402 Proceedings of Boy al Society of Edinburgh. [sess. 
it in P, and if PQ be an infinitesimal element of the integral curve 
at P, we can in genexal find two and only two isoclinals contiguous 
to p which pass through Q, and they in general touch the envelope 
on opposite sides of P. Therefore if we start from P to draw the 
integral curve, we have a choice of two directions at Q, each 
differing infinitesimally from p ; and these in general give two 
distinct branches of the integral curve as the sign of the variation 
of p is different in the two. The envelope is therefore a locus of 
cusps for the integral curves. Now Cayley has shown that if 
<j>(xyp) = 0 is an integral algebraic function of p, the isoclinal family 
has in general an envelope which is given by the ^-discriminant 
of <f>. The p-discriminant is therefore in general a locus of cusps 
on the integral family. 
Conversely, if E is a locus of cusps for the integral family, it is 
part at least of the envelope of the isoclinal family ; for two 
contiguous isoclinals intersect at Q. 
If, however, the direction p is also the direction of the isoclinal 
at P, and therefore of the envelope, the direction p does not cross 
the envelope and there is no discontinuity in the variation of p for 
the integral curve. The direction of the isoclinal family is given 
by 
therefore the condition for the p-discriminant being an envelope 
locus for the integral family is 
<f>x+P<t> y = ° .... (2) 
and this condition is in general both necessary and sufficient. 
Several other cases now present themselves. The ^-discriminant 
