346 
CALCULUS 
Fig. 83 
at each one of its points, it is tangent to the vector which pertains 
to that point. It would seem likely that such a curve exists through 
each point of S. For, if we start at any point (*„, y 0 ) of S and go 
along the vector at that point to a point (x u y,) near by ; if from here 
we proceed along the vector pertain 
ing to this latter point to a point 
(x 2 , y 2 ) a little further on; and if 
x we continue this process, we are 
thus led to a broken line, whose 
slope at any one of its points* 
differs but slightly from the value of f(x, y) at that point. It is 
natural to expect this line to approach a certain curve as its limit 
when its vertices increase in number, the greatest distance between 
two successive vertices approaching 0. On introducing a suitable 
restriction on f(x, y) (a reason for which will appear when we study 
singular solutions) it turns out that this is the case; i.e. that there 
is a curve, , , N 
y=4>(*), 
toward which all these broken lines converge, and that the slope of 
this curve at each point is that of the vector pertaining to this point.. 
Analytically this means that the function <j>(x) satisfies the given 
differential equation, or 
<£'(*) =/!>,</> (*)]• 
The condition to be imposed on f(x, y) may be stated in the form 
that f y (x, y) = df/dy shall exist and be continuous throughout S. 
This condition is somewhat more restrictive than is needed, but it 
includes the cases of importance which arise in practice. Moreover, 
when this condition is fulfilled, the solution is unique; i.e. the 
neighborhood of an arbitrary interior point of S is swept out just 
once by a one-parameter family of curves, no two of which have a 
point in common. 
We note that the solution depends on an arbitrary constant, y 0 . 
At first sight it might appear as if it depended on two arbitrary 
constants, x 0 and y 0 . It does ; and still there is only a one-parameter 
family of solutions involved, for we get all the solutions which 
course the neighborhood of the point (x 0 , y 0 ) by holding x 0 fast and 
allowing y 0 alone to vary. For example, the right lines which have 
* At a vertex, the slope of one of the lines abutting on it is just right, by 
construction. The slope of the other line is not far wrong. 
MM