of a homogeneous fluid mass that revolves upon an axis. 89 
First, let the attracted point be in the surface, in which 
case r = R : then, are the only quantities that con- 
stantly retain the same values in all the bodies. These 
quantities remain unchanged, because the line r, or R, always 
makes the same angles with the axes of the co-ordinates. 
We therefore have, 
V(R) _ 1? ± — 1 
R 1 — ^ ' }R’ R’ R j ’ 
the letter F being the mark of a function. Hence, 
V(R) = R*xF.{£, ± 
Again, let us put, 
a = r.y ; b = r V 1 — p/.Cos.xa; c — y\/i — y 2 . Sin. ; 
and [x will be the cosine of the angle which the line r, or R, 
makes with the axis of a ; and -ar the angle which the pro- 
jection of the same line upon the plane of b and c makes 
with the axis of b : then, when r— R, we have, 
V (R) = R* x F. | y, V 1 — . [x 3 . Cos. sr,Vi — y \ Sin. y^ • 
Secondly, suppose that the attracted point is placed within 
each of the bodies ; then the quantities common to them all 
are these, viz. Hence, 
j R r r r j ’ 
V (r) f r a b c ■) 
~~R^~ — r • \ R> T’ T’ V J 
Consequently, 
V(r) = R'xF.Jfffi}, 
V (r) = R* x F . [x, V 1 — [x 2 . Cos. ar, vT — p- 2 . Sin. itrj . 
In order to have a more exact notion of this function, we may 
suppose it to be expanded in a series of the powers of the 
fraction ^ : then, 
MDCCCXXIV. 
N 
