PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 671 



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



9 f V\ 

 may be written in the form A„ . A„ . ^r- ( — 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^ii 9^u 9'^u 9v 9v 9'^v 9hi 9^v 



9x 9y 9x^ 9x . 9y 9y^ 9x 9y 9x^ 9x . 9y 9y^ 



by p, q, r, s, t, p', q', r', s', t', respectively, we have, since u and v are 



orthogonal, 



pp' + qq' = 0; (19) 



so 



tht — — — 9x ,9K ^_J_ (^ ^ I ?^ ?^\ 

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



p^r -{- 'i p q s -\- qH 



"u 



(23) 



, 9h^ _ 9K 9x 9/i„ 9y _ 1 /'9/iu . 5" , 9h^ dv 



/'9J^ ,9i^,9K ^\ 



9v 9x ' 9v ^ 9y ' 9v h^ \ 9x 9x 9y ' 9y J 



p'[pq(r-t) + s(q'-p')] 



Since, however, h^ = />^ + q"^, K?' = p'^ + q'^ ; q^ . li^ = p'"- . //„-, 



and ^-^ = ± ivir^^9'-Vy-Vni) . ^.^^^ 



9o /i„ . At/ 



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



pqr -{■ (q" — p") s — p q t = 0. (25) 



