126 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
[330 
or 
(2m 2 — m — 1) A 2 + (2m 2 — 3m + 1) y 2 — (2m 2 — 2m) A (x + 2Vd) = 0, 
and therefore 
A 2 + (2m — 1 )y 2 + (2m 2 — 2m) A (x 4- 2Vd) = (2m 2 — m) (^L 2 + y 2 ). 
Hence the term in { } is 
= (2m 2 — m) (A 2 + y 2 ) (A + yp); 
or, what is the same thing, it is = (4m 2 — 2m) A Vd {A 4 yp). Hence, restoring for 
A 2 + (2m — 1) y 2 its value 2mB, we find 
@ J2jn-^a (A + yp) _ 
or 
U' 2m — 1 . . 
Ij B ^ ^ " 
But writing U 1 , U 2 to denote the values corresponding to 4 Vd, — Vdl respectively, we 
have 
jj,i _ (2m ^1) U 1 + yp _j_ Vd), 
U\ = ^ ^ 2 (j) lx + yp — Vd), 
and thence 
TT , TT , (2m —l) 2 IfU* (/ . 
U 1 U 2 = Bjl {(■ mx + yp) - a } 
_ (2m — l) 2 TJJJ 2 
BA 
y [y (P* ~ 1) + 2m«p|. 
But we have 
and thence 
and moreover 
where 
and we thence find 
U' = P' + Qf Vd + ^- = 27 d WD + QU' + 2P'Vd), 
U\U' 2 = - 4 X D {(2Q'd + QDJ - 4F 2 d}, 
U, U 2 = P 2 - Q 2 d = A,A 2 (BA)™-', 
AA 2 = m?a? — d = — y n -, 
B 2 B 2 = (mx 2 4 iff — d = y 2 [y 2 + (2m — 1) x 2 }; 
- A{ (2 Q'o + Q D ')»_4P't]} 
= (2m — l) 2 A X A 2 (BxBf)™- 2 y 2 [y (p 2 — 1) 4 2mxp] 
= — (2m — l) 2 y‘ 2m ~ 1 [y- + (2m — 1) # 2 ] m ~ 2 {y(p 2 — 1) + 2mxp).