330] 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
129 
The equation is that of a cone of the second order, meeting the plane of zx in the 
lines z — 0, z = 2x V« 2 — b 2 (and therefore such that its sections parallel to the plane of 
xy are parabolas), and meeting the plane of yz in the lines z = ±y \/a 2 — b 2 (the 
origin being at the vertex of the cone or conical point of the surface). 
Returning to the original origin, and to the equation of the surface written in the 
form 
z 2 + z (a 2 + b 2 — x 2 — y 2 ) + a 2 b 2 — b 2 x 2 — a 2 y 2 = 0, 
calling this for a moment z 2 + 2Bz + 0 = 0, the differential equation is G' 2 - 4BB'C' + 4<CB' 2 =0; 
or, substituting, this is 
(b 2 x + a 2 yp) 2 — (a 2 + b 2 — x 2 — y 2 ) (x + yp) (b 2 x + a 2 yp) + (a 2 b 2 — b 2 x 2 — a 2 y 2 ) (x + yp) 2 = 0 ; 
or, reducing, this is 
(a 2 — b 2 ) xy [xy (p 2 — 1) — (a 2 — b 2 — x 2 + y 2 ) p) = 0, 
or say 
xy [xy (p 2 — 1) — (a 2 — b 2 — x 2 + y 2 )p] = 0, 
where the factor xy arises from the level lines (z + b 2 = 0, y— 0) and (z + a 2 = 0, x — 0). 
Throwing out this factor, the equation becomes 
xy (p 2 — 1) — (a 2 — b 2 — x 2 + y 2 ) p = 0, 
which is satisfied identically by z + b 2 = 0, y = 0, x 2 = a 2 — b 2 . The first derived equation is 
(xp + y) (p 2 - 1) + 2 (x - yp)p = 0, 
which for the values in question gives 
p(p 2 + 1) = 0, 
where the factor p = 0 corresponds to the section y = 0 by the plane z + b 2 = 0: and 
taking the conical point for origin, and observing that the polar of the line x = 0, 
y = 0 in regard to the tangent cone is z - x Va 2 - b 2 = 0, then writing the equation of 
the tangent cone in the form 
(z-x \/a 2 - b 2 ) 2 - (a 2 - b 2 ) (x 2 + y 2 ) = 0, 
the two tangent planes through (x = 0, y = 0) are given by the equation x 2 + y 2 = 0; and 
for these planes we have p 2 +1 = 0. The factor p 2 +1=0 determines therefore the 
directions of the envelope at the conical point. 
VIII. 
In verification of the equation 
z = \k (x 2 + y 2 ) + b ax (x 2 + y 2 ) 
for a quadric surface in the neighbourhood of the umbilicus, I remark that, staiting 
from the equation 
or 
a? y 2 z 2 
Li-J =1 
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