PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 671 



Since the divergence of any vector V which has the u curves for lines 



9 ( V\ 

 may be written in the form h u . h„ . -=- ( — J as well as in the form (3), 



the condition stated in equation (18) is at once obtained. 

 If we denote the quantities 



9u 9u 9 2 u 9 2 u 9 2 u 9v 9v 9 2 v 9 2 v 9 2 v 



9x 9y 9x 2 9x . 9y 9y 2 9x 9y 9x 2 9x . 9y 9y 2 



by Vi 9> r > s > *> ?'■> Q'i r '> s '> *■'> respectively, we have, since u and v are 

 orthogonal, 



pp' + qq' = 0; (19) 



whence by differentiation we get 



p'r + pr' + q's + q s' = 0, (20) 



p's + ps' + q't + qt> = 0. (21) 



w v. 9x h x 1 p p 



We have, moreover, ■=- = 7- . cos (x, u) = 7- . -f- = 7-., > 



9u h u h u h lt h 2 



'u '*« '*u 



1 • -i 1 9y q 9x p' 9y a 1 ,_„ 



and, similarly, ?f = h' ^T = h > T=h> ( 22 ) 



9u h" dv h- 9v hJ v ' 



so 



th t — — — ^4.^ 3}L_ J_ (3K 9u 9h u 9u\ 



9u 9x 9u 9y 9u h 2 \ 9x ' 9x 9y ' 9y ) 



_ p 2 r -f 2pq s + q 2 t 

 - p ' 



"u 



9h u _ 9h u 9x 9h u 9y _ 1 (9h n _ 9v t 9h u dv" 



(23) 



/9% t 9v 9h u dv\ 



\ 9x 9x 9y ' 9y J 



9v 9x 9v 9y ' 9v h 2 \ 9x 9x 9y ' 9y 



= p'[pq(r-t) +s(q 2 -p 2 )\ t 

 qh a h 2 



Since, however, h 2 = p 2 + q 2 , " h 2 = p' 2 + q' 2 ; q 2 . h 2 — p' 2 . h 2 , 



and ^ = ± {pqr + (q 2 -p 2 ) S -pqi)^ 



9v h v . h 2 v ' 



Equation (18) is equivalent, therefore, to the equation 



pqr + (q 2 — p 2 ) s — p q t — 0. (25) 



