PEIRCE. — THE GRAPHICAL SUPERPOSITION OF LINES OF FORCE. 153 



and 





If in these equations the arbitrary function C is made equal to unity, 

 the conditions degenerate into the familiar definitions of any two 

 pairs of conjugate functions. 



In order that a single function (/?) may exist the partial derivatives 

 of which with respect to x and y shall be equal, respectively, to 



da , ^ da 



it is necessary and it is sufficient that a and t, should satisfy the 

 condition 



or 



C [ ^ da\ d f ^ da\ ^ , ^ 



dt da dt da ^ ^, , . , ^ 



In order that /a may exist, t, and X must satisfy the et^uation 



H.dJ^_^dJj_\_^^_^^^^^^ 



dx dx dy dij \ J • \. • J 



If loge t, be represented by s7, the last two equations take the 

 forms 



^ o "^ a 



Cm ca . Cu Ca „„ , , „ . , 



S-a:; + ey-aJ+^W = "' (^4) 



dm d\ dm dX _o/vN 



s7-r.+ij-r, + ^^^^ = '>' , (2.^) 



and, if each of these be differentiated with respect to x and with 

 respect to y, m may be eliminated from the resulting equations and a 

 necessary condition for a and A obtained, which may be stated in the 

 form of the determinantal equation — 



