198 THEORY OF NEWTONIAN FORCES. [FT. I. CH. IV. 



equating the coefficients of dq l} dq 2 , we obtain 

 du .dv 6 



O -- I"* 0~ =7 > 



( 12 ) % fyi i 

 9w . 9v . <f> 



_ I n _ __ n r 



o * ^ T~ 



oq 2 cq 2 hz 

 Now eliminating <, 



, N , \du .dv} ., (du . dv} 



(13) Mo- +*5rr !BS *iio" +*V-M 



and equating the real parts on each side, and the imaginary parts 

 in like manner we obtain 



/ \ z, d u i d v i d v i 3u 



(14) h>2^- = ni^, h 2 ^-=h l . 



2 dq 2 l dqS *dq 2 l dq, 



Solving for the derivatives of v 

 , . _dv _h 2 du dv _h l du 



~^~h^ 87 2 ~r 2 g^' 



differentiating respectively by q 2 and q lt and adding 



<ty 2 

 Solving for the derivatives of u 



. . du hi dv du h 2 dv 



(I7) -^,= 



differentiating and adding 



that is, the functions u and v satisfy the same equation as the 

 potential and flux-functions V and /*. Such a pair of functions, 

 forming a set of orthogonal coordinate lines on the surface q 3 , may 

 accordingly be taken for the potential and flux-function. If we 

 have a second pair of functions u', v' such that 



du t *+W***M'd* t 



we have as before 



7 + idv' 



du + idv <f> ' 



