fluid; and sup- 
 posing these attractions to depend upon the same function of the distance, he 
 concludes that even if the attraction of the capillary tube be of a less intensity than 
 that of the water, provided the intensity of the latter attraction be not twice as 
 great as that of the former, the water will still rise in the tube (p. 121). Clairaut 
 supposed that the attraction was sensible only at very small distances (p. 113). 
 
 Shortly afterwards Segnef* introduced the supposition that forces of attraction of 
 both the particles of the solid and of the fluid vanished at sensible distances. He 
 concluded that these forces gave' a constant tension to the free capillaiy surfaceis, and 
 
 • Phil. Tram., 1711 and 1712. i Commentarii Soc. Reg. Sei. Gottingensis. T 1 
 b Old. 1718 and 1719. 1751. 
 
 • ThSorie de la Figwre de la Terre, 1743. Chapitre z. 
 
THEOEETICAL EXPLANATIONS OF CAPILLARY ACTION. 5 
 
 thence he tried to calculate the forms of sessile drops of fluid with a view to com- 
 pare them with their measured forms. But in his calculations he took into account 
 only the curvature of the vertical sections made by a plane passing through the axis 
 of the drop. His measurements of the actual forms appear not to have been very 
 precise. 
 
 An important paper on the Cohesion of Fluids was read before the Koyal Society 
 by Dr Young* in which he pointed out the necessity of taking into account the 
 curvatures of both of the principal sections of the drop, and clearly propounded the 
 true principles on which the solution of the problem must depend. He arrived at 
 the conclusions (1) that the tension of a free capillary surface would be constant, and (2) 
 that the angle of contact between a given solid and fluid surface would also be con- 
 stant. He attempted to derive these hypotheses from physical considerations, but it 
 is not easy to follow his reasoning. Even the editor of his works. Dean Peacock, 
 observes on his Analysis of the Simplest Forms that "In th.e original Essay, the 
 "mathematical form of this investigation and the figures were suppressed, the reasoning 
 "and the results to which it leads being expressed in ordinary language : even in its 
 "altered form the investigation is unduly concise and obscure'"". And respecting the 
 appropriate angle of contact. Young confesses that "the whole of this reasoning on the 
 "attraction of solids is to be considered rather as an approximation than as a strict 
 "demonstration"". This may in part be urged as a reason why Laplace ** did not 
 more fully recognise the value of Young's labours. And although many of their 
 results agreed, the processes by which they arrived at them were very different, except 
 that they were much on a par in respect to the constaaicy of the angle of contact, 
 which Laplace did not deduce mathematically from his theory. Very good accounts 
 of Laplace's Theory were given by Petit ^ and Pessuti?, while it was attacked by 
 others, as Young ^, Brunacci'', Poisson* and others. 
 
 Gauss'' by a new and striking mathematical investigation obtained the same 
 differential equation to the form of capillary surfaces as La-place had done, and also 
 supplied the defect of his work by obtaining an expression for the angle of contact 
 of the fluid with the solid. Like Laplace he supposed the fluid to be homogeneous 
 and incompressible. Bertrand ' has published a Memoir on Capillary Action, with a 
 view to make known the method of Gauss, as well as some simplifications of which 
 it is susceptible. 
 
 In 1831, Poisson published his important work, the Nonvelle Theorie de I' Action 
 Gapillaire. He strongly objects, to Laplace's Theory because he has omitted in his 
 calculations to take account of a physical circumstanccj the consideration of which 
 was essential; that is, the rapid variation of density which the liquid suffers near 
 
 » Deo. 20, 1804. ^ Quarterly ^Review and Works, Vol. i., p. 436. 
 
 " Works, Vol. I., p. 420 (note). - " BmgnaUUi, T. ix., 1816. 
 
 ■= lb. p. 434. ' Nouvelle Theorie, 1831. 
 
 4 Mic. Gil. Supp. au X Livre, 1806, 1807. ^ Princip. Gen. Theo. Fig. Fluid. Gott. 1830. 
 
 " Journal de I'^cole Polyteehnique. Cahier xti, Dove's Repertorium, Bd. v., p. 49. 
 
 1813. ' Liouville xiii., p. 185. 
 ' Mem. Soc. Ital. T. xiv. 
 
6 THEORETICAL EXPLANATIOl^JS OF CAPILLARY ACTION. 
 
 its free surface, and near the solid against which it rests, "sans laquelle les ph^no- 
 "mfenes capillaires n'auraient pas lieu"*. But he, in fact, arrives at a differential equa- 
 tion of precisely the same form as Young, Laplace and Gauss, It must be confessed 
 that Poisson is probably quite right in supposing a rapid variation of density near 
 the free surface of a fluid, and he has done good service in shewing how this sup- 
 posed variation of density near the free surface of fluids may be taken account of in 
 the mathematical treatment of Capillary Action. The reader may be further referred 
 to a M4imoire sur la Theorie de I' Action MoUculaire, par Jean Plana ^ 
 
 • Nouvelle TMorie, p. 6. ' Turin Memoires, 2 SSrie, T. xiv. 
 
CHAPTER II. 
 
 EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. 
 
 Many attempts have been made in recent times to test by experiment these 
 theoretical explanations of capillary phenomena. For this purpose Haiiy and Tremery' 
 at the request of Laplace made some experiments to determine the elevation of water 
 and of oil of oranges, and the depression of mercury in capillary tubes. Their results 
 appear to have satisfied Laplace that the elevation or the depression of a fluid in 
 capillary tubes varied inversely as the diameter of the tube. A tube of one milli- 
 metre in diameter gave a mean elevation of 13"°" "569 for water, and of 6"™ -7389 
 for oil of oranges, and a mean depression of 7"™ '333 for mercury. 
 
 In the SuppUment A la TMorie de V Action Gapillaire, Laplace found the fol- 
 lowing expression for the approximate thickness {q) of a large drop of fluid resting 
 on a horizontal plane '' : 
 
 , 1 - cos' -^ 
 
 1 /2 . ot' ^ """ 2 
 
 2 „ 7 • =r 
 aal sm -^ 
 
 For comparison, Gay-Lussac measured the thickness of a drop of mercury one 
 decimetre (21) in diameter resting upon a perfectly horizontal glass plane, and found 
 it to be 3°™-378 at a temperature 12''-8C. In calculating the value of q Laplace 
 
 neglects the value of — j because it is an insensible quantity. He then supposes 
 
 - = 13 square millimetres, and or' = 152 grades ==136'*'8 as determined by some previous 
 
 experiments, and substituting finds g- = 3"™ -39664, instead of the measured thickness 
 3""" -378. 
 
 Gauss merely refers to the results of Laplace, and gives the value of his a' 
 
 which is equivalent to the ^ of Laplace, equal 3 '25 square millimetres. 
 ' Supp. au X Livre, p. 52, -53. '' P- 64. 
 
8 EXPEBIMENTAL TESTS OE THEORIES OF CAPILLARY ACTION. 
 
 Poisson* obtains the following expression for the approximate theoretical thick- 
 ness (k) of a drop of fluid resting on a horizontal plane : 
 
 2 ^ Si'cosl-^ ^^ 
 
 Here the a, and to' of Poisson are respectively the a/ - and tt - ct' of Laplace. 
 
 Keferring to a previous experiment, Poisson writes a" cos w' = 4'5746 for a tempera- 
 ture of 12°-8C., and for a first approximation, he uses only the first term in (o). 
 Thus 
 
 ^''= (a V2 cos j) = a" (1 -f cos o)'), 
 or F cos &)' = (a" cos ft)') (1 -f cos ft)'). 
 
 And writing for k, 3°™ -378, the experimental thickness of a drop of mercury of 
 radius ^=50"°", at a temperature 12°-8C., as found by Gay-Lussac, he obtains 
 
 (3-378)'' cos ft)' = a" cos w (1 + cos ft)') = 4-5746 (1 + cos a), 
 
 which gives cos a>' = cos 48° nearly, or a' = 48° nearly, and a" cos a>' = a' cos 48° = 4-5746 
 
 now gives a or a/- = 2"™ -6146. 
 
 a? . .... 
 
 In the next place the term — only is neglected, because it is insensible : 
 
 Z' = Z + (V2-l)a = 50 -l-l-083^ 51-083; and A; =3"^ -378. 
 
 Substituting in (o), <o' is found to be 45° 30', which gives by the help of the 
 equation a" cos ft)' = 4-5746, a^ or -=6-5262 square millimetres, and a or a /- = 2'^-5547. 
 
 Avogadro*" made numerous experiments to clear up some doubtful points relative 
 to capillary action. He carefully examined how far any air or moisture commonly 
 supposed to adhere to the interior of glass tubes might affect the depression of 
 mercury. With this object in view, he exhausted the air, and heated the glass tube 
 when the mercury was not in contact with it, and he found that the depression of 
 the mercury in the tube was precisely the same after as it was before these pre- 
 cautions were taken. 
 
 In order however to determine the capillary constant a* or -, for mercury, he 
 
 made use of a tube of copper 20"™ long, and 2"™ -80 in diameter", well amalgamated 
 
 ' Nouvelle TUorie, p. 217. i> Accad. Fit. e Mat. Torino, T. 40 (1836). 
 
 « Ibid. p. 221. 
 
EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. 9 
 
 in the interior, and found it to be 5"56 square millimetres*, and, therefore, a or 
 A / - = 2'"'° 'SS?. Then substituting this value of a" in Poisson's'' formula 
 
 ^=-V + 6-b[6H|(1-&0^-|], 
 
 and making h = ■t""" '69, a = radius of tube = 0°"" -9525, according to Gay-Lussac's ex- 
 periment, he obtained h = cos to' = 0-8440 or m = 32°-5 ° = (180° — l^T'o) nearly, instead 
 of 45°'5 given by Poisson. 
 
 Substituting these two values a'' = 5'56 and w'=32'''5 in Poisson's expression (o), 
 for the theoretical thickness of a large drop of mercury quoted above, he obtains 
 3""" •235 instead of the measured thickness S'^'-STS. Upon this he remarks that the 
 smallness of this difference which corresponds to considerable differences in the values 
 of a? and of cosw', shews that this observation was little adapted to give, by its 
 combination with the depression of mercury in capillary tubes, exact values of these 
 quantities. 
 
 Avogadro then determined to measure the depression of mercury in a capillary 
 tube, so that he might obtain a value of m determined entirely from his own ex- 
 periments. His glass tube had a radius of 0"""'80*. He adopted a depression of 
 gram .j25^ that being the mean of a great number of careful observations. The tempera- 
 ture was between 10° C. and 14° C. This depression is rather less than that found by 
 Gay-Lussac quoted above, when allowance is made for difference in the radii of the 
 tubes with which they experimented. Substituting as before he finds 
 
 w' = 40° 21' = (180° -139° 39'). 
 
 In the next place Avogadro substitutes the value of cos a/ just found = 07621 
 and a''=6-56, in Poisson's formula (o) quoted above, and finds 3'""''154 for the thick- 
 ness of a large drop of mercury instead of Gay-Lussac's measured thickness 3™° '378. 
 
 Desains^ has deduced from Danger's experiments' a* or - = 6'7144, which gives 
 
 a or */- = 2"^ -5912 and a = 37° 52' 33" = (180° - 142° 7' 27"), which values appeared 
 to satisfy best the whole of the experiments. He states however that for different 
 sorts of mercury a or */- varied from 2"™ "55 to 2"""' "61, and to' from 38° to 45° 
 or from (180°— 142°) to (180' -135°). Desains also obtained from experiments with 
 large drops of mercury a or ^1 = 2"^ -621 and o,' = 41°36'30"=(180°-138°23'30"). 
 
 Still more recently Quincke has made very numerous experiments with a view 
 to determine the capillary constants for a variety of fluids, and also for metals at 
 
 » P. 221. 
 
 ^ Nouvelle TMorie, p. 147." 
 ' Acead. Fis. e Mat. p. 223. 
 
 B. 
 
 " P. 227. 
 
 • Ann. de Ch. Ph. [3] T." li. (1857). 
 
 ' Ann. de Ch. Ph. [3] T. xxiv. p. 501. 
 
10 EXPERIMENTAL TESTS OF THEOEIES OP CAPILLARY ACTION. 
 
 a temperature just above the melting point. He found that the values of a or ^ - 
 
 decreased for the same drop of mercury*, according to the time it had stood in 
 position. He also found that w' varied from 38° to 45<'^ or from (180° - 142°) to 
 (180° - 135°). But other results were obtained far beyond these limits. For the 
 
 mean value of a or ^- he adopted 2""" -8°, and some of his experiments gave as 
 
 high a value as 2°"" '9, both of which differ considerably from the previously received 
 value 2°™-6. 
 
 In 1868-9 Quincke published*^ the results of some experiments made to deter- 
 mine the capillary constants at the common surfaces of two fluids incapable of 
 mixing. In this case he pursued methods of experimenting in some respects similar 
 to those I had suggested in my application to the Eoyal Society in 1855. But 
 the value of Quincke's results is very much diminished by the manner in which 
 he carried out his experiments, and by his mode of determining the theoretical 
 forms of sessile drops of fluid. Thus Quincke's method requires the measurement, 
 with great precision, of the height of the vertex of a large drop above the largest 
 horizontal section of the drop. But in my experiments I have found that only a 
 rough approximation to this quantity can be obtained directly by the most careful 
 measurement. The theoretical forms of Quincke are much the same as those of 
 Segner, for in the calculations of both, one of the two principal radii of curvature 
 is supposed to be infinite. There is also a further objection to the use of large 
 drops of fluid, which Quincke's methods of calculation necessitated, because they 
 change their form slowly when a change in their volume is made. But only a 
 slight change in the volume of a small drop will give a marked change in its form. 
 
 The favourite method of testing the theories of capillary action has been by 
 the measurements of the heights to which fluids rise in capillary tubes. In cases 
 where the fluid wets the solid, there is only one constant, a, to be determined, as 
 the angle tu' = 0. But experiments of this kind are very liable to be vitiated by 
 irregularities in the bore of the tubes, or by impurities adhering to the inner 
 surface of fine tubes, which do not admit of being cleaned. The layer of fluid 
 which lines the tubes must make a sensible reduction in the radii of the finer 
 capillary tubes. And the theoretical expressions for the height of the fluids in 
 these tubes are approximations which are not strictly applicable to tubes of large 
 diameter used in experiments of this kind. 
 
 Some recent writers on capillary action have disputed the correctness of the 
 results arrived at by the earlier experimenters. Thus Simon ° has concluded from 
 numerous experiments of his own that the elevation of water in capillary tubes is 
 very far from- varying inversely as their diameters, and that the height to which 
 water rises between parallel plates compared with that which takes place in tubes, 
 instead of being as 1 : 2, is as 1 : 3, or rather as 1 to tt. 
 
 » Fogg. Ann. Bd. cv., p. 35 (1858). i Fogg. Ann. Bd. cxx'xix. 
 
 " P. 45. = P. 47. " Ann. de Ch. Ph. [3] T. xxxviii. (1851). 
 
EXPERIMENTAL TESTS OF THEORIES OF CAPILLARY ACTION. 
 
 11 
 
 Bfede* comes to the conclusion that the depression of mercury and the elevation 
 of water in glass tubes do not respectively vary inversely as the diameters of the 
 tubes exactly, and that the thickness of the substance of the tubes has a sensible 
 effect, or, in other words, that the molecular attractions are not insensible at sensible 
 distances. 
 
 Wolf* afterwards concluded from his experiments that the elevation of the same 
 fluid in capillary tubes, all circumstances being alike in other respects, depends 
 upon the nature of the tube. 
 
 Laplace and Poisson considered that the only effect of a change of temperature 
 was to change the elevation of a capillary column according to the change in density. 
 Thus Laplace " says "L'^l^vation d'un fluids qui mouille exactement les parois d'un 
 " tube capillaire, est, a diverses temp&atures, en raison directe de la density du 
 " fluide, et en raison inverse du diamfetre interieur du tube." And Poisson ^ obtains 
 for the elevation (h) of a fluid in a capillary tube of radius a 
 
 h = 
 
 ^gpa. 
 
 i: 
 
 Rr*dr. 
 
 He then supposes that by a change of temperature h, p and R are respectively 
 
 changed into h', p and R', neglecting the change in a. And having found -j-— ^® 
 
 remarks " L'exp^rience montre, en effet, que pour un m^me liquide h, diff^rentes 
 "temperatures, I'^l^vation du point G croit proportionellement a la density ; ce qui 
 " donne lieu de croire que la force repulsive de la chaleur, ou du moins, sa 
 " variation, que nous avons n^glig^e, n'a qu'une influence insensible sur I'int^grale 
 
 "f Rr^dr." 
 Jo 
 
 Very careful experiments have been carried out by Frankenheim and Sondhauss, 
 and afterwards by Brunner, to determine how far the height of the capillary column 
 depends upon the temperature. Frankenheim " found that the height to which 
 water rises in a capillary tube 1™" '0 in radius at a temperature f C. is 
 
 15"™ -336- 0-02751* -0-000014f between -2''-5 and 93°-4C., 
 
 and Brunner' finds it to be 
 
 15"™-33215-00286396« from 0" to 82''C. 
 
 Hence it appears that the elevation of fluids decreases with an increase of 
 temperature much more rapidly than would be expected according to the theories 
 of Laplace and Poisson. 
 
 In the foregoing sketch of the progress of experiments made to determine 
 capillary constants I have given attention chiefly to those where mercury was 
 
 « Savans Etr. Brux. T, xxv. (1853), 
 
 b Ann. de Ch. Ph. [3] T. xlix. 
 
 ' Supp. Th. de VAcHon Capillaire, p. 39. 
 
 ■1 Nmivelle TMorie, p. 106. 
 
 " Pogg. Ann. Bd. lxxii. (1847). 
 
 f Disquisitio Phys. Exp., p. 34, 35 (1846). 
 
 2—2 
 
12 EXPERIMENTAL TESTS OP THEORIES OF CAPILLARY ACTION. 
 
 the' fluid employed. Every experimenter finds that changes of form are constantly 
 going on in capillary surfaces from one cause or another. Still something more 
 definite is desirable in the results. But as the experiments have been conducted 
 apparently with every precaution, it does not appear probable that any new experi- 
 ments of the same kind would lead to better results. When «' is determined by 
 reflection its value must be obtained for a point at a short distance from the 
 junction of the solid and fluid surfaces. The experiments on the thicknesses of 
 large drops of fluid are not satisfactory because the theoretical expression is not 
 exact, and because the thickness of the drop varies so slowly in large drops. Also 
 the approximate theoretical thickness is given in terms of two unknown quantities 
 a and m. 
 
 During the time when I was able to use the Cambridge University Library, 
 I made copious extracts from numerous papers on this subject, but it does not 
 appear necessary for me to allude further to them in this place, especially as the 
 late Professor Challis has published a very good and elaborate report on Capillary 
 Action*. For numerous references to the works of early writers on the subject, 
 reference may be made to the articles " Capillaritat," " Cohasion" and " Tropfen" in 
 Gehler's Physihalisches Worterhuch. Recent experiments will be found referred to 
 in Fortschritte der Physik 1845, &c. and in Jahresbericht, 1847, &c. von Liebig, 
 Kopp, u. Will. See also the article on Capillary Action in the 9th edition of the 
 Encyclopcedia Britannica by the late Professor Clerk Maxwell. 
 
 * Brit. Ass. Report, 1834. 
 
CHAPTER III. 
 
 ON THE CALCULATION OF THE THEORETICAL FORMS OF DROPS OF 
 FLUID, UNDER THE INFLUENCE OF CAPILLARY ACTION, WHEN SUCH DROPS 
 ARE BOUNDED BY SURFACES OF REVOLUTION WHICH MEET THEIR RESPECTIVE 
 AXES AT RIGHT ANGLES. 
 
 We have already stated that various methods of obtaining the difiFerential equa- 
 tion to the surface of fluid under the action of capillary forces have been given by 
 Laplace and other writers on Capillary Action. The form of the eqiiation obtained by 
 these different methods is, however, in all cases the same. 
 
 Perhaps the simplest way of obtaining the equation in question is to consider 
 the fluid to be in equilibrium under the action of gravity and of a uniform surface 
 tension. 
 
 Let T be this uniform tension, R and Bl the principal radii of curvature at 
 any point of the surface of the fluid, 'p the fluid pressute at that point. 
 
 Then i + i^ = f-' 
 
 If z be the vertical coordinate of the point measured downwards, o- the density 
 of the fluid, and g the force of gravity, then 
 
 •p = g<yz + C, where (7 is a constant. 
 
 When two difierent fluids are separated by the capillary surface, p is the dif- 
 ference of the pressures in the two fluids at their point of meeting, aud a is the 
 difference of the densities of the fluids. 
 
 When a drop rests upon or hangs from a horizontal plane surface, the remaining 
 surface of the drop being free, this free surface will evidently be one of revolution 
 about a vertical axis, and it will meet the axis at right angles. 
 
 Take the axis of revolution as the axis of z, and the point in which it meets 
 the free surface as the origin. 
 
14 CALCULATION OF FORMS OF DROPS. 
 
 Let X be the horizontal and 2 the vertical coordinate of any point in a 
 meridional section of the surface of the fluid, p the radius of curvature of the 
 meridional section at that point, and ^ the angle which the normal to the surface 
 makes with the axis of revolution. 
 
 Then the length of the normal terminated by the axis is -. — r • and we have 
 ^ •' sm<f)' 
 
 sin <^ ' 
 and the above found equation becomes 
 
 1 sin <j> _G + gaz 
 
 p~ir~~T~- 
 
 Let 6 be the radius of curvat ur e at the ori gin, so that at that point we 
 have^ both 
 
 Hence 
 and the equation becomes 
 
 p = b, and limit (—. — r ) = b. 
 Vsm <p/ 
 
 T b 
 
 1 sin^_2 OCT 
 p^ X ~h^ T''' 
 
 P (^\ T \b) ■ 
 
 I^et ~Y- ^^ called /3, which is an abstract number. Also let s be the length of 
 
 the arc of the meridional section, measured from the origin and terminated at the 
 point under consideration. 
 
 Then ds = pd<^, 
 
 dx= p cos ^d^, 
 dz = p sin <^d<f> ; 
 
 <l 
 
 d^ 
 
 h 
 
 = (^) sin ^d<^. 
 
 For the sake of simplicity, we will write x, z, p and s instead of - , - ^ and - 
 
 b ' b' b b' 
 
CALCULATION OF FORMS OF DROPS. 15 
 
 which amounts to taking the quantity h as the unit of length, and we may at any 
 time re-introduce the quantity 6 by writing 
 
 r , r > f and y instead of x, z, p and s. 
 
 Thus simplified, our equation becomes 
 
 1 sin (f) „ ^ 
 p X 
 
 Also when = 0, we have ^ = 0, p = l and limit f^ — r) = l, hence the form of the 
 
 curve depends on the single parameter /3. The magnitude of the curve, or its scale, 
 is proportional to 6. 
 
 The same equation is applicable to the case of hanging dro ps, but in that case 
 z IS to be measured upwards from the vertex, and j8 will be n egative. 
 
 dx 
 
 and sin ^ = 
 
 the above ■ equation is equivalent to 
 
 ' g.liW^'^"' 
 
 a , differential equation of the 2nd order. The two arbitrary constants, which enter 
 into the integral of this equation are to be' determined by the condition that when « = 0, 
 
 ^ = 0, and ^5- = 1. 
 
 xdx 
 
 We are unable either to find the general relation between x and z, by means of 
 this equation, or to express "these two quantities in terms of a third variable. 
 
 We may, however, as in \ all cases where the differential equation to a curve is 
 given, develope the increments of the coordinates in series proceeding according to 
 ascending powers of the increment of the quantity chosen as the independent variable. 
 Thus we can trace a small portion of the curve starting from a known point, and 
 then we naay make the point which terminates this portion a new starting point for 
 tracing another small portion, and so on successively until any required portion of 
 the curve has been traced. 
 
 For instance, suppose the given equation to bit 
 
 df~-^[dt'^'V' 
 
 where / denotes any function of the -quantities ^, y and t. 
 
16 CALCULATION OF FORMS OF DROPS. 
 
 Then by repeated differentiations of this equation, and by substitution of the 
 value of 5 in the successive results, we may find the general values of the higher 
 differential coefiBcients 
 
 W df 
 in terms of -£ , V and t. 
 
 Hence if, for a given value t^ of t, we know that 
 
 2/ = 2/o and I = (|)^, suppose, 
 
 we can find the values of C| and the higher differential coefficients of y, which corre- 
 
 ar 
 
 spond to f = f„. 
 
 Let these values be denoted by f^j , f ^j , &c. 
 
 Therefore if t, = t„ + Bt^, and if y, and (-£) be the values of y and ■£ which 
 correspond to t = t^, we have by Taylor's theorem 
 
 The increment St^ must be taken so small as to render these series convergent. 
 
 The values of y^ and (-^j being thus known, we may find (j^j , (-^A , &c., by 
 the same formulae as before ; and then if 
 
 and if y^ and f-^j be the values of y and -^ which correspond to i = ij,, we may simi- 
 larly find y^ and (-tt) , and the same process may be repeated as often as we please. 
 
 A similar process may be employed if we have any number of simultaneous 
 differential equations, and the same number of dependent variables, such as, for 
 instance, the following : 
 
 ;7i=/(«> y, t), 
 
 dt 
 
 dy 
 dt 
 
 = F{x, y, t). 
 
CALCULATION OF FOEMS OF DROPS. 17 
 
 The method fails if any of the differential coefficients employed become infinite 
 in the interval over which the integrations extend, and therefore the independent 
 variable should be so chosen that no infinite or very large values of the differential 
 coefficients will be introduced. 
 
 The intervals adopted should be so small that a few of the terms of the series 
 will suffice to. give the results with all the accuracy that is desired. 
 
 After a few points of the curve, in the neighbourhood of the starting point, 
 have been determined by the foregoing or some equivalent method, it will usually 
 be found more convenient to determine other points of the curve in succession by 
 making use of a series of successive values of the differential coefficient which is 
 given immediately by the differential equation, rather than by employing the values 
 of the successive differential coefficients of higher orders which are fovind by means 
 of the several derived equations. 
 
 To fix the ideas we will suppose, with especial reference to our present problem, 
 that the given differential equation is one of the first order, say 
 
 Let ... t_^, f_g, f_j, f_j, i„, fj, &c. be a series of values of the independent variable 
 t, forming an arithmetical progression with the common difference eo, 
 
 I^et ... 2/.„ 3/.3, y_„ y_„ y,, y^, &c. , 
 
 denote the. corresponding values of y, and let 
 
 -l-^ 9-3. 9'-2. ?-i. ^o. ?i. &c. 
 be the corresponding values of q, or oi —, 
 
 and suppose to to be so small that the successive differences of these values of q 
 soon become small enough to be neglected. 
 
