292-293] GENERAL PROBLEM OF SLOW MOTION. 527 



The functions u' t v, w f may be expanded in series of solid har- 

 monics, and it is plain that the terms of algebraical degree n in 

 these expansions, say u n ', v n ', w n f , must separately satisfy (2). The 

 equations VX/ = 0, V 2 v n ' = 0, V 2 w/ = may therefore be put in 

 the forms 



d fdvn dw n '\ _ d fdun dwn\ 

 d \ dx d ) dz \ dz dx } ' 



d fdwn dv n '\ _ d fdv^ _ du n '\ 

 dz \dy dz ) dx \ dx dy ) ' 



(3). 



dz 



A ( d ^L _ dw *L\ =(. " - ^ 



dx V dz dx I dy V dy dz 



Hence 



dw n ' _ dv^ _ d/Xn du n ' _ dw n ' _ d% n dv n f du^ __ dxn A x 

 dy dz ~ dx ' dz dx "~ dy ' dx ~ ~dy ~~ dz '"^ '' 



where % n is some function of x, y, z ; and it further appears from 

 these relations that V 2 ^ w = 0, so that % n is a solid harmonic of 

 degree n. 



From (4) we also obtain 



dy y dz dx 1 dy dz 



- j^ (%< + yv n f + zw n f ) . . . (5), 

 with two similar equations. Now it follows from (1) and (2) that 



V 2 (xUn + yn + aWn) = .................. (6), 



so that we may write 



+ yOn + ZWn = n+1 .................. (7), 



where <f> n+1 is a solid harmonic of degree n + 1. Hence (5) may be 

 written 



(8). 



. - - 



dx dy 1 dz 



The factor n + 1 may be dropped without loss of generality ; and 

 we obtain as the solution of the proposed system of equations : 



