of a homogeneous fluid mass that revolves upon an axis. 147 
This expression is to be substituted in the equation ( A ) : and 
it is to be observed that JJ~ d p! d d i= 2 nr c ? ; and that is 
of the same order with a. Hence we get, 
C—~a 2 — ~ cf y -{- d a . JJ y' d\fl dm 1 
+ ofH.jjy' .& ] dfld-v' 
+ *■ + f (1-^*) } 
+ t/a.ffy'.d-* dp' dw' 
-j- & c. 
This is the approximate equation of the surface, when the 
equation ( A ) is alone taken into account ; and it is to be 
proved that this equation cannot subsist unless it belong to 
an elliptical spheroid of revolution. 
In the first place, the nature of the function y must depend 
upon the integrals by which its value is expressed. But all 
the integrals are independent of the angles that determine 
the position of the attracted point in the surface, unless so far 
as those angles enter into the expressions C^, C^ 2 \ C^ 3 \ &c. 
which are all functions of y. Now y is a function of [x, 
V 1 — [x 3 Sin. OT, 1/1 — p- 2 . Cos. nr : and hence it follows, that 
y and y' are functions of three rectangular co-ordinates of a 
point in the surface of a sphere. 
In the second place, every function of three rectangular 
co-ordinates is susceptible of an arrangement, by which it 
will be converted into a series of the same integrals contained 
in the foregoing equation. Let 
f —V 1 — q e y — |— g s : 
then, as is well known, we shall have, 
