PEIRCE. — LINES OP CERTAIN PLANE VECTORS. 675 



u — a x + y \/k 2 — a 2 -\- if/ (a), (33) 



subject to the condition 



= x u + df' (a). 



The equation h u = k is also equivalent to the equation, 



tt . 8 ± = i. (34) 



The complete integral of (35) is 



u 



u =a\ -\ (- c, 



a 



and its general integral * may be found by eliminating a between the 

 equations, 



u = a\ + ^ + <f> (a), = X — -„ + <]> / (a). (35) 



a ar 



If u is to be harmonic while h u is expressible in terms of u, u is of the 

 form </>(A) + i/'O".), where A = x + yi, fx = x — yi. Since 



we must have 



4 *' (A) • f (» = 4/ [0 (A) + ^ f»],] (36) 



and if we differentiate both sides of this equation with respect to A and /x 

 we shall get 



4>» (A) • f 0*) = ^ (A) •/' [> (A) + ^ Wl 



,/>'(A) • v" 0*) = *'00 •/' WW + *G0]> 



whence 



[^(x)j»-[^o*)f < ; 



Since the first member of (37) involves A only, and the second member 

 fA only, we may equate each member to a constant, — k' 2 , and consider 

 separately the cases where k is or is not zero. 



* Forsyth, Differential Equations, p. 307. 



