350 



Mr. T. Craig on Steady Motion in 



depend upon zonal harmonics ; merely generalizing his solu- 

 tion, I am led to write for S the series 



aOM) rtjSi $2^2 



+ 



+ 



r r 



bi$i Z> 2 S ; 



, 63S3 , 



3 "*" j& " 1 " 



or simply 



" 



(25) 



S t - being a solid spherical harmonic of degree i. For brevity, 

 make 



Oi + bii* 



then 



E,= 



S=2R,R 



(26) 



In order to find the values of the constants, it is necessary 

 to take into account the conditions to be fulfilled at the sur- 

 face of the sphere, i. e. at the finite bounding surface of the 

 fluid. These are 



= > for r = a. 

 = 0.) 



v 

 w=0, 



(27) 



We must first determine the function U, V, W, however, 

 before these boundary conditions can be introduced. The fol- 

 lowing method is due to Borchardt, and is given in the Mo- 

 natsber. der Berlin. Akad. for 1873. If we have a function s 

 of x, y, z satisfying the equation 



V« 2 = 0, (28) 



and four other functions connected with s by the relations 



• (29) 



s = s + x 



ds ds 

 dx y dy 



ds - 

 dz 



ds 



dy 



ds 



~ y d7 





ds 

 S * = X dz 



ds 

 dx 





ds 



s ^d7v- 



ds 

 dy' 



J 



