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



Thus at all points of the locus of ultimate intersections we have 

 dx = ~7.d%+ ~- clr] = 



and dij = ^Ld^ + ^ k drj = 0. 



These equations give x = const, and y = const., and thus the locus 

 of ultimate intersections consists of a set of discrete points. 

 Moreover this set of points is such that the coordinates of each 

 of them make the expression with which we started infinite. 



Thus we have, in general, that if two consecutive members of 

 one of the two families intersect, then all their points of inter- 

 section are points of the character under discussion, and it follows 

 that all the members of the family pass through this set of points. 

 The only exception to this is that the curves of the £ family may 

 possibly have for a locus of ultimate intersections the locus of 

 the points for which </> (x, y) vanishes, and the curves of the rj 

 family may possibly have for a locus of ultimate intersections the 

 locus of the points for which the same expression becomes infinite. 



The question now arises whether these special loci, should 

 they exist, belong to the generalization of the singular points of 

 an equipotential system. This depends on how we define that 

 generalization. If we define it with the aid of our original ex- 

 pression, we see that these loci do not belong to it if they arise 

 from the vanishing of 1\ or h 2 , but if they take their origin from 

 either of these quantities becoming infinite then they do so 

 belong. If on the other hand we define it with the aid of the 

 expression 



d£ , .dr> 



Jh A 



dx + idy ' 



then the exact contrary is the case. 



It does not follow that either of the two families should be 

 such that all its members pass through the points under dis- 

 cussion, although it is true that if any two belonging to one 

 family do so, all belonging to that family will. Also it is evident, 

 on account of the orthogonal property, that if one of _ the families 

 be such that all its members pass through the said system of 

 points, then those of the other family do not do so, only one 

 member of that family passing through any one of the points. 

 It is however conceivable that there may be cases in which the 

 members of one family pass through some of the points, and the 

 members of the other family through others of them. 



