[330 
330] ON DIFFERENTIAL EQUATIONS AND UMBILICI. 125 
= 0, f— m; we 
Putting for shortness 
A = mx + Vn, 
B = mot? + y 2 + x VD, 
where it will be remembered that 
□ = m 2 x 2 + y 2 , 
then we have 
2mB = A 2 + (2m — 1) y 2 . 
G 
— i V 2 m — 1 ’ 
The integral equation may be written 
h=P+QVU= U = AB™- 1 ; 
and we have 
U A + 1 ' B ~ VD ’ 
if 
[A! , ^B'\ 
® = V □ + ( m -• 
But we have 
A' >/□ = mV□ + m 2 x + yp = mA 4-yj?, 
Ay + VD)} 2w " T i 
i?' Vn = (2m# + 2yp) VD 4- □ 4- x (m 2 x + yp) 
= 2m 2 x 2 + y 2 + xyp 4- (2mx 4 2y/:>) V □ 
= H 2 4- pyx + 2VO, 
and 
1 2m 
B ~ A 2 + (2m - 1) </ 2 ’ 
and the value of © thus is 
mA+yp A‘-+py(x + 2an) 
®- - A +(2m ¿m) A , + {2m -l)f 
_ 1 n l(m4 + «/») [4* +(2m - 1);/ ! ] + (‘¿ill- - -2m)[A- + Ap,j(x + 2v'd)]), 
A [A 2 + (2m-\)y 2 ] 1 
where the expression in { } is 
= (2m 2 — m) A (A 2 4- y 2 ) 
+ yp {A 2 4- (2m - 1) f + (2m 2 - 2m) A(x + 2VO)}. 
Here the coefficient of is = (2m 2 - m) (¿ 2 + y 2 ); in fact we have identically 
A 2 + y 2 - 2A VO =0, 
and thence 
(2m 2 - 3m 4- 1) (A 2 + y 2 )-% (2m -1) (m - 1) A VQ = 0, 
that is 
(2m 2 - m - 1) ^l 2 + (2m 2 - 3m + 1) y 2 - (2m — 2) A {A + (2m — 1) VD] = 0; 
that is