116 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. [330 
Hence if we represent the roots X, Y in the form P ± Q V□ , so that P = -B, 
Q \/Q = VP- — AC, Q 2 being the greatest square factor of B-— AC, then 
( .x - ry=*Q>a. x\ r = F ± (q Vd + Æ), 
X'T = P” - A (2QU + QOJ ; 
and the derived equation is 
n = -Q 2 {4OP' 2 - (2 Q□ + QD') 2 } = 0. 
If B, C, &c. are functions of the coordinates (x, y), the equation z 2 + 2Bz + C = 0 
(z an arbitrary constant) represents a series of curves in the plane of xy ; but if we 
consider z as a coordinate, then the equation represents a surface, and the curves in 
question are the orthogonal projections on the plane of xy of the sections of the surface 
by the planes parallel to the plane of xy. To fix the ideas, the plane of xy may be 
taken to be horizontal, and the ordinates z vertical. 
Writing the equation in the form 
(z + B) 2 — (B 2 — (7) = 0, 
we see that the surface contains upon it the curve z + B — 0, B 2 — C = 0, which is the 
line of contact with the circumscribed vertical cylinder : such curve may be termed the 
envelope, or, when this is necessary, the complete envelope. The equation of the 
surface has however been taken to be (z — P) 2 — Q 2 □ = 0 (viz. it has been assumed 
that P = —P, B 2 — C=Q 2 □); the envelope thus breaks up into the curve, z — P= 0, Q = 0, 
taken twice, and the curve z — P = 0, D = 0; the former of these is in general a nodal 
curve on the surface, and it may be spoken of as the nodal curve ; the latter of them 
is the reduced or proper envelope, or simply the envelope. And the terms nodal curve 
and envelope may also be applied to the curves Q = 0 and □ = 0, which are the pro 
jections on the plane of xy of the first-mentioned two curves respectively. There is 
however a case of higher singularity which it is proper to consider : suppose that 
Q and □ have a common factor K, say Q = KB, □=A’V ; the complete envelope 
Q 2 D=P 2 A" 3 V =0 here breaks up into the nodal curve R= 0 twice, the cuspidal curve 
K — 0 three times, and the reduced or proper envelope y = 0 once. 
Reverting for a moment to the form (z + X)(z+Y)= 0, the derived equation 
il = — (X — F) 2 X'Y' = 0 is satisfied by (X — F) 2 =0; this equation, or say the equation 
of the envelope, being in fact the singular solution of the differential equation. This 
assumes however that the differential equation is given in the form in which it is 
immediately obtained by derivation from the integral equation, without the rejection 
of factors which are functions of the coordinates (x, y) only; it is proper to consider 
the reduced equation obtained by rejecting such factors. Thus if X and Y are rational 
functions, the reduced form is X'Y'= 0, which is no longer satisfied by the equation