of a homogeneous fluid mass that revolves upon an axis. 9 7 
now if we put p for the angle opposite to the side whose 
cosine is f, we may write dy dty in place of dfd*j', making 
y and vary between the same limits as ^ and -a r. This is 
allowable ; not that we must conceive the two fluxions as 
continually equal to one another, but because the total sum, 
between the prescribed limits, is, in either case, equal to the 
whole surface of the sphere. If now we substitute the value 
of given in § 2, the foregoing expression will become, 
V ) 
(-o* .. rpL± 
2 . 4 . 6 . . . 2 i JJ d 
and the integral is, 
(-0 
-• X 2 7T X 
x dy x dp : 
2 . 4 . 6 ... 2 i 
V* / 
a quantity which, being divisible by 1 — y 2 , is evanescent at 
both the limits of y. 
Omitting what has been proved to be evanescent in V (r), 
and collecting into one sum all the parts multiplied by r 2 , and 
separating them from the rest, we get 
V (r) ■ dv! dsr' + x {—g+JJ log.R'xC 
+ rJ/R'&>d/d*' 
—JJ 
h - rrd^dp'dn' 
zJJ R* 
— &c. 
which expression may be thus written in finite terms, viz. 
V(r) = r‘f— + {_ y +// log.R'xC (2 > dv.'dm' 
as will be evident by expanding -j , and performing the in- 
O 
MDCCCXXIV. 
