444 
ON THE ATTRACTION OF AN ELLIPSOID. 
[75 
Also from the equation 
+ 18 — IX) (co 2 + m8 — mX) (co 2 + n8 — nX) 
differentiating with respect to X, and writing X = P, 
(eo 2 + 18 — IP) (w 2 + m8 — mP) (co 2 + n8 — nP) 
or, as this may be written, 
PQP 
k(P — Q)(P — R) 
Pa 2 
; + 
rrPb 2 
+ 
?rC J 
k ((&) 2 +18 — IP) 2 (<ö 2 + m8 — mPy (co 2 + n8 — nP) 2 j ’ 
(CO 2 +18-IP) <y + m8 -mP)(co 2 + n8-nP) \ 2 ;p . 2 + —— PV| + 7 a ^ DV 
v /v '((y+ZS-ZP) 2 (ft) 2 +mS-mP) 2 (o) 2 +nS-nP) : 
and from the values first written down, for B, G, F, we obtain (B — mP) (G — nP) — F 2 
= (co 2 +m8)(co 2 +n8)—m 2 b 2 (co 2 +n8)—n 2 c 2 (co 2 +m8) — mP(co 2 + n8—n 2 c 2 ) — nP(co 2 +m8—m 2 b 2 ) + mnP 2 
=(co 2 + m8 — mP) (co 2 + n8 — nP) — m 2 b 2 (co 2 + n8 — nP) — n 2 c 2 (co 2 + m8 — mP) 
l 2 a 2 (co 2 + m8 — mP) (co 2 + n8 — nP) ^ , A . 
= 5 a>*+ 18 —IP >the last reduction being effected by means of the equation 
1 Pa 2 m 2 b 2 n 2 c 2 
co 2 +18 — IP co 2 + m8 — mP co 2 + n8 — nP 
Hence 
PQR {(B - mP) (C - nP) - F 2 Y 
K (P-Q)(P-R) 
la 
(co 2 +l8—lP)^(co 2 +m8 —mPy(co 2 +n8—nP) 2 
l 3 a 2 
+ 
rrPb 2 
+- 
n A c 2 
(co 2 +l8—IP) 2 (a> 2 +m8 — mP) 2 (co 2 +n8—nP) 
4 
Substituting this value, and multiplying out the fractions in the denominator, 
V = 47via x 
f (w 2 (or + 18 — IPy (co 2 + m8 — mP)~ (co 2 + n8 — wP)" dco 
J Pa 2 (co 2 +m8—mP) 2 (co 2 +n8—nP) 2 +m 3 b 2 (co 2 +n8—nP) 2 (co 2 +l8—IP) 2 +rPc 2 (co 2 +l8—IP) 2 (co 2 +m8—mP) 2 ’ 
the reduction of which integral has been already treated of in the former part of 
this present memoir.