[330 
330] ON DIFFERENTIAL EQUATIONS AND UMBILICI. 119 
3er than unity, 
VQ + &c., 
irrational ones 
to a cuspidal 
t-VÜ, so that 
gular curve on 
gh the origin 
for this value, 
the lines y = 0, 
he origin are 
obtained from 
nt in question, 
) values given 
hat the curve 
□ = 0 is not 
) longer three 
r gives, as in 
values j)= ±i 
ned from the 
>elongs to the 
ar solution of 
curvature will be P, Q; and if P, Q w T ere branches of an indecomposable curve, then 
P, Q would also be branches of an indecomposable curve, and we should have P a 
branch of two different indecomposable curves, which is of course impossible. In the 
case of an umbilicus, the two curves P and Q coincide together; or, as we may 
express it, the curves of curvature through an umbilicus are the duplication of a single, 
in general indecomposable, curve; and in general this curve has at the umbilicus a 
trifid node. I use this expression to denote a point at which there are three distinct 
tangents, or, more accurately, three distinct directions of the curve: an ordinary triple 
point is of necessity a trifid node, but not conversely. The umbilicus of an ellipsoid 
or other quadric surface is a peculiar exceptional case. 
In support of the foregoing conclusions, consider a surface having an umbilicus at 
the origin, and take z = 0 as the equation of the tangent plane at that point; the 
equation of the surface in the neighbourhood of the umbilicus will be 
z = \k (x 2 + y 2 ) + £ (ax? + 3bx-y + Sexy 2 + dy 3 ) ; 
so that, writing as usual p and q for the first, and r, s, t for the second, differential 
coefficients of z, we have 
p = kx 4- | (ax 2 + 2bxy + cy 2 ), 
q = ky + 2 + 2cxy + dy 2 ), 
r = k + ax + by, 
s — bx + cy, 
t = k + cx + dy. 
The differential equation of the curves of curvature projected on the plane of xy is 
(iO" K 1 + ^ S ~ Pqt ^ + Tx^ 1+ q ^ V ~ ^ ^ ~ 1 +P ^ S ~ Pqr J = ° ’ 
and substituting therein the foregoing values of p, q, r, s, t, but attending only to 
the terms of the lowest order in (x, y), and using moreover in the sequel p in the 
place of ^, the equation becomes 
CLOG 
(bx + cy) (p 2 l) + [(a-c)«+ Q>-d)y\p = 0; 
which may be taken as the differential equation of the curves of curvature at and 
in the neighbourhood of the umbilicus. The equation is satisfied identically by the 
values x = 0, y = 0, which correspond to the umbilicus; and to find p, we have to 
differentiate the equation, and then substitute these values of x and y\ we thus obtain 
(b + cp)(p 2 - 1) + [(a — c) + (b - d)p]p= 0, 
or, what is the same thing, 
p (a + 2bp + cp 2 ) ~(b + 2cp + dp 2 ) = 0, 
a surface are 
2 are the two 
iwo curves of 
a cubic equation for the determination of p.