108] CONNECTED WITH THE THEORY OF ATTRACTIONS, 
it is easy to see that 
37 
(X 2 + v 2 )(P + u 2 ) = S 4 , (\ 2 - 6 2 ) (l 2 -f 2 ) = 8*, 
and 
;= eti 
b 2 a _ B 2 a, 
¥+P 2 “ 
_ v g2 /i _. 0 
p + v? v > li~fi ~ 
Proceeding to express the single integrals in terms of the new constants, we have in 
the first place k 2 — &k 2 , where 
k 2 = 
ai 
+...; 
or if we write 
we have 
/2 7 ! 
k 2 = 7 ^ L _+ 
P + u 2 ^-/i 2 
aa, + bb, ... = ll, cos m, 
211, cos ft) 
Hence also % = S 4 j, where 
{P + W 2 ) 2 (4 2 -/a 2 ) 2 (¿ 2 + *■) (¿> 2 -//) ' 
j=-,—^,^-k 2 - ^ 
(k 2 ~fff (¿ 2 + w 2 ) 2 ’ 
whence 
1 1 2111 cos to 
— 1 = , h ^ 7T- + 
*/ /o i ..o! 1 7 o _/*o * 
¿ 2 +W 2 ^8 ~/ x 2 (P + U 2 ) (l, 2 ~f 2 ) 
= (P + u 2 ) (l, 2 -f 2 ) № + U * 
where p 2 —P + l 2 — 211, cos co, that is 
p 2 = (a- a,) 2 + (b- b,) 2 + ...; 
consequently dps 2 + - v 2 = 8 4 n, where n is given by 
fl S 2 (p 2 + u 2 -f 2 ) g u? 
11 = 
(l, 2 -f 2 ) 2 (P + u 2 ) (l, 2 -f 2 ) (P + v?f 
and it is clear that e will be the positive root of 
0 
(P 2 + V? —fl) . 
■ C / 7 „ / 7 _ C 
are 
(h 2 ~fi 2 ) 2 (P + O (k 2 ~fi 2 ) (P + uj • 
It may be noticed that, in the particular case of u — 0, the roots of this equation 
(irp _ /2\ n 2_ f '2\ 
0, and • ■ Consequently if p 2 — f 2 and l 2 — f 2 are of opposite signs, 
Vi 
we have e = 0 ; but if p 2 — f[ 2 and l, 2 — fp are of the same sign, e 
(P 2 -.fi 2 )(h 2 -fi 2 ) 
PfP