1890.] related to Families of Curves. 59 



Thus, if ds be the length of an elementary arc, we have 

 ds' = d,' + dy> = gdf + f v d v J + {|df + |*,f 



2. We proceed to discuss the properties of the loci of ulti- 

 mate intersections of the two families of the orthogonal system. 

 Suppose that P is the point (x, y), and that Q is a neighbouring 

 point on the curve of the family r\ which passes through P ; then 

 by the preceding article we have 



*M©'+®1* 



v 



If this vanish independently of the value of d%, then two con- 

 secutive curves of the family £ cut at the point P. The vanishing 

 of the said expression requires that either 



dx = dy_ 



dec , .dy _ 



The first of these conditions has reference to the existence of 

 a real locus of ultimate intersections, as is otherwise evident. The 

 equations 



--0 and ^-0 

 8 £-U and af _u, 



if expressed in terms of x and y f would denote two curves which 

 would in general intersect in a set of discrete points. If however 

 the expressions dx/dg and dy/dg had a common factor, then two 

 branches, one belonging to each curve, would coincide, and con- 

 sequently the curves of the family £ would have a locus of ultimate 

 intersections not consisting wholly of discrete points. We can 

 however show that there are restrictions on the form of such a 

 factor should it exist ; for if this factor be not such that if equated 

 to zero it secures that cf> (x, y) shall at the same time vanish, then 

 it follows that at all points for which 



^ = and A=0. 



. 01; 0% 



We have also 



3 / = and 9-0. 



07) 07) 