 Let t^t^ + tm, 
 
 and suppose that we have already found the values of 
 
 • ■■ y_4. y-3. 2/-ii> 2/-1 up to ^0 
 and therefore also those of ... q_^, q_^ q_^, q_^ up to g'^, 
 
 and that the successive differences of these quantities are taken according to the 
 following scheme : 
 
 n q ■ . 
 
 4 q^ 
 
 Ag-3 
 
 ... 
 
 
 ... 
 
 3 ?.3 
 
 A?-, 
 
 AV. 
 
 A'?.. 
 
 . . . &c. 
 
 2 g-. 
 
 A?-, 
 
 AV> 
 
 A'?, 
 
 ■ ^*q^ &c. 
 
 1 ?-. 
 
 A?„ 
 
 A=?„ 
 
 
 
 ?0 
 
 
 
 
 
 E, 
 
X8 CALCULATION OF POEMS OP DROPS. 
 
 Then the general value of q found by the ordinary formula of interpolation, 
 for any value of n, will be 
 
 , = ,„+ A,„ ^4- A=,„ !i|-t^H- A',„ ^i|±^+ AV„^^^ 
 provided that n be taken between limits for which this series remains convergent. 
 
 Hence the general value of y will be 
 
 y = jqdt = Q)jgdn, 
 
 or, substituting the above value of q, and adding a constant to the integral so as. 
 to. make y=ya when n= 0, 
 
 3/=^„ + eo|g„n + Ag4VA-gJ^)^. + AV„/ '-(--^^^^,\^f^) cZ. + &c.}. 
 
 where all the integrals are supposed to vanish when n = 0. 
 
 If, in particular, we put m = -l, and substitute the several values of the 
 definite integrals 
 
 1-172 '^'^' I 1.2.3 ^ "'■^^- 
 
 we shall have, by changing the signs throughout, 
 
 2'o-y-. = -|^,-^A2„--A=^„-^A»j,-^A^^„- — A'g„-g^A«y„ 
 
 275 , 33953 ^ 8183 ., ) 
 
 24192 ^« 3628800 ^»~ 1036800 ^""'^''•J- 
 
 Similarly, putting «=1 and substituting the values of the definite integrals 
 we shall have 
 
 + -525^.,, 1070017 .,_^ 2082753^, , ] 
 ^ 17280 ^ ^« + 3628800 ^ ^«~^ 7257600 ^ ^o + '^°j • 
 
 It will usually be found expedient to choose « so small as to render it unnecessary 
 to proceed beyond the fourth order of differences. 
 
 The series last found gives the value of y^ in terms of quantities which are 
 all supposed to be already known, that is, the value of the variable y which was 
 previously known for values of the independent variable extending as far as « = « 
 now becomes known for the value t=t,+co, or at the end of an additional interval J 
 
CALCULATION OP FORMS OP DROPS. 19 
 
 It will be remarked, however, that the coefficients of the series above found 
 for 3/0 — y_j, after the first two terms, are much smaller and diminish . much more 
 rapidly than the corresponding coefficients of the series for y^ — y„. Hence by taking 
 into account the same number of terms of the series in the two cases, the value 
 of 2/0 — y_j will be determined with much greater accuracy than that of y^ — y^. 
 
 In what has gone before, the successive values of y up to y„ are supposed to 
 be already known, and therefore the equation which gives the value of y„— y^^ may 
 be regarded as merely supplying a verification of former work. If, however, we 
 suppose that the value of j/, is only approximately known, while the successive values 
 as far as y_^ have been found with the degree of accuracy desired, we may use the 
 equation for y^ — y_^ to give the corrected value of y„, in the following manner. 
 
 Suppose that (y„) is an approximate value oi y^, and let y„={y^ + ''}> where r] \s 
 so small that its square may be neglected. 
 
 Also let {q^ be the corresponding approximate value of q^, found from the equation 
 
 by putting 2/ =(?/„) and t=t^. 
 
 Then we may put q„ = (g-J + hq, 
 
 where h denotes the value of the partial differential coefficient -~ or -^7' — - found 
 
 ay dy 
 
 by substituting (y„) for y and t^ for t after the differentiation. 
 
 Let ACg-j), A^ (§'„), A'(g'„), A*.(g'„), &c. denote the values of the successive dif- 
 ferences formed with the approximate value {q^ and the known values q_^, q_^, &c. 
 which immediately precede it, then we have 
 
 AVo = A' (?„)+&'?, 
 
 A\ = A\q'>) + H 
 &c. = &c. 
 
 But, by the equation before obtained, 
 
 f 1a 1 a2 1 as 19 A4 3 .5 863 ., 
 
 275 , 33953 ., 8183 .^ _„ 1 
 
 24192^'' 3628800^° 1036800^° }' 
 
 Or, substituting for y„, q^, Aq^, A^q^ &c. their values in terms of tj and known 
 quantities, 
 
 (yo)-y-i + '; = »{(?o)-|A(3„)-lA=fe)-^A»(?„)-^A*fe)-&c.} 
 
 3—2 
 
20 CALCULATION OF FORMS OF DROPS. 
 
 Hence if e denote the excess of the quantity 
 
 r 1 1 1 IQ ) 
 
 - {(2„) - 1 ^ (2o) - ^ A=(?o) - 4 ^' (?o) - 720 ^' ^^"^ - *'•[ 
 
 over the quantity (i/o) - y_^, we shall have 
 
 or 7] = 
 
 which determines rj, and therefore l/o = (j/o) + V> and g'„ = (?o) + *'? ^°*^ become known. 
 If in finding e we stop at the term involving A*(g'„), we shall have 
 
 6 
 
 V = 
 
 ^-720'*'* 
 
 and ki] — 
 
 1-720'"* • 
 
 It will be observed that the coef3Bcient of wh in the denominator of these 
 expressions is the same as that of wA'g',, in the expression for 2/1.— 2/o- 
 
 This is no mere coincidence, as it is easy to shew that, generally, the coefficient 
 of any term foA'q^, 
 
 in the expression for y^ — y^, is equal to the sum of the coefficients of the terms 
 involving 
 
 (oq^, wAg-j, (oh.%, (Sfc. . . . mA'g',, 
 
 in the expression for y^ — y_,. 
 
 Hence if in finding e we also include the term involving A° (g„), we shall similarly 
 have 
 
 6 . 
 
 7) = 
 
 1 95 ■ 
 
 and kT} = - 
 
 1-288'"* 
 
 An approximate value of y„ may always be found from the series of values 
 — y-i> y-s> V-v 1/-1 previously calculated, by taking^ the successive dififerences of four 
 or fiv-e of the , last^ terms of the series, and assuming that the last difference so 
 found remains constant. 
 
CALCULATION OP FORMS OF DROPS. 21 
 
 The numerical operations will be greatly facilitated by the use of Tables which 
 exhibit the values of 
 
 720^^' 160^?' 60480^2, &c. 
 
 for given values of A*q, A% A\ &c. 
 
 Such Tables have been formed by Mr Bashforth for this purpose, and are given 
 at the end of this Chapter. 
 
 Having made these preliminary observations on the general method of finding 
 successive small portions of a curve by means of its differential equation, we will 
 now proceed to apply the method to the problem under consideration, viz. to the 
 tracing of the curve formed by a meridional section of a drop of fluid, by means 
 of the equation above found 
 
 1 sin <f> „ ^ 
 
 p X 
 
 First, suppose ^ to be taken as the independent variable. 
 
 The above equation may be regarded as giving p ag a function of the co- 
 ordinates X and s, and these latter quantities are to be found by the integration 
 of the equations 
 
 dx 
 
 dz . . 
 
 Also X and z vanish with ^, and p is initially = 1. 
 
 We will first find the form of the curve in the neighbourhood of the origin 
 by developing p and the coordinates x and z in series of ascending powers of 0. 
 
 Instead of employing the general method described at the outset, it will be 
 found more convenient, in this particular case, to proceed as follows : 
 
 Assume, as we evidently may do, 
 
 p = 1 + &,<^" + 6,.^* + &,f + 1,-^' + ^„f° + &c., 
 
 where h^, \, &c. are constants to be determined, then 
 ^=pcos<^ = p|l-^2<^=+5-27374f-l-27:6<^' 
 
22 CALCULATION OF FORMS OF DROPS. 
 
 Substitute the assumed value of p and integrate, therefore 
 
 * = [<^ - 6 "^^ + l70 '^' - 5040 '^' + 362880 ^' " 39916800 *" + '^°- J 
 
 + J.g^'-ro^' + 4^'-6^0^'+44M20*"-H 
 
 + &C., &c. , 
 
 Similarly 
 
 4 = /' l*^ - 6 ^' + 120 "^^ - 50i0 '^' + 362880 "^^ " 39916800 '^" + '^'j ' 
 and therefore 
 
 ^=B^'-^'^'+ 7^0 "^°-iok'^'+ 362^0 '^"-4790^ 
 
 + ^'[i^^-3^^^+9io^^-5ok^'°+435i560'^'^-H 
 + ^^G^°-r8'^'+r4)^"-60i80^" + H 
 
 + &c., &c. 
 Also, we find 
 ^ = 1 - \<f>' + (6/ - 6J .^^ - (J/ - 2hp, + \) f + (6/ - 36,^6, + 6/ + 26,6, - 6,) </,"> 
 
 - >/ - 46/6, + 36,6,^ + 36/6, - 26,6, - 26,6, + 6 J .^" + &c. 
 
CALCULATION OF FOEMS OF DKOPS. 23 
 
 Also 
 
 
 
 
 
 a 
 
 X ' 
 
 and 
 
 sin<^ 
 
 1- 
 
 -u- 
 
 "*'120 
 
 f. 
 
 1 
 
 5040 
 
 and from above 
 
 • 
 
 
 
 
 
 "^^ + 362880 "^^ "39916800 "^^ + *°-' 
 
 r^ - (^ ^») ^^-»- (iro-^^^+^>^- (5-^-168^^ 44^^-^«) ^° 
 
 "^U62880 6480 '^"^216 * 18 "'^9 7^ 
 
 ~ U99I68OO 443520 ""'"7920 *~ 264 «"'" 22 « 11 'V "^ 
 + &c., &c. 
 
 Hence, by performing the division indicated, we may find 
 
 * 
 
 / 2, 4, 22,3,1,, 16, 68,, 
 ' + (,4725 *» - 4725 ^« "'" 135 ^" + 81 ^" "^ 4725 ^-1575 ^"^^ 
 
 "'■V93555 ""''42525 ' 14175 " 1215 " 243 " 
 
 , 52 8 7 , , 16 , „ 4 , 8 
 • "'"155925^* "14175^"''* "'"525 ''"''* ■^135''^^* "525^* 
 
 - 25 ^"^*' + 1485 ^« "" 945 ^^^^ ~ 21 ^"'^^ "^ 35 ^*^» + 297 ^- 
 
 2 
 + &C., &c. 
 
 + ^K\-Yi^.)r 
 
 Substitute these expressions in the equation 
 
 i + ^ = 2 + ^., 
 p X 
 
24 CALCULATION OF FORMS OF DROPS. 
 
 and equate the coefficients of corresponding powers of <f>, and we shall find successively 
 
 h- " 3 1183 
 
 '^«~~6760'^ 128 '^ 9216^' 
 
 .* ^»- ~ 8960'^'^ 18432'^ ■^92160'^ ^81920 '^' 
 
 _ 233 1 ^ 1469 
 
 '^"~ 14515200 '^ 36288 '^ 442368^ 
 
 _ 104513 „, . 4882031 „ 
 5529600 '^ 88473600'^^' 
 
 which gives the value of p in terms of <p, as far as ^'°, 
 
 Again, substituting these values of b^, h^, &c., in the expressions for - , a; and z, 
 we shall obtain 
 
 , (J_ o 31_ , 401 8431 A ,, 
 
 "^V8960'^ 30720 '^ 92160'^ 737280^/^ 
 
 233 ^ 17 „, . 1517 o.. 
 
 /_233_ 17__ n,. 
 
 U4515200''^ 725760 '^ ' 2211840' 
 7409 _, . 522091 
 
 + 
 
 2764800 
 
 522091 N 
 '^+88473600^;'^ 
 
 to the 10th order in ^; 
 
 ^ 1.362880 ^ 4032'^ ^55296 '^ ^ 138240 '^ ^ 737280 '^ y"^ 
 
 _/ 1 .103 565 3937 
 
 V39916800 ^ 14515200 '^ ^ 2322432 ^ ^ 2211840 '^ 
 
 121447 ^ , 443821 .A ^„ 
 ^+88473600^;'^ 
 
 "'■ 22118400 
 to the 11th order, and 
 
CALCULATION OP FORMS OP DROPS. 25 
 
 U0320 "^ 46080 '^ 9216 '^ "^ 73728 '^ / ^ 
 
 + ( ^ 4. 11 o, 157 ^. , 2987 , 6799 N ,„ 
 V3628800 "*" 241920 '^ "^ 184320 '^ "^ 691200 ^ 819200 '^ / ^ 
 
 -(: 
 
 1 269 7993 „ 3551 
 
 479001600 174182400*^ ' 139345920^ ' 5308416 
 
 , 724007 ^ 4882031 A ;, 
 265420800 '^ "^ 1061683200 '^ ^ '^ 
 to the 12th order. 
 
 It is hardly necessary to remark that in these expressions the coefficient of each 
 power of <l> thus found is exact, and not merely approximate. 
 
 Also if s denote the length of the arc of the curve measured from the origin, 
 V 80640 '^ ^ 165888 '^ ^ 829440 '^ ^ 737280 '^ J ^ 
 
 + 1 
 
 + ; 
 
 / 233 1 1469 104513 ^ 
 U59667200 '^ 399168 '^ "^ 4866048 '^ "^ 60825600 '^ 
 
 443821 
 
 /S')<^" 
 
 88473600 ' 
 to the 11th order in <f). 
 
 In order that the terms in these series which involve higher powers of ^ may be 
 insignificant, ^ must not exceed a certain limiting value which will, of course, depend 
 on the value of /3. The larger the value of /3, the smaller will be this limiting 
 value of ^. 
 
 To find the values of the coordinates for larger values of <f>, we must proceed 
 step by step according to the method described above, <f) being taken for t, and x 
 and z in turn taken for y, the value of ^ being increased at each step by a given 
 small quantity. 
 
 Let ft) be the circular measure of the interval between two consecutive values of 
 <j), then CO must be so chosen that the series above found will give sufficiently accurate 
 values of the coordinates throughout several, say four or five such intervals. 
 
 Suppose ...^_5, 4'-i> 4'-s> <^-2> ^-1' ^0 *° ^® ^ series of consecutive values of </>, with 
 the common difference co, and let 
 
 •■•^_6) *_4> ^-3! ^-2) ^-t.) "^O 
 
 B. 
 
26 CALCULATIOIf OF POEMS OF DROPS, 
 
 be the corresponding values of the coordinates, and 
 
 •••P_5, P-i, P-i, P-V P-l' Pn 
 
 the corresponding radii of curvature. 
 
 The equations to be integrated are 
 
 dx . 
 
 ^ = pcos4>, 
 
 dz . , 
 
 ^ = psin0, 
 
 where - + — 2 = 2 + Bz. 
 
 p X 
 
 Suppose that the values of the coordinates, and consequently those of the radius 
 of curvature, have been calculated for the successive values of <^ up to <f)_^, and we 
 wish to find the values of the same quantities for ^ = 0„. 
 
 In the first place, we may obtain an approximate value of p^ in the following 
 manner. 
 
 Tabulate the calculated values of log /a, and form their successive differences 
 according to the following scheme : 
 
 logp-s 
 
 Alog/3-4 A'log/j_3 
 
 logP-4 A'log/)_3 AMogp_, 
 
 ^logp-s A^ogp., 
 
 log/>_3 AHog/)., AMogp.^ 
 
 Alogp_, A'log|0_, 
 
 logP-2 A'logjo.j 
 
 Alogp.i 
 logP-i 
 
 If (o is taken sufficiently small, the differences as we proceed to higher orders 
 will rapidly diminish, and it will generally be easy by inspection of the two or three 
 last calculated fourth differences, to fix upon an approximate value of the fourth 
 difference A* log /a, immediately succeeding. 
 
 Call this approximate value A* log (pj, and by successive additions form A' log (/)„), 
 AMog(/j„), Alog(|0„) and log(jo„), thus 
 
 A'log/3_, 
 AHogp., Anog(/,,) 
 
 Alogp_, A^ogCp.) 
 
 log/'-. Anog(p„) 
 
 Alog(p„) 
 
 log (po) 
 
CALCULATION OP FORMS OF DROPS. 27 
 
 Form the values of 
 
 ...dx_^, dx_^, dx_^, dx_^, dx_^, 
 .■.dz_^, dz_^, dz_^, dz_^, de_^, 
 and of their successive differences, according to the following scheme : 
 
 <»/3_5 cos 0_g = dx_^ ^^dx_^ A*dx_, 
 
 Adx_^ A'da;_3 
 
 (op_^ cos <f)_^ = dx_^ A^dx_^ A.*dx_, 
 
 Adx_^ A!'dx_^ 
 
 asp_^ cos ^_3 = (Za;.3 A^dx_^ A*dx_ 
 
 Adx_^ A'da!_j 
 
 ft)p_2 cos </).2 = dx_^ A^(fa:_j 
 
 Adx^^ 
 o)p_j cos ^_, = dx_^ 
 
 and 
 
 «P-s 
 
 i sin ^-5 
 
 = dz^^ 
 
 Adz , 
 
 A'dz^ 
 
 A'dz, 
 
 A'dz,, 
 
 fop^ 
 
 , sin ^_^ 
 
 = dz_^ 
 
 Adz, 
 
 A'dz_, 
 
 A'dz, 
 
 ^'dz,. 
 
 ^P.z 
 
 sin ^.3 : 
 
 = dz^ 
 
 
 ^'dz,. 
 
 
 A'dz_, 
 
 Adz_^ A^dz,^ 
 
 <B/>_2 sin ^_3 = dz^^ A^dz_^ 
 
 Adz_j 
 (op_^ sin ^_j = cZ^_j 
 
 If p„ were known, we might similarly form 
 
 dx^ = mp^ cos ^„ and dz^ = wp^ sin ^„, 
 
 and the successive differences 
 
 Adx^, A^dx^, A'dx^, A*dx^, &c., 
 Arfa„, A''J3„, A^dz^, A*dz^, &c., 
 and then we should have, by what has been already proved, 
 
 «„ - «_i = rfa-o - 2 ^<^*o - 12 ^"'^'^o ~ 24 ^''^*».~ 720 ^'^^° ~ '^*'-' 
 and z„ - z_^ =dz^--^Adz^—^ A^dz, - ^^ A^dz^ - ^ A^dz^ - &c. ; 
 
 and when a;„ and 5^„ had thus been found, we should have the equation 
 
 l+!i5i. = 2 + ;3., 
 in verification of the value which had been used for />„. 
 
 4—2 
 
28 CALCULATION OF FORMS OF DROPS. 
 
 Now, let {dx„) and (d^„) be approximate values of da;, and dz„ respectively, given by 
 
 (dsB^) = CO (pa) cos 0„, 
 (dz,) = CO (p,) sin 4>^, 
 
 and let the successive differences found by employing (dx,) instead of dx^, and (dz„) 
 instead of dz^, be denoted by 
 
 A(&„), A'(_dx,), A^dx,), A'(dx^),&c., 
 
 and A{dz,), A^dz,), A'{dz,\ A'{dz,),&c., 
 
 respectively, and suppose that (xj and (aj are given by the equations 
 
 W - ^-x = ('^^o) - J A (&J - 1 A^dx,) - 1 A\dx,) - ^ A* (<i^o) - &c., 
 Also let [pj be found from the equation 
 
 and suppose that this gives [p„] = (p„) (1 + e), 
 
 where e is a very small known quantity. 
 
 Then if the true value of p, = (/>,) (1 + ,,), the correction of the value of (dx,), 
 and therefore also that of the values of A (dx^ A^ (dx,), A' {dx,). A* {dx,), &c. will be 
 
 ■nco (p,) cos cj)„ 
 
 and the correction of the values of (dz,), A{dz,), A'(dz,), A'(dz,), A* (dz,), &c. will be 
 
 V(o (p„) sin ^0. 
 
 Hence if we stop at the terms which involve differences of the 4th order, we 
 shall have 
 
 / N 251 , , 
 «o - (a^oj = 720 '''" ^P"^ '^^^ ^o' 
 
 and 
 
 
 ^o-(«o) = 
 
 72o'?"Wsin^„. 
 
 Hence, since 
 
 
 Po ^a 
 
 and 
 
 
 
 we find 
 
 1 
 
 Po 
 
 ~[pr ""•^» 
 
 "1 1 " 
 
 + /3[«o-(^,)] 
 
 
 
 251 
 
 = 720'""^' 
 
 o„) sin ^„ 
 
 'cos d),, „"1 
 
CALCULATION OF FORMS OF DROPS. 29 
 
 but _= (1_^)_ nearly, 
 
 Po (Po) 
 
 and u;r^}^'-'^' ^^^^^^' 
 
 1 251 
 
 and therefore r} : 
 
 251 
 
 , nearly. 
 
 cos<^o_,_^1 
 
 Hence rj is found, and therefore the values of a?„ and z^, which were required, 
 become known. 
 
 In practice, the following slight modification of the above process will be found 
 convenient. 
 
 Suppose the assumed value of log(/3o) to be increased by 100 units of the last 
 place of decimals employed, then while calculating the values of (dx^), {dz^, (x^), 
 {Zf,) and the consequent value of [pj, note at the side of the work, the changes 
 which would be severally caused in each of these quantities by such an augmen- 
 tation of log(/j(|). It may be remarked that the changes in (a;,,) and («„) will be 
 
 251 
 
 f^^ times the changes > in (dx^) and (dz^) respectively, when we stop at terms 
 
 251 
 involving A*, and that f^^ may be conveniently put under the form 
 
 720 
 
 1 
 
 3 
 
 1 + 
 
 ^ f 1 - -V 
 
 20 V 12/ 
 
 Now suppose that an increase of 100 units in log(po) causes a diminution of 
 
 fi units in log[|0„], and that the excess of log[/3(,] above log(p„) is A, of the same 
 
 units, then the correction to be applied to the assumed value log(/3„) will be 
 
 . 100 , 
 
 \— — such units, 
 
 100 + /i 
 
 and the correction to the value of log [/>„] will be 
 
 — — — ^— such units, 
 100 + /X 
 
 and the proportionate changes required in the values of (dx„), {dz^, («„) and {z^ 
 
 will be at once found. 
 
 If in finding (a;,) and {z^ we include the terms which involve differences of 
 
 251 
 the 5th order, the fraction =^^, which occurs in the above, should be replaced by 
 
 288 3 \ 96/ ' 
 
 We may, of course, change the value of to whenever the more or less rapid 
 rate of diminution of the successive differences shews that it is expedient to increase 
 or diminish the interval. It is only necessary, by selection from or interpolation 
 between the values already calculated, to find the coordinates for a few values of ^ 
 separated from each other by the newly chosen interval. 
 
30 CALCULATION OF FORMS OF DROPS. 
 
 It may be remarked that when, by means of the appropriate series, we have 
 found the values of p for a sufficient number of small values of (jj, we can form 
 the corresponding values of dx and dz, and thence derive the corresponding values 
 of X and z by the process of integration before explained, without the necessity of 
 employing the series for finding those quantities. 
 
 A numerical example of the work will be given later on. 
 
 When X and z, and therefore also p, are known for a given value of 0, we 
 can find the volume of the portion of the drop terminated by the horizontal plane 
 a,t distance s from the origin, without any further integration. 
 
 For if F be this volume, we have 
 
 dV^'TTx'djs. 
 
 Also from the differential equation, when the radius of curvature at the vertex 
 is taken for the unit of length, 
 
 1 sind) „ - 
 
 - + ^ = 2 + ^z, 
 
 p X 
 
 J ^, J, aj dp , COS d>d<h sin d>dx 
 
 and therefore adz = — ^ H ^ — T- ^ — . 
 
 p X or 
 
 Hence '^^~fi| a^ + a;cos^(?(^-sin<^(?a;[, 
 
 and, integrating the first term of dV by parts, 
 
 ^ — q\ 2 I — h/a; cos <}) d(j) — f sin (f> dxY 
 
 = Q \ J(x cos <f> d<ji + sin (j) dx)> , 
 
 since dx = p cos (j) d<p, 
 
 ir "^ [^ • A '^'^ (1 sinrf)) 
 
 ^\p ) ^ \p a; j' ^ 
 
 no correction being required, since Y vanishes with x. 
 
 If now the radius of curvature at the vertex be equal to 6 instead of being 
 unity, we must replace x and P by | and ^ respectively, and F by ^. 
 Hence we have, in this case, 
 
 7r6V 
 
 F = 
 
 or, replacing - by its equivalent ?+^_!^, 
 
 Jl _ sin <^\ 
 
 \p -^Y' 
 
 b 
 
 y_ irly'ay' (2 2sin<^ /Ss) 
 
 /8 16 ~^+"f;' 
 
CALCULATION OF FORMS OF DROPS. 31 
 
 The expression just found for V is, I believe, due to Bertrand, who gives it 
 in Liouville's Journal, Tome xiiL p. 185. 
 
 This result may also be found very simply, without employing the differential 
 
 equation, thus : 
 
 1 
 The portion of the drop of which the volume is V is kept in equilibrium by 
 
 (1) its own weight, 
 
 (2) the pressure of the contiguous fluid on the opposite side of the bounding 
 plane^ 
 
 (3) the tension across the circle in which the bounding plane meets the 
 surface of the fluid. 
 
 This tension acts at each point in a direction which makes the same angle (j) 
 with the horizontal plane. Hence if T be the tension per unit of length, since iirx 
 is the circumference of the circle before mentioned, the. resolved part of the tension 
 in a vertical direction is 
 
 iirx T sin </>. 
 
 Also the weight is g<TV, and the total pressure of the contiguous fluid is 
 
 irx^p,' 
 where p is the pressure at any point of the bounding plane, so that 
 
 _, /I sin d>\ 
 
 Hence gaV^ 2'n-xTsin A = '7rx'T (- + ?^) , 
 
 \p X J 
 
 and 
 
 •K^ t\ _ sin^ N 
 g<y \p X ) ' 
 
 which is identical with the former result, since as we have seen 
 
 If the constant /3 be positive, the process above explained, in which ^ is taken 
 as the independent variable, may be carried on to any extent that may be desired, 
 and the radius of curvature p of the meridional section of the drop will always 
 remain finite. But if ^ be negative, that is if the drop hang from the lower side 
 of a horizontal plane, then if we proceed further and further from the vertex, the 
 
 value of - , which at the vertex itself is supposed to be unity, will continually 
 
 diminish, until at a certain point it vanishes and then becomes negative. The point 
 of the curve at which p thus changes sign by passing through infinity is a point 
 of inflexion, where ^ attains a maximum value. Hence all differential coefiicients 
 taken with respect to ^ as the independent variable become infinite at the point 
 in question and have large positive or negative values at points near it. 
 
