21 
636] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 
by changing the constant, viz. this is 
c 2 — 2 cxy — (1 — x 2 — y 2 ) = 0. 
The two discriminants are here each = (x 2 — 1) (y 2 — 1), and we have 
{x 2 — 1) {y 2 — 1) = 0 
as a true singular solution. The curves are in fact the system of conics (ellipses and 
hyperbolas) each touching the four lines x = l, x= — l, y = 1, y = — 1. 
Ex. 3. 
(1 — y 2 )p 2 — 1 = 0, that is, (1 — y 2 )dy 2 — dx* = 0. 
This is an extremely interesting example : the curve is the orthogonal trajectory 
of the system of sinusoids y = sin (x + c), which is the integral of Example 1 ; and we 
thus at once see that the real portion of the curve is wholly included between the 
lines y = — 1, y= +1, being an infinite continuous curve, having a series of equidistant 
cusps alternately at the one and the other line, and obtained by the continued 
repetition of the finite portion included between two consecutive cusps on the same 
line. The discriminant of the differential equation equated to zero gives y 2 — 1=0, 
the equation of the two lines in question; but this does not satisfy the differential 
equation, and it is consequently not a singular solution; by what precedes, it appears 
that it is, in fact, a cusp-locus. 
We thus see that the curves which represent the integral equation have no real 
envelope; but it is to be further shown that there is no imaginary envelope, and that 
the curve obtained by the elimination of the parameter is, in fact, made up of a 
(imaginary) node-locus and of the foregoing cusp-locus. 
The curve is properly represented by taking x, y each of them a one-valued 
function of the parameter 0, viz. we may write 
y — cos 0, 
x = c + \Q — J sin 20. 
In fact, these values give 
^ = — sin 0, = £ (1 — cos 20) = sin 2 0, 
and therefore 
1 - 1 
P sin 0 V(1 — y*)’ 
that is, (1 — y 2 )p 2 — 1 = 0, the differential equation. 
It is obvious that to a given value of the parameter there corresponds a single 
point of the curve; and it is to be shown that, conversely, to a given point of the 
curve corresponds in general a single value of the parameter.