extensive Class of Spheroids. 
% 
the whole sphere : then the difference of these values will be 
the quantity proposed to be investigated. 
With regard to the first value of V, it is to be observed that 
R is here the radius of the inner, and p that of the outer sur- 
face; therefore ( 6 ), 
bi0 = ±i-JJ{- jn-M 
But I say that /Q^ . df . d W' = o, when the fluent is extended 
between the proper limits : for pt, and y are the cosines of 
the three sides of a spherical triangle, and vr ' — -nr is the 
angle of the same triangle opposite to the side whose cosine 
is y ; and if we put to denote the angle opposite to the side 
whose cosine is then since the fluxion of the spherical sur- 
face may be either df . dvd or dy . d ^/ ; therefore, when the 
fluents are extended to the whole surface of the sphere, we 
shall have 
. dy . dty = 2?r .f Q (0 . dy : 
but . dy, between the limits y — — 1 and y = 1, is = o 
(No. 2) : therefore JQ ^ . df . dvr = 0. 
Consequently the preceding expression of will become 
simply 
fO __ ff Ol^. dju . . dm. ' 
— i-z r ;_2 
and the value of V, relative to the shell of matter between 
the spheroid and sphere will be expressed by this series, viz. 
V = i f/if- R’) • dfi . dTz'-rjjR. dp- . dv> . Q (,) 
- >\JJ log. R . dp' . dw' . Q (a) 
+ ’*■ 
—J— &c. 
K 
MDCCCXII. 
