802 PROFESSOE BOOLE OX THE COMPARISON OF TRANSCENDENTS, MHTH 
Now/(o-) is to vanish when a falls without the hmits 0 and 1, and s as falling between 
the limits 0 and co is to be positive. But from the expression for c, it appears that when 
(P" 
5=0 (r=0, and when s=oo (r= 7 v+ 7 i +72 J also s increases with s (Art. 42.), and therefore 
^3 
(p IP (p 
passes over the value 1 when w+Ti+T^ > 1 ? but reaches not that value otherwise. 
Aj il'2 A 3 
The former condition is realized when the attracted point is external. Eepresenting 
by n the value of s for which o- = l, we have 
— z=.'2ihjiji^'7r§a 
, / a^s b^s c^s \ 
» (l + /^r 
fj being the positive root of the equation 
b'^s 
1 + h^s 1 -1- ^ 2 ^ t P h^s 
(14.) 
(15.) 
When the attracted point is internal we have only to substitute co for >] in the upper 
limit of integration; for all positive values of <7 then satisfy the condition (r<l, and 
positive values only are admissible. 
Both these cases may also be derived from the more general theorem deducible fr'om 
(5.), Art. 42, 
dxdydz 
' {a—xY-\- {b~yf-^ {c—z)^} 
where ;; is the positive root of the equation 
X = ^hJiJi^T^a 
C’’ j If/ 1C, bh c-s \ 
i ds.s-^f ( hh + —^ +-- - - 3 + — 
\ \ 1+Ai 5 l + Ip/isS/ 
Jo (l+A?s)i(l+/^l5)^(l + /4)5^ 
h^s-h 
b^s 
1 P hiS 1 P 7^2^ 1 P h^s 
= 1. 
(17.) 
When h approaches to 0 this root approaches to the positive root of the equation (15.) if 
(p IP (p ^ (p ]p (p 
Ti + p+Ti is greater than 1 , hut tends to 00 if 75 + 71+75 is equal to or less than 1 . 
Al Ag /ig hi A2 A3 
Ify’(o-) = l, the expressions are easily reducible to elliptic functions and agree ulth 
known results. 
Lastly, let the law of force be that of the inverse fourth power of the distance and 
equal to g’. The expression for the attraction on an external point (u, 5, c) is 
i ^ 
3 da ’ 
where 
V 
-fc 
dxdydz 
{ {a—xY+{b~yY+{c—zY)^ 
^ Jo 
ds.S^j\<T) 
(1 P7+)^(1 P /+)-(] p/+)^ 
^/(°') ^ d,f{(r) 
3 da 3 da d[<r) 
2Qas df{a-) 
3(lpA?s)' %) 
2^as ds df{<T) 
' 3{l+/ils)' d{(r)‘ ds 
(16. 
therefore 
