I remark that we may without loss of generality write d = 0: but to simplify 
the investigation, I suppose in the first instance that we have also b = 0; this comes 
to assuming that one of the three planes ax 3 + Sbx 2 y + Sexy 2 + dy 3 = 0 bisects the angle 
formed by the other two planes. The differential equation consequently is 
or, putting for shortness 
cy(p- — l) + (a — c)xp=0-, 
- = — 2m, 
y (p 2 — 1) + 2mxp = 0, 
which is the differential equation previously considered. Hence, writing now h in the 
place of z, the equation of the curve of curvature in the neighbourhood of the 
umbilicus is 
h = (mx + Vd ) (mac? + y 2 + Vd)™ -1 , = P + Q Vd, 
where □ = m 2 at + y 2 ; or, what is the same thing, the equation is 
A 2 - 2PA + P 2 - Q 2 d = 0 ; 
and the equation (in the neighbourhood of the umbilicus) of the curve through the 
umbilicus is 
P 2 - Q 2 d = - y 2m {y 2 + (2m - 1) x 2 }™- 1 = 0 ; 
so that the umbilicus is a trifid node. In the case however of an ellipsoid or other 
quadric surface, we have m = 1, so that the equation of the curve of curvature in the 
neighbourhood of the umbilicus is 
h = x + VV + y 2 , 
or, what is the same thing, 
A 2 — 2hx — y 2 = 0 : 
and for the curve through the umbilicus, in the neighbourhood of the umbilicus, the 
equation is y 2 = 0, so that there is only a single direction of the curve of curvature. 
The differential equation gives, however, at the umbilicus p(p 2 +l) = 0; the value 
p = 0 is that which corresponds to the curve of curvature; the other two values 
p = ± i correspond to the curve (pair of lines) x 2 + y 2 = 0, which is the envelope of 
the curves of curvature, or, more accurately, the envelope of the projections of the 
curves of curvature on the tangent plane at the umbilicus. 
BlcicJcheath, October 17, 1863. 
IY. 
The differential equation for the curves of curvature in the neighbourhood of an 
umbilicus was obtained in a form such as 
(bx+ cy)(p 2 - l) + 2(fx + gy)p=ti ;