32 CALCULATION OF POEMS OF DROPS. 
 
 This circumstance makes it necessary, when /3 is negative, to choose a dififerent 
 independent variable. 
 
 Suppose now that s, the length of the arc measured from the vertex, is taken 
 as the independent variable. 
 
 The equations to be integrated are 
 
 dd) = - ds, 
 P 
 
 dso = cos ^ ds, 
 dz = sin ^ ds, 
 
 where the value of - in terms of x, z and ^ is given, as before, by the equation 
 P 
 
 1 sin rf) a . n 
 p X 
 
 Also to determine the constants of integration, we have, when « = 0, 
 
 a; = 0, ^ = 0, <i = 0, and - = 1. 
 
 P 
 
 We must first find the form of the curve in the neighbourhood of the vertex, 
 by developing <^, x and z in series of ascending powers of s. 
 
 We have already found s as well as x and z in series of ascending powers df ^, 
 and by means of Lagrange's theorem it is easy to transform these series so as to 
 obtain the required series in powers of s. 
 
 From the expression of s in terms of <Jf>,.we find by transposition, 
 
 _/_ 1 g. 53 , 2011 6799 \ 
 
 V 80640 '^ 165888 '^ 829440 ^ 737280 ^ j^ 
 
 ■ ,1 233 1 fl. , 1469 104513 ^ 443821 \ 
 
 "^ U59667200 '^ 399168 '^ "^ 4866048 ^ "^ 60825600 '^ "^ 88473600 '^ )^ 
 
 - &c., 
 
 or <j) = s + F{ifi), suppose, 
 
 which is in the proper form for the application of Lagrange's theorem. 
 
 Hence, we have 
 
CALCULATION OF FORMS OF DROPS. 33 
 
 dijc dz 
 Also if in the values oi x, z, -rj , -j^ in terms of <h, we change di into s, and 
 
 denote the results by 
 
 (^), (.), (g) and (g), 
 
 we have, by the same theorem, 
 
 In this way, we obtain 
 
 /I 629 487 1 N 
 
 ^ \80t)40 '^ ^ 5806080 '^ 5806080 '^ 737280^/' 
 
 + ( ^^3 7 17539 2n_ 1_ N 
 
 ^ V159667200 '^ ^ 2851200 '^ ^ 851558400 '^ 47308800 '^ ^ 88473600 ^ J 
 
 + &c., &c. 
 
 a; = 5 - g s' + (^ J20 - 40 ^j *' ~ (5040 ~ 280 ^ "^ 2688 ^ r ' 
 
 ( 1 __i B + ^^ff '^-~B^)s^ 
 
 1,362880 4480 ^ ^ 725760 '^ 82944 '^ / 
 
 / 1 1 p. 26617 , 1273 7 N 
 ^ ^ 319334400 ^ 15966720 '^ ^ 2703360 '^ / 
 
 + 
 
 V39916800 118800 ^ ' 319334400 
 + &c., &c. 
 
 f 1 61 3 151 ^3 1 3^ 3 
 
 """l 40320 '^64512'^ 107520 ^ 73728 ''^ T 
 
 / 1 19_ „ 14849 ^, 809 1 „A , 
 
 U628800 403200 '^ '*' 58060800 '^ 7257600 '^ "^ 7372800 ^ J * 
 
 ■ ^ 1 , 2477 ^ 2917 , 1264267 ^3 
 V 479001600 1916006400 '^ 121651200 '^ "^ 30656102400 '^ 
 
 42137 „4 1 o5 \ li 
 
 6812467200 "^ 1061683200 
 + &c., &c. 
 B. 
 
34 CALCULATION OF FORMS OF DROPS. 
 
 Also, we have 
 
 p ds 
 
 /I 629 487 1 A 
 
 "^ Vb960 ^ "^ 645120 ^ 645120 ^ "^ 81920 ^ / 
 
 / 233 7 17539 271 11 \ 
 
 U4515200^"^ 259200 '^ "^ 77414400 '^ 4300800 '^ 88473600^7 
 + &c., &c. 
 
 As before, it may be remarked that the coefficient of each power of s thus 
 found is exact, and not merely approximate. 
 
 We may also find these series for <j), x and z in terms of s independently, in 
 the following manner : 
 
 Assume, as we evidently may do, 
 
 - = 1 + c/ + c/ + c/ + c/ + cy + &c., 
 
 therefore </, = j J = s + _ c/ + - c/ + ^ c/ + ^ c/ + — cj' + &c., 
 
 since and s vanish together. 
 Hence we may find 
 
 ( 1 1 1 ,, j_ _j. _i \ , 
 
 U0320 360 ^^ "^ 36 ^^ "^ 30 ''^ 15 ^^^^ 7 ''v * 
 
 COS( 
 
 + 
 
 / 1 ]__ 1_ 2__L s 1 1 
 
 1,3628800 15120 °^ "*" 432 "^ 162 ''^ "^ 600 ^* ~ 30 ''^^^ 
 + 5{)'^'-42'''' + 2l'»''«+l4'" 
 
 + &c., &c., 
 and 
 
 "^ 1,362880 21 60 '^^ ^108 ''^ 1 62 ''^ "•■ 120 ''<' "" 15 ^^''^ ~ 14 ^» + 9 ''s j *° 
 ~ U99I68OO ~ 120960 ''^ "^ 2160 ''^'' ~ 324 ^''° + 3600 ''^ ~ 90 '^''^' 
 
 + &c., &c. 
 
 + 90'^'''^ + 50'^'-r68'»+^''^''» + ^^a-n 
 
 10 
 
 \ 
 
CALCULATION OF FORMS OF DROPS. 35 
 
 And therefore 
 
 I 3240 "» + 324 "'^ "*" 270 '^* 135 "^^^ 63 ''v * 
 
 1 
 
 + 
 
 362880 
 
 / 1 1_^ 1_ , 1 
 
 V39916800 166320 ''^"'' 4752"'^ 17!: 
 
 1 , 1 J_ 1 \ 
 
 "*" 550 "^ ~ 462 "'= "•■ 231 ''''^<' ^ 99 W 
 
 3 J_ ]_ 
 
 1782"'' "^6600''' 330 "^'^^ 
 
 + &c., &c.. 
 
 and 
 
 
 V40320 ~ 576 ''^ "^ 144 ''^ ''' 80 ^' 56 "V * 
 
 / 1 1_ 1 , 1 3 1 1 
 
 ''"(,3628800 21 600 ''2 "^ 1080 "" 1620 "" ^ 1200 *'' 150 "^''^ 
 
 ~140'''' + 96''«r 
 
 / 1 ]^_ '1 2__1 8 1 
 
 U79OOI6OO 1451520 "^ 25920 "^ 3888"^ ''■43200^' 
 
 1 1 , J_ 2_^_ 1 
 
 1080 "^"^ 1080 ^^ "' "^ 600 "' 2016 "» '*' 252 """^ 
 
 ^ 216 '^ 132 'V 
 + &c., &c. 
 
 Hence, we may find by division 
 
 ( 2 52 , 2 3 J6_ _J72 _^ ,1 \ 
 
 4 32 , 532 3 52 16 4 , 
 ■^ ' " 93555 '^^ ~ 66825 "^ 467775 "^ 155925 ''^ 3465 '''''' 567 ''^ ''^ 
 
 4725 " 14175 '^ 567 ' 4725 ' 4725^' 63 «^9 V 
 ,,4 296 BIN 
 
 24 , 4_ 296 __8_ 
 
 1925"* 1485"^ 10395"^"^ 297" 
 
 + &c., &c. 
 
 5—2 
 
36 CALCULATION OF FORMS OF DBOPS. 
 
 Substitute these expressions in the equation 
 
 p X 
 and equate the coefficients of corresponding powers of s, and we shall find successively, 
 
 ^^""48^ + 192^"' 
 
 « 5760*^ 1920 "^ 9216' 
 
 « ~ 8960 '^ 645120 '^ 645120 '^ ^ 81920 '^ ' 
 
 233 7 17539 ^^1 . 11 . 
 
 '» ~ 14515200 '^ ^ 259200 '^ ^ 77414400 '^ 4300800 '^ ^ 88473600 '^ ' 
 
 which agree with the coefficients of the several powers of s in the value of - which 
 has been already found in another way. 
 
 Also by the substitution of these coefficients in the expressions for x and z 
 given above, we shall obtain the same values of x and z as those which have been 
 before found. 
 
 By means of the above series, we may determine the values of x, z and ^ 
 for given values of s in the neighbourhood of the origin. As in the case where ^ 
 was taken as the independent variable, in order that the terms of these series which 
 involve higher powers of s may be insignificant, s must not exceed a certain 
 limiting value which will, of course, depend on the value of ^. The larger the 
 value of /3, the smaller will be this limiting value of s. 
 
 In order to find the values of x, z and ^ for larger values of s, we must 
 proceed step by step, as in the former case, the value of s being increased at each 
 step by a given small quantity, w suppose. 
 
 The interval q> should be so chosen that the series above found will give 
 sufficiently accurate values of x, z and ^ throughout several, say four or five such 
 intervals. 
 
 The process to be followed is exactly similar to that explained before, except 
 that in this case there are three quantities x, z and ^ to be determined by inte- 
 gration instead of the two x and z. 
 
 It is this circumstance only which makes it preferable to employ as the 
 independent variable in the case where this method is applicable, viz. when /9 is 
 a positive quantity. 
 
 The present method is equally applicable whether /S be positive or negative. 
 
CALCULATION OF FORMS OP DROPS. 37 
 
 Now, suppose ...s_5, s_4, s_^, s_^, s_^, s^ to be a series of consecutive values of s, 
 with the common difference co, and let ...^.j, ^.j, ^_g, ^_j, (/>_,, ^„ be the correspond- 
 ing values of ^, and 
 
 ■•■i''_5) ^-it ^_|) '''_2) ^_i> 3!oi 
 •••■^-6) ^-4) ^_3> ^_21 ^-11 ^0' 
 
 the corresponding values of the coordinates, and 
 
 •■•P-S> P-i, P-3, P-i, P.i,Po> 
 
 the corresponding radii of curvature. 
 The equations to be integrated are 
 
 ds p' 
 
 dx . 
 
 ^=cos.^, 
 
 dz 
 
 T- = sm 0, 
 
 ds 
 
 1 1 sin <f) „ „ 
 
 where - -| r = 2 + Bz. 
 
 p X 
 
 Suppose that the values of 0, x and z, and * consequently also the corresponding 
 values of the radius of curvature p, are known for the successive values of s up to s_,, 
 and we wish to find the value of each of these quantities for s = s„. 
 
 In the first place, we may obtain an approximate value of — in the following 
 
 Po 
 
 manner. 
 
 Tabulate the calculated values of - , and form their successive differences accord- 
 
 P 
 ing to the following scheme: 
 
 ± 
 P-B 
 
 a' 
 
 A^l 
 
 P-3 
 
 A'l 
 
 ... 
 
 1 
 
 P.4 
 
 A^ 
 
 P-3 
 
 A'-l 
 
 A^i 
 
 P-4 
 
 A^A 
 
 P-2 
 
 P-2 
 
 1 
 
 P-3 
 
 P-2 
 
 A^^ 
 
 P-3 
 
 A^ 
 
 A^-L 
 
 A»i- 
 
 P-i 
 
 
 P-^ 
 
 P-i 
 
 
 1 
 
 
 P-i 
 
 
 
 P-2 
 
 aJ- 
 
 P-i 
 
 
 
 
 1 
 
 
 
 
 
 P-l 
 
 
 
 
 
38 CALCULATION OF FORMS OF DROPS. 
 
 If 0) is taken sufficiently small, the dififerences as we proceed to higher orders 
 will rapidly diminish, and it will generally be easy by inspection of the two or three 
 last calculated fourth differences to fix upon an approximate value of the fourth 
 
 difference A*— immediately succeeding. 
 Po 
 
 Call this approximate value A* [ — J , so that the approximate is distinguished from 
 
 \Po' 
 
 the true vahxe by being inclosed in a parenthesis, and by successive additions form 
 
 1_ 
 
 P-2 A 
 
 A-i 
 
 
 
 © 
 
 Form the values of 
 
 ... dcj)_^, dj>_^, J0.3, d^_^, d(f>_^, 
 
 ... dx_^, dx_^, dx_^, dx_^, dx_^, 
 
 ... dz_^, dz_^, dz_^, dz_^, dz_^, 
 
 and of their successive differences, according to the following scheme : 
 
 O) 
 
 — = d^_, ^^^ A^d0 
 P^ A#., A»d,^.3 
 
 — =#-4 A^C?<^_3 b.'d<^^ 
 
 ''- Ld^_, A'd<^_, 
 
 — = <^<^-3 A»d<^_, A^c^d) , 
 P-^ A#., A=#., 
 
 — = #_a A^drf) , 
 ''- A#_. 
 
 w cos (^_5 = cZa;_5 ^^dx_^ A.*dx_. 
 
 Adx_^ A'dx_^ 
 
 o) cos (f)_^ = (Za;.^ A'da;,, A*dx_. 
 
 Adx_^ A^dx^ 
 
 CO cos ^_3 = (Za;_3 A'di^.^ A*da;_ 
 
 Adx^ A'dx_^ 
 
 ft) cos ^_2 = dx_^ ^dx_^ 
 
 Adx_^ 
 ft) cos ^_j = d^_j 
 
CALCULATION OP FORMS OF DROPS. 39 
 
 and 
 
 CO sin (j)_^ = dz_^ ^^dz_^ ^^^.^ 
 
 o) sin ^_^ = dz_^ ^dz^ ^.*dz , 
 
 Mz_^ A'dz_, 
 
 <a sin 0_3 = dz_^ AVa_jj A.*dz_^ 
 
 Adz_^ AWs.j 
 
 <a sin <f) „ = dz^ AWs , 
 
 to sin ^_, = dz_^ 
 If — and ^d were known, we might similarly form 
 
 Po 
 
 d(j}g = - , dxg = to cos <f>^ and dzj = c* sin ^j, 
 
 and the successive differences 
 
 M4>„ AV^„, Ay<^„, A*(^^„, &c., 
 AcZ«„, A^t^a;,,, A'c?a;„, A^c^aj^, &c., 
 AcZiT^, A'i^o, A'<?^„, A*&„, &c., 
 
 and then we should have, by what has been already proved, 
 
 </.„ - <^., = <?<^„ - 1 A#„ - i A''^. - 1 AV<^„ - ^ A*cZ.^„ - &c., 
 
 a?„ - a;., = (^^0 - 2 AcZ^o - j^ A^'dx, - ^^ A^dx^ - ^ A^t?a;„ - &c., 
 
 ^0 - •^-1 = «?^o - 2 ^^^0 - 12 ^''^^» ~ 24 '^°^^'' ~ 720 '^^'^^o - &c. ; 
 
 and when ^„, »„ and z^ were thus found, </>„ ought to agree with its assumed value, 
 and the values of ^„, a;„ and z^ should satisfy the equation 
 
 Po «0 
 
 which thus affords a verification of the value which was used for — . 
 
 Po 
 
 Now, let (cZ^o) ^® ^^ approximate value of d<p„, given by 
 
 (d<]>,) = CO (- 
 
 ^Po' 
 
 and let the successive differences found by employing (d<f>^ instead of dcp^ be denoted by 
 
 A (#„), A\dcl>X AX#o), AXdcp,), &c., 
 and suppose that (^J is given by the equation 
 
 (•^o) - </>-x = (cZ^o) - 1 A (#„) - 1 AXtZ,^„) - ^ A^((^,/,„) - ^ AXc^0„) - &c. 
 
40 CALCULATION OF FORMS OF DROPS. 
 
 Also, let (dx,) and (dz^) be approximate values of dx„ and dz^ respectively, given by 
 
 (c?«J = w cos (0„), 
 
 (dz,) = (o sin ((^„), 
 
 and let the successive differences found by employing {dx^ instead of dx^, and {dz^ 
 instead of dz^, be denoted by 
 
 A {dx,), A\dx,), A\dx,), A\dx,), &c., 
 and A (dz^, A^[dz^, A'{dz^), A\dz^), &c., respectively, 
 
 and suppose that {x„) and [z„) are given by the equations 
 
 (X,) - x_, = (dx,) - 1 A (dx,) - ^ A\dx,) - ^ A\dx,) - ^ A\dx,) - &c., 
 {z,) - z_, = (dz,) -\a {dz,) - i AXdz,) - i AXdz,) - ^ A%dz,) - Scci 
 
 Also let 
 
 "1 
 
 .Po^ 
 
 be found from the equation 
 and suppose that this gives 
 
 £] = ©-• 
 
 where e is a very small known quantity. 
 
 Then, if the true value of - =f-j+,y, the correction of the value of (d^„), and 
 therefore also that of the values of A(d^„), A'(d^„), A'(c?0„), AXdcfi,), &c., will be cor,. 
 
 Hence, if we stop at the terms which involve differences of the 4th order, we 
 shall have 
 
 9o - (9o) = 720 "''• 
 
 251 
 
 Wherefore cos ^„ = cos (^J - ^^q '"'? ^^" ("^o) 
 
 251 
 
 and sin <^„ = sin ((^„) + ^ w'? cos (^„). 
 
 Hence the correction to be applied to the values of (dx^), A (dx.), A^dxX AXdx ) 
 A\dx,), &c., will be 
 
 -72o'»'7Sin(0„), 
 
CALCULATION OF FORMS OF DROPS. 
 
 41 
 
 and the correction to be applied to the values of {dz^, A {dzX ^^{dz^, ^Xdz^), 
 A%dz„), &c., will be 
 
 720 « '7 cos (^„). 
 
 Whence if, as before, we stop at the terms which involve differences of the 4th 
 order, we shall have 
 
 /251 Y 
 ^0 - K) = - (^720 "J '^ ^"^^ ^^»^' 
 
 Hence, since 
 
 and 
 we find 
 
 /251 \* 
 ^0 - W = ^720 "j '' ^°® ^^»^' 
 
 sin (</)„) f/251 V . ,,J 
 /251 \^ 
 
 cos («^o) 
 
 but 
 
 Hence 
 
 1 ri-I _ sin (<^„) f /231 V • ij.\l 251 
 {—\='q, by supposition. 
 
 or 
 
 L „/251 \7sin(6.)V 251 cos (A.) ^ /251 V /jl\1 
 
 which gives 17. 
 
 Whence the values of — , 0(„ «„ and 5, become known. 
 
 251 
 
 If terms involving differences of the 5th order be included, the coefficient j^^ 
 
 95 
 in the above expressions must be replaced everywhere by hod- 
 
 As in the former case, the following slight modification of the above process will 
 be found convenient in practice. 
 
 B. 6 
 
42 CALCULATION OF FORMS OF DROPS. 
 
 Suppose (-) the assumed value of - to be increased by 100 units of the last place 
 of decimals employed, then while calculating the values of (#„), (<^o). ('^^o). (^^o)> K). (^o) 
 and the consequent value of [-1 , note at the side of the work the changes which 
 
 . /1\ 
 
 would be severally caused in each of these quantities by such an augmentation of \^-J . 
 
 As before/ it may be remarked that if we stop at the terms involving A* the 
 
 changes in {j>), W and (s„) will be ,^ times the changes in (d^„), {dx,) and {dz,) 
 
 251 
 
 respectively, and that j^ may be conveniently put under the form 
 
 251 
 
 If we also include the terms involving A', the coefficient y^Q ™"^* ^® replaced 
 
 1- 95 1/. 1\ 
 
 Now suppose that an increase of 100 units in (-A causes a diminution of /i units 
 
 in — I , and that the excess of 
 LPoJ 
 
 rection to be applied to the assumed value f — j will be 
 
 — above ( — ) is X of the same units, then the cor- 
 .PoJ W 
 
 and the correction to the value of 
 
 100 X , 
 
 —r- such units, 
 
 100 + /ti 
 
 1 
 
 Poi 
 XfM 
 
 will be 
 such units, 
 
 100 +/i 
 
 and the proportionate changes required ia the values of (d<j)„), ((/>„), (c^a;„), (dz^), («„) 
 and (zo) "will be at once found. 
 
 A numerical example of the method, when s is taken as the independent 
 variable, is given hereafter. 
 
 As before, we may, if it is found convenient, increase or diminish the interval 
 between the successive values of s. 
 
 It may be remarked, as before, that when, by means of the appropriate series, 
 
 we hav^ found the values of — for a sufficient number of small values of s, we can 
 
 form the corresponding value of d^, and thence derive the corresponding values of 
 d> by integration, and again by means of these we can find the corresponding values 
 of dec and dz, and thence derive by integration the corresponding values of x and z 
 without employing the series for those quantities, unless we choose to do so as a 
 means of verification. 
 
CALCULATION OF FORMS OP DROPS. 43 
 
 In the foregoing investigation, ^ is supposed to be expressed in the circular 
 measure, but in order to find cos <j) and sin from the tables, (p must be converted 
 into degrees, minutes and seconds. This can be readily done by means of special 
 tables for the purpose. 
 
 Or we may, if we choose, express dcj) at once in seconds by multiplying bj'' 
 206264'8, the number of seconds in the unit of circular measure, and thence find 
 the number of seconds in <f) by integration, without passing through the circular 
 measure. Thus if (f>" denote the number of seconds in ^, and if 7 denote the 
 number 206264-8, we shall have 
 
 deb" = - 7 cZs = - rJW, 
 
 P P 
 
 where log 7 = 5'3144251. 
 
 In conclusion, it may be worth while to say a few words in order to point 
 out the distinction between the method of integration above explained and that 
 which is commonly known under the name of "Integration by Quadratures." In 
 this latter method, we have to find y from the equation 
 
 where f{t) is a known function of t. 
 
 If we regard q as the ordinate of any point of a curve corresponding to the 
 abscissa t, then 1/ will be the area included between the curve, the axis of t, the 
 ordinate q, and some fixed ordinate. 
 
 In this case the values of q can be found, a priori, for any given values of t, 
 whereas in the more general case already treated of, where 5' is a function of y 
 as well as of t, the unknown quantities y and q must be found simultaneously, 
 and therefore we can only proceed step by step. 
 
 As the simpler case is included in the more general one, we may, of course, 
 still employ the same formula of integration that we have already obtained, but it 
 will be more advantageous to use a slightly different one. 
 
 If we denote the successive values of q by 
 
 •••9'«-2. 9'.-.. ^n, &c., 
 and if the corresponding values of y be denoted by 
 
 ••■2/n-2. 2/n-l. !/n' ^^■' 
 
 and if, in a notation similar to that already employed, the successive differences of 
 the quantities q be represented as in the following scheme: 
 
 2n+l 
 
 A?„ ^ A'g„,, A^g,,,, A\,, A\^, 
 
 ^\.. , AV„,, AVn« ^^ AV„« &c., 
 
 A?„« A\^, A^?n.3 A'g„^ A\,, 
 
 6—2 
 
44 CALCULATION OF FORMS OF DROPS. 
 
 then, as is before proved, we shall have 
 
 Vn ~ yn-i~h'^^ between the limits t = t^ + {n—1) a and t=-t„ + na 
 ( 1a 1 a2 1 as 19 a4 3 ., 863 .^ 
 
 275 , 33953 ^ 8183 ^ _ x, \ 
 
 24192^" 3628800^" 1036800^" ]' 
 
 If we transform this series into one which contains only such differences as, in the 
 above scheme, occur in the same horizontal lines as q^ and ^q^, we shall find 
 that the coefficients of the successive differences ultimately diminish much more 
 rapidly' than before. 
 
 When the differences of higher orders than the 9 th are neglected, it is- readily 
 shewn that the above series is equivalent to 
 
 r 1 A 1 AS 1a, 11 A4 11 A« 
 
 J/n - ^'n-i = « |9'. - 2 ^?» - 12 ^ 2»« + 24 ^'*+' "^ 720 ^"^^ ~ 1440 ^"*^ 
 
 - 191 .5 191 , 2497 .8 _ 2497 , _ «. 1 
 
 60480 ^»+»"'' 120960 ^'■+="^3628800 ^-*-* 7257600 ^»« J" 
 
 Similarly, by repeated applications of this formula, we have 
 
 191 e 191 2497 8 2497 ^^ ) 
 
 60480 ^"+2 ■•" 120960 ^»« ■*" 3628800 ^"^ 7257600 ^"^ ~ '^j ' 
 &c., &c. 
 
 191 AC-. 191 A7 2497 8 2497 ^^ o 1 
 
 60480 ^"^ "^ 120960 ^""^ "*" 3628800 ^"^ 7257600 ^'"+'> ~ ' j ' 
 
 Adding all these equations, and observing that 
 
 A?n + Ag„., + &c. + Ag',„,, = g„ -5„, 
 A'^«« + ^\ + &c. + A'q^^ = Ag„„ - A^„„, 
 A'?,.. + A'2„ + &c. + A'q^ = A»^„,, - A^g„,, , 
 &c. = &c. 
 
 ^\^ + AVs + &c. + A'-g^ = A»g„,, - A=g^„, 
 we obtain 
 
 yn-ym = W^t between the limits t = % + ma and « = i^4-nca^ 
 
 = " |?»Hi + g„,« + &c. + q,,, + ?„ - I (?„ - g,„) 
 
CALCULATION OF POBMS OF DROPS. 45 
 
 ~ 1440 ('^*^"+2 ~ ^'?m J - go480 (^°^«+8 - ^^Qm+s) + 120960 ^'^°5'»+3 - ^'9mJ 
 
 2497 , '>497 ) 
 
 + 3628800 ^^ ^"^ - ^'^"•-*) - 7257600 ^^'^-^^ ~ ^'^™«) " '^°j ' 
 
 in the first line of which expression 
 
 ^rr^i + ?™« +&c. + q^-^ (q^ - qj 
 may be replaced by 
 
 i (?« + 2«) + g'™^ + ff„« + &c. + q^_,. 
 
