of a homogeneous fluid mass which revolves upon an axis. 127 
positive, will be the same in quantity, at points diametrically 
opposite on the sphere, at which points [/, Sin. vr and Cos. -a/, 
are different only in their signs. And because, 
y = p V' 1 f . v' 1 — - /A 1 * Cos. ( ■sr — 73 ) , 
it is obvious thaty will likewise have the same value and dif- 
ferent signs at any two points diametrically opposite on the 
sphere ; and the same property will belong to every func- 
tion of y that contains only the odd powers. Now we have 
R'. C W d[d d sr'= — R' . C W d 0' Sin f Y dvr' ; 
and, as & increases from 0 to tt, and 73 from 0 to 27 r, it is ob- 
vious that the fluxions will be the same in quantity, but will 
have different signs, at any two points diametrically opposite 
on the sphere; because the sign of C {1) , which is an odd func- 
tion of 7, alone changes. Wherefore the integral will decrease 
just as much in one hemisphere as it increases in the other; 
and being extended to the whole sphere, it will be equal to 
zero. In the same manner it is proved that 
i) 
dyidd 
i — 2 
t' 
whenever i is an odd number. Thus all the equations we 
are considering, in which i is an odd number, are satisfied by 
the supposition that R' is an even function of p „ V 1 — pi * .. 
Sm tht 7 , V 1 — ju/ 1 . Cos ■&'. 
It remains to consider the cases when i is an even number, 
