j=?|;=?3 (^> 



258 Mr. A. E. H. Love on the Oscillations of a Rotating 



By Helmholtz's equation of vortex motion^ ? being inde- 

 pendent of vt, y, z, 



xxGHCG 



and 



so that 



h' = k" (8) 



Now a'x + li'y is the acceleration of the particle at [x, y, z) 

 parallel to the axis x, and h'x + h'y is the acceleration parallel 

 to the axis y, and thus the moment of the effective forces 

 about the axis z is proportional to //. Hence /i' = 0. 



Now, by integration, 



Vi -pIp = i {a'x^ + by + c'z^) + const. . . (9) 

 Suppose the liquid to be subject to its own attraction ; then 



where 

 or 



Hence the pressure is given by 



^//c) = const.-i[(7pA + aO^'+ (7pB+&')/+ (7pC + c')^'].(12) 



The surface x^ jc^ + y'^ jJ? + z^ jc^ = 1 can be free if 



; i(7|oA + a')a2 = i(7pB + 60^2^i(7pO + c'>' = o-say, (13) 



and then the pressure at any point is given by 



plp = V^I(j + <7{l-x^la^-y'lb'-z'lc''). . . (14) 

 4. Now 



or 



aa =a — C'a—zQila -^ — j-:—lra , „ , ,.,,.. ; 



