Attraction of Spheroids differing little from a Sphere. 381 
SES UW me 
where the function U" satisfies the equation 
g) bh Sep eet tal nate 
du du 1-2" dw 
Now suppose the attracted pomt to be upon the surface of 
the spheroid, and suppose } to be the radius of a sphere differing 
little from the spheroid, and passing through the attracted point. 
The value of V’ will consist of two parts, one of which (B) is the 
sum of the products of each particle of the sphere by the reciprocal 
of its distance from the attracted point, and the other (C) is the 
similar sum for all the particles of the excess of the spheroid 
above the sphere, that excess being supposed negative when the 
surface of the spheroid is below that of the sphere. Suppose 
now the point to be raised dr above the surface; B will be increased 
by a quantity which is ultimately proportional to 57, and which we 
shal] call B’.dr; and C will be pau bus a quantity ultimately 
SE ee oh lip f GAOL 
equal to = dr. To find the value cee OMtice f be the distance 
of a particle dm of the spheroidal excess, an the attracted point ; 
if y be the angle made by the radii drawn to the attracted point 
and the particle dm, f? = 2b(1—cos y). When the point is raised 
dr above the surface, f?=6+ dr\?—-2b.6+dér.cosy+%; or if we 
reject the term 67)’, 
f° =20b* (1—cos y) + 26.87 (1—cos y) =f? (1 ae 
and consequently, 
sie : or 
Jar Cart ae 
and therefore 
ém dC or 
SF =0 7 mor + Lora (1-3) .€ 
3c 2 
