1902 - 3 .] Isoclinal Lines of a Differential Equation. 
To find the integral curve we have 
407 
p = -(x + y) ± Jx\x - 1) 
= -x-y±(x- \x 2 ) 
to a first approximation. Hence to same order of approximation 
y= -i« 3 , y= - x 2 . 
Example (4) : 
(y-p) 2 = « 3 - 
The ^-discriminant is a locus of cusps for the isoclinal family, 
and is therefore a locus of ramphoid cusps for the integral family. 
To find the integral family we have 
p-y= ± x* 
hence y — ae x ± e x j e~ x x% 
which to a first approximation is 
y — a + ax + \x 2 ± f x % , 
a ramphoid cusp, except when a = 0, i.e ., when the direction of 
the integral curve is the same as that of the isoclinal. 
Example (5) : 
p 2 — 2 xp + x 2 — y 3 = 0 . 
The ^-discriminant y — 0 is a locus of cusps for the isoclinal family 
and at the origin the direction of the integral curve lies along it, 
hence the origin is a tac-point. A first approximation gives 
putting 
p = x y = ^x 2 
p = x + m y = \x 2 + v ; 
and neglecting terms not required for the second approximation 
we get 
67 = + 
J-2" 
hence 
