of a homogeneous fluid mass that revolves upon an aocis. 91 
to any material body, whatever be its form. The body may 
consist of parts not connected by any mathematical law ; or, 
which is the same thing, it is not necessary that the equation 
of its surface be subject to the law of continuity. 
2. The co-ordinates of the molecule d m being x , y, z, let 
R' denote its distance from the origin of the co-ordinates ; 
and put, 
x = R'. ; y = R V'i — ^ . Cos. nr'; z = R' V'i - [x'\ Sin. nr'; 
then, since we likewise have, 
a = r. y.; b = r. 1/1 — ifl • Cos. nr; c — r\/i — ^\Sin. ot; 
and, 
/= v'o — *)• + ( b—y r + ( c - *)• ; 
we shall get, 
y — \ V 1 — . V l — [j/ 2 . Cos. ( nr — nr'), 
f=\ / r 2 — 2 r R' . y R' 2 . 
It now becomes necessary to expand y in a series of the 
T R 
powers of , or of — . Much has already been written on 
this expansion. The coefficients have been exhibited in va- 
rious forms, and many remarkable properties which they 
possess have been very diligently explored. It would not, 
therefore, Jae necessary to add any thing upon this subject, 
unless it be possible to give to the same quantities a new 
and more simple form of expression, useful in the present 
investigation. 
If we suppose, 
{i + C 
(I) R' 
+ C 
(2) 
R ' 1 
T 
the following differential equation has already been proved 
in the Philosophical Transactions for 1812, viz. 
