Mr. Ivory on the Attractions 
360 
M 
cos. 2 p 5 sin. 2 p cos. 7 q , sin. 2 p sin. 2 q 
~W~ ' ~ * F 1 ' 5 
XT _ b sin. p cos. q , c sin. /> sin. q 
™ * it' 2 "T” ' "ftii 3 
D 
1 k n 
c . 
k" z 
then 
R‘- 2 .^.R-£ = 0. 
This equation has two roots, viz. 
p ± v'n* + md|+ n _ 
and, because D is always positive when the attracted point is 
within the solid, as is here supposed, both these roots are real 
quantities, whatever be the angles p and q. Conceive the line 
R to be produced to meet the surface of the ellipsoid again 
below the plane of y and %, then, if the produced part be de- 
noted by R', it is plain that R and R' will be the two roots of 
the above equation : and because R', although in an opposite 
direction, has the same angular position as R, we may substi- 
tute R'for R, in the expression for : thus, 
f (dR!\ 
[1) 
A 1 " = *//{%' cos.y> sin. % p -{- cos. % p sm.p^dp . dq. 
Therefore, by adding together the two values of A^, and. 
taking half the sum, we get 
A 
or 
(>: 
-//ft 1 ? +9', 
cos .p sin.^+scos.^ sm.p^dp.dq. 
rfiiM 
A (l) = cos .p sin. + 2 cos. % p si n.p^dp . dq : 
the limits of this fluent being, as before, from p = o top = 
and from q = 0 to q = 2^. 
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