1902 - 3 .] Isoclinal Lines of a Differential Equation. 403 
may be a locus of double points on the isoclinal family — in general 
a node locus. In this case the isoclinals, and therefore the integral 
curves, in general cross the ^-discriminant. For, if P and Q have 
the same meaning as formerly, we find the same phenomena 
appearing on the integral curve as in the case of an envelope, with 
this difference, that we also have a point Q' on the other side of 
the ^-discriminant from Q at which also two integral curves or two 
branches of the same integral curve diverge. 
As the latter is a higher order of singularity it is less general. 
(It must be noticed that the two isoclinals contiguous to p which 
pass through Q' are not in general the same as the two which pass 
through Q.) The conditions for a node locus on the isoclinal 
family, and therefore for a tac-locus on the integral family, are 
<k = 0 <*>, = 0 .... (3) 
in addition <£ = 0 and <£ p = 0, and these are in general sufficient. 
Now, in general, if an isoclinal pass through a point P, there is one 
and only one curve contiguous to it which passes through a given 
point which is contiguous to P, but if P is a tac-point of the 
integral family, and Q is a point contiguous to P on both the 
integral curves that touch at P, since the rate of variation of p is 
in general different for the two integral curves, two contiguous 
isoclinals must pass through Q ; and similarly for Q' on the other 
side of P from Q, and therefore two isoclinals must pass through 
P ; and as the p of both integral curves is the same at P, these 
two isoclinals must be branches of the same curve, therefore the 
conditions (3) are also in general necessary. 
The branches of the isoclinal curve at a node divide the plane 
/ 
