292-293] GENERAL PROBLEM OF SLOW MOTION. 



527 



The functions u, v , w 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 , must separately satisfy (2). The 

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

 the forms 



dx 



d fdVn d?ln\ _ d 



dy V dx dy ) ~ dz \ dz 



dz \ dy dz J dx \ dx dy 



d fdiin dw n \ _ d idwn dv n 



dx \ dz dx ] dy \ dy dz 



(3). 



Hence 



_ 

 dy dz ~ dx dz 



dw^ _ dxn dv n _ du n _ dxn 

 dx ~ dy dx dy ~~ dz &quot; 



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

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

 degree n. 



From (4) we also obtain 



- ^ (Win + yv n + ZW n *) . . . (5), 



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

 V 2 (a?u n + yn + *w n ) = .................. (6), 



so that we may write 



OM n + yVn -f ZWn = &amp;lt;/&amp;gt; ?l+1 .................. (7), 



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

 written 



( + 1) Un = *! + g d X- _ /Xn ..... (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 : 



