Attraction of Spheroids differing litile from a Sphere. 383 
rind 
Mr. Ivory proceeds to shew that the equation = =- a cannot 
be considered to be demonstrated by Laplace’s process, and that 
in a particular instance it is actually erroneous. 
Perhaps it is doubtful whether in the elucidation of nice and 
difficult points like that before us, it is advisable for any one to 
confine himself entirely to analysis. By considering the expressions 
with reference to their geometrical. signification, I think we shall 
be able to satisfy ourselves that there can. be no error in the 
process of Laplace. 
It is allowed that the expression for the variation of C is 
correct, for all that part of the excess of the spheroid above the 
sphere, whose distance from the attracted point is much greater 
than dr. Suppose then a small circle described on the sphere 
with a radius very much smaller than the radius of the sphere. 
and which may be made as small as we please: and suppose ér 
to. be taken very much smaller than the radius of this circle. 
Since there is no doubt that the equation is true for that part 
of C which is not included within this circle, it will be proved 
to be true for the whole if we can shew—1* that for the matter 
included within this cirele, C may be made as small as we please, 
by diminishing the radius of the circle, and the. magnitude of ér: 
—2™ that the variation of C dependent on 67, or its proportion to ér, 
may be made as small as we please by the same process. Now 
if we suppose two planes making a small angle to pass. through r, 
cutting out a small pyramid from the matter between the sphere 
and spheroid; since the angle which is made by the tangent 
planes of the sphere and spheroid, though small, is finite, their 
distance from each other, or the thickness.of the included matter, 
is nearly proportional to the distance from the attracted point; 
the breadth also is proportional to the. same distance: hence a 
section of the pyramid is proportional to the square of the dis- 
