192 
MESSES. A. E. H. LOVE AND F. B. PIDDUCK 
where 
and we have 
_ ( r-rp (s-sp 
i (r' + s') (r + s) ’ 
X(r', S ')=f 
Jac \ocr au j 
1+ ^ 
= I Y —-L (d--1) du 
Jac <t ^ fda\ ~ \du 
\duj 
= -5H 
- */. 
Acer 
23. Determination of the Coefficients B. —The integral | ( Y/cr)du may be evaluated 
Jac 
approximately by assuming, as in Article 21, that the equation of the locus, of which 
AC is an arc, is of the form 
(r-RO/Zj - B^+B^+B 3 ^ 3 +..., 
where S stands for (s — Sp/Z^ Then along AC we have 
u- U x = S 1 {(B 1 -1)^+BA 2 + B : /+...}, 
du = Z x {(B 1 -l)+2B 2 <S+3B 3 «S a +...} dS, 
I = (AqT E +80 (q-D (»- +210 (7 A (y f + . 
o- cr* [ (r' + Spcr (A + Sp 2 cr 2 J 
Also any inverse power of a can be expanded in powers of 0 by means of the equations 
o- = Z x {1 + (B x + 1 ) S + BA 2 + B :i <b +...}, 
<r~ K = 2 rCi- f 
(T —Z, . K (k+ l) f<T — N J 2 
which give 
+ 
2 ! 
rr~ K — V - K 
U - 
d 2 
i-UB 1+ i)j-{,b 2 -Ao_1) ( b 1+ i) 
- [*,-,(.+ 1 ) (B. +1) B 2 + ill±Pi£±2) (B 1 + 1 )»| s> 
- t-B.- AnP b 2 >-« (<+i) (b 1+ 1 ) b 8 + ^+iH..-+2) (B,+If B 2 
(B. + l)‘ i*- 
_ y ( /c +f)(' c + ^)( /c + S) 
4! 
To obtain the expression for X on r = R x we have to put B x for r', so that 
r-r' = 2 1 (B 1 ^+B/ + B 3 ^+...), 
