1879.] the Capillary Phenomena of Jets. 95 



in which «{ n is an absolute constant. The corresponding velocity- 

 potenrial is 



It is instructive to verify these results by the formula? applicable to 

 steady motion. The resultant velocity q at any point is approxi- 



Ly equal to — m 3 and 



dz 



matel 1 



— =— 1'+^ L'_IL\ / COS fCZ COS »0. 



At the surface we have approximately r=a, and 



io-=iu — _p A: r_f_ — ^ ^_ cos kz cos 



ikJ\tka) 



Thus by the hydro dynamical equation of pressure, with use of (25), 

 since v=pk~ 1 , 



Pressure = const. + 7«a _2 T (n~ — 1 + 7 L -~a~) cos &3 cos nO . (32). 



The pressure due to superficial tension is T (R-, -1 + Ro -1 ), if R l5 R , 

 are the radii of curvature in planes parallel and perpendicular to the 

 axis ; and from (30) 



— 1^ -1 =— £=— & 2 7„ cos n6 cos kz 

 dz 2 



R 1 " 1 = r 1 + - — — =a 1 + 7, i a 2 (/z 2 — l)cos /£0 cos fes ; 

 dd'~ 



so that 



Pressures const. + ^ n a~~(n~ — 1 + Z. ,2 cr)cos nO cos kz. 



Thus the pressure due to velocity is exactly balanced by the capillary 

 force, and the surface condition of equilibrium is satisfied. 



Appendix II. 



"We will now investigate in the same manner the vibrations of a 

 liquid mass about a spherical figure, confining ourselves for brevity to 

 modes of vibrations symmetrical about an axis, which is sufficient for 

 the application in the text. These modes require for their expression 

 only Legendre's functions P„ ; the more general problem, involving 

 Laplace's functions, may be treated in the same way, and leads to 

 the same results. 



The radius r may be expanded at any time t in the series 



r=a + a l P 1 (u)+ . . . + «,P ;j (") + . . (33), 



