55 1 
figure of the earth. 
in the spheroidal surface of equal density is assumed to 
be a f i + *. 1— / 3 — (^- + A ) -P * — ^ /4 )- 
(3.) Let V be the sum of the products of each particle by 
the reciprocal of its distance from any point : let p be the 
sine of the latitude of that point, r its distance from the 
centre : also let Q and w be the latitude and longitude of that 
point, 6' and d those of any other point. Then it is easily 
found that their distance 
= V {r 3 — 2rR(sin Q.sinQ'-}- cos 6 . cos Q'. cos d — «) + R 2 } = 
1/ {r*— QrRfap -\-Vi — p* V 1 —p 2 . cos d — d) + R 2 } • 
let this = z. Suppose now the heterogeneous spheroid 
divided into wedges by planes passing through the axis, and 
suppose each of these wedges divided into pyramids whose 
vertices are at the centre : let Sd be the angle of two planes, 
and the angle made by the two surfaces of a pyramid 
which cut these planes ; and suppose the pyramid divided 
into frustra, the length of one being £R. Then the solid 
content of this frustrum is ultimately R cos &'$d. ^R = 
R 2 < 5 R . Sd. tip ultimately ; and if p be its density, the product 
of the particle into the reciprocal of its distance from the 
given point is ultimately •— R * R •**>•*£ . Consequently, to 
find the sum of those products for the spheroid, we must 
integrate with respect to R, d, and f : or, which amounts 
to the same, we must integrate ~ with respect to 
a , d , and p ! : that is, we must take J Sjfyj ~ • 57 *> or in 
• I prefer this notation, as it does not necessarily carry with it the idea of 
infinitely small quantities. 
