358 



the fiirtlier integration of tlie difïerential equation can only be earned 

 out by successive approximations or a development in series. 



In the case that tlie drop or bubble deviates little from the spher- 

 ical shape, ?/ is small compared to h '). In first approximation ?/ 

 may thus be neglected by the side of A, i. e. we may put // = 0; as a 

 second approximation a circular meridian section is then obtained; 

 if the ex[)ressiou for // corresponding to this as a function of x is 

 substituted in a, a first deviation from (he sphere is found as a 

 third approximation, etc. ''). * 



which is also found directly, when, for instance by applying the so called "weiglit- 

 method", Ihe rise in a capillary tube is calculated. The contradiction found by 

 A. Ferguson (I'hil. Mag., (6), 28, (191 4) p. 128) between the result of the integration 

 of the differential equation and that of the application of the weight-method is 

 merely due to an error of computation in the approximation of equation (2), owing 

 to which Ferguson's formula (7) is incorrect. 

 Equation (2) can also be written as follows 



k 

 x sill (f> z= \ k.v* (h \ y) — f.— ''• • (^ ) 



where v = tt xhj — u represents the volume arising by the rotation of the surface 

 OAA"0. (2) gives: 



2,71X0 sin (f =r 71 (ft,—/',) .'/.c' (/' -f .v) — (f'l— .",) .^^'i • • (3') 



which expresses for instance, that the resultant of the forces acting along the 

 edge of a section of a hanging drop makes equilibrium with the hydrostatic pressure 

 on the section and the weight of the -portion below it, in other words the surface 

 tension does not balance the weight of a hanging drop alone, a fact which may 

 also be derived from a simple consideration of the equilibrium (cf. on this point 

 Th. Lohnstein, Ann. d. Phys., (4) 20 (1906) p. 23S). 



2 

 1) Hence Eq is also small compared to h or to -^ , that is kRJ^ is a small 



number. 



') Cf. for instance A. Winkelmann, Handb. der Physik, 2e Aufl. 1 (2j, 1143- 

 1144, 1908. 



Putting ij = R(^—y R^^^^-\-z, where z is considered infinitely small as compared 



v' 



to ?/, and supposing that z' is also infinitely small compared to ?/', sm if = --^=.=zz 



may be developed in a series, which gives, if s^ represents the first approximation 

 of s: 



R* _ ._. ...... R.-\- yR.'-^' 



as is also found by Ferguson (loc cit.) although in a somewhat circuitous manner. 

 This expression, however, does not hold near x = Eq, as z\ is there no longer 

 infinitely small with respect to ij\ but of the same order ol magnitude ^viz. of 

 the order (kR^^^) h. ; this fact has been overlooked by Ferguson (loc. cit.). 



