PEIRCE. — THE GRAPHICAL SUPEKPOSITION OF LINES OF FORCE. 151 



value of the gradient ratio, hjh^, obtained from the given values of 

 a and /?, is the function li of ./■ and y ; then 



and if, for {da)- j {dyY in this equation, we substitute the value 



obtained from (1), it appears, since the gradient of a real function 

 cannot vanish, that 





If for {dafl(dxy^ in (7), we substitute its equivalent 



derived from (1), we shall learn that 



Either the upper signs or the lower signs must be used in (9) and (11). 

 If now we treat the eciuation 



V = «Vv (12) 



in a similar way we shall obtain the e(|uations 



d\ ^ fd,j.\ 



