[330 
330] ON DIFFERENTIAL EQUATIONS AND UMBILICI. 127 
= 0, 
Hence, the derived equation being 
Q s |(2«'D + eO')“-4F'D) =0, 
). 
the last preceding equation becomes 
Q~ \Z\y 2m ~ l \y- + (2m — 1) x 2 } m ~ 2 \y (p- — 1) + 2mxp] = 0. 
e, restoring for 
Here, besides the factor Q 2 corresponding to the nodal curve, and the factor □ 
corresponding to the cuspidal curve, we have the factors y-™- 1 and [y 2 + (2m — l)# 2 } m ~ 2 ; 
and, rejecting all these, the differential equation in its reduced form is 
y ( p 2 — 1) + 2 'mxp = 0 ; 
and the required verification is effected. The occurrence of 
Q 2 □¿/ 2m-1 [if + (2m — 1) x 2 } m ~~ 2 
respectively, we 
as a factor in the complete derived equation would give rise to some further investi 
gations, but I will not now enter on them. 
I remark however that if m = l, viz. if the integral equation be const. = x J r V# 2 4-y 2 , 
or say z = x + V^c 2 + y 2 , or, what is the same thing, 
z 2 — 2 zx — y 2 = 0, 
then observing that y 2 + (2m — l)# 2 is here =x 2 +y 2 which is = □, so that 
□ [y n - + (2m — 1) ¿t, ,2 } m—2 = □ . D -1 = 1, 
the differential equation in its complete form is 
y(p 2 y+2px-y) = 0- 
so that we have here the factor y which divides out. The last-mentioned result is 
most readily obtained directly from the equation 
n = Q 2 (2Q'D + QD? “ = °> 
which is the derived equation corresponding to the integral equation z = P + Q VO. 
We in fact have P = x, Q = 1, □=a? 2 + y 2 , and the derived equation thus is 
(x + ypf - (x 2 + y 2 ) = 0, 
that is, y (p 2 y + 2px — y) = 0. 
I mention also, in connexion with the foregoing investigation, the integral equation 
z = x + *J2x 2 - y 2 , or z 2 - 2 zx — x 2 + y 2 = 0, 
for which the derived equation in its complete form is 
(2x — ypf — (2x 2 — y 2 ) = 0, 
or, what is the same thing, y 2 p 2 — 4>xyp + 2x 2 + y 2 = 0, and for which therefore there is 
no factor to divide out. 
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