S OF [636 
636] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 23 
i + ~ i sin 2a, where 
In particular, taking c x = c, the intersections are given by 0 — rir + /3, where /3 is 
any root of the equation 2/3 — sin 2/3=0; viz. we have thus an infinite number of 
intersections lying on each of the lines y = cos rir cos /3. If /3 = 0, the intersections lie 
is 
on the two lines y — 1, y = — 1 respectively; if /3 be an imaginary root of the 
equation 2/3 — sin 2/8 = 0, then they lie on the imaginary lines y = cos rir cos /3. But by 
1 = a; and, taking the 
in general satisfied; 
the parameter 6. If, 
iy rir, then the last- 
le given point of the 
er; these values are 
y be equal, viz. this 
± 1, x = c + ^rir), and 
are on each of the 
what precedes, it is clear that in the former case the intersections are nothing else 
than the cusps on the lines y = 1, y = — 1; and in the latter case nothing else than 
the nodes on the lines y = cos m cos /3; viz. there is no proper envelope, but instead 
thereof we have lines of cusps and of nodes. 
Ex. 4. 
(1 — y 2 )p 2 — (1 — x 2 ) = 0, 
that is, 
(1 — y 2 ) dy 2 — (1 — x 2 ) dx 2 = 0. 
I have not examined this; the curve is the series of orthogonal trajectories of 
the conics of Example 2, and the integral equation may be represented by y = cos 0, 
x = cos </>, where c = (29 — sin 26) — (2cf> — sin 2</>), 
nation 2/3 — sin 2/3 = 0, 
Equating to zero the discriminant of the differential equation, we have (1— y 2 ) (1 — x 2 )=0, 
2r7T, 
viz. the four lines x = 1, x = — 1, y = 1, y = — 1; this is not an envelope, but a locus 
of cusps. 
s a and 2rv — a of 
two lines y — + 1, 
ition 2/3 — sin 2/3 = 0, 
y = cos T7T cos /3; and 
different imaginary 
, we cannot directly 
minant in regard to 
curve by the con- 
and thus determine 
c, c x , and let 6, 0 X 
tersection; we have 
+ 6, but we cannot 
to any given value 
iy root of the equa-