408 Proceedings of Royal Society of Edinburgh. [sess. 
Example (6) : 
y~p = x n . 
The a?-axis has contact of ( n - l) th order, hence, as the direction of 
the integral curve at the origin is also along the ic-axis, the tangent 
there has contact of the n th order. Integrating we get 
y = - e x I x n e~ x dx 
to a first approximation. 
Example (7) : 
The />-disciiminant is y 2 (y 2 - -^f) = 0. 
The directions of the integral curves at the origin are 
p = 0 and '£>— - 1 . 
The direction of the isoclinal curve corresponding to^> = 0 is 
V= ±x; 
the corresponding part of the ^-discriminant is therefore y 2 = 0, 
and the integral family has a tac-point of the second species. The 
integral curves at the origin are 
y — - x + \x 2 + J# 3 , y = \x 2 ± \x z . 
It must be noticed that, although the direction of the integral 
curve lies along one branch of the , isoclinal family, it is not the 
branch to which that direction belongs, and so there is no 
inflexion. 
( Issued separately April 4 , 1903 .) 
