﻿732 Mr. G. Breit on the Effective 



The surface w = constant gives a spheroid of revolution 

 whose equation is 



a 2 cosh 2 u a 2 sinh 2 u 



as is seen by eliminating v from (3) ; and the surface 

 v = constant gives a hyperboloid of revolution whose 

 equation is 



= 1, (5) 



as is seen by eliminating u from (3). 



The two sets of surfaces represented by (4) and (5) are 

 orthogonal because (2) is a con formal transformation. 

 Also the planes 6 = constant are perpendicular to both (4) 

 and (5). Thus the co-ordinates (u, v, 6) are orthogonal. 



It is readily shown that the Laplacian in these co- 

 ordinates is 



— ^— (coshw^— )H ^— [cosv^r — ) 



cosn u on \ ou J cos vov\ ov > 



cosh 2 ^ — cos 2 v ~d 2 V n 

 cosh 2 ti cos 2 v ' ~d@ 2 



In particular, if V is independent * of 6, 



v 2 v = 



cosh u ~ftu 



l , av\ ,13/ av\ ,„ 



I cosn u -^—- -f ^- ( cos v^~ ). (6) 



\ ou/ COS v ov \ ov J 



* This expression may be derived by remembering that if 



are three orthogonal co-ordinates of such a kind that the differentials of 

 length corresponding to the three differentials 



dx\, dx 2 , dx 3 



are 



dxi dx-2 dxz 



then 



V'V = MA | fc g- (,4 f) -*=£■ (A g) 



See W. E. Byerly, ' Fourier Series and Spherical Harmonics/ p. 289, 

 equation (6). 



