1902 - 3 .] Isoclinal Lines of a Differential Equation. 401 
selves in curves, namely the lines of force. These lines of force 
are the integral curves of the differential equation. 
A differential equation of the first order <fi(xyp) = 0 besides de- 
fining the integral family, also defines a family of curves obtained 
by regarding p as an arbitrary constant in </> = 0, and these curves 
have the property that all the integral curves, that intersect any 
particular curve of the second family, have the same direction ( i.e ., 
the same p) at the points of intersection. The latter family has 
been called the “Loci of Contacts of Parallel Tangents,” by Hill 
(Proc. Lond. Math. Soc., x ix., 1888, p. 561), but, at the suggestion 
of Professor Chrystal, I propose to use in this paper the more 
convenient term “Isoclinal Family.” This family gives a method 
of describing any integral curve ; for if, beginning at any arbitrarily 
chosen point, we draw an infinitesimal line in the direction specified 
by one of the isoclinal curves passing through that point, we in 
general come to another isoclinal giving a new direction differing 
infinitesimally from the original direction, and so on. Of course, 
as the starting-point is arbitrary, the integral curve must be 
developed on both sides of it ; also it must be noticed that the 
direction specified by the isoclinal is not in general the direction 
of the isoclinal itself. How it is evident that there is in general 
one and only one isoclinal that passes through the point 
(x + dx , y+pdx) and is also contiguous to the isoclinal p 
through (x , y) and similarly for the point (x - dx , y - pdx) ; also 
in general the isoclinal (p + dp) lies wholly on one side of the 
isoclinal p in the neighbourhood of (x , y), and at such a point the 
y of the integral curve and its first differential co-efficients are in 
general synectic functions of x. This is, however, not in general 
the case in the neighbourhood of the envelope. For, in general, 
one of a family of curves does not cross the envelope of the family 
in the neighbourhood of the point of contact. Thus the three 
contiguous curves p - dp ,p and p + dp all lie on the same side of 
the envelope, and all touch it. Moreover the direction p will not 
in general be that of the envelope, but will cross it. But as 
there is no contiguous isoclinal across the envelope to indicate a 
new direction, the integral curve cannot cross the envelope, and 
must therefore have either a cusp or a stop point. If the envelope 
is the curve indicated in figure 1 by E and the isoclinal p touches 
