PEIRCE. — THE GRAPHICAL SUPERPOSITION OF LINES OF FORCE. 155 



or else that a and A have the same level curves: this last case, as 

 being uninteresting, may be left out of account. 



Sometimes (28) is more convenient than the unexpanded form of 

 the same condition which follows immediately if we solve (24) and 

 (25) for dm/dx and dzj jdy, 



da 



d\ 



(2!)) 



(30) 



and equate the derivative with respect to i/ of the second member of 

 the first e(|uation to the derivative with raspect to x of the second 

 member of the other. This process yields the relation. 





t-"^^^>-|-^^^"^' 



da ^dX 

 dx dy 



da dX 

 dy dx 



(31) 



and it is possible to check the fact that (28) and (31) are equivalent 

 by a straightforward but somewhat laborious comparison of the two. 



If a and A satisfy (31), a function I exists which satisfies (22) and 

 (23), functions (i and fx exist which satisfy (17) and (20), and (a, /8), 

 (A, fx), (a + A, /8 + i«) are orthogonal pairs of functions. 



If, for instance, both a and A represent values in the xy plane of 

 harmonic space functions ( T, \V) the level surfaces of which are sur- 

 faces of revolution about the x axis, so that 



1 d f dV\ a-F ^ 

 y'yyV'^)-^-^ = ^ 



(32) 



with a similar equation for W, 



1 da 



V^ (a) = - 



y ^y 



^''W=--.-^. 



1 aA 

 y .%' 



(33) 