 Also, by substituting for the dififerences of odd orders in the series for 2/„-y„_i, 
 viz. by putting 
 
 &c. &c., 
 we obtain 
 
 Vn - y«-i = » { J fe + ?„- J - ^ (A^?„« + A'qn) + ^ (A\^, + A*g^,) 
 
 191 2497 1 
 
 - 120960 (^°^- + A'?„ J + ^25^^00 (^'?- + ^'?-) " &c.| , 
 
 and similarly, by substituting for the differences of even orders in the series for 
 
 2/«-2/m' viz. by putting 
 
 A'?^i = A?„^, - Aq-^, A\^^^ = Ag„^, - A?„, 
 A'g,.« = A'^,^ - A'^„^,„ A'q^ = A^g^ - A=g„^„ 
 &c. &c., 
 
 we obtain 
 
 2/« - 3^^ = « {2 (ff- + ffJ + 2»« + ?„+2 + &c. + q,_. 
 
 - 24 (^?" + A?„«) + 24 (A?„ + Aq^^J 
 
 + liio (^'^«« + ^'O - 1^ (A^?^« + A'q^J 
 
 mlQl 
 ~ 120960 ^ ^"+' "^ ^''+'' 120960 ^ ^"^^ "^ ^"^^^ 
 
 9497 2497 
 
 + 7^7600 ("^'^"^ + '^'^''-) - 7257600 (^'^™« + ^'^-^^ " '^'•. 
 
 When, by means of the method before explained, we have found a series of 
 successive values of q, viz. 
 
 ?m. ffm+l. &C-. 9'n-l' ?n. 
 
 together with the differences of odd orders which are immediately contiguous to 
 the horizontal lines through q„ and q„, we may advantageously employ the formula 
 just obtained in verification of the value of y„ — 2/„ previously found. 
 
46 
 
 CALCULATION OF FORMS OF DROPS. 
 
 Weddle's approximate formula for the area of a curve which is divided into 
 6 portions by 7 given equidistant ordinates*, viz. 
 
 2/6-2/0= |^{?o+9'. + ?*+26+^(9'i + ?5) + 6?3l. 
 has likewise been found to afford a convenient means of verification. 
 
 Note. In the reference made at the top of p. 31 to Bertrand's paper, the page 
 referred to should be 208 instead of 185, the latter being the page at which the paper 
 begins. 
 
 EXAMPLE OF THE METHOD, WHEN IS TAKEN AS THE 
 INDEPENDENT VARIABLE. 
 
 Suppose that /8 = 6, and that the values of x and z, and also that of p, have 
 been already calculated for values of ^ at intervals of 2|° from 0° to 321", and 
 that we wish to find the values of the same quantities for ^ = 35°. 
 
 Here 
 
 CO = the circular measure of 2^°, 
 = 0-04363323, 
 log o) = 8-6398174. 
 
 In the first place calculate a table giving the logarithms of w cos <f>, to sin ^ 
 and sin <^ for values of <^ at intervals of 2^°. Thus for = 35° the calculation 
 will be 
 
 (0 8-6398174 8-6398174 
 
 cos^ 9-9133645 sin^ 97585913 
 
 8-5531819 
 
 8-3984087 
 
 The following is a portion of the table. 
 Table A. 
 log (&) cos ^) log (to sin (f) 
 
 Boole's Finite Differences, p. 39. 
 
 log (sin ^) 
 
 30° 
 
 8-5773480 
 
 8-3387874 
 
 9-6989700 
 
 32| 
 
 8-5658466 
 
 8-3700339 
 
 9-7302165 
 
 35 
 
 8-5531819 
 
 8-3984087 
 
 9-7585913 
 
 37i 
 
 8-5392841 
 
 8-4242645 
 
 9-7844471 
 
 40 
 
 8-5240714 
 
 8-4478849 
 
 9-8080675 
 
CALCULATION OF FORMS OF DROPS. 
 
 47 
 
 Next, collect in a table the values of log/9 for the successive values of up 
 to ^ = 32^°, and find their differences to the 4th order, thus 
 
 
 
 Table I. 
 
 
 
 <^ 
 
 log/) 
 
 A 
 
 A' 
 
 A= 
 
 A* 
 
 
 
 22i 
 
 9-8858160 
 
 - 198034 
 
 
 + 2502 
 
 
 25 
 
 9-8660126 
 
 - 199279 
 
 -1245 
 
 + 1869 
 
 - 633 
 
 271 
 
 9-8460847 
 
 - 198655 
 
 + 624 
 
 + 1449 
 
 - 420 
 
 30 
 
 9-8262192 
 
 - 196582 
 
 + 2073 
 
 (+ 1049) 
 
 - (400) 
 
 32| 
 
 9-8065610 
 
 (-193460) 
 
 (+ 3122) 
 
 
 
 35 
 
 (9-7872150) 
 
 
 
 
 
 In order to find an approximate value of log p for (f> = 35°, assume a value for 
 the 4th difference immediately following those already found in the table, and by 
 means of this form successively the approximate values of the 3rd, 2nd and 1st 
 differences and of the log p for <f} = 35°. These values are placed in parentheses to 
 indicate that they are only approximate. 
 
 In the above case, we have assumed the next 4th difference to be — 400. 
 
 Collect in a table the values of dx = pm cos ^ for the successive values of ^ up 
 to ^ = 32^°, forming them by means of the known values of log p and the logarithms 
 in the 1st column of Table A. 
 
 Add to this table the approximate value of dx for ^ = 35°, . formed by means of 
 the approximate value of logp just found, and find the differences of these quan- 
 tities to the 4th order, thus 
 
 
 
 Table II. 
 
 
 
 <t> 
 
 dx 
 
 Adx A'dx 
 
 A'dx 
 
 A*da; 
 
 22I 
 
 -03099194 
 
 + 2802 
 - 194464 
 
 + 2314 
 
 -968 
 
 23 
 
 -02904730 
 
 + 5116 
 - 189348 
 
 + 1453 
 
 -861 
 
 27i 
 
 •02715382 
 
 + 6569 
 - 182779 
 
 + 868 
 
 -585 
 
 30 
 
 •02532603 
 
 + 7437 
 - 175342 
 
 (+ 406) 
 
 (- 462) 
 
 32i 
 
 •02357261 
 
 (+ 7843) 
 (- 167499) 
 
 
 
 35 
 
 (•02189762) 
 
 
 
 
48 
 
 CALCULATION OF FORMS OF DROPS. 
 
 For <f) = 35° the calculation will be 
 
 log(p) 9-7872150 
 log(fflcos0) 8-5531819 
 
 + 100 
 
 log 8-3403969 
 
 (dx) -02189762 + 50-3 
 
 The change of (dx) in units of the last decimal place, which would be produced 
 by an increase in log (p) of 100 units in the last decimal, is placed at the side. 
 
 Similarly, collect in a table the values of dz = pa sin tji for the same values of 
 <j>, forming them by means of the logarithms in the 2nd column of Table A. Add 
 to this table the approximate value of dz for (f) = 35°, and find the differences, as 
 before, to the 4th order, thus 
 
 
 
 Table 
 
 III. 
 
 
 
 <!> 
 
 ds 
 
 Adz 
 
 A'dz 
 
 AWa 
 
 A*dz 
 
 22°J 
 
 •01283728 
 
 + 70770 
 
 - 13145 
 
 + 1416 
 
 + 79 
 
 25 
 
 •01354498 
 
 + 59041 
 
 - 11729 
 
 + 1348 
 
 - 68 
 
 274 
 
 ■01418539 
 
 + 48660 
 
 - 10381 
 
 + 1262 
 
 - 86 
 
 30 
 
 •01462199 
 
 + 39541 
 
 - 9119 
 
 (+ 1126) 
 
 (- 136) 
 
 32| 
 
 •01501740 
 
 (+ 31548) 
 
 (- 7993) 
 
 
 
 35 
 
 (•01533288) 
 
 
 
 
 
 For <}) = 35° the calculation will be 
 
 log{p) 9-7872150 +100 
 log (g) sin ^) 8-3984087 
 
 log 8-1856237 
 
 (dz) -01533288 + 35-3 
 
 the change of (dz) for an increase of 100 units in log (p) is placed at the side. 
 
 Collect in two other tables the successive values of w and s which have been 
 already computed, and form the differences of these quantities to the 4th or 5th 
 orders, by which means any error of consequence that may have crept into the work 
 will at once become apparent. 
 
CALCULATION OF FORMS OF DEOPS. 
 
 49 
 
 From the values of {dx), (dz) and their differences, and from the known values 
 of a; and 3 for ^ = 22^", find tlie approximate values of x and z ior (p = 35°, thus 
 
 <^ = 32r, X 
 
 •45780303 
 
 
 <b = S5\{dx) 
 
 •02189762 
 
 
 - i A {dx) 
 
 83749,5 
 
 50,3 
 
 - tV ^Xdx) 
 
 - 658,6 
 
 2,5 
 
 - ^ ^\dx) 
 
 -16,9 
 
 - ,2 
 
 -r^^^\dx) 
 
 + 12,2 
 
 3) 52,6 
 
 </. = 35",(x) 
 
 •48053156 
 
 17,5 
 
 For <j) = 32^", z -12246288 
 
 
 ^ = 35", (dz) •01533288 
 
 
 - i A(dz) -15774 
 
 35,3 
 
 - iV ^\dz) 666,1 
 
 1,8 
 
 - ^V ^"(dz) - 46,9 
 
 -,2 
 
 -r%^\dz) 3,6 
 
 3) 36,9 
 
 (z) -13764424,8 
 
 12,3 
 
 6 
 
 6 
 
 /3 (z) -82586549 
 
 74 
 
 In order to prevent an accumulation of small errors, the quantities involving 
 A, A", A' and A* are carried to one place of decimals beyond those which are 
 ultimately retained. 
 
 At the side are calculated the changes of (x) and (s), in units of the 8th place 
 of decimals, which would be required if log (p) were increased by 100 units of the 
 7th place of decimals. 
 
 These changes are found by multiplying the corresponding changes of (dx) and 
 (dz) already found by 
 
 720 3 1 ^ 20 \ 12yj ■ 
 
 Next, from (x) and (s) find j^ and log [p] by the formula 
 
 LrJ 
 
 ^ = ^^^«-'^*' 
 
 /8 being here = 6. 
 
 Table (A), log sin 35° 9-7585913 
 
 log (x) 9-6817219 1,5 
 log 0-0768694 - 1,5 
 
 N°. 1-1936290 
 
 2 + 6 (z) = 2-8258655 
 
 subtract 1-1936290 
 
 1 
 
 [p] 
 
 1-6322365 
 
 log PI 0-2127831 
 
 7,4 
 -4, 
 
 11,4 
 3,0 
 
 or hg[p] 9-7872169 -3,0 
 
 log (p) 9-7872150 
 Difference 19 
 
 The numbers placed at the side are the changes which would be caused by 
 the increase in log (p) before mentioned. 
 
 B. 7 
 
50 CALCULATION OF TORMS OF DROPS. 
 
 It should be remarked that in this last calculation the quantity 2 + 6 (^) is 
 only carried to 7 places of decimals, whereas in the above calculation 6(z) was given 
 to 8 places. 
 
 Consequently the change of 2 + 6 (2) or that of 6 (z) before found must be divided 
 by 10, in order to reduce it to units of the 7th decimal. 
 
 "We see that the value thus found for log [p] exceeds the assumed value of 
 log(/3) by 19 units in the 7th decimal place. 
 
 Hence the correction to be applied to log(p) will be 
 
 19 X r— K = 18,4 such units, 
 J.Uo 
 
 and the corrected value of hgp for = 35" will be 
 
 •9-7872168, 
 which may now be added to the numbers in Table I. 
 
 Similarly, the corrections to be applied to the values of (dx) and (dz) will be 
 50,3 X -184 = 9,3 and 35,3 x -IS* = 6,5 
 respectively, in units of the 8th decimal, so that the corrected values will be 
 
 d.« = -02189771 and d^= -01533294, 
 which may be added to the numbers in Tables II. and III. 
 
 The successive differences of log (p) will of course require the same correction as 
 log(/3) itself, and similarly for the differences of {dx) and (dz). 
 
 Also, the corrections to be applied to the values of (x) and («) will be 
 
 17,5 X -184 = 3,2 and 12,3 x "184 = 2,3 
 
 respectively, in units of the 8th decimal, so that the corrected values will be 
 
 a; = -48053159 and ir = -13764427, 
 
 which may be added to the Tables of the collected values of x and s respectively. 
 
 The provisional values of log (p), {dx) and {dz) and of their respective differences, 
 which in the foregoing example have been inclosed in parentheses, may in the actual 
 work be merely written in pencil, so that, when they have served their purpose, they 
 may be easily effaced, and then replaced by the corrected values written in ink. 
 
 The Volume V of the portion of the drop corresponding to ^ = 35", is at 
 once found by the formula 
 
CALCULATION OF FORMS OF DROPS. 
 
 51 
 
 thus. 
 
 - 1-6322367 
 P 
 
 !Ei 1-1936290 
 
 0-4386077 
 
 log 9-6420762 
 
 so" log 9-3634438 
 
 •TT log 0-4971499 
 
 9-5026699 
 
 jSlog 07781513 
 
 log 8-7245186 
 
 V -05302963 
 
 In this as -well as in the former part of the -work of this example, b is 
 supposed to be unity, so that in the general case the quantities above denoted by 
 
 p, X, z and V will be replaced by ^ 
 
 ^ , |, I and p respectively. 
 
 The following shews the application of the formula of correction 
 
 6 
 
 
 V- 
 
 ,251 , ,, . 
 1+720'"^^°)^'" 
 
 ^<Po 
 
 
 this example. 
 
 
 
 COS(/)„ 
 
 9-91336 
 
 loga;„ 
 
 9-68172 
 
 
 
 9-36344 
 0-54992 
 3-5475 
 6- 
 
 251 
 
 2-39967 
 
 
 
 9-6475 
 
 720 
 
 2-85733 
 
 
 log 
 
 0-97989 
 
 
 9-54234 ■ 
 
 
 iPoT 
 
 to 
 
 m 
 
 9-57443 
 9-75859 
 8-63982 
 9-54234 
 
 8-49507 
 
 ' 
 
 
 
 
 •031266 
 
 
 
 
 
 1-C 
 
 31266 
 
 Hence 
 
 V = 
 
 1-031 
 
 nearly. 
 
 which agrees very well with the result found by the other method. 
 
 7—2 
 
52 
 
 CALCULATION OF FORMS OF DROPS. 
 
 EXAMPLE OF THE METHOD, WHEN s IS TAKEN AS THE 
 INDEPENDENT VARIABLE. 
 
 Suppose that it is required to calcxilate the theoretical form of a pendent drop 
 of fluid, where yS = — 0-5. 
 
 As in the former example, we will suppose that 6 = 1. 
 
 First, putting /S = — 0'5 in the general formula for - , we obtain for points near 
 
 the origin 
 
 - = 1 - 0-1875 s= + 0-01692,7083 s* - 0-00248,2096 s' 
 P 
 
 + 0-00028,3075 / - 000003,3537 s" + &c., 
 
 which series is sufficient to give — to 9 or 10 places of decimals, provided s do 
 
 not exceed 0-4. 
 
 The value of - is the same for corresponding positive and negative values of s, 
 
 so that if we put s = 0, s = + 0-1, s = + 0-2, s = + 0-3 and s = + 0-4 in succession, 
 we may obtain 
 
 s 
 
 
 1 
 
 P 
 1- 
 
 + 0-1 
 ±0-2 
 ±0-3 
 ±0-4 
 
 0-99812,67 
 0-99252,69 
 0-98326,03 
 0-97042,33 
 
 Also <f> = j-ds = s- 0-0625 s' + 0-00338,54166 s' - 000035,458o s' 
 
 + 0-00003,1453 s' - 0-00000,3049 s" + &c. 
 
 Similarly, putting /3 = — O'o in the series for x and z respectively, we obtain 
 
 x = s- 0-1666 s" + 0-02083,3 s' - 0-00244,9157 s' + 0-00029,4734 s» 
 - 0-00003,5199 s" + &c., 
 
 z = 0-5s^- 0-05729,16 s* + 0-00716,14583 s» - 0-00085,03747 s' 
 + 0-00010,17165 s" - 0-00001,21847 s" + &c. 
 
 From which we may find 
 
 s 
 
 <f> (in circ. measure). 
 
 (j} (in deg. &c.) 
 
 X 
 
 z 
 
 
 
 00 
 
 0° 0' 0" 
 
 0-0 
 
 0-0 
 
 ±01 
 
 ± 0-09993,753 
 
 ± 5 43 33-596 
 
 ± 0-09983,354 
 
 0-00499,428 
 
 ±0-2 
 
 ± 0-19950,108 
 
 ± 11 25 50051 
 
 . ± 0-19867,330 
 
 0-01990,879 
 
 ±0-3 
 
 ± 0-29832,065 
 
 ±17 5.33-051 
 
 ± 0-29555,009 
 
 0-04454,110 
 
 ±0-4 
 
 + 0-39603,409 
 
 ± 22 41 27-896 
 
 ± 0-38954,273 
 
 007856,212 
 
CALCULATION OF FORMS OF DROPS. 53 
 
 Now, let us suppose that the values of p, ip, x and z have been already calcu- 
 lated for s=0'l, s = 0'2 and s = 0'3, and that we wish to find the values of the same 
 quantities for s = 0'4 by the foregoing method of integration. 
 
 Here we have w = O'l. 
 
 From the given values of - we may find the corresponding values of d<f>=(o-, 
 and their successive differences, as shewn in the following Table : 
 
 s 
 
 d^ 
 
 Arf0 
 
 A'd^ 
 
 A'd<f> 
 
 A*dcl> 
 
 A'dcj, 
 
 -0'3 
 
 0-09832,603 
 
 92,666 
 
 
 
 
 
 -0-2 
 
 0-09925,269 
 
 55,998 
 
 -36,668 
 
 -0,597 
 
 
 
 -01 
 
 0-09981,267 
 
 18,733 
 
 - 37,265 
 
 -0,201 
 
 + 0,396 
 
 +,006 
 
 0-0 
 
 0-1 
 
 - 18,733 
 
 -37,466 
 
 + 0,201 
 
 + 0,402 
 
 -,006 
 
 0-1 
 
 0-09981,267 
 
 - 55,998 
 
 - 37,265 
 
 + 0,597 
 
 + 0,396 
 
 (-,018) 
 
 0-2 
 
 0-09925,269 
 
 - 92,666 
 
 - 36,668 
 
 (+0,975) 
 
 (+0,378) 
 
 
 03 
 
 0-09832,603 
 
 (-128,359) 
 
 (-35,693) 
 
 
 
 
 0-4 
 
 (0-09704,244) 
 
 
 
 
 
 
 If we supply another line of differences by supposing the 6th difference to be 
 constant, we shall obtain the quantities included in parentheses in the above Table, 
 
 and the corresponding assumed value of .-^ for s = 0-4 will be 0-97042,44. 
 
 From the values of (d^) and its differences, and the known value of <J3 for s = 0-8, 
 find the approximate value of <f> ior s= 0-4, thus 
 
 For s = 0-3, -29832,065 
 
 s = 0-4, d(l) -09704,244 
 
 - i Adcj) + 64,179,5 
 
 - rVAWd!) + 2,974,4 ^ . . ^ , , . , , 
 
 _ 1 A'fJrh — 040 fi ^ ^^ units of 8th decimal place 
 
 -■^A'd<f> - ,010 ^ 
 
 -jhs^'d,^ + ,000 ,4 3)^ 
 
 (^) -39603,413 ^ 
 
 or 22° 41' 27"-903 0"-068 
 
 The changes placed at the side correspond to an increase in j— of 100 units 
 in the 7th decimal place. 
 
 As the interval a is rather large, we have taken into account the terms in A". 
 
.54 
 
 CALCULATION OF FORMS OF DROPS. 
 
 "With this value of (0) we calculate the corresponding values of {dx) and 
 {dz), thus 
 
 logcos(^) 9-9650125 -0,6 \ogsm{j>) 9-5863199 +3,4 
 
 log CO 9- log 0) 
 
 9^ 
 
 8-9650125 
 
 9; 
 
 8-5863199 
 {dz) -03857624 +3,0. 
 
 {dx) -09225980 - 1,2 
 
 The small quantities at the side are the changes of the quantities opposite to 
 which they stand, in units of the last decimals respectively employed, which would 
 be caused by an increase of 0"-06S in {<f), or by an increase of 100 units in the 7th 
 
 decimal place of -p-- . 
 {p) 
 
 With these values of {dx) and {dz) and the previously calculated values of dx 
 
 and dz for s=0-l, s = 0-2 and s = 0-3, we may form the following Tables: 
 
 s 
 
 dx 
 
 Ldx 
 
 6?dx 
 
 ^''dx 
 
 A'cZic 
 
 t^'dx 
 
 -0-3 
 
 -09558,313 
 
 243,344 
 
 
 
 
 
 -0-2 
 
 -09801,657 
 
 148,449 
 
 - 94,895 
 
 - 3,660 
 
 
 
 -01 
 
 •09950,106 
 
 49,894 
 
 - 98,555 
 
 - 1,233 
 
 + 2,427 
 
 + 39 
 
 0- 
 
 -1 
 
 - 49,894 
 
 - 99,788 
 
 + 1,233 
 
 + 2,466 
 
 -S9 
 
 01 
 
 -09950,106 
 
 - 148,449 
 
 - 98,555 
 
 + 3,660 
 
 + 2,427 
 
 (-181) 
 
 0-2 
 
 •09801,657 
 
 -243,344 
 
 - 94,895 
 
 (+ 5,906) 
 
 ( + 2,246) 
 
 
 0-3 
 
 ■09558,313 
 
 (- 332,333) 
 
 (- 88,989) 
 
 
 
 
 0-4 
 
 (-09225,980) 
 
 
 
 
 
 
 s 
 
 dz 
 
 ^dz 
 
 ^'dz 
 
 L'dz 
 
 AV0 
 
 ^'dz 
 
 -0-3 
 
 --02939,154 
 
 957,351 
 
 
 
 
 
 -0-2 
 
 -•01981,803 
 
 984,091 
 
 26,740 
 
 -13,119 
 
 
 
 -0-1 
 
 - -00997,712 
 
 997,712 
 
 13,621 
 
 - 13,621 
 
 -302 
 
 502 
 
 -0- 
 
 -0 
 
 997,712 
 
 
 
 - 13,621 
 
 
 
 502 
 
 01 
 
 + -00997,712 
 
 984,091 
 
 -13,621 
 
 -13,119 
 
 + 502 
 
 (476) 
 
 0-2 
 
 •01981,803 
 
 957,351 
 
 - 26,740 
 
 (- 12,141) 
 
 (+978) 
 
 
 0-3 
 
 •02939,154 
 
 (918,470) 
 
 (-38,881) 
 
 
 
 
 0-4 
 
 (-03857,624) 
 
 
 
 
 
 
CALCULATION OF FORMS OF DROPS. 
 Hence we find x and a for s = 0-4, thus 
 
 55 
 
 For s = 0-3, x = 
 
 •29555009 
 
 For s = 0-4, dx = 
 
 •09225980 
 
 — \/^dx 
 
 166166,5 
 
 - ^A'dx 
 
 7415,7 
 
 - T^A'dx 
 
 - 246,1 
 
 -a^^'dx 
 
 - 59,3 
 
 -ih^'dx 
 
 3,4 
 
 (X) 
 
 •38954269 
 
 -0,4 
 
 For s = 0-3, z = 
 
 = -04454110 
 
 For 5 = 0-4, ds 
 
 •03857624 
 
 - |Ac?^ 
 
 - 459235 
 
 - ^\'dz 
 
 3240,1 
 
 - ^^A'dz 
 
 505,9 
 
 -h%A'dz 
 
 -25,8 
 
 - liv^'dz 
 
 - 8,9 
 
 (z) 2) 
 
 -07856210 +1,0 
 
 -0z 
 
 -03928105 0,5 
 
 The changes in (x) and (z) are found by multiplying those in (dw) and (dz) 
 respectively, by -^^^ or ^ nearly. 
 
 Next, with these values of (<f)), (x) and («), find -^ by the formula 
 
 LrJ 
 
 
 1 9,o/^N siiiW 
 
 8 being hi 
 
 ;re =-0-5. 
 
 
 .g sin (<^) 
 log (x) 
 
 9-5863199 + 3,4 
 
 9-6905551 0,0 
 
 9-9957648 3,4 
 
 •9902955 7,7 
 
 
 2 + /3(^) 
 subtract 
 1 
 
 w 
 
 = 1-9607189,5 
 •9902955 
 
 -0,05 
 
 7,7 
 
 N". 
 
 0-9704234,5 
 
 -7,75 
 
 
 
 
 but i. 
 
 (p) 
 
 . difference 
 
 = 0-9704244 
 
 
 
 
 .• 
 
 9,5 
 
 
 It will be seen that in the above we have expressed the change of 2 + /S (z) 
 in units of the 7th decimal place instead of the 8th decimal as before, so that 
 the number found before has been divided by 10. 
 
 The value thus found for 
 units in the 7th decimal place. 
 
 T^^ is less than the assumed value of 7^ by 9,5 
 W (P) ^ 
 
 Hence the correction to be applied to pr will be 
 
 -9,5 
 
 100 
 
 107,75 
 
 = — 9 such units, 
 
 and the corrected value of - will be 0-9704235. 
 
 P 
 
 Similarly, the correction to be applied to the value of (^) will be 
 
 - -09 X "-068 = - "-006, 
 so that the corrected value of ^ will be 22° 41' 27"-897. 
 
56 
 
 CALCULATION OF FORMS OF DROPS. 
 
 Also the corrections to be applied to the values of (x) and (z) respectively, 
 will be 
 
 - -09 X - 0,4 and - "09 x 1,0 
 
 in^units of the 8th decimal place, both of which are insensible. 
 
 These results agree very well with the more accurate ones found before by 
 the use of series. 
 
 The Volume V is found by the formula 
 
 (-/3) \ X pj 
 thus, 
 
 sin<^ 
 
 X 
 
 0-9902954 
 
 1 
 P 
 
 0-9704235 
 
 00198719 
 
 log 8-2982394 
 
 a;" log 9-1811102 
 
 IT 0-4971499 
 
 7-9764995 
 
 (-^)log 9-6989700 
 
 log 8-2775295 
 
 V -0189465 
 
 Application of the formula of correction 
 
 r,-/95 V/sin(<^„)V, 95 cos(<^„) ^ / 95 V , , J 
 ^ = "11+^288") (-(^J +288'"-(^-^(288'"j^°^(M 
 to this example. 
 
