PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 675 



u — ax^y ^Jk^ — a^ -\- if/ (a), (33) 



subject to the condition 



The equation A„ = ^- is also equivalent to the equation, 



— . — r= 1 (34) 



The complete integral of (35) is 



a 



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

 equations, 



u = aX + ^ + cj> (a), = X - ^ + (^' (a). (35) 



Cv Cv 



If M is to be harmonic while /t,^ is expressible in terms of m, u is of the 

 form <^(A) + '/'(i"-)? where X =^ x -\- yi, [x. ^=^ x — yl. Since 



dA dfji 

 we must have 



4<^'(X)-,/,'0.) = 4/[c/>(A) + ^(/x)],] (36) 



and if we differentiate both sides of this equation with respect to X and /a 

 we shall get 



<^"(A) • ^'(/z) = c/>' (A) •/' [<A (A) + ^ (/x)], 



</>'(A) • ^" (/x) = ^'(/x) •/' [<^(A) + ^(^)], 

 whence 



[<^'(A)J^-[f(^)p- ^'^^ 



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

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

 separately the cases where k is or is not zero. 



* Forsytli, DiffL-rciitial iMjuations, p. 307. 



