v (v 2 + 2 m — 1) 2 m - Id 2m -1 d 2 + 2m — 1 v v + i V 2m — 1 v — i V 2m — 1 
giving 
= 
D n _ m 
2 m- 1’ 2m-1' 
The integral equation thus is 
2m m-1 
const. = (m« — VO) -1 (m# + V[Ij) 2m_1 |(m«+ i V2m — 1t/+VD)( m# — i V 2m — T y + VG )} 2m 1 
where □ = m 2 x 2 + ?/ 2 ; or, observing that 
{mx + i V2m — 1 y + VD) (m* — i V2m — 1 y + VD) 
= {mx 4- VO) 2 + y 2 
= 2m {mx? + y 2 + x V □), 
the integral equation is 
1 m—1 
const. = {mx + VO) 2 — 1 {mx? + y 2 +x VD) 2m-1 , 
or, what is the same thing, 
const. = {mx + VG) {mx 2 + y 2 + x VCD)” 1-1 , 
the result given in the former part of the present paper. 
VI. 
I annex the following a posteriori verification of the solution 
const. = {mx + V □ ) {mx 2 + y 2 + x V □ ) m—1 
of the particular equation 
y {p 2 — 1) + 2mxp = 0.