 ^ log 9-51833 
 a 9- 
 
 ^^^<f°^ log 9-99576 
 8-51833 
 
 i.^h 
 
 0))= log 7-03666 
 cos(^„) 9-96501 
 
 8-51833 
 
 8-51409 
 
 
 -/S 9-69897 
 
 cos ((/.„) 9-96501 
 
 2 
 
 
 6-70064 
 
 8-48334 
 
 7-02818 
 
 
 
 (x) 9-59055 
 
 2 log 0-30103 
 
 
 
 8-89279 
 
 7-32921 
 
 
 
 •078125 
 
 
 
 
 •002134 
 
 
 
 
 •000502 ■ 
 
 
 
 
 •080761 
 
 Hence 
 
 1? = 
 
 1-081 ' 
 
 which very nearly agrees with the result found before by the other method. 
 
TABLES 
 
 FOB FACILITATING THE CALCULATION OF 
 
 li^* AA^ &c. 
 720 ' 160 ' 
 
58 
 
 TABLES. 
 
 Table shewing the value of A* which corresponds to each unit in the value of 
 
 19_A^ L_ 
 
 720 37-8947 ' 
 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 o 
 
 I 
 
 2 
 
 3 
 
 
 
 379 
 
 758 
 
 1137 
 
 38 
 
 417 
 796 
 
 "75 
 
 76 
 
 455 
 
 834 
 
 1213 
 
 114 
 
 493 
 
 872 
 
 1251 
 
 152 
 
 531 
 
 909 
 
 1288 
 
 189 
 568 
 
 947 
 1326 
 
 227 
 
 606 
 
 985 
 
 1364 
 
 265 
 
 644 
 
 1023 
 
 1402 
 
 303 
 
 682 
 
 1061 
 
 1440 
 
 341 
 
 720 
 
 1099 
 
 1478 
 
 
 
 I 
 2 
 3 
 
 4 
 
 5 
 6 
 
 1516 
 
 1895 
 2274 
 
 1554 
 1933 
 2312 
 
 1592 
 1971 
 
 2349 
 
 1629 
 2008 
 2387 
 
 1667 
 2046 
 2425 
 
 1705 
 2084 
 2463 
 
 1743 
 2122 
 2501 
 
 1781 
 2160 
 2539 
 
 1819 
 2198 
 
 2577 
 
 1857 
 2236 
 2615 
 
 4 
 
 5 
 6 
 
 7 
 8 
 
 9 
 
 2653 
 3032 
 3411 
 
 2691 
 3069 
 3448 
 
 2728 
 3107 
 3486 
 
 2766 
 
 3145 
 3524 
 
 2804 
 3183 
 3562 
 
 2842 
 3221 
 3600 
 
 2880 
 
 3259 
 3638 
 
 2918 
 
 3297 
 3676 
 
 2956 
 
 3335 
 3714 
 
 2994 
 
 3373 
 3752 
 
 7 
 8 
 
 9 
 
 lO 
 
 II 
 
 12 
 
 3789 
 4168 
 
 4547 
 
 3827 
 4206 
 4585 
 
 3865 
 
 4244 
 4623 
 
 3903 
 4282 
 4661 
 
 3941 
 4320 
 4699 
 
 3979 
 4358 
 4737 
 
 4017 
 4396 
 
 4775 
 
 4055 
 4434 
 4813 
 
 4093 
 4472 
 4851 
 
 4131 
 4509 
 4888 
 
 10 
 II 
 12 
 
 13 
 
 14 
 15 
 
 4926 
 
 5305 
 5684 
 
 4964 
 
 5343 
 5722 
 
 5002 
 5381 
 5760 
 
 5040 
 
 5419 
 5798 
 
 5078 
 
 5457 
 5836 
 
 5116 
 
 5495 
 5874 
 
 5154 
 5533 
 5912 
 
 5192 
 5571 
 5949 
 
 5229 
 5608 
 5987 
 
 5267 
 5646 
 6025 
 
 13 
 14 
 IS 
 
 16 
 
 18 
 
 6063 
 6442 
 6821 
 
 6101 
 6480 
 6859 
 
 6139 
 6518 
 6897 
 
 6177 
 6556 
 6935 
 
 6215 
 6594 
 6973 
 
 6253 
 6632 
 7011 
 
 6291 
 6669 
 7048 
 
 6328 
 6707 
 7086 
 
 6366 
 
 6745 
 7124 
 
 6404 
 
 6783 
 7162 
 
 16 
 
 17 
 18 
 
 19 
 20 
 
 21 
 
 7200 
 
 7579 
 7958 
 
 7238 
 7617 
 7996 
 
 7276 
 
 7655 
 8034 
 
 7314 
 
 7693 
 8072 
 
 7352 
 
 7731 
 8109 
 
 7389 
 7768 
 8147 
 
 ■7427 
 
 ■ 7806 
 
 8185 
 
 7465 
 7844 
 8223 
 
 7503 
 7882 
 8261 
 
 7541 
 7920 
 8299 
 
 19 
 20 
 21 
 
 22 
 23 
 24 
 
 8337 
 8716 
 
 9095 
 
 8375 
 8754 
 9133 
 
 8413 
 8792 
 91 7 1 
 
 8451 
 8829 
 
 9208 
 
 8488 
 8867 
 9246 
 
 8526 
 8905 
 9284 
 
 8564 ■ 
 
 8943 
 9322 
 
 8602 
 8981 
 9360 
 
 8640 
 9019 
 9398 
 
 8678 
 ■9057 
 9436 
 
 22 
 
 23 
 24 
 
 25 
 26 
 27 
 
 9474 
 
 9853 
 10232 
 
 9512 
 
 9891 
 
 10269 
 
 9549 
 
 9928 
 
 10307 
 
 9587 
 9966 
 
 10345 
 
 9625 
 10004 
 10383 
 
 9663 
 10042 
 10421 
 
 9701 
 10080 
 10459 
 
 9739 
 10118 
 10497 
 
 9777 
 10156 
 
 10535 
 
 9815 
 10194 
 
 10573 
 
 25 
 26 
 27 
 
 19 
 The units of =^ A* are placed at the top of the Table, and the tens at the side. 
 
TABLES. 
 
 59 
 
 Table shewing the value of A^ which corresponds to each unit in the value of 
 
 A A== ^ A^ 
 
 160 53-3333 
 
 
 o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 o 
 
 I 
 
 2 
 
 3 
 
 o 
 
 533 
 1067 
 1600 
 
 53 
 
 587 
 
 1120 
 
 1653 
 
 107 
 640 
 
 "73 
 1707 
 
 160 
 
 693 
 1227 
 1760 
 
 213 
 
 747 
 1280 
 1813 
 
 267 
 800 
 
 1333 
 1867 
 
 320 
 
 853 
 
 1387 
 
 1920 
 
 373 
 
 907 
 
 1440 
 
 1973 
 
 427 
 960 
 
 1493 
 2027 
 
 480 
 1013 
 
 1547 
 2080 
 
 
 I 
 2 
 3 
 
 4 
 5 
 6 
 
 2133 
 2667 
 3200 
 
 2187 
 2720 
 3253 
 
 2240 
 2773 
 3307 
 
 2293 
 2827 
 3360 
 
 2347 
 2880 
 
 3413 
 
 2400 
 2933 
 3467 
 
 2453 
 2987 
 3520 
 
 2507 
 3040 
 
 3573 
 
 2560 
 
 3°93 
 3627 
 
 2613 
 
 3147 
 3680 
 
 4 
 5 
 6 
 
 7 
 8 
 
 9 
 
 3733 
 4267 
 4800 
 
 3787 
 4320 
 
 4853 
 
 3840 
 
 4373 
 4907 
 
 3893 
 4427 
 4960 
 
 3947 
 4480 
 
 5013 
 
 4000 
 
 4533 
 5067 
 
 4053 
 4587 
 5120 
 
 4107 
 4640 
 5173 
 
 4160 
 
 4693 
 5227 
 
 4213 
 
 4747 
 5280 
 
 7 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 5333 
 5867 
 6400 
 
 5387 
 5920 
 
 6453 
 
 544° 
 5973 
 6507 
 
 5493 
 6027 
 
 6560 
 
 5547 
 6080 
 6613 
 
 5600 
 
 6133 
 6667 
 
 5653 
 6187 
 6720 
 
 5707 
 6240 
 
 6773 
 
 5760 
 6293 
 6827 
 
 5813 
 6347 
 6880 
 
 10 
 II 
 12 
 
 13 
 14 
 15 
 
 6933 
 
 7467 
 8000 
 
 6987 
 7520 
 8053 
 
 7040 
 
 7573 
 8107 
 
 7°93 
 7627 
 
 8160 
 
 7147 
 7680 
 8213 
 
 7200 
 
 7733 
 8267 
 
 7253 
 7787 
 8320 
 
 7307 
 7840 
 
 8373 
 
 7360 
 
 7893 
 8427 
 
 7413 
 7947 
 8480 
 
 13 
 14 
 15 
 
 i6 
 
 17 
 i8 
 
 8533 
 9067 
 9600 
 
 8587 
 9120 
 
 9653 
 
 8640 
 
 9173 
 9707 
 
 8693 
 9227 
 9760 
 
 8747 
 9280 
 
 9813 
 
 8800 
 
 9333 
 9867 
 
 8853 
 9387 
 9920 
 
 8907 
 9440 
 9973 
 
 8960 
 
 9493 
 10027 
 
 9013 
 
 9547 
 10080 
 
 16 
 
 17 
 18 
 
 The units of t^^' are placed at the top of the Table, and the tens at the side. 
 
 8—2 
 
60 
 
 TABLES. 
 
 Table shewing the value of A' which corresponds to each unit in the value of 
 
 863 .„ 1 .„ 
 
 60480 
 
 A'' = 
 
 70-0811 ' 
 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 o 
 
 
 
 70 
 
 140 
 
 210 
 
 280 
 
 350 
 
 420 
 
 491 
 
 S6i 
 
 631 
 
 
 
 I 
 
 701 
 
 771 
 
 841 
 
 911 
 
 981 
 
 1051 
 
 1121 
 
 1191 
 
 1261 
 
 1332 
 
 I 
 
 2 
 
 1402 
 
 1472 
 
 1542 
 
 1612 
 
 1682 
 
 1752 
 
 1822 
 
 1892 
 
 1962 
 
 2032 
 
 2 
 
 3 
 
 2102 
 
 2173 
 
 2243 
 
 2313 
 
 2383 
 
 2453 
 
 2523 
 
 2593 
 
 2663 
 
 2733 
 
 3 
 
 4 
 
 2803 
 
 2873 
 
 2943 
 
 3013 
 
 3084 
 
 3154 
 
 3224 
 
 3294 
 
 3364 
 
 3434 
 
 4 
 
 5 
 
 3504 
 
 3574 
 
 3644 
 
 3714 
 
 3784 
 
 3854 
 
 3925 
 
 3995 
 
 4065 
 
 4135 
 
 5 
 
 6 
 
 4205 
 
 4275 
 
 4345 
 
 4415 
 
 4485 
 
 4555 
 
 4625 
 
 4695 
 
 4766 
 
 4836 
 
 6 
 
 7 
 
 4906 
 
 4976 
 
 5046 
 
 5116 
 
 5186 
 
 5256 
 
 5326 
 
 5396 
 
 5466 
 
 5536 
 
 7 
 
 8 
 
 5606 
 
 5677 
 
 S747 
 
 5817 
 
 5887 
 
 5957 
 
 6027 
 
 6097 
 
 6167 
 
 6237 
 
 8 
 
 9 
 
 6307 
 
 6377 
 
 6447 
 
 6518 
 
 6588 
 
 6658 
 
 6728 
 
 6798 
 
 6868 
 
 6938 
 
 9 
 
 lO 
 
 7008 
 
 7078 
 
 7148 
 
 7218 
 
 7288 
 
 7359 
 
 7429 
 
 7499 
 
 7569 
 
 7639 
 
 10 
 
 n 
 
 7709 
 
 7779 
 
 7849 
 
 7919 
 
 7989 
 
 8059 
 
 8129 
 
 8199 
 
 8270 
 
 8340 
 
 II 
 
 12 
 
 8410 
 
 8480 
 
 8550 
 
 8620 
 
 8690 
 
 8760 
 
 8830 
 
 8900 
 
 8970 
 
 9040 
 
 12 
 
 13 
 
 9HI 
 
 9181 
 
 9251 
 
 9321 
 
 9391 
 
 9461 
 
 9531 
 
 9601 
 
 9671 
 
 9741 
 
 13 
 
 14 
 
 981I 
 
 9881 
 
 9952 
 
 10022 
 
 10092 
 
 10162 
 
 10232 
 
 10302 
 
 10372 
 
 10442 
 
 14 
 
 15 
 
 10512 
 
 10582 
 
 10652 
 
 10722 
 
 10792 
 
 10863 
 
 i°933 
 
 11003 
 
 11073 
 
 "143 
 
 15 
 
 The trnita of 
 
 863 
 60480 
 
 A° are placed at the top of the Table, and the tens at the side. 
 
TABLES. 
 
 61 
 
 Table shewing the value of A' which corresponds to each unit in the value of 
 
 A'. 
 
 '^'.A' = ^^ 
 
 24192 
 
 87-970f> 
 
 
 O 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 o 
 
 o 
 
 88 
 
 176 
 
 264 
 
 352 
 
 440 
 
 528 
 
 616 
 
 704 
 
 792 
 
 
 
 I 
 
 88o 
 
 968 
 
 1056 
 
 "44 
 
 1232 
 
 1320 
 
 1408 
 
 1496 
 
 1583 
 
 1671 
 
 I 
 
 2 
 
 1759 
 
 1847 
 
 1935 
 
 2023 
 
 2111 
 
 2199 
 
 2287 
 
 237s 
 
 2463 
 
 2551 
 
 2 
 
 3 
 
 2639 
 
 2727 
 
 2815 
 
 2903 
 
 2991 
 
 3°79 
 
 3167 
 
 3255 
 
 3343 
 
 3431 
 
 3 
 
 4 
 
 3519 
 
 3607 
 
 369s 
 
 3783 
 
 3871 
 
 3959 
 
 4047 
 
 4135 
 
 4223 
 
 4311 
 
 4 
 
 5 
 
 4399 
 
 4487 
 
 4574 
 
 4662 
 
 475° 
 
 4838 
 
 4926 
 
 5014 
 
 5102 
 
 519° 
 
 5 
 
 6 
 
 5278 
 
 5366 
 
 5454 
 
 5542 
 
 5630 
 
 5718 
 
 5806 
 
 5894 
 
 5982 
 
 6070 
 
 6 
 
 7 
 
 6158 
 
 6246 
 
 6334 
 
 6422 
 
 6510 
 
 6598 
 
 6686 
 
 6774 
 
 6862 
 
 6950 
 
 7 
 
 8 
 
 7038 
 
 7126 
 
 7214 
 
 7302 
 
 739° 
 
 7478 
 
 7565 
 
 7653 
 
 7741 
 
 7829 
 
 8 
 
 9 
 
 7917 
 
 8005 
 
 8093 
 
 8181 
 
 8269 
 
 8357 
 
 8445 
 
 8533 
 
 8621 
 
 8709 
 
 9 
 
 10 
 
 8797 
 
 8885 
 
 8973 
 
 9061 
 
 9149 
 
 9237 
 
 9325 
 
 9413 
 
 9501 
 
 9589 
 
 10 
 
 II 
 
 9677 
 
 9765 
 
 9853 
 
 9941 
 
 10029 
 
 10117 
 
 10205 
 
 10293 
 
 10381 
 
 10469 
 
 II 
 
 12 
 
 I0SS7 
 
 10644 
 
 10732 
 
 10820 
 
 10908 
 
 10996 
 
 1 1 084 
 
 11172 
 
 1 1260 
 
 1 1 348 
 
 12 
 
 The units of 
 
 275 
 24192 
 
 A' are placed at the top of the Table, and the tens at the side. 
 
62 
 
 TABLES. 
 
 Table shewing the value of A° which corresponds to each unit in the value of 
 
 33953 
 3628800 
 
 A» = 
 
 106-87715 
 
 A'. 
 
 
 O 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 o 
 
 o 
 
 107 
 
 214 
 
 321 
 
 428 
 
 534 
 
 641 
 
 748 
 
 85s 
 
 962 
 
 
 
 I 
 
 1069 
 
 1176 
 
 1283 
 
 1389 
 
 1496 
 
 1603 
 
 1710 
 
 1817 
 
 1924 
 
 2031 
 
 I 
 
 2 
 
 2138 
 
 2244 
 
 2351 
 
 2458 
 
 2565 
 
 2672 
 
 2779 
 
 2886 
 
 2993 
 
 3°99 
 
 2 
 
 3 
 
 3206 
 
 3313 
 
 3420 
 
 3527 
 
 3634 
 
 3741 
 
 3848 
 
 3954 
 
 4061 
 
 4168 
 
 3 
 
 4 
 
 4275 
 
 4382 
 
 4489 
 
 4596 
 
 4703 
 
 4809 
 
 4916 
 
 5023 
 
 5130 
 
 5237 
 
 4 
 
 5 
 
 5344 
 
 5451 
 
 5558 
 
 5664 
 
 5771 
 
 5878 
 
 5985 
 
 6092 
 
 6199 
 
 6306 
 
 5 
 
 6 
 
 6413 
 
 6520 
 
 6626 
 
 6733 
 
 6840 
 
 6947 
 
 7°54 
 
 7161 
 
 7268 
 
 7375 
 
 6 
 
 7 
 
 7481 
 
 7588 
 
 7695 
 
 7802 
 
 7909 
 
 8016 
 
 8123 
 
 8230 
 
 8336 
 
 8443 
 
 7 
 
 8 
 
 8550 
 
 8657 
 
 8764 
 
 8871 
 
 8978 
 
 9085 
 
 9191 
 
 9298 
 
 9405 
 
 9512 
 
 8 
 
 9 
 
 9619 
 
 9726 
 
 9833 
 
 9940 
 
 10046 
 
 10153 
 
 10260 
 
 10367 
 
 10474 
 
 10581 
 
 9 
 
 The units of 
 
 33953 
 3628800 
 
 A' are placed at the top of the Table, and the tens at the side. 
 
 Table shewing the value of A° which corresponds to each unit in the value of 
 
 8183 ,„ 1 
 
 1036800 
 
 A'' = 
 
 126-7017 
 
 A^ 
 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 • 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 
 
 127 
 
 253 
 
 380 
 
 507 
 
 634 
 
 760 
 
 887 
 
 1014 
 
 1 140 
 
 
 
 I 
 
 1267 
 
 1394 
 
 1520 
 
 1647 
 
 1774 
 
 1901 
 
 2027 
 
 2154 
 
 2281 
 
 2407 
 
 I 
 
 2 
 
 2534 
 
 2661 
 
 2787 
 
 2914 
 
 3041 
 
 3168 
 
 3294 
 
 3421 
 
 3548 
 
 3674 
 
 2 
 
 3 
 
 3801 
 
 3928 
 
 4054 
 
 4181 
 
 4308 
 
 4435 
 
 4561 
 
 4688 
 
 4815 
 
 4941 
 
 3 
 
 4 
 
 5068 
 
 519s 
 
 5321 
 
 5448 
 
 5575 
 
 5702 
 
 5828 
 
 5955 
 
 6082 
 
 6208 
 
 4 
 
 5 
 
 6335 
 
 6462 
 
 6588 
 
 6715 
 
 6842 
 
 6969 
 
 7095 
 
 7222 
 
 7349 
 
 7475 
 
 5 
 
 6 
 
 7602 
 
 7729 
 
 7856 
 
 7982 
 
 8109 
 
 8236 
 
 8362 
 
 8489 
 
 8616 
 
 8742 
 
 6 
 
 7 
 
 8869 
 
 8996 
 
 9123 
 
 9249 
 
 9376 
 
 95°3 
 
 9629 
 
 9756 
 
 9883 
 
 10009 
 
 7 
 
 8 
 
 10136 
 
 10263 
 
 10390 
 
 10516 
 
 10643 
 
 10770 
 
 10896 
 
 11023 
 
 11150 
 
 11276 
 
 8 
 
 9 
 
 1 1403 
 
 "53° 
 
 11657 
 
 11783 
 
 11910 
 
 12037 
 
 12163 
 
 12290 
 
 12417 
 
 I2S43 
 
 9 
 
 The units of 
 
 8183 
 1036800 
 
 A° are placed at the top of the Table, and the tens at the side. 
 
CHAPTEE IV. 
 
 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 The coordinates ^ and -r for the curves represented by Laplace's differential 
 
 equation were calculated by the method of Professor Adams for values of ^, 5", 10°, 
 
 15° 175», 180", and for values of i3, i, i |, ^; f , 1 ; IJ, 2, 2J; 3, 4, 5, 6, 7, 8; 
 
 10, 12, 14, 16 ; 20, 24, 28, 32 ; 40, 48, 56, 64, 72, 80, 88, 96 and 100. For ;S=1 the 
 calculations were made by Professor Adams, for /3 = 10 by Professor W. G. Adams, 
 and for the values of /3, J, ^, 3, 6, 16 and 32 by myself The calculations for the 
 remaining positive values of ^ were made by Dr C. Powalky, who was recommended 
 for the work by the late Professor Encke. 
 
 Afterwards the values of j and y corresponding to ^ = 5°, and for the suc- 
 cessive values of ^, 0, 8, 16, 24, 32, 40... 88 and 96 were arranged in order and 
 differenced. Then the values of ^ and y corresponding to the same value of <j}, and 
 
 to the values of /S, 36, 44... 92 were found from the above by interpolation, arranged 
 in order with the values of the same quantities for the values of ^, 0, 4, 8, &c. 
 
 already calculated, and the whole differenced. Next the values of j and r corre- 
 sponding to ^ = 5° and to the values of /3, 18, 22, 26, 30, 34, 38, 42, &c. were 
 found from the above by interpolation, arranged in order with the values of the 
 same quantities for values of /3, 0, 2, 4, 6, 8,. 10, &c. already found, and the whole 
 
 differenced. And in the same manner the values of y and ^ corresponding to (j) = b" 
 
 were found by interpolation for the values of ^, 9, 11, 13, 15, 17, 19, &c., arranged 
 in order with the values of the same quantities already found for the remaining 
 integral values of y8 up to 100, and the whole dififerenced. 
 
 Further, the same process was gone through for values of ^, 10°, 15°, 20°.... 
 175° and 180°. 
 
 Finally the results were arranged as they have been given in Table II., with ^ 
 for their argument, and differenced to test the accuracy of the work. 
 
64 COMPAEISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 Valuea of ts '«^ere calculated by the formula already given (p. 30) for the same 
 values of B for -which y and j were calculated, and the results have been given in 
 
 
 
 Table III. 
 
 Values of ? and f corresponding to values of <^, 15°, 30", 45", 60°, 90°, 120°, 
 
 
 
 135°, 140°, 145° and 150° were taken from Table II., and, the intermediate values 
 having been supplied by interpolation, the results have been given in Table V. This 
 Table is useful in calculating the theoretical forms of drops corresponding to given 
 values of ^8. 
 
 Other Tables were formed by interpolation of values of r, f and p inter- 
 mediate to those given in the above-mentioned Tables, but they are at once too 
 extensive and yet too incomplete for publication in their present form. In this way 
 
 Tables of values of | , r and tj were formed corresponding to values of /S, 0-0, 
 
 01, 0-2, 0-3 49-8, 49-9, 500, for values of (f>, 135°, 136°, 137°, 138° 153°, 154°, 
 
 155°, which were extremely useful, as will be explained hereafter, in deducing the 
 capillary constants from the measured forms of drops of mercury. But these Tables 
 would have been still more convenient if they had been formed so as to give 
 
 log -T , log T and log ^ instead of t j t and tj, . 
 
 The values of r and v vary so rapidly for low values of /3 and high values 
 
 of <b that they could be obtained by interpolation correct only to four places of 
 decimals from /3 = to /3 = 1'9. Beyond that they were calculated to five places 
 
 Y 
 of decimals, while the values of t, were calculated, unfortunately, to only four places 
 
 of decimals throughout. 
 
 The rectangular coordinates of points in the outlines of drops of mercury were 
 measured by the help of a microscope moveable on vertical and horizontal slides by 
 micrometer screws. The microscope was focused by motion along a third slide parallel 
 to the line of collimation. These three slides were parallel to three rectangular axes, 
 two of which were horizontal. 
 
 There was found to be a difficulty in arranging the cross lines of the microscope, 
 so as to obtain a correct outline of the drop of fluid, because it would be difficult 
 to judge when the crossing of the micrometer lines was exactly on the contour of 
 the drop. Much labour was expended on the construction of a position micrometer, 
 where the intersection of the middle of one micrometer line with a side of the other 
 line was to be the centre, about which they turned. At each observation the micrometer 
 
COMPARISON OP CALCULATED AND MEASURED FORMS OF DROPS. 65 
 
 lines were to be turned so that the above-mentioned side of the micrometer line 
 became a tangent to the contour of the drop, while the other line was a normal 
 at that point. But this could not be considered a satisfactory arrangement, because 
 there would be some error of excentricity, and the microscope would be liable to be 
 slightly disturbed by the application of the force necessary to bring the cross lines 
 into their proper position. This contrivance was therefore abandoned in favour of a 
 more simple arrangement suggested to me by the late Professor W. H. Miller. This 
 was to use two pairs of parallel and equidistant spider lines, one pair being horizontal 
 and the other pair vertical, so as to form by their intersection a small square in 
 the centre of the field of view (Fig. 1). This arrangement appears to be quite 
 satisfactory, for it is easy to judge when a small arc of the contour of a drop of 
 fluid passes through the middle of this small central square. 
 
 The screws used to measure the vertical and horizontal coordinates of points in 
 the contour of a drop of fluid were originally formed of one piece of metal. Great 
 care was taken to obtain a screw of uniform pitch throughout, of about 63 turns to 
 the inch. When however the screws were mounted, it was found that although each 
 one was tolerably uniform, the two screws differed sensibly in their pitch. By the use 
 of a micrometer ruled to the one-hundredth of an inch, the exact rate of both screws 
 at every point was determined. In this way tables of the value of every turn were 
 calculated for both the vertical and horizontal screws. In the following measure- 
 ments of the forms of five drops of mercury made in 1863, the original readings 
 of the screws are given as they were entered in the observing book, as well as 
 their values in inches obtained by the help of the above-mentioned Tables. 
 
 The coordinates of numerous points in the contours of these five drops of 
 mercury, which vary considerably in size, were measured, because it was desired to 
 find whether the theoretical forms would agree satisfactorily with the true forms of 
 drops of mercury. In Fig. 2, the theoretical forms of these drops are given on a 
 large scale, and an attempt has been made to indicate by a cross the position of 
 some of the points measured. 
 
66 
 
 COMPAKISON OF CALCULATED AND MEASURED FORMS OP DROPS. 
 
