PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 677 



functions of a single parameter, are contained as, of course, they should 

 be, in this general integral. 



It is evident that every family of isothermal lines which are the 

 curves of a function u which satisfies (38) is a set of straight lines which 

 pass through a point. 



Transformation of the Equation h u =f(v). 



Given a function which satisfies (10) or (27), there always exists a 

 function which has the same lines, and a gradient expressible in terms of 

 the orthogonal function alone. The lines of all functions which satisfy 

 these equations are therefore those of functions which satisfy an equation 

 of the form 



?P = (42) 



or p 2 r+ 2pqs + q 2 t = 0. (43) 



If we take advantage of the Principle of Duality and make p = x', 

 q = y',px-\-qy — z = z', we shall get the transformed equation 



x' 2 ■ t' — 2 x'y' • s' + y' 2 ■ r> = 0, (44) 



and if then we put, m = — 2 log (x 2 + y 2 ), n = tan -1 [ - ), the result is 



which is equivalent to Fourier's familiar equation for the linear flow of 



heat. 



Qh 



If u is to be harmonic, while -=-^ = 0, we may write 



u = <f> (x + yi) + xb (x — yi), 



and substitute this value in equation (43). 

 The resulting equation is 



O' (x + yi)} 2 -^"(x- yi) + ty> (x - yi)Y • 0" (x + yi) = 0, (46) 



'^"(x-yl) ^"(x + yi) 



