of homogeneous Ellipsoids. 349 
^ __ CrC dx . dy . dz . ( a — x) 
JJJ | ( a _ ar )2 + (6—^)* + (£_*)* j|- 
g __ rrr dx . dy .dz. (b—y) 
JJJ | («— * *) 2 + ip— y) z + (c— z ) z ( 1 ) 
Q rr r dx . dy . dz . (c—z) 
JJJ | {a — *) 2 + (6— (c— z) 2 j|- 
where the several triple fluents must be extended to all the 
molecules that compose the mass of the solid.* 
The expressions of A, B, and C, just found, are all integra- 
te with respect to one of the variable quantities they contain. 
Thus A is integrable with respect to x : Let x' be the greatest 
value of x ( y and z remaining constant ) on the positive side 
of the plane of y and z, and x" the greatest value, on the ne- 
gative side of the same plane ; then, the integration being 
performed, we shall get 
A = JJ dy .dz . 
{ j (*-*')* + {b- y y+ {(«+*")* + (b — y) z -f- (c-s) 4 ji}* 
In this expression of A, the fluxion under the sign of double 
integration denotes the attraction which a prism of the matter 
of the solid, whose length is x' -f- x" and its base dy . dz , exerts 
on the attracted point, in the direction of the length of the 
prism. 
If the plane, to which x is perpendicular, bisect the solid, as 
is the case of the principal sections of solids bounded by finite 
surfaces of the second order, then x'^=x " : and as x ' is nothing 
more than what x becomes at the surface of the solid, if we 
now suppose x, y, z to be three co-ordinates of a point in the 
surface, and, for the sake of brevity, put 
* Mecan. Celeste, Tom. I. p. 3. 
Zz 
MDCCCIX. 
