[330 
330] 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
121 
but to simplify 
= 0 ; this comes 
Disects the angle 
is 
[ now h in the 
ourhood of the 
ve through the 
lipsoid or other 
mrvature in the 
! umbilicus, the 
e of curvature. 
= 0; the value 
ler two values 
;he envelope of 
ejections of the 
bourhood of an 
and it was only because this equation did not appear to be readily integrable, that I 
considered, instead of it, the particular form 
y (p 2 — 1) + 2mxp = 0. 
But the general equation can be integrated; and the result presents itself in a 
simple form. For, returning to the differential equation 
and assuming 
or 
we have 
and we may write 
Assuming also 
(bx + cy) (p 2 - 1) + 2 (fx + gy)p = 0, 
bx + cy _ — 2v 
fx+gy~ v 2 -\’ 
(bx + cy) (v 2 — 1) + 2 (fx + gy) v = 0, 
1 = -, or (p — v) (vp + 1) = 0, 
v — 1 V 
p — v = 0. 
V 
y = ux, or 11 — —, 
* ' X 
the relation between u and v is 
or, as this may be written, 
b + cu _ — 2v 
f+gu~ v 2 -l’ 
^_1 + 2¿±Í!% = 0, 
b + CU 
giving 
v = ~ (/+ gu) - V(6 + cu)- + (/+ guf 
b + cu 
where for convenience the radical has been taken with a negative sign. We have more 
over 
j. b (v 2 - 1) + 2/v 
c (v 2 — 1) + 2gv' 
The equation p — v = 0, substituting for y its value ux, then becomes 
du 
x + u — v = 0 ; 
ax 
or, as this may be written, 
dx du 
1__— 
x u — v 
= 0; 
C. V. 
16