1881.] 



On Pendent Drops. 



365 



It will be observed that with wide tubes the inclination which the 

 extreme element of the curve makes with the horizontal base of the 

 tube gradually increases as the drop is expelled, till it separates ; but 

 with narrower tubes it passes through a maximum value, which may 

 be greater than 90°, after which it diminishes again. 



Professor Quincke has shown that the angle of contact between a 

 given liquid and solid in air is definite and constant. This, however, 



Taking the vertex as origin, and the axis of revolution as the axis of ?/, the equa- 

 tions to the generating curves of the two liquids may be written — 



Ti(- + -V)-^=I%i ft, 



Vpi pi/ Ri 



T 2 (± + ±)-^=V 2 y 2 ...... (ii), 



\ p 2 P 2 / R 2 



where and T 2 are the surface tensions. 



p x p\ and p 2 p' 2 are the principal radii of curvature at any point, 



and R 2 are the radii of curvature at the vertices, 

 D x and D 2 are the densities of the liquids. 



If these curves are symmetrical, the ratio between all corresponding lines must be 

 the same, and remembering that p\ and p" 2 , the radii of the curvature due to revolu- 

 tion, are the normals drawn from the curve to the axis, and therefore corresponding 

 lines in the plane of the paper, we have — 



J*! = Pi = _Pj = JL = j£ 

 R 2 P-2 P r 2 y' 



whence, writing E. 1 = R 2 K, p! = p 2 K, &c, in (i), and dividing by (ii), we have — 



T (JL + J: 



^PaK V 2 K E ? K ' Dj 



\P2 



y 



1 2 >y ~D. 2 y 



+ ■ 



p2 p' 2 Ba/ 

 Tt 1 D lir 



t 2 x k = d1 k ' 



whence K 2 = FFTT' 



±2 D l 



The empirical formula of the note to page 364 contains the experimental verification 

 of this result, and shows in addition that the limit of stability is reached at corre- 

 sponding stages by drops that develop similarly, or that this limit does not depend 

 on the absolute dimensions of the curve. M. Dupre's tacit assumption that this will 

 be the case seems to me hardly legitimate. 



M. Dupre makes extremely sagacious use of the relative weight of drops whose 

 forms while hanging are similar, for finding the surface tension. 



My observations, however, go to show that the depths of residual liquid are not 

 usually proportional to the other linear dimensions, and that any such use is not 

 certain to lead to results that are more than approximately correct. 



2 c 2 



