122 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
[330 
or, what is the same thing, 
dx dv — du dv 
But 
where 
4- — — 0. 
x v — u v — u 
b (v 2 — 1) + 2 fv 
v — u = v 4— — — — 
V 
c (y 2 — 1) 4- 2gv c (v 2 -1)4- 2gv ’ 
V = v [c (v 2 — 1) + Zgv] +b‘ 2 (v — I) + 2fv 
= (6 + cv) (v 2 — 1) + 2 (/+ gv) v, 
and the differential equation takes thus the form 
dx dv — du [c (v- — 1) + 2gv] dv 
+ ■ 
x v — u 
0; 
and hence, writing 
and 
V=(b + cv) (v 2 —1)4-2 (/4- gv) v = c (v — a.) (v — /3) (v — y), 
c (v 2 —1)4- 2gv _ c (v 2 — 1) 4- 2gv 
V 
ABC 
= 1 1 
c {v — a) (v — /3) (v — 7) v — a v — /3 v — 7 ’ 
so that 
A = 
c (a 2 -1)4- 2ga 
c (a 2 - 1) 4- 2goc+ 2 {f+(b + g)u + ccL 2 } ’ 
with the like values for B and C—values which are such that A 4- B 4- C = 1 
integral equation is 
const. = x (v — u) (v — a)~ A (v — ¡3)~ B (v — y)~ c , 
or 
, substituting for v — u its value, = ——-, 
5 ’ c(v 2 -l) + 2gv 
But 
const. = x {c (v 2 — 1) 4- 2gv}~' (v — a) l ~ A (v — ¡3) l ~ B (v — 7) 1 ~ c . 
- (/+ gu) -*JU m 
v = 
b + cu 
if for shortness U=(b + cuf 4- (/4-gu) 2 , and thence 
2 _ 2 (/4- gu) 2 + (b + cu) 2 + 2 ( f+ gu) V U 
(b + cu? 
and 
c (v 2 — 1) 4- 2gv = 
2 (cf- bg) (/4- gu + 'dU) 
v — a 
(b 4- cu) 2 
_ — (/4- gti) — V U — a (b 4- cu) 
b 4- c u. 
, &c. 
,—the