58 Mr Brill, On certain Points specially [Feb. 24, 



is independent of the ratio dy : dx. Now it is evident that there 

 must be certain points or loci for which the inverse of this 

 expression vanishes, and therefore the expression itself becomes 

 infinite. These points or loci may be considered as a generaliza- 

 tion of the singular points of equipotential systems. 



I do not propose to discuss the general case, but to confine 

 myself to a large and important series of cases in which the ratio 

 of the two lis is some definite function of x and y, which function 

 of x and y is the same for all systems of a particular class. I 

 shall write hjl\ = <f> (x, y), and instead of the expression given 

 above shall consider the expression 



<f> (x, y) d% + idy 

 dx + idy 



The fact that this expression has a value independent of the 

 ratio dy : dx, requires the existence of the relations 



and the value of the expression may be written in either of the 

 forms 



Moreover it follows that the value of the expression 



dx + idy 

 cp (x, y) d% + idy 



is independent of the value of the ratio dg : dn, and this requires 

 the existence of the relations 



dx , , .dy , dy , , .dx 



^ = <l>(x,y) f v and f^-t&y)^, 



and its value may be written in either of the forms 



dy .dx 1 (dx . dy 



dn % dn 4>(x,y)\di; 9f 



It is also to be remarked that the relations given above lead 

 to the equation 



dx dx dy dy 

 d% dn di; dy 