 No. I 
 
 Ori 
 
 ginal readings 
 
 Readings converted 
 
 The same 
 
 when the 
 
 origin of 
 
 
 ""fllrnlntpr 
 
 
 in turns of the screws 
 
 into inches 
 
 
 coordinates is at the vertex 
 
 
 ^•cliUUlaLCt. 
 
 
 z" 
 
 < 
 
 < 
 
 z' 
 
 ^/ 
 
 < 
 
 z 
 
 ^. 
 
 ^. 
 
 «^ 
 
 z 
 
 X 
 
 
 
 
 incli 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 
 inch 
 
 inch 
 
 13-282 
 
 23-416 
 
 30-288 
 
 0-2491 
 
 0-4410 
 
 0-5705 
 
 0-0925 
 
 0-0650 
 
 0-0645 
 
 i44"-48 
 
 0-0925 
 
 0-0647 
 
 13-300 
 13-400 
 13-500 
 
 23'398 
 23-270 
 23-170 
 
 30-340 
 30-440 
 30-540 
 
 ■2495 
 •2513 
 ■2532 
 
 -4406 
 •4382 
 •4363 
 
 -5715 
 •5733 
 •5752 
 
 ■0921 
 -0903 
 -0884 
 
 -0654 
 •0678 
 -0697 
 
 -0655 
 •0673 
 •0692 
 
 i4o°-o 
 i35°-o 
 
 ■091 1 
 -0892 
 
 •0666 
 -0686 
 
 13-600 
 
 23-072 
 
 30-625 
 
 ■2551 
 
 •4345 
 
 -5768 
 
 -0865 
 
 •0715 
 
 -0708 
 
 
 
 
 13-800 
 14-000 
 
 22-970 
 22-879 
 
 30-772 
 30-876 
 
 •2588 
 •2626 
 
 -4326 
 -4308 
 
 ■5796 
 -5816 
 
 •0828 
 •0790 
 
 -0734 
 -0752 
 
 ■0736 
 •0756 
 
 I20°-0 
 
 ■0822 
 
 -0740 
 
 14-200 
 
 22-797 
 
 30-95° 
 
 -2663 
 
 -4293 
 
 •5830 
 
 •0753 
 
 -0767 
 
 •0770 
 
 
 
 
 14-400 
 
 22-735 
 
 31-003 
 
 •2701 
 
 -4281 
 
 •5840 i 
 
 •0715 
 
 ■0779 
 
 -0780 
 
 
 
 
 14-600 
 
 22-680 
 
 31-033 
 
 ■2738 
 
 -4271 
 
 -5845 
 
 -0678 
 
 -0789 
 
 -0785 
 
 r\r>P'r\ 
 
 -0623 
 
 
 14-960 
 
 22-672 
 
 31-064 
 
 -2806 
 
 -4269 
 
 -5851 
 
 -0610 
 
 •0791 
 
 -0791 
 
 90 
 
 -0791 
 
 15-200 
 
 22-687 
 
 31-034 
 
 •2851 
 
 ■4272 
 
 •5845 
 
 •0565 
 
 ■0788 
 
 •0785 
 
 
 
 
 15-400 
 
 22-712 
 
 30-993 
 
 •2888 
 
 •4277 
 
 •5837 
 
 -0528 
 
 -0783 
 
 -0777 
 
 
 
 
 15-600 
 
 22-762 
 
 30-952 
 
 •2926 
 
 -4286 
 
 •5830 
 
 -0490 
 
 •0774 
 
 •0770 
 
 
 
 
 15-800 
 
 22-840 
 
 30-890 
 
 •2963 
 
 -4301 
 
 -5818 
 
 •0453 
 
 •0759 
 
 •0758 
 
 
 
 
 i6-ooo 
 
 22-925 
 
 30-826 
 
 -3001 ; 
 
 ■4317 
 
 -5806 
 
 ■0415 
 
 ■0743 
 
 -0746 
 
 
 
 
 16-200 
 
 23-010 \ 
 
 30-720 : 
 
 -3039 
 
 ■4333 
 
 -5786 
 
 •0377 
 
 •0727 
 
 •0726 
 
 eo'-o 
 
 
 
 16-400 
 
 23-128 
 
 30-590 
 
 •3076 
 
 -4356 
 
 -5762 
 
 -0340 
 
 -0704 
 
 ■0702 
 
 ■0370 
 
 -0720 
 
 16-600 
 
 23-250 
 
 30-447 
 
 -3114 
 
 •4378 
 
 •573s 
 
 -0302 
 
 •0682 
 
 •0675 
 
 
 
 
 16-300 
 
 23-418 
 
 30-274 
 
 •3151 
 
 •4410 
 
 -5702 
 
 ■0265 
 
 •0650 
 
 -0642 
 
 ^r-O-^ 
 
 ■0238 
 
 •0618 
 
 17-000 
 
 23-620 
 
 30-090 
 
 •3189 
 
 -4448 
 
 -5667 
 
 •0227 
 
 -0612 
 
 •0607 
 
 45 
 
 17-200 
 
 23-860 
 
 29-865 
 
 •3226 
 
 ■4493 
 
 ■5625 
 
 •0190 
 
 -0567 
 
 ■0565 
 
 -.«o-„ 
 
 
 -0462 
 
 17-600 
 
 24-437 
 
 29-278 
 
 -3301 
 
 -4602 
 
 -5514 
 
 -OI15 
 
 -0458 
 
 •0454 
 
 30 
 
 •0120 
 
 i8-ooo 
 
 25-390 
 
 28-316 
 
 •3376 
 
 •4782 
 
 ■5333 
 
 •0040 
 
 •0278 
 
 •0273 
 
 T F-0.^ 
 
 
 
 18-214 
 
 *#* 
 
 *** 
 
 -3416 
 
 •5060 
 
 •5060 
 
 •0000 
 
 •0000 
 
 ■0000 
 
 15 
 
 -0033 
 
 ■0251 
 
 "Weight 4-57 grs. /? = 2-334 a =119-6 (o=U4°-48 6 = 0-09878 in. Temp. 40° F. 
 Error in calculation of V= + 0000 035 cubic inch. 
 
COMPAEISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 67 
 
 No. II 
 
 Original readings 
 
 Readings converted 
 
 The same 
 
 when the 
 
 origin of 
 
 
 ""'q Ipiilaf/aH 
 
 in turns of the screws 
 
 
 nto inches 
 
 
 coordinates is at the vertex 
 
 
 v.. d.i\. Uld. LCL 
 
 
 2" 
 
 < 
 
 < 
 
 z' 
 
 x; 
 
 < 
 
 z 
 
 x^ 
 
 ■^. 
 
 ^ 
 
 z 
 
 X 
 
 
 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch ' 
 
 inch 
 
 
 inch 
 
 inch 
 
 13-023 
 
 24-710 
 
 34-330 
 
 0-2443 
 
 0-4654 
 
 0-6467 
 
 0-1054 
 
 0-0907 
 
 0-0906 
 
 i48''-28 
 
 0-1054 
 
 0-09065 
 
 13-100 
 13-200 
 13-300 
 
 24-603 
 24-489 
 24'3S6 
 
 34-464 
 34-596 
 34-678 
 
 ■2457 
 -2476 
 ■2495 
 
 -4634 
 •4612 
 
 •4587 
 
 -6492 
 
 •6517 
 -6532 
 
 -1040 
 -I02I 
 -1002 
 
 ■0927 
 -0949 
 
 ■0974 
 
 •0931 
 -0956 
 -0971 
 
 i45°-o 
 i4o°-o 
 
 i35°-o 
 
 -1044 
 -1028 
 -1009 
 
 •0921 
 
 •0943 
 -0963 
 
 13-400 
 
 24-260 
 
 34-762 
 
 •2513 
 
 ■4569 
 
 -6548 
 
 •0984 
 
 -0992 
 
 -0987 
 
 
 
 
 13-600 
 13-800 
 
 24-146 
 24-040 
 
 34-904 
 35-010 
 
 •2551 
 ■2588 
 
 •4547 
 ■4527 
 
 •6575 
 •6595 
 
 •0946 
 •0909 
 
 -1014 
 •1034 
 
 -1014 
 •1034 
 
 I20°-0 
 
 -0938 
 
 -1018 
 
 14-000 
 
 23'975 
 
 35-086 
 
 -2626 
 
 ■4515 
 
 •6609 
 
 -0871 
 
 •1046 
 
 -1048 
 
 
 
 
 14-400 
 
 23-867 
 
 35-178 
 
 -2701 
 
 -4495 
 
 -6626 
 
 -0796 
 
 •1066 
 
 •1065 
 
 
 
 
 14-732 
 
 23-848 
 
 35-203 
 
 -2763 
 
 •4491 
 
 -6631 
 
 -0734 
 
 -1070 
 
 -1070 
 
 9o°-o 
 
 •0734 
 
 -1070 
 
 15-000 
 
 23-866 
 
 35-192 
 
 -2813 
 
 -4495 
 
 ■6629 
 
 -0684 
 
 •1066 
 
 ■1068 
 
 
 
 
 15-400 
 
 23'9So 
 
 35-105 
 
 •2888 
 
 •4510 
 
 -6613 
 
 -0609 
 
 ■1051 
 
 •1052 
 
 
 
 
 15-800 
 16-200 
 
 24-072 
 24-270 
 
 34-989 
 34-782 
 
 •2963 
 -3039 
 
 -4533 
 -4571 
 
 •6591 
 •6552 
 
 -0534 
 ■0458 
 
 -1028 
 -0990 
 
 -1030 
 •0991 
 
 6o''-o 
 
 -0463 
 
 ■0993 
 
 16-600 
 17-000 
 
 24-534 
 24-906 
 
 34-515 
 34-162 
 
 -3114 
 -3189 
 
 -4620 
 -4691 
 
 -6501 
 -6435 
 
 •0383 
 -0308 
 
 -0941 
 -0871 
 
 •0940 
 -0874 
 
 45°-o 
 
 •0315 
 
 -0878 
 
 17-400 
 
 25-361 
 
 33-685 
 
 -3264 
 
 •4776 
 
 •6345 
 
 -0233 
 
 •0785 
 
 •0784 
 
 3o°-o 
 
 •0170 
 
 •0687 
 
 17-800 
 
 25-978 
 
 33-078 
 
 ■3339 
 
 -4893 
 
 -6230 
 
 •0158 
 
 -0668 
 
 •0669 
 
 i8-ooo 
 
 26-368 
 
 32-688 
 
 -3376 
 
 •4966 
 
 -6157 
 
 -OI2I 
 
 -0595 
 
 -0596 
 
 
 
 
 18-200 
 
 26-867 
 
 32-165 
 
 •3414 
 
 -5060 
 
 •6058 
 
 •0083 
 
 •0501 
 
 •0497 
 
 i5»-o 
 
 -0051 
 
 -0394 
 
 18-400 
 
 27-540 
 
 31-478 
 
 ■3451 
 
 -5187 
 
 -5929 
 
 •0046 
 
 -0374 
 
 -0368 
 
 18-600 
 
 28-637 
 
 30-348 
 
 ■3489 
 
 •5394 
 
 •5716 
 
 -0008 
 
 -0167 
 
 •015s 
 
 
 
 
 18-642 
 
 *** 
 
 *** 
 
 ■3497 
 
 •5561 
 
 •5561 
 
 •0000 
 
 -0000 
 
 ■0000 
 
 
 
 
 Weight 9-523 grs. 
 
 j8 = 6-44 a=126-2 o)=148»-28 6 = 0-159r6in. Temp. 37" F. 
 Error in calculation of F= + 0-000 044 cubic inch. 
 
 9—2 
 
68 
 
 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 No. Ill 
 
 Original readings 
 in turns of the screws 
 
 Readings converted 
 into inches 
 
 The same when the origin of 
 coordinates is at the vertex 
 
 Calculated 
 
 z" 
 
 xr 
 
 < 
 
 / 
 
 x; 
 
 < 
 
 Z 
 
 ^J 
 
 X, 
 
 "^ 
 
 z 
 
 X 
 
 12-570 
 
 12-900 
 13-000 
 13-400 
 14-300 
 
 i6-ooo 
 17-000 
 1 8 'OOO 
 18-580 
 
 22-563 
 
 22.282 
 22-232 
 22-024 
 21-863 
 
 22.418 
 
 23-358 
 
 25-225 
 
 *** 
 
 34-683 
 35-05° 
 
 35-112 
 35-326 
 35-480 
 
 34-928 
 
 33-985 
 
 32-132 
 
 *** 
 
 inch 
 0-2358 
 
 -2420 
 
 •2438 
 
 -2513 
 •2682 
 
 •3001 
 -3189 
 •3376 
 -3485 
 
 inch 
 0-4249 
 
 ■4196 
 •4187 
 -4147 
 •41 1 7 
 
 -4222 
 -4399 
 ■4751 
 •5400 
 
 inch 
 0-6533 
 •6602 
 -6614 
 •6654 
 •6683 
 
 -6579 
 -6401 
 -6052 
 •5400 
 
 inch 
 0-1127 
 
 •1065 
 •1047 
 -0972 
 -0803 
 
 •0484 
 •0296 
 ■0109 
 •OOOO 
 
 inch 
 0-1151 
 
 -1204 
 •1213 
 
 -1253 
 •1283 
 
 •1178 
 
 •lOOI 
 
 •0649 
 
 •OOOO 
 
 inch 
 0-1133 
 
 •1202 
 •1214 
 
 •1254 
 •1283 
 
 •1179 
 
 ■lOOI 
 
 •0652 
 
 •OOOO 
 
 i4i''-7i 
 i4o°'oo 
 
 135°-°° 
 
 I20°^00 
 
 9o°^oo 
 6o''^oo 
 
 45°-°° 
 3o''-oo 
 
 i5°-oo 
 
 inch 
 0-1127 
 
 •II2I 
 
 •lioi 
 •1026 
 
 •0813 
 •0527 
 -0367 
 •0206 
 •0065 
 
 inch 
 0-II42 
 
 -I150 
 •1171 
 •1228 
 
 •1283 
 •1202 
 •1078 
 •0865 
 -0516 
 
 Weight 14:^725 grs. 
 
 /3=ll-0 a=118-3 *)=141''-71 6 = 0^21572iii. Temp. 39" F. 
 Error in calculation of F= + ©•000 057 cubic inch. 
 
COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 69 
 
 No. IV 
 
 Original readings 
 
 Read 
 
 ings converted 
 
 The same 
 
 when the 
 
 origin of 
 
 Calculated 
 
 in turns of the screws 
 
 into inches 
 
 
 coordinates is at the vertex 
 
 0" 
 
 < 
 
 < 
 
 z' 
 
 < 
 
 < 
 
 Z 
 
 *. 
 
 *. 
 
 <^ 
 
 z 
 
 X 
 
 
 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 
 inch 
 
 inch 
 
 12-985 
 
 19-918 
 
 33-865 
 
 0-2435 
 
 0-3750 
 
 0-6379 
 
 0-1168 
 
 0-1311 
 
 0-1318 
 
 i4o°-o 
 
 O-I168 
 
 0-1315 
 
 13-000 
 13-100 
 
 19-900 
 19-812 
 
 33-918 
 34-010 
 
 •2438 
 •2457 
 
 '3747 
 
 -3730 
 
 -6389 
 -6406 
 
 •1165 
 ■II46 
 
 -1314 
 -1331 
 
 -1328 
 ■1345 
 
 i35°-oo 
 
 -1148 
 
 •1337 
 
 13-200 
 
 19725 
 
 34-093 
 
 -2476 
 
 -3714 
 
 -6422 
 
 -II27 
 
 -1347 
 
 •1361 
 
 
 
 
 13-300 
 
 19-642 
 
 34-167 
 
 "2495 
 
 -3698 
 
 -6436 
 
 -II08 
 
 -1363 
 
 -1375 
 
 
 
 
 13-400 
 
 i9"S53 
 
 34"230 
 
 •2513 
 
 -3681 
 
 -6448 
 
 -1090 
 
 •1380 
 
 -1387 
 
 I20°-00 
 
 -1072 
 
 -1394 
 
 i3-5°° 
 
 19-500 
 
 34-30° 
 
 -2532 
 
 •3670 
 
 -6461 
 
 -1071 
 
 -1391 
 
 -1400 
 
 
 13-600 
 
 19-440 
 
 34-362 
 
 -2551 
 
 •3660 
 
 -6472 
 
 -1052 
 
 -1401 
 
 •1411 
 
 
 
 
 13-800 
 
 19-360 
 
 34-412 
 
 •2588 
 
 •3645 
 
 -6482 
 
 •1015 
 
 ■1416 
 
 •1421 
 
 
 
 
 14-000 
 
 19-294 
 
 34-486 
 
 -2626 
 
 -3633 
 
 -6496 
 
 -0977 
 
 -1428 
 
 -1435 
 
 
 
 
 14-200 
 
 19-243 
 
 34-532 
 
 -2663 
 
 ■3623 
 
 -6505 
 
 -0940 
 
 •1438 
 
 -1444 
 
 
 
 
 14-400 
 
 i9'i95 
 
 34-548 
 
 -2701 
 
 -3614 
 
 -6508 
 
 •0902 
 
 ■1447 
 
 •1447 
 
 
 
 
 14-600 
 
 19-182 
 
 34-560 
 
 -2738 
 
 -36II 
 
 6510 
 
 •0865 
 
 ■1450 
 
 -1449 
 
 90°-oo 
 
 •0856 
 
 -1450 
 
 14-800 
 
 19-200 
 
 34-560 
 
 •2776 
 
 -3615 
 
 -6510 
 
 -0827 
 
 -1446 
 
 -1449 
 
 15-000 
 
 19-226 
 
 34-545 
 
 -2813 
 
 ■3620 
 
 -6507 
 
 -0790 
 
 •1441 
 
 •1446 
 
 
 
 
 15-200 
 
 19-277 
 
 34-520 
 
 •2851 
 
 •3629 
 
 -6502 
 
 •-0752 
 
 -1432 
 
 -1441 
 
 
 
 
 15-400 
 
 19-307 
 
 34-456 
 
 •2888 
 
 -3635 
 
 -6490 
 
 -0715 
 
 -1426 
 
 -1429 
 
 
 
 
 15-600 
 
 i9'3S3 
 
 34-388 
 
 -2926 
 
 ■3644 
 
 -6477 
 
 -0677 
 
 •1417 
 
 •1416 
 
 
 
 
 15-800 
 
 i9'445 
 
 34-322 
 
 -2963 
 
 ■3661 
 
 •6465 
 
 -0640 
 
 -1400 
 
 -1404 
 
 
 
 
 16-000 
 
 i9'539 
 
 34-228 
 
 -3001 
 
 -3679 
 
 -6447 
 
 -0602 
 
 •1382 
 
 -1386 
 
 60" -00 
 
 -0567 
 
 -1367 
 
 16-200 
 
 19-652 
 
 34-107 
 
 -3039 
 
 -3700 
 
 -6424 
 
 -0564 
 
 •1361 
 
 •1363 
 
 
 16-400 
 
 19-760 
 
 33-975 
 
 ■3076 
 
 -3721 
 
 -6399 
 
 •0527 
 
 ■1340 
 
 •1338 
 
 
 
 
 i6-6oo 
 
 19-913 
 
 33-832 
 
 -3114 
 
 -3749 
 
 -6373 
 
 -0489 
 
 -1312 
 
 -1312 
 
 
 
 
 i6-8oo 
 
 20-050 
 
 33-687 
 
 ■3151 
 
 -3775 
 
 •6345 
 
 •0452 
 
 -1286 
 
 -1284 
 
 
 
 
 17-000 
 17-200 
 
 20-240 
 20-442 
 
 33-496 
 .33-282 
 
 -3189 
 •3226 
 
 -38 1 1 
 ■3849 
 
 -6309 
 -6269 
 
 -0414 
 
 -0377 
 
 •1250 
 -1212 
 
 -1248 
 -1208 
 
 45°-oo 
 
 -0402 
 
 -1239 
 
 17-400 
 
 20-653 
 
 33-036 
 
 •3264 
 
 -3889 
 
 -6223 
 
 -0339 
 
 -II72 
 
 •I162 
 
 
 
 ■ 
 
 17-600 
 
 20-902 
 
 32-792 
 
 -3301 
 
 •3936 
 
 -6177 
 
 -0302 
 
 -II25 
 
 •II16 
 
 
 
 
 17-800 
 
 21-204 
 
 32-516 
 
 -3339 
 
 -3993 
 
 -6125 
 
 -0264 
 
 -1068 
 
 -1064 
 
 30° -00 
 
 -0234 
 
 -1016 
 
 18-000 
 
 21-548 
 
 32-178 
 
 -3376 
 
 -4058 
 
 -6061 
 
 -0227 
 
 •1003 
 
 -1000 
 
 
 18-200 
 
 21-927 
 
 31-786 
 
 -3414 
 
 -4129 
 
 -5987 
 
 -0189 
 
 -0932 
 
 -0926 
 
 
 
 
 18-400 
 
 22-342 
 
 31-320 
 
 -3451 
 
 •4208 
 
 -5899 
 
 -0152 
 
 -0853 
 
 -0838 
 
 
 
 
 18-600 
 
 22-880 
 
 30-816 
 
 ■3489 
 
 -4309 
 
 -5804 
 
 -01 14 
 
 -0752 
 
 -0743 
 
 
 
 
 18-700 
 
 23-170 
 
 30-494 
 
 •3507 
 
 •4363 
 
 •5744 
 
 -0096 
 
 •0698 
 
 •0683 
 
 i5"-oo 
 
 -0078 
 
 -06^1 
 
 18-800 
 
 23-517 
 
 30-134 
 
 •3526 
 
 -4429 
 
 -5676 
 
 -0077 
 
 •0632 
 
 •0615 
 
 '-'"O* 
 
 18-900 
 
 23'95o 
 
 29-762 
 
 -3545 
 
 •4510 
 
 •5606 
 
 -0058 
 
 -0551 
 
 -0545 
 
 
 
 
 19-000 
 
 24-436 
 
 29-205 
 
 ,-3564 
 
 •4602 
 
 -5501 
 
 •0039 
 
 -0459 
 
 -0440 
 
 
 
 
 19-100 
 
 25'i65 
 
 28-522 
 
 -3582 
 
 -4739 
 
 -5372 
 
 •0021 
 
 -0322 
 
 -03 1 1 
 
 
 
 
 19-210 
 
 *** 
 
 4S-** 
 
 •3603 
 
 -5061 
 
 -5061 
 
 -0000 
 
 •0000 
 
 -0000 
 
 O'OO 
 
 -0000 
 
 -0000 
 
 Weight 19-77 grs. 
 
 /3=17-5 a=116-9 oj=140°-00 6 = 0-27358in. Temp. 38°r. 
 Error in calculation of F— + O'OOO 012 cubic inch. 
 
70 
 
 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 No. V 
 
 Original readings 
 
 Readings converted 
 
 The same 
 
 when the origin of 
 
 Calculated 
 
 in turns of the screws 
 
 into inches 
 
 
 coordinates is at the vertex 
 
 2" 
 
 < 
 
 < 
 
 z' 
 
 x; 
 
 < 
 
 Z 
 
 ^1 
 
 ^. 
 
 <^ 
 
 z 
 
 X 
 
 
 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 inch 
 
 
 inch 
 
 inch 
 
 17732 
 
 21-555 
 
 36-516 
 
 0-3326 
 
 0-4059 
 
 0-6878 
 
 0-1174 
 
 0-1411 
 
 0-1408 
 
 i39°-4i 
 
 0-1174 
 
 0-14095 
 
 i7'8oo 
 
 21-492 
 
 36-603 
 
 ■3339 
 
 •4047 
 
 -689s 
 
 ■I161 
 
 -1423 
 
 ■1425 
 
 i35''-o 
 
 •1156 
 
 
 1 7 '900 
 
 21-398 
 
 36-702 
 
 •3357 
 
 -4029 
 
 -6914 
 
 ■1143 
 
 -1441 
 
 -1444 
 
 .1429 
 
 iS'ooo 
 
 21-333 
 
 36-782 
 
 -3376 
 
 -4017 
 
 •6929 
 
 -1124 
 
 •1453 
 
 -1459 
 
 
 
 
 l8'200 
 
 1 8 '400 
 
 21-189 
 21-068 
 
 36-910 
 37-005 
 
 •3414 
 -3451 
 
 -3990 
 -3967 
 
 -6953 
 -6971 
 
 ■1086 
 -1049 
 
 -1480 
 -1503 
 
 •1483 
 -1501 
 
 I20^'00 
 
 •1082 
 
 •i486 
 
 i8'6oo 
 
 20-986 
 
 37-094 
 
 -3489 
 
 -3952 
 
 •6987 
 
 -loii 
 
 •1518 
 
 -1517 
 
 
 
 
 19-000 
 
 20-910 
 
 37-200 
 
 ■3564 
 
 -3937 
 
 -7007 
 
 -0936 
 
 -1533 
 
 -1537 
 
 
 
 
 i9'336 
 
 20-868 
 
 37-214 
 
 •3627 
 
 ■3929 
 
 -7010 
 
 •0873 
 
 -1541 
 
 -1540 
 
 
 
 -0868 
 
 
 19-800 
 
 20-905 
 
 37-194 
 
 •3714 
 
 ■3936 
 
 •7006 
 
 -0786 
 
 ■1534 
 
 -1536 
 
 90 "OO 
 
 ■1540 
 
 20-209 
 
 21-006 
 
 37-095 
 
 •3789 
 
 -3955 
 
 -6988 
 
 •07 1 1 
 
 -1515 
 
 -1518 
 
 
 
 
 20-600 
 
 21-141 
 
 36-938 
 
 •3864 
 
 •3981 
 
 -6958 
 
 •0636 
 
 •1489 
 
 -1488 
 
 6o''-oo 
 
 ■0581 
 
 
 21-000 
 
 21-368 
 
 36-736 
 
 -3939 
 
 •4024 
 
 -6920 
 
 •0561 
 
 ■1446 
 
 -1450 
 
 -1459 
 
 2 1 -400 
 
 21-638 
 
 36-443 
 
 •4014 
 
 ■4075 
 
 -6865 
 
 •0486 
 
 ■1395 
 
 ■1395 
 
 . -0.- -. 
 
 
 
 2 1 -800 
 
 2 2-00O 
 
 36-084 
 
 •4089 
 
 •4143 
 
 -6797 
 
 •041 1 
 
 ■1327 
 
 •1327 
 
 45 00 
 
 •0417 
 
 -1331 
 
 22-20O 
 
 22-460 
 
 35-640 
 
 •4164 
 
 •4230 
 
 •6713 
 
 •0336 
 
 ■1240 
 
 ■1243 
 
 
 
 
 22-600 
 
 23-031 
 
 35-052 
 
 •4239 
 
 •4337 
 
 -6603 
 
 ■0261 
 
 ■1133 
 
 ■1133 
 
 ^„0,„ -. 
 
