ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
117 
[330 
that P = — B, 
z* + 2Bz + G = 0 
xy; but if we 
1 the curves in 
s of the surface 
of xy may be 
0, which is the 
be termed the 
quation of the 
been assumed 
z-P= 0, Q = 0, 
general a nodal 
latter of them 
rms nodal curve 
2h are the pro- 
lively. There is 
: suppose that 
mplete envelope 
3 cuspidal curve 
erived equation 
ay the equation 
equation. This 
in which it is 
t the rejection 
aper to consider 
1 Y are rational 
ay the equation 
330] 
(X — Yf = 0. In the before-mentioned case where the roots are P + Q FD (or 
(X — F) 2 = Q 2 D), P, Q, and □ being rational functions of (x, y), the derived equation 
O = ~Q 2 {4QP' 2 - (2Q'B + QD') 2 } = 0 
divides out by the factor Q-, but it does not divide out by □ ; the reduced form is 
therefore 
4DP' 2 —(2Q'D + QQ') 2 = 0, 
which is not satisfied by Q — 0, while it is still satisfied by □ — 0 (since this gives also 
□ ' = 0); that is, the nodal curve Q = 0 is not a solution of the differential equation, 
but we still have the singular solution □ = 0, which corresponds to the reduced or 
proper envelope. In the case Q = KR, □ = V of a cuspidal curve, the above form 
of the differential equation becomes 
4P V P' 2 - {3KK'R V + IP (2V R' + V 'P)} 2 = 0, 
which divides out by K; and, when reduced by the rejection of this factor, it is no 
longer satisfied by the equation K = 0, which belongs to the cuspidal curve; that is, 
neither the nodal curve R = 0 nor the cuspidal curve K = 0 is a solution of the 
differentia] equation, but we still have the singular solution y = 0, which corresponds 
to the reduced or proper envelope. It would appear that the conclusion may be 
extended to singularities of a higher nature, viz. the factor corresponding to any 
singular curve which presents itself as part of the complete envelope divides out from 
the derived equation; and such singular curve does not constitute a solution of the 
reduced equation, but we have a singular solution corresponding to the reduced or 
proper envelope. 
II. 
Consider the differential equation 
y (p 2 — 1) + 2mxp = 0, 
where, to fix the ideas, to > or = 1; the integral equation may be taken to be 
2 = (mx + VtoV + y 1 )(mx 2 + if + aVm 2 x 2 + y 2 ) m_1 ; 
or rather, writing for shortness □ = to 2 # 2 + y\ and putting 
z = (mx + VO )(rroc 2 + y 2 + x VD ) m_1 = P + Q VD, 
the integral equation is 
(z - Pf-Q 2 □ = o, or £ 2 - 2Pz + P 2 - Q 2 D = 0, 
where 
P 2 - Q 2 \3 = (to 2 « 2 - □) {(to« 2 + y 2 ) 1 - « 2 D} m ~ 1 = - y im [y 2 + (2to - 1) « 2 } m_1 . 
In the particular case to = 1 the equation is 
2 = « + V« 2 + y 2 , or z 2 — 2zx — y 2 = 0.