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 hu=f(y). 



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 



^ = (42) 



or 



pV+ 2p9S + r< = 0. (43) 



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

 q =1 y', px -\- q7j — z ^= z' , we shall get the transformed equation 



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



and if then we put, ?« = — 2 log (x^ + y^), n = tan""^ [ - ], the result is 



9m - 9n'' ^*^^ 



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



heat. 



9h 

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



dti 



?< — </, (x + iji) + il/(x — yi), 



and substitute this value in equation (43). 

 The resulting e({uation is 



[<!>' (^ + yiYr • ^" (^ - yO + [«A' (^ - yOl' •>" (^ + yf) = o, (46) 



^"(x-yQ _ _ <i>"{x-\-yi) 

 l^'{x-'yi)y [0'(x + yO]'^* ^^ 



