332 Mr. Gr. F. 0. Searle on the Steady Motion 



the surface of the ellipsoid, shall vanish at infinity, and shall 

 satisfy (7). We see at once that if f (x, y, z) satisfies 

 V 2 /=0. then / (x/ si a, y, z) satisfies (7). Now from 

 electrostatics we know that 



U \/(a' 2 + \Xb 2 + \)(c 2 + X) 

 where X is connected with x, y, z by the relation 



x 2 %r z 2 



+ */fT + ;3r7^= 1 > 



a' 2 + \ b 2 + X r c 2 + \ 

 satisfies v 2( £ = 0. 

 Hence 



AdX 



J A \/{a' 2 + X)(b 2 + X){c 2 + X) ' " v 



where X is connected with x, y, z by the relation 



^i^T+3^T=l| • • ' ( 9 ) 



(a l2 + X) b 2 + X c 2 + \ 

 satisfies (7). 



Writing a 2 for aa' 2 , (8) and (9) become 



W=C AdX (10) 



J A \/{a 2 + aX){b 2 + X)(c 2 + X) ' v ; 



T 2 V 2 " 2 



+ ^ + 7^TY. = 1 - ■ ' • (11) 



a 2 + aX b 2 + X c 2 + \ 



This value of ¥ is constant over the surface of the 

 ellipsoid a, b, c, for X=0 at all points of this surface ; it also 

 vanishes at infinity, and it satisfies (7). It is therefore the 

 value of ^F required. To find the constant A we make a have 

 its proper value qjAtrbc at the end of axis a. 



Now 



K K 



at the end of the axis. 



But by (5) E i=-^- 



Again, at x = a, y = z = we have d\/dx=2a/a and con- 

 sequently 



dV = dVdX = A^ 2a 



dx d\ dx ~ abc 'a' 



Hence A= ^ 



2K' 