 
 •1106 
 
 23-000 
 
 23-788 
 
 34-312 
 
 •4314 
 
 -4480 
 
 -6463 
 
 •0186 
 
 ■0990 
 
 -0993 
 
 30 00 
 
 ■0247 
 
 23-400 
 
 24-786 
 
 33-256 
 
 •4389 
 
 -4668 
 
 •6264 
 
 •01 1 1 
 
 •0802 
 
 •0794 
 
 T F-0.«« 
 
 ■0086 
 
 
 23-800 
 
 26-546 
 
 31-570 
 
 -4463 
 
 •5000 
 
 •5946 
 
 -0037 
 
 •0470 
 
 •0476 
 
 15 00 
 
 •0707 
 
 23'99S 
 
 **# 
 
 *** 
 
 •4500 
 
 •5470 
 
 -5470 
 
 •0000 
 
 -0000 
 
 ■0000 
 
 
 
 
 
 
 
 
 Weight ? 
 
 ^=2 
 
 4-023 
 
 a=119 
 
 9 0) 
 
 = 139"- 41 
 
 6 = ( 
 
 3-31646 i 
 
 n. Te 
 
 mp. 49° 
 
 F. 
 
 The theoretical forms of these five drops of mercury have been drawn to a large scale in Fig. 2, 
 ■\vhere the measured points are indicated by small crosses. 
 
COMPARISON OV CALCULATED AND MEASURED FORMS OF DROPS. 71 
 
 The agreement between theory and experiment appears to be so far satisfactory. 
 And if on more exact comparison any slight discrepancy between theory and experi- 
 ment should become apparent, it will be known that this is not due to any error 
 in the calculated forms. 
 
 In adapting a theoretical form to the measured form of a drop of mercury, it 
 would be sufficient to secure its passing through the vertex A (Fig. 4) and the two 
 points B, G, for which ^ = 90°, if it was possible to measure AO correctly. But this 
 can be accomplished practically only with sufficient accuracy to give a rough first 
 approximation to the value of /3, by finding OC-r-AO and referring to Table I. 
 This value of /3, if erroneous, must be corrected by trial till a curve is found from 
 Table II., which passes through JD and E, the extremities of the base, or till two 
 curves are found for consecutive values of yS, one of which fe,lls outside, and the other 
 within BE. Then by proportional parts the exact value of /3 required can be found. 
 
 Let BG=2R, DE=2r, and AN = H. The following example will explain how 
 the values of the capillary constants are obtained by means of these quantities. 
 
 For the drop No. V. 2iJ = 0-3081 inch, ir= 0-1174 inch, and 2r = 0-2819 inch. 
 Having found by the help of Table I. and by trial that the proper value of /3 
 lies between 24-0 and 24" 1, we proceed to find b' the radius of curvature at the 
 vertex corresponding to y8' = 240. From Table II., when ^ = 90", we find that 
 
 ^ = 5 = ^^i^ = 0-48692 and therefore &' = ^^MI^, which gives log 5' = 9-50020. 
 bo b U-4oDyz 
 
 That will suffice to secure a curve which passes through the vertex A and has the 
 
 correct width BG. We wish in addition to secure a curve which passes through 
 
 the two points D, E at the base of the drop, or through two points d, e near the 
 
 base. 
 
 log r = 9-14907 log 5"= 9-06967 
 
 log 6' = 9-50020 log 6' = 9-50020 
 
 H 
 
 log p= 9-64887 log y = 9-56947 
 
 and therefore J = 0-44552 and ^ = 0-37108. 
 
 And to find the theoretical form of this drop we use the manuscript Tables 
 
 z H 
 
 above referred to^ For /3' = 24-0 the Table gives t; = 0-37108 =77 corresponding to 
 
 </)=139°-36. And for the same value of <j), | =0-44608-0-00050=0-44558. Hence 
 
 ^-^ = and r>-% 000006, <^ = 139°-36. 
 
 00 00 
 
 ' See note » on next page. 
 
72 
 
 COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 
 
 Again, corresponding to /3" = 24-l we find in the same manner as before 
 log &"= 9-50070, which gives ^ = 0-44501 and | = 0-37066. And for ^"=24-1 Uhe Table 
 
 gives ^=0-37066 corresponding to </, = 139"-58, and p = 0-44481. Here we have 
 |-p = 0; |,-|, = + 0-00020; </, = 139»-58. 
 
 The required value of /3 therefore falls between 24-0 and 24-1, and its exact value 
 may be found as follows : 
 
 iS' = 24-0 gives log V = 9-50020 ; error in ^, = - 0-00006 ; and ^ = 139°-36 
 
 ^" = 24-1 gives log h" = 9-50070 ; 
 
 p = + 0-00020; and 0" = 139 -58 
 
 Diff. + 0-1 
 
 + 0-00050 
 
 + 0-00026 
 
 + 0-22 
 
 Hence by taking proportional parts we must find such values of h^' , Slog 6' 
 and S0' as will make the error in t? vanish. 
 
 jS'= 24-0 gives log&'= 9-50020; error in | =-0-00006; and 0'= 139''-36 
 
 SiS'=+ -023 „ S log &' = + 0-00012; ,; „ + 0-00006 ; „ Sf = + 0-05 
 
 „ ^ = 139-41 
 
 Hence ;S = 24-023 
 
 log&= 9-50032; 
 
 
 
 Hence J = 0-31646 inch, a = ^,=^^^^ = 119-94, and ^/? =0-1291 inch. Also 
 
 the volume =&'x 0-2072 = 0-0065667 cubic inch, and the corresponding weight is 22-535 
 grains. 
 
 . Nearly the same results might be arrived at by using the values of ^ and 7 
 
 in Table H. for /3 = 24 and /S = 25, only the differences would correspond to a 
 difference of 1 instead of 0-1 in the value of /3, and to a difference of 5° instead of 
 1° in the value of <^. 
 
 Note ». 
 
 p=24'0 
 
 ^ 
 
 X 
 
 "6 
 
 z 
 
 ISS" 
 139 
 140 
 
 A 
 
 •44747 loq 
 
 •44608-^^9 
 
 •44468 "" 
 
 &c.. 
 
 A 
 •36942 ^2„ 
 ■37065 + i^^ 
 •37185 + ^''" 
 
 Note •>. 
 
 ^=24-1 
 
 •> 
 
 X 
 
 6 
 
 z 
 & 
 
 138» 
 139 
 140 
 &c. 
 
 A 
 
 •44700 ,„„ 
 
 •44561-^1^ 
 
 •44421-1*" 
 
 &c. 
 
 A 
 
 •36874 ,,„„ 
 
 •36997 + i^^ 
 
 •37117 + 1''" 
 
 &o. 
 
COMPARISON OF CALCULATED AND MEASURED FORMS OF DROPS. 73 
 
 It is evident that the above calculations would have teen facilitated if the 
 Tables referred to in the note had been calculated for log ^ , log y and log ^ rather 
 
 than for ^, r and p, as has been already remarked. 
 
 The coordinates at the points of the theoretical curve at which the tangent is 
 
 inclined to the horizon at angles of 15°, 30", 45", 60°, 90°, 120°, 135° &c,, are found 
 
 by the help of Table V. for values of /3, 0-0, Ol, 0% OS 46-5, 46-6, 467. 
 
 For instance for (f) = 135°, 
 
 73' = 24-0 ; ^ = 0-45156; 
 
 6 b 
 
 = 0-36554 
 
 + S/3'= 0-023 gives -11 -15 
 
 X 
 
 ;g= 24-023 ; ^ = 0-45145 ; ^= 0-36539 
 
 and 6 = 0-31646 inch. 
 
 Therefore x = hx 0-45145 = 0-1429 inch, 
 
 and 2 = 6x0-36539 = 0-1156 inch. 
 
 DETEKMINATIONT OF CAPILLARY CONSTANTS OF MERCURY IN 
 
 CONTACT WITH GLASS. 
 
 The great impediment to the exact determination of capillary constants arises 
 from the changes that usually take place at capillary surfaces when left undisturbed 
 for some time. All careful experimenters have recognised this difficulty. It seemed 
 therefore best to place a drop of mercury in position and to take measures of 2R, 2r 
 a,nd If as opportunity offered. Drops weighing 4, 8, 12, 16, 20 and 24 grains were 
 used, because it was expected -that, if a. and a were not really constant for mercury 
 resting on glass, some indication of the manner in which they varied would thus 
 be made manifest. The mercury was obtained as being pure from a leading philo- 
 sophical instrument maker about 1862. When any experiment was to be made, a 
 sufficient quantity was taken from this store, and after having been used, it was 
 treated as waste. Also the same glass plate table was used in all the experiments. 
 The glass plate was cleaned with blotting paper or with the pith of the stalk of 
 the artichoke. And after this, either the same or a fresh drop of mercury was 
 placed in position and vibrated. In the following tables of experiments the operation 
 of cleaning the glass and replacing the same drop of mercury is indicated by a dotted 
 
 line But a change in the mercury used is denoted by a line — across the table. 
 
 The reading of the thermometer is given and also the time during which the 
 drop had been in position when the measurement was made. The experiments were 
 carried on in a small workshop built in a garden apart from other buildings. The 
 
 B. 10 
 
74 DETEEMINATION OF CAPILLARY CONSTANTS 01" MEECUEY. 
 
 observing table rested on supports driven into the ground which were independent 
 of the brick floor. There were public roads, used chiefly for light traffic, on two 
 sides at the distances of 60 and 60 yards. The slow changes in the forms of drops 
 of fluid appear to arise, (1) from some small change that takes pla,ce in the tension 
 of the enveloping surface, (2) from changes of temperature between night and day, 
 and (3) from slight tremors arising from passing vehicles, &c. The calculation of 
 the capillary constants was carried on as the experiments were made. After all had 
 been completed the reductions of the instrumental observations into inches were 
 carefully examined, and the calculations of all the 145 experiments were repeated, 
 so that the residts given in the following tables may be considered to be quite correct. 
 
 The variation in the value of the capillary constants deduced from drops of 
 mercury of the same size was much greater than was expected. But, when the 
 mean values of « and a derived from each form of drop were compared, the agree- 
 ment was surprisingly close. Hence so far as these experiments go the form of sessile 
 drops appears to be that indicated by the Theories of Young, Laplace, Gauss and 
 Poisson. 
 
 Finally the values of a, w, and V were calculated from the mean values of 
 2i2, 2r, and H for each size of drop of mercury. The results are given on the last 
 page for comparison with the means of the values of a and a derived from each experi- 
 ment for each size of drop of mercury. 
 
 In order to carry out the original scheme, as sketched in the Introduction, many 
 more experiments should be made, particularly for the purpose of finding the effects 
 of variation of temperature on the values of the capillary constants. 
 
 The calculations for negative values of /8 should also be greatly extended, so that 
 the intervals between them might be readily filled up by interpolation, as we have 
 done in the case of positive values of /3. 
 
 The measuring instrument in its present form appears to be satisfactory. The 
 microscope descends in a vertical direction by its own weight and is raised by the 
 screw. A screw of about 50 turns to the inch is very suitable for experimenting -with 
 mercury. But a quicker motion will become desirable when experiments are made 
 with a drop of one fluid immersed in another fluid, as the drops may be then much 
 larger. 
 
 All documents connected with these calculations now in my possession will be 
 carefully preserved, and every assistance will be afforded to any person who may under- 
 take the completion of the work. 
 
 MiNTINQ ViCAEAGE, Oct. 1883. 
 
DETERMINATION OF CAPILLARY CONSTANTS OP MERCURY. 
 
 75 
 
 Drop op 4 Grains op Mercury. 
 
 No. of 
 Observa- 
 tion 
 
 H 
 
 H 
 
 r 
 
 ? 
 
 a 
 
 (U 
 
 J\ 
 
 Temp. 
 F 
 
 Error in 
 V 
 
 Hours 
 
 in 
 position 
 
 
 I 
 
 24 
 25 
 26 
 
 27 
 
 28 
 29 
 
 30 
 31 
 
 117 
 
 118 
 119 
 
 120 
 
 121 
 122 
 
 123 
 
 124 
 
 125 
 126 
 127 
 128 
 
 129 
 
 130 
 
 131 
 132 
 
 133 
 Means 
 
 inch 
 0-07460 
 
 inch 
 0-09050 
 
 -09150 
 -08950 
 ■08940 
 
 -08950 
 
 inch 
 0-05990 
 
 ■06040 
 •06170 
 •06185 
 
 ■06165 
 
 •05965 
 ■06070 
 
 •06050 
 •06045 
 
 i^925 
 1^922 
 
 2-394 
 2'3i6 
 
 2^292 
 
 2^005 
 2^457 
 2^400 
 2^424 
 2^126 
 
 2-6'JO 
 
 2^549 
 2^472 
 
 2^638 
 2^633 
 
 2^755 
 
 ll7^oo 
 
 145-43 
 
 145-65 
 146-67 
 
 145-48 
 145-68 
 
 inch 
 0-1308 
 
 •1321 
 
 •1237 
 •1248 
 
 m. metres 
 3^321 
 
 3^356 
 3^142 
 3-169 
 
 60° 
 
 cubic inch 
 + -000002 
 
 ? 
 
 
 ■0753s 
 ■07635 
 ■07610 
 
 114-54 
 130-70 
 128^52 
 
 127^90 
 
 61 
 58 
 59 
 
 61 
 
 -1- -000039 
 + -000045 
 + -000036 
 
 -1- "000034 
 
 + "000005 
 + •000013 
 
 
 
 12 
 18 
 
 
 -07600 
 
 ■1251 
 
 3"i76 
 
 ? 
 
 
 -07475 
 •07570 
 
 -09040 
 -08870 
 
 -08930 
 •08895 
 
 120-03 
 135-42 
 
 132-84 
 134-18 
 
 146-70 
 148-11 
 
 ■1291 
 •1215 
 
 3^279 
 3-087 
 
 60 
 58 
 
 \ 
 13 
 
 
 -07580 
 -07565 
 
 148^52 
 148-47 
 
 •1227 
 -1221 
 
 3^n7 
 3-101 
 
 60 
 62 
 
 + ^00002 2 
 + ^000014 
 
 8 
 
 22 
 
 
 ■07525 
 
 ■0759s 
 ■07585 
 
 •08950 
 
 •06115 
 
 123-59 
 142-61 
 138-41 
 
 135-47 
 
 144-68 
 
 147-73 
 146-56 
 
 147-97 
 
 151-40 
 150-96 
 
 -1272 
 
 -1184 
 -1202 
 
 3^231 
 3-008 
 3-053 
 
 63 
 
 62 
 64 
 
 4- •OOOOIO 
 
 -1- •000005 
 -1- -000007 
 
 1 
 
 
 -08760 
 -08790 
 
 ■■•o88'75 
 
 •06140 
 ■06160 
 
 •06090 
 
 13 
 15 
 
 
 ■07585 
 
 ■07535 
 •07545 
 
 -07620 
 
 •1215 
 
 •1180 
 •1182 
 
 3-086 
 
 66 
 
 -1- -000017 
 
 — -000009 
 
 - -000005 
 
 3 
 
 
 •08795 
 -08800 
 
 •08760 
 
 •05940 
 ■05965 
 
 143-56 
 143-09 
 
 144-77 
 
 2-997 
 3-003 
 
 64 
 67 
 
 34 
 36 
 
 
 •06140 
 
 148-57 
 
 •1175 
 
 2-985 
 
 66 
 
 4- -000014 
 
 21 
 
 
 -07380 
 -07430 
 
 •07475 
 •07490 
 -07470 
 
 -07460 
 
 -07480 
 -0751S 
 •07545 
 ■07550 
 
 0-07531 
 
 •08850 
 •08810 
 •08770 
 •08760 
 •08925 
 
 -08970 
 
 -08940 
 -08880 
 •08880 
 •08850 
 
 ■05860 
 ■05960 
 
 ■0595s 
 •05975 
 •05970 
 
 •05965 
 
 •06035 
 •06040 
 •06045 
 •06040 
 
 2^157 
 2^263 
 2^48 1 
 
 2^525 
 
 2^I72 
 
 2^075 
 
 2^104 
 
 2^305 
 2-391 
 2^467 
 
 129^85 
 132-60 
 
 139-85 
 141-02 
 127-39 
 
 123-57 
 
 148-16 
 147-20 
 
 149^17 
 149-16 
 147-32 
 
 146-77 
 
 •1241 
 -1228 
 -1196 
 •1191 
 ■1253 
 -1272 
 
 3-152 
 3-119 
 3-038 
 3-025 
 3-183 
 
 3-231 
 
 63 
 62 
 61 
 
 63 
 61 
 
 61 
 
 — -000048 
 
 — -000037 
 
 — -000030 
 
 — -000027 
 
 — -OOOO 11 
 
 — -000008 
 
 
 
 24 
 48 
 
 71 
 88 
 
 
 
 
 124-15 
 131-33 
 133-73 
 136-53 
 
 145-62 
 147-09 
 
 147^94 
 148-49 
 
 147^52 
 
 -1269 
 •1234 
 -1223 
 
 •I2IO 
 0-1233 
 
 3-224 
 
 3-134 
 3-106 
 
 3-074 
 
 61 
 
 59 
 
 57 
 58 
 
 — -000005 
 
 — -000002 
 -1- -000007 
 + -000005 
 
 38 
 61 
 96 
 III 
 
 
 0-08890 
 
 ©•06041 
 
 132-03 
 
 3-131 
 
 
 
 
 
 
 
 
 
 
 10—2 
 
76 
 
 DETERMINATION OP CAPILLARY CONSTANTS OF MERCURY. 
 
 Drop op 8 Grains of Mercury. 
 
 No. of 
 Observa- 
 tion 
 
 R 
 
 H 
 
 r 
 
 ^ 
 
 a 
 
 0) 
 
 J\ 
 
 Temp. 
 P 
 
 Error in 
 V 
 
 Hours 
 
 in 
 position 
 
 2 
 3 
 
 4 
 
 32 
 33 
 
 34 
 
 134 
 
 135 
 136 
 
 137 
 138 
 
 139 
 140 
 
 141 
 142 
 
 143 
 144 
 
 145 
 Means 
 
 inch 
 
 0^09980 
 
 •loooo 
 
 inch 
 
 o-ioioo 
 
 •10090 
 
 •10360 
 
 -10210 
 -10070 
 
 •lOIIO 
 
 inch 
 
 0^08535 
 
 ■08535 
 
 ■08345 
 
 -08380 
 -08400 
 •08410 
 
 ■08315 
 •08350 
 •08340 
 
 •08320 
 •08330 
 
 •08335 
 •08335 
 
 •08390 
 -08420 
 -08445 
 •08435 
 •08430 
 
 5-226 
 
 5'37o 
 4^456 
 
 4-878 
 5-700 
 5^534 
 
 4-892 
 5^521 
 S'567 
 
 4-874 
 5-101 
 
 5-079 
 5-228 
 
 5-280 
 5-423 
 5*464 
 5-522 
 5-321 
 
 127-96 
 129-57 
 
 117-69 
 
 124-28 
 134-61 
 132-23 
 
 12374 
 131-53 
 132-05 
 
 123-59 
 126-22 
 
 126-52 
 128-51 
 
 129-29 
 130-89 
 
 131-35 
 132-20 
 129-77 
 
 128-44 
 
 144-45 
 145-10 
 
 i45'89 
 
 146-11 
 148-62 
 148-09 
 
 148-18 
 
 149-83 
 iSo;25 
 
 147-94 
 148-82 
 
 148-19 
 148-67 
 
 147-51 
 147-44 
 147-06 
 
 147-39 
 
 14675 
 
 147-57 
 
 inch 
 
 o'i25o 
 
 -1242 
 
 -1304 
 
 •1268 
 -1219 
 -1230 
 
 -1271 
 
 •1233 
 •1231 
 
 •1272 
 •1259 
 
 •1257 
 •1248 
 
 -1244 
 -1236 
 ■1234 
 -1230 
 •I241 
 
 0^1248 
 
 m. metres 
 3-175 
 3-156 
 
 60° 
 60 
 
 cubic inch 
 
 -f -OOOOO I 
 
 -t- "000017 
 -1- ^000030 
 
 ? 
 
 . ? 
 
 ? 
 
 •09905 
 
 ■09915 
 ■09990 
 ■09990 
 
 •09945 
 ■looio 
 -IOOI5 
 
 -09940 
 
 •09975 
 ■09950 
 
 ■09960 
 
 •09960 
 •09980 
 
 ■09985 
 •09985 
 •09965 
 
 0-09969 
 
 3-311 
 
 3-222 
 3-096 
 3-124 
 
 3-229 
 3^132 
 
 3'i26 
 
 3^231 
 3^i97 
 
 3^194 
 3^169 
 
 3^159 
 3^140 
 3^134 
 3-124 
 
 3-153 
 3-171 
 
 58 
 
 62 
 61 
 62 
 
 + ^000004 
 -1- ^000008 
 -H •000017' 
 
 + -000037 
 + -000043 
 + -000046 
 
 + -000034 
 + -000045 
 
 + -000026 
 -1- -000023 
 
 -1- -000013 
 ■¥ •000016 
 + "000014 
 4- •OOOOII 
 
 -1- -000008 
 
 I 
 
 18 
 22 
 
 5. 
 21 
 29 
 
 12 
 22 
 
 
 10 
 
 -10295 
 -10180 
 -10180 
 
 -10290 
 •10265 
 
 •10230 
 -10200 
 
 -10150 
 -10120 
 
 -lOIOO 
 
 -10090 
 
 -I0I20 
 O-IOI76 
 
 58 
 57 
 57 
 
 58 
 58 
 
 56 
 58 
 
 55 
 56 
 57 
 58 
 58 
 
 13 
 23 
 37 
 46 
 61 
 
 0-08392 
 
 
 
 
 
DETERMINATION OF CAPILLARY CONSTANTS OP MERCURY. 
 
 11 
 
 Drop op 12 Grains op Mercury. 
 
 No. of 
 Obser- 
 vation 
 
 R 
 
 H 
 
 r 
 
 /3 
 
 a 
 
 (D 
 
 7! 
 
 Temp. 
 F 
 
 Error in 
 V 
 
 Hours 
 
 in 
 position 
 
 5 
 
 6 
 
 7 
 
 35 
 36 
 
 37 
 38 
 
 39 
 
 40 
 
 41 
 42 
 
 43 
 44 
 45 
 46 
 
 47 
 Means 
 
 inch 
 0-11720 
 
 •I1735 
 •II763 
 
 ■11700 
 •I1810 
 
 inch 
 
 0-10900 
 
 •10920 
 
 •10785 
 
 inch 
 
 o-ioiio 
 
 •10230 
 
 •10288 
 
 -10020 
 -IOII5 
 
 8-728 
 
 8-331 
 9-044 
 
 8-267 
 10-592 
 
 124-99 
 121-50 
 
 126-53 
 
 121-70 
 136-68 
 
 133-39 
 133-07 
 
 120-94 
 134-08 
 130-85 
 131-47 
 
 126-ir 
 124-84 
 131-22 
 130-81 
 133-35 
 
 147-27 
 144-48 
 144-43 
 
 148-32 
 150-66 
 
 150-91 
 150-20 
 
 147^83 
 149-89 
 148-58 
 148-10 
 
 inch 
 
 0-1265 
 
 •1283 
 
 •1257 
 
 •1282 
 •1210 
 
 m. metres 
 3'2i3 
 3-259 
 3^193 
 
 3^256 
 3-073 
 
 3-110 
 3^"4 
 
 3-266 
 3-102 
 3-140 
 3-133 
 
 3-199 
 3-215 
 3-136 
 3-MI 
 3-111 
 
 3-166 
 
 59° 
 
 59 
 
 59 
 
 cubic inch 
 
 - -OOOOOI 
 
 H- -oooori 
 
 - -000013 
 
 -t- -000022 
 
 - -000009 
 
 -1- -000035 
 + -000016 
 
 + -000032 
 -1- -000006 
 
 - -000005 
 
 - -000015 
 
 -1- -000027 
 
 - -000025 
 
 - -000043 
 
 - -000014 
 
 - -000009- 
 
 ? 
 2 
 20 
 
 -11020 
 -10700 
 
 62 
 64 
 
 63 
 64 
 
 62 
 
 63 
 62 
 
 64 
 
 63 
 63 
 63 
 64 
 65 
 
 1 
 
 2 
 
 38 
 
 -I1830 
 -11810 
 
 -10800 
 -10785 
 
 -11030 
 •10750 
 -10790 
 •10760 
 
 ■10105 
 -10115 
 
 10-183 
 10-073 
 
 8-200 
 
 10-215 
 
 9-642 
 
 9-726 
 
 8-925 
 8-588 
 
 9-509 
 9-546 
 9-969 
 
 -1225 
 •1226 
 
 2 
 
 9 
 
 \ 
 12 
 
 13 
 16 
 
 2 
 
 7 
 21 
 
 45 
 145 
 
 ■I1710 
 •11810 
 -II770 
 •I1770 
 
 •10050 
 
 ■IOI35 
 •1013s 
 
 •IOI60 
 
 -10005 
 •10025 
 
 -10035 
 
 -10040 
 •10025 
 
 •1286 
 •1221 
 •1236 
 •1233 
 
 ■1259 
 •1266 
 ■1235 
 
 •1237 
 •1225 
 
 -11740 
 ■I1675 
 -II710 
 -11740 
 ■11765 
 
 0-I1754 
 
 ■10955 
 •10915 
 
 -10787 
 
 -10820 
 
 •10790 
 
 o'io844 
 
 150-17 
 148-15 
 
 149-54 
 150-04 
 151-26 
 
 o-ioioo 
 
 128-85 
 
 148-74 
 
 0-1247 
 
 
 
 
 
 
 
 1 
 
 Drop of 16 Grains of Mercury. 
 
