of a homogeneous fluid mass that revolves upon an axis. 131 
But, because p'= Cos. 9 ', the three last integrals will become,, 
ff Sin. 8 9 ' Cos. 9 ' Sin. «/ Log. S . d 9 ' da/ 
ff Sin. 2 9 Cos. 9 ' Cos. m Log. S . d 9 ' d ™ 1 
//Sin. 3 9 ' Sin. zd Cos. vs* Log. S . d 9 ' dm' : 
and, attending to the expression of S, it will follow that, in 
the two first integrals, if we suppose zd to remain constant 
and 9 ‘ to vary from o to 180°; the fluxions will be equal, 
but will have different signs, at equal distances from o and 
180°: wherefore the integrals, taken between the prescribed 
limits, are evanescent. In the third integral, if we suppose 
9 ’ to remain constant and tar' to vary from o to 360°, the flux- 
ions will be equal, but will have different signs, at equal dis- 
tances from o and 180° in the first semicircle: and at equal 
distances from 180° and 360° in the second semicircle : 
wherefore the whole integral is evanescent. Rejecting there- 
fore the three last terms of the value of K, and representing 
the three remaining integrals by L, M, N, we shall get, 
k+tO-iO-It— f») V 
+ (— — + ■ ( 1 — f*’ ) S' n -’ ® 
+ (T-T*+f) .(i-f’JCM.--. 
This is the denominator in the formula for R 2 , and it is en- 
tirely similar to the expression of S. Wherefore the two 
values of R 2 and R ,E are alike in point of form ; and the figure 
of the fluid mass that corresponds to the given rotatory 
velocity will be determined by making them coincide. 
We are now to conclude that a homogeneous fluid mass 
cannot be in equilibrio by the attraction of its particles and 
a centrifugal force of rotation, unless it have the figure of an 
