of a homogeneous fluid mass that revolves upon an axis. 93 
Now, by combining these equations, after having taken the 
fluxions of each a proper number of times, all the intermediate 
quantities between <p ^ and <p^ may be made to disappear ; 
and we shall finally obtain, 
* (0) - 
(W 
4 (i) 
1.2.3 
. 2 1 
7 x =- = 0 : 
dy‘ 
and by restoring the expressions that and stand for, 
K (0 
f 
a d l d l) 
i 0-v a )‘ 
r(*) (—O' d v 
^ 1 . 2.3 ... 2 * 
dy 
But, from the series equal to C (,) , we get 
Wherefore, 
d * CP ( • X 
— =1.3. 5. ..(22 l). 
c («)_ (-0* w £ (1 — y 1 ) 1 
2.4. 6 ... 21 dy * 
From this very simple expression, the most remarkable pro- 
perties of the coefficients of the expansion of -j, are very 
readily deduced. 
3. We may suppose that the indefinitely small molecule 
dm is a parallelopiped, of which the height is equal to d R # ; 
and the length to the small line described by the 
motion of R' perpendicular to the plane of y, z ; and the 
breadth to R* dm' V 1 — [ji/ 2 , the small line described by the 
motion of R' parallel to the same plane. The volume of the 
molecule is therefore equal to d R' x - 7 - - x R' d^' 1/1 u/ 9 • 
V 1 — [t! z r ’ 
and, if g be put for its density, we shall have 
d m = g R' 2 d R ' d d ts . 