 No. of 
 Obser- 
 vation 
 
 R 
 
 H 
 
 r 
 
 /8 
 
 a 
 
 (D 
 
 v/i 
 
 Temp. 
 F 
 
 Error in 
 V 
 
 Hours 
 
 in 
 position 
 
 8 
 9 
 
 10 
 ir 
 
 48* 
 49* 
 
 SO* 
 
 51* 
 52* 
 
 53 
 
 54 ■ 
 
 55 
 56 
 
 Means 
 
 inch 
 
 o^i3298 
 
 ■13290 
 
 •13185 
 •13215 
 
 ■13275 
 •13378 
 
 ■13375 
 
 ■13315 
 •13338 
 
 inch 
 
 0-11305 
 
 -11283 
 
 •I 1370 
 ■II290 
 
 •II420 
 ■II250 
 ■II245 
 
 -II285 
 ■I1240 
 
 •II525 
 -IIII5 
 
 -11240 
 -II2IO 
 
 inch 
 
 0-11645 
 
 ■11642 
 
 ■ •"535 
 •11575 
 
 ■11675 
 ■11750 
 •11760 
 
 ■I 1 745 
 -11725 
 
 -11718 
 ■11823 
 
 -11695 
 -11665 
 
 0-11689 
 
 i4^7o3 
 14-809 
 
 13-448 
 14-228 
 
 i3^3.55 
 15^584 
 i5^53o 
 
 14^529 
 i5^3ii 
 
 "•853 
 16-922 
 
 15^472 
 15-758 
 
 127-57 
 
 i28^i9 
 
 124-10 
 127-08 
 
 148-23 
 148-20 
 
 147-70 
 147-90 
 
 146-32 
 147-94 
 147-63 
 
 146-25 
 147-57 
 
 143-46 
 147-72 
 
 148-31 
 148-89 
 
 inch 
 
 0-1252 
 
 -1249 
 
 m. metres 
 3-180 
 
 3^173 
 
 3-225 
 3^187 
 
 3-229 
 3^154 
 3^156 
 
 3"i94 
 3^158 
 
 3^341 
 3-100 
 
 3^150 
 3"i33 
 
 3^183 
 
 59° 
 59 
 
 55 
 51 
 
 67 
 66 
 66 
 
 cubic inch... 
 -t- -ooooio 
 4- -000012 
 
 6 
 
 7 
 
 •1270 
 •125s 
 
 - -000031 
 
 - '000033 
 
 + -000014 
 -1- -000028 
 -t- •000021 
 
 
 2 
 
 \ 
 3 
 6 
 
 123-77 
 129-73 
 
 129-57 
 
 126-50 
 129-38 
 
 115-62 
 134-30 
 
 130-06 
 131-48 
 
 127-49 
 
 -1271 
 •1242 
 ■1242 
 
 •1257 
 ■1243 
 
 •1315 
 •1220 
 
 65 
 63. 
 
 62 
 62 
 
 63 
 
 63 
 
 - •000008 
 
 — ^000007 
 
 + '000053 
 + '000037 
 
 + '000028 
 -1- .'000009 
 
 II 
 21 
 
 \ 
 47 
 
 16 
 
 19 
 
 •13225 
 ■13415 
 
 ■13338 
 •13325 
 
 ■1240 
 ■1233 
 
 0-13306 
 
 O-II29I 
 
 147-39 
 
 •1253 
 
 
 
 
 
 
 
 
 Weight of this drop -was 16'12 grains. 
 
78 
 
 DETERMINATION OE CAPILLARY CONSTANTS OF MERCURY. 
 
 Drop of 20 Grains of Mercury. 
 
 No. of 
 Obser- 
 vation 
 
 H 
 
 H 
 
 r 
 
 ^ 
 
 a 
 
 b) 
 
 J\ 
 
 Temp. 
 F 
 
 Error in 
 V 
 
 Hours 
 in 
 
 position 
 
 12 
 
 13 
 14 
 
 15 
 16 
 
 17 
 
 57 
 58 
 59 
 
 60 
 61 
 
 62 
 
 63 
 64 
 
 65 
 
 66 
 67 
 68 
 
 69 
 
 70 
 71 
 
 72 
 73 
 74 
 75 
 76 
 
 Means 
 
 inch 
 0-14655 
 
 ■14405 
 "14420 
 
 ■14325 
 "I4420 
 
 ■14435 
 
 ■14563 
 ■14590 
 ■14803 
 
 ■14458 
 ■14508 
 
 ■14528 
 
 •14543 
 ■14520 
 
 ■14513 
 
 -14650 
 
 ■14775 
 -14790 
 
 inch 
 o"ll425 
 
 "II910 
 "I1880 
 "I2100 
 ■I1810 
 "I1820 
 
 inch 
 0-13040 
 
 24*243 
 i7"o63 
 
 i7'373 
 15-018 
 18^438 
 18^387 
 
 19-386 
 19-836 
 26-497 
 
 18-967 
 20-595 
 
 20-184 
 20"937 
 20-671 
 20-330 
 
 133^07 
 
 116-94 
 
 117-71 
 
 III-IO 
 I2I-II 
 120-69 
 
 147-93 
 
 147-47 
 147-28 
 
 147-16 
 149-04 
 148-76 
 
 145-71 
 145-50 
 148-00 
 
 149-11 
 149-26 
 
 149-58 
 150-17 
 
 150-27 
 
 149-25 
 
 146-98 
 147-38 
 147-17 
 
 148-99 
 145-48 
 146-04 
 
 149-82 
 
 149-04 
 148-94 
 
 147-31 
 147-59 
 
 148-05 
 
 inch 
 0-1226 
 
 m. metres 
 3^114 
 
 Ji° 
 
 SI 
 51 
 SI 
 51 
 52 
 
 cubic inch 
 
 — -000033 
 
 — ^000013 
 
 — ■000016 
 + ^000002 
 
 — "000046 
 
 — "00003 1 
 
 H- "000020 
 + "000027 
 + "000054 
 
 — "000036 
 
 — "000048 
 
 + "OOOOOI 
 
 — "cioooo8 
 
 — "000023 
 
 — "000029 
 
 12 
 
 2 
 
 3 
 ? 
 
 ? 
 ? 
 
 -12710 
 "12740 
 "12605 
 "12680 
 ■12705 
 
 "12980 
 ■13023 
 "13200 
 
 ■1308 
 ■1304 
 ■1342 
 ■1285 
 -1287 
 
 -1283 
 -1278 
 -1214 
 
 3-322 
 
 3-311 
 3-408 
 
 3-264 
 3-270 
 
 3-258 
 3-246 
 3-083 
 
 3-251 
 3^200 
 
 3-219 
 3-195 
 
 3"200 
 
 3-211 
 
 -II725 
 
 ■11695 
 •11370 
 
 121-60 
 122-44 
 
 135-72 
 
 I22-II 
 126-03 
 
 124-50 
 126-38 
 126-03 
 125-18 
 
 6i 
 61 
 62 
 
 \ 
 
 I 
 
 47 
 
 -11785 
 -I1665 
 
 "I1730 
 "11685 
 ■11695 
 •11695 
 
 "12720 
 "12788 
 
 •12785 
 ■12785 
 
 •12755 
 "12790 
 
 -1280 
 "1260 
 
 62 
 
 63 
 
 61 
 
 63 
 64 
 
 63 
 
 63 
 61 
 60 
 
 4 
 
 19 
 
 24 
 
 29 
 
 "1267 
 ■1258 
 "I260 
 "1264 
 
 -1252 
 "1218 
 "1216 
 
 "I2II 
 •I24I 
 ■1237 
 
 ■1235 
 
 •1238 
 
 "I2I5 
 •1240 
 ■1246 
 
 -11580 
 •11380 
 "11360 
 
 ■11380 
 "II480 
 ■11472 
 
 ■11535 
 -11530 
 ■11382 
 ■11505 
 ■11550 
 
 •13045 
 •13193 
 "13220 
 
 ■13160 
 
 •13233 
 •13223 
 
 •12905 
 
 •12935 
 "13028 
 
 •13043 
 •13015 
 
 22-031 
 25-840 
 26-186 
 
 127-50 
 134-71 
 135-24 
 
 136-29 
 129-80 
 130-62 
 
 3-181 
 
 3-095 
 3-089 
 
 3-077 
 3-153 
 3-143 
 
 3-138 
 
 3^144 
 3^087 
 3^150 
 3^164 
 
 -1- "000027 
 -1- "000037 
 
 -H "000040 
 
 I 
 12 
 
 23 
 
 -14805 
 
 ■14758 
 "14768 
 
 26-769 
 23"662 
 24-065 
 
 23-175 
 22-958 
 
 25-358 
 22-982 
 
 22"48l 
 
 61 
 64 
 63 
 
 -i- "OOO061 
 
 + "000063 
 + "000068 
 
 13 
 
 25 
 36 
 
 3 
 
 7 
 
 47 
 49 
 S3 
 
 "14618 
 ■14615 
 
 ■14675 
 "14648 
 ■14640 
 
 131-05 
 130-54 
 135-42 
 130-02 
 128-86 
 
 65 
 65 
 66 
 66 
 67 
 
 — "000020 
 
 — "000023 
 
 — "000046 
 
 — "OOOOII 
 
 "000000 
 
 0-14593 
 
 0-11621 
 
 •12935 
 
 126-95 
 
 o^i256 
 
 3^191 
 
 
 
 
 
 
 
DETERMINATION OF CAPILLARY CONSTANTS OP MERCURY. 
 
 79 
 
 Drop op 24 Grains op Mercury. 
 
 No. of 
 
 
 
 
 
 
 
 
 n. 
 
 
 
 
 Obser- 
 
 R 
 
 H 
 
 r 
 
 j8 
 
 a 
 
 b) 
 
 
 11 
 
 Temp. 
 
 Error in 
 
 
 vation 
 
 
 
 
 
 
 
 'V 
 
 a 
 
 F 
 
 V 
 
 position 
 
 
 inch 
 
 inch 
 
 inch 
 
 
 
 
 
 inch 
 
 m. metres 
 
 
 cubic inch 
 
 
 l8 
 
 0-16085 
 
 O^IlSoo 
 
 0'I4550 
 
 33^541 
 
 127^29 
 
 145-29 
 
 0-1254 
 
 3-184 
 
 54" 
 
 + -000241 
 
 22 
 
 19 
 
 ■1605s 
 
 •I1840 
 
 •14550 
 
 32-000 
 
 125-23 
 
 144-36 
 
 •1264 
 
 3-210 
 
 55 
 
 + -000231 
 
 23 
 
 20 
 
 21 
 22 
 
 •16060 
 
 •I1820 
 '•11825 
 
 ....".L4.S.70 
 -14085 
 
 •13745 
 
 .3.?;.2.33 
 32-264 
 28-249 
 
 125-47 
 
 144-07 
 
 152-70 
 
 ■1263 
 
 -1265 
 
 ...3.:.i°7... 
 
 3-147 
 3-212 
 
 58 
 
 "56 
 
 60 
 
 + •000225 
 + -000072 
 
 .35 
 
 17 
 
 •15870 
 
 130-29 
 
 •15638 
 
 •I20IO 
 
 125-08 
 
 — -000050 
 
 8 
 
 23 
 
 77 
 
 •15645 
 •15835 
 
 •II925 
 
 •13730 
 •14243 
 
 29-889 
 30-584 
 
 128-08 
 
 153-63 
 
 •1250 
 -1259 
 
 3-174 
 
 60 
 
 — -000078 
 
 32 
 
 •I183O 
 
 126-26 
 
 146-41 
 
 3-197 
 
 67 
 
 + -000024 
 
 78 
 
 •15970 
 
 ■11583 
 
 -14380 
 
 37-232 
 
 134-90 
 
 147^49 
 
 •1218 
 
 3-093 
 
 65 
 
 + -000014 
 
 14 
 
 79 
 
 ■15923 
 
 •II582 
 
 •I44S3 
 
 34-856 
 
 132-01 
 
 144-44 
 
 •1231 
 
 3-126 
 
 65 
 
 - -000039 
 
 24 
 
 80 
 81 
 
 ....75923 
 •15845 
 
 ■11685 
 
 ■14455 
 •I42I5 
 
 3.i'.75o 
 32-230 
 
 128-58 
 
 143^97 
 147-64 
 
 J1247 
 ■1245 
 
 3;.i68 
 
 66 
 66 
 
 + -000016 
 — -00000 1 
 
 38 
 3 
 
 •II770 
 
 128-97 
 
 82 
 
 •15893 
 
 ■II71O 
 
 •14248 
 
 34-217 
 
 131-48 
 
 148-28 
 
 •1233 
 
 3-133 
 
 65 
 
 + -000008 
 
 8 
 
 83 
 
 •15908 
 
 ■I1615 
 
 •I42IO 
 
 37-078 
 
 135-72 
 
 150-00 
 
 •1214 
 
 3-083 
 
 63 
 
 — -000020 
 
 19 
 
 84 
 85 
 
 •15898 
 
 •I161O 
 •II950 
 
 •14208 
 -I40I3 
 
 36-928 
 
 27-527 
 
 135-67 
 123-08 
 
 149-82 
 
 •1214 
 .....^..„.„.„ 
 
 3-084 
 
 63 
 64 
 
 -•000033 
 - -000061 
 
 31 
 
 •15675 
 
 147-51 
 
 3-238 
 
 h 
 
 86 
 
 ■15840 
 
 •II7IO 
 
 •14280 
 
 32-457 
 
 129-43 
 
 146-13 
 
 •1243 
 
 3-157 
 
 64 
 
 — -000038 
 
 16 
 
 87 
 
 ■15908 
 
 •11675 
 
 •14323 
 
 34-352 
 
 131-45 
 
 146-95 
 
 ■1234 
 
 3-133 
 
 65 
 
 + -o'oooos 
 
 24 
 
 88 
 
 •15945 
 
 •II570 
 
 •14325 
 
 37-580 
 
 135-84 
 
 148-28 
 
 •1213 
 
 3-082 
 
 63 
 
 — -000010 
 
 60 
 
 89 
 
 •15938 
 
 •I1670 
 
 •14343 
 
 34-991 
 
 I3I-99 
 
 147-22 
 
 ■1231 
 
 3-127 
 
 62 
 
 + -000032 
 
 . 72 
 
 90 
 
 ■15790 
 
 ■II9OO 
 
 •14320 
 
 27-462 
 
 121-18 
 
 143-09 
 
 •1285 
 
 3-263 
 
 61 
 
 + -000009 
 
 86 
 
 91 
 92 
 
 •15855 
 
 •I1825 
 
 •14330 
 •13950 
 
 30-109 
 28-350 
 
 125-10 
 
 144-80 
 149-24 
 
 •1264 
 
 3-212 
 3-219 
 
 61 
 61 
 
 + -000037 
 — -000049 
 
 87 
 
 1 
 
 •15683 
 
 ■II947 
 
 124-51 
 
 •1267 
 
 93 
 
 •15660 
 
 •I1895 
 
 ■13985 
 
 28-357 
 
 124-94 
 
 148-05 
 
 •1265 
 
 3-214 
 
 61 
 
 — -000098 
 
 15 
 
 94 
 95 
 
 ■15640 
 ■15795 
 
 •0930 
 •I 1805 
 
 •13990 
 
 •14040 
 
 .?.7.:34o 
 32^352 
 
 123-28 
 
 147-28 
 150-46 
 
 •1240 
 
 3-235 
 
 61 
 "61 
 
 — -000100 
 
 17 
 6 
 
 130-00 
 
 3-151 
 
 - -000018 
 
 96 
 
 •15805 
 
 ■11750 
 
 •14065 
 
 33^373 
 
 131-55 
 
 150-37 
 
 -1233 
 
 3-132 
 
 64 
 
 — -000037 
 
 26 
 
 97 
 
 •15780 
 
 •I1815 
 
 •14053 
 
 31-695 
 
 129-10 
 
 149-76 
 
 •1245 
 
 3-161 
 
 64 
 
 - -000033 
 
 31 
 
 98 
 
 99 
 100 
 
 lOI 
 
 ......!'.57So 
 
 •15820 
 
 •II775 
 
 •14065 
 -14220 
 -I4I20 
 
 "■■-14163 
 
 .3L;5.92 
 
 3.°;soo 
 
 29-962 
 
 30-681 
 
 129-42 
 
 148-88 
 146-60 
 
 .14.7;47 
 147-21 
 
 -1243 
 
 ■■■■r258 
 
 •1258 
 
 ■1254 
 
 3-158 
 
 ...3.;.i.96. 
 
 ■3-196 
 
 3-184 
 
 62 
 63 
 65 
 
 — -000081 
 + -000012 
 
 — -000040 
 
 — -000033 
 
 45 
 
 •I183O 
 
 "'•iTsis 
 
 126-35 
 
 4 
 
 126-35 
 
 1 
 
 7 
 
 •15785 
 
 ■11810 
 
 127^24 
 
 67 
 
 102 
 
 •15820 
 
 •11755 
 
 •I4I95 
 
 32-145 
 
 129-23 
 
 147^53 
 
 •1244 
 
 3-160 
 
 61 
 
 — -000027 
 
 22 
 
 103 
 
 104 
 
 ■15830 
 
 •11755 
 
 •14200 
 
 32-333 
 
 30-860 
 
 129-39 
 
 147-66 
 1 47 '94 
 
 ....■.^.?.4.3 
 •1249 
 
 ...3:1.58... 
 3-172 
 
 62 
 ■62 
 
 — -000020 
 
 — -000076 
 
 31 
 
 •IS745 
 
 ■11790 
 
 •14095 
 
 128-21 
 
 h 
 
 loS 
 
 •15785 
 
 •11785 
 
 •I4I45 
 
 31-339 
 
 128-40 
 
 147^76 
 
 •1248 
 
 3-170 
 
 63 
 
 — -000040 
 
 12 
 
 106 
 
 •15730 
 
 •11850 
 
 •I4I20 
 
 29-209 
 
 i25"44 
 
 146-69 
 
 -1263 
 
 3-207 
 
 63 
 
 — -000063 
 
 26 
 
 107 
 
 •15810 
 
 •11795 
 
 •14250 
 
 30-517 
 
 126-54 
 
 145-75 
 
 •1257 
 
 3-193 
 
 64 
 
 — ^000020 
 
 36 
 
 108 
 
 •15820 
 
 •11740 
 
 •14200 
 
 32-372 
 
 129-62 
 
 147 -48 
 
 •1242 
 
 3-155 
 
 61 
 
 - ^000033 
 
 50 
 
 ■109 
 
 •15785 
 
 •I 1 740 
 
 •I42IO 
 
 31-391 
 
 i28^49 
 
 146-34 
 
 •1248 
 
 3-169 
 
 61 
 
 - ^00007 2 
 
 74 
 
 no 
 
 •15770 
 
 •11775 
 
 •14220 
 
 30-263 
 
 126^73 
 
 145-54 
 
 •1256 
 
 3-191 
 
 61 
 
 — ^000066 
 
 98 
 
 III 
 
 112 
 
 •15765 
 ■15770 
 
 •11760 
 
 •14225 
 
 3.°:.34.8 
 29-'76i 
 
 126-96 
 
 145-37 
 147-61 
 
 J?..?55. 
 •1261 
 
 3-188 
 3-202 
 
 63 
 "64 
 
 — -000081 
 
 - -000013 
 
 146 
 I 
 
 ■11870 
 
 •I4I20 
 
 125-82 
 
 113 
 
 •15850 
 
 •11710 
 
 •14250 
 
 33"077 
 
 130-32 
 
 147-13 
 
 •1239 
 
 3-147 
 
 61 
 
 — -000025 
 
 10 
 
 114 
 
 •15820 
 
 •11710 
 
 •14220 
 
 32-692 
 
 130-16 
 
 147-11 
 
 •1240 
 
 3-149 
 
 58 
 
 - -000051 
 
 34 
 
 "S 
 
 •15830 
 
 •I 1 700 
 
 •I42IO 
 
 33^261 
 
 130-95 
 
 147-66 
 
 •1236 
 
 3-139 
 
 61 
 
 - -000045 
 
 59 
 
 ri6 
 
 Means 
 
 •15850 
 
 •11725 
 
 •14260 
 ©•I4I99 
 
 32^668 
 
 129-63 
 
 146-82 
 147-46 
 
 ■1242 
 
 3-155 
 3-169 
 
 61 
 
 - -000019 
 
 81 
 
 0-15825 
 
 o^ii778 
 
 128-52 
 
 0-1248 
 
 
 
 
 
 
80 
 
 DETERMINATION OF CAPILLARY CONSTANTS OF MERCURY. 
 
 Summary of Mean Eesults for each weight of Drop of Mercury. 
 
 Weight 
 
 Laplace's 
 
 Error in 
 
 (1) 
 
 Error in 
 
 J' 
 
 Error 
 
 J'- 
 
 Error 
 
 Drop 
 
 a 
 
 a 
 
 
 
 V a 
 
 
 'V a 
 
 
 Grains 
 
 
 
 
 
 
 
 inch 
 
 inch 
 
 m. metres 
 
 m. metre 
 
 4 
 
 132-02 
 
 + 3-31 
 
 147-52 
 
 -0-27 
 
 0-1233 
 
 -•0015 
 
 3-131 
 
 -0-038 
 
 8 
 
 i28"44 
 
 - 0-27 
 
 147-57 
 
 -0-22 
 
 ■1248 
 
 
 3-171 
 
 + 0-002 
 
 12 
 
 128-85 
 
 + o"i4 
 
 148-74 
 
 + 0-95 
 
 •1247 
 
 — "OOOI 
 
 3-166 
 
 - 0-003 
 
 i6 
 
 127-49 
 
 — I"22 
 
 147-39 
 
 - 0-40 
 
 •1253 
 
 + "0005 
 
 3-183 
 
 + 0-014 
 
 20 
 
 126-95 
 
 -i"76 
 
 148-05 
 
 + 0-26 
 
 "I256 
 
 + "0008 
 
 3-191 
 
 + 0-022 
 
 24 
 
 Means 
 
 128-52 
 
 — 0-19 
 
 147-46 
 
 -0-33 
 
 "1248 
 
 
 3-169 
 
 
 128-71 
 
 147-79 
 
 0-1248 
 
 3-169 
 
 
 
 
 
 Values of a, m, &c., deduced from the mean values of E, H, and r 
 FOR each size of Drop of Mercijry. 
 
 Weight 
 Drop 
 
 R 
 
 H 
 
 r 
 
 j8 
 
 a 
 
 (1) 
 
 J\ 
 
 Error in V 
 
 Grains 
 
 4 
 
 8 
 12 
 16 
 20 
 24 
 
 Means 
 
 inch 
 
 0-07531 
 
 -09969 
 
 -II7S4 
 •13306 
 
 •14593 
 •15825 
 
 inch 
 0-08890 
 -10176 
 "1 0844 
 "11291 
 •11621 
 •11778 
 
 inch 
 
 0-06041 
 
 -08392 
 
 -lOIOO 
 
 "11689 
 
 •12935 
 "14199 
 
 2^334 
 
 5-236 
 
 9-328 
 
 I4"68i 
 
 21-433 
 31-796 
 
 131-96 
 i28"4i 
 128-88 
 127-32 
 126-87 
 128-55 
 
 147-51 
 147-56 
 148-76 
 147-40 
 148-07 
 147-46 
 
 inch 
 
 0-1231 
 
 "1248 
 
 "1246 
 
 •1253 
 "1256 
 "1247 
 
 m. metres 
 3-127 
 3-170 
 3-164 
 3-183 
 3-189 
 3-168 
 
 3-167 
 
 cubic inch 
 + 0-000003 
 + "000021 
 + "000002 
 + "000025 
 + "00000 1 
 — "000013 
 
 128-67 
 
 147-79 
 
 0-1247 
 
 
 
 
 
 The forms of these six drops are given in Fig. 3 on a large scale; 
 
U/<^ = 90° 
 
 /3 
 
 •0 
 
 •I 
 
 •2 
 
 ■3 
 
 •4 
 
 •5 
 
 •6 
 
 7 
 
 •8 
 
 ■9 
 
 o 
 
 I 
 
 2 
 
 i-ooooo 
 •15466 
 
 •24507 
 
 •02 1 80 
 •16546 
 •25248 
 
 •04149 
 
 ■17576 
 •25967 
 
 •05942 
 •18562 
 •26666 
 
 •07589 
 •19508 
 •27345 
 
 •09115 
 •20418 
 •2"8oo6 
 
 •10542 
 •21294 
 •28650 
 
 •11880 
 •22138 
 -29278 
 
 •13140 
 •22953 
 ■29890 
 
 •14333 
 
 •23742 
 •30488 
 
 3 
 
 4 
 5 
 
 1-31072 
 
 •36278 
 
 •40615 
 
 •31643 
 
 •36745 
 •41012 
 
 •32201 
 •37204 
 •41403 
 
 •32748 
 ■37656 
 •41789 
 
 •33283. 
 •38100 
 •42169 
 
 •33807 
 •38535 
 ■42544 
 
 •34320 
 •38963 
 •42914 
 
 ■34824 
 ■39386 
 •43278 
 
 ■35318 
 ■39802 
 
 •43638 
 
 •35803 
 -4021 1 - 
 
 ■43993 
 
 6 
 
 7 
 8 
 
 I '44344 
 •47621 
 
 •S°SSo 
 
 •44690 
 •47928 
 •50827 
 
 •45032 
 •48232 
 •51101 
 
 •45369 
 •48533 
 •51371 
 
 •45702 
 •48830 
 •51640 
 
 •46032 
 '49124 
 •51906 
 
 •46358 
 •49415 
 -52169 
 
 •46679 
 •49703 
 •52430 
 
 •4^996 
 •49988 
 •52689 
 
 •47310 
 -50270 
 •52946 
 
 9 
 
 lO 
 
 II 
 
 i'532oo 
 •55621 
 •57852 
 
 •53452 
 •55851 
 •58065 
 
 ■53702 
 •56080 
 
 ■58277 
 
 ■53949 
 •56307 
 •58488 
 
 •54194 
 •56533 
 •58698 
 
 •54437 
 •56758 
 •58906 
 
 •54678 
 •56981 
 •59112 
 
 •54917 
 •57202 
 
 •59317 
 
 •55154 
 •57421 
 •59520 
 
 •55389 
 •57638 
 -59722 
 
 12 
 
 13 
 14 
 
 I-S9923 
 •61856- 
 •63667 
 
 ■60122 
 •62042 
 •63842 
 
 •60320 
 •62227 
 •64016 
 
 •60517 
 •62411 
 •64189 
 
 •60712 
 •62594 
 
 •64'36i 
 
 •60906 
 •62776 
 ■64532 
 
 •61099 
 -62957 
 -64702 
 
 •61290 
 •63136 
 •64871 
 
 •61480 
 •63314 
 ■65039 
 
 •61669 
 
 ■63491 
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 140 
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 95 
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 105 
 
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 125 
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 140 
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 155 
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 20 
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 35 
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 65 
 
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 80 
 85 
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 95 
 
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 115 
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 125 
 
 130 
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 140 
 
 145 
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 155 
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 170 
 
 175 
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Cambttlige : 
 
 rP.INTED BY C. J. CLAY, M.A. AND SON, 
 AT THIC UNIVERSITY PKESS. 
 
Scaler. 
 
 01 OZ 03 04 00 OS 07 OS Off /O II I'J 13 l-l IS JO II 13 IS 20 Incll, 
 
 PIG. 1 
 
 ric4. 4 . 
 

 ':^i'r 
 
 WM:- 
 
 
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