98 Mr. Ivory on the figure requisite to maintain the equilibrium 
tegrations with respect to d r. By comparing this expression 
of V(r) with the foregoing one, we get, 
K = — — +ff log. R' X C (2) <V dm. 
From this expression it follows that K has the same value 
in all similar homogeneous bodies. Suppose another body, 
similar and homogeneous to the first, and having the axes of 
the co-ordinates similarly placed : let R' and R', be lines 
drawn to the surface from the origin of the co-ordinates, and 
making the same angles with the axes ; then, K' denoting 
the like quantity in the second body as K in the first, we 
have 
K = — — -j- log. R' x C ^ dfid ■ ut\ 
K f = — "i" Jf l°g- R/ x ^ d dnj' ; 
wherefore, 
K — K' = ff log. -51 x C <2) rf,/ dW. 
But the two bodies being similar, and R' and R', lines simi- 
larly drawn in them, it follows that will remain un- 
changed, when p/ and -sr' vary. Consequently, 
K — K' = log. A- x fj C (2) dp' d W = o. 
4 . Having now laid down the properties of attraction to 
which we shall have occasion to refer, we are next to con- 
sider the conditions necessary to the equilibrium of a fluid 
mass. These were first reduced to a uniform mode of calcu- 
lation by Clairaut, in his Theory of the Figure of the earth. 
They are investigated in all the great treatises of rational 
mechanics ; in the Mecanique Analytique of Lagrange, the 
Mecanique Celeste of Laplace, and the Mecanique of Poisson. 
