406 Proceedings of Royal Society of Edinburgh. [ sess . 
The p-discriminant is easily found to be 
x 2 = 0 
and the directions of the isoclinal curve at the origin are given by 
y 2 + 3 xy - \x 2 = 0 , 
hence 
y = - ix or \x . 
The direction of the integral curve at the same point is along 
the x-axis, i.e., in the region not containing the ^-discriminant, 
hence the integral curves should have opposite curvatures. 
A first approximation at the origin gives 
p 2 + 3px - \ x 2 = 0 , 
hence 
2 / = i( - 3 ± i)x 2 . 
The two "branches have opposite curvatures as predicted. 
Example (2) : 
p 2 + 2(x + y)p + \x 2 + 2 xy + y 2 = 0 . 
The direction of the integral curves at the origin is along the 
x-axis. The jp-discriminant is x 2 = 0, and the directions of the 
isoclinals are given by 
y— - f-z or -\x . 
Hence the integral curves have a tac-point of the second species. 
A first approximation gives 
p 2 + 2px + \x 2 , 
hence 
Example (3) : 
p 2 4- 2{x + y)p + 2 xy + y 2 + x 3 = 0 . 
The /9-discriminant is 
x 2 (x - 1) = 0 
where x 2 corresponds to the tac-locus. 
The directions of the isoclinals are given at the origin by 
y{y + %x) = 0, 
hence, as the direction of the integral curve is along the z-axis, 
there is an inflexion on one branch. 
