CEETAIN APPLICATIONS TO THE THEOET OF DEFINITE INTEOEALS. 801 
I think that this example very clearly shows, that, regarding the integral 
as made up of elements some of which are finite and perfectly determinate in value, the 
others infinitesimal and subject to the conditions of ordinary integration, we can inter- 
pret the general theorem (4.), Art. 38, in accordance with known truth. In determining, 
as we have done, the value of the element each break in the discontinuous 
function, we attach no new signification to a difierential coefficient. We regard that 
element as the limiting value to which As-^f[(T) approaches as As approaches to 0. In 
the present instance this value is finite. Usually it is 0. 
The following examples, which are adopted with some improvements from my previous 
memoir, will illustrate the more important applications of (5.), Art. 42. 
Let n — S, ^=^, h=0, and let us substitute z for ^i, x^, and a, b, c for «i, a. 2 , ; 
then 
_rrr ^ hi) 
J J J { {a—xY+ 
the limits being given by the conditions 
V 
( 11 .) 
The value of V becomes 
V 
?! I I 
hV hV hi^ ■ 
= — hJi^h^Ty 
Jo 
ds.s' 
wherein 
b^s . c^s 
'\+h\s^\ + hWl + hls 
(12.) 
(13.) 
Now the attraction, according to the law of nature, of the ellipsoid whose equation is 
~2 ,.2 -2 
-4-^-1-— =1 
l2\ i2\ t2 -*-9 
^2 ^3 
and whose internal density is expressed by the function 
^ U. /»2 /4/ 
dY 
upon the point (a, b, c), \vill Observing that, in the value of V given in (12.), 
a only appears as involved in ff, we have 
d dff d d / d\~^ . d(T„. 2as . 
da dada-^^ da\d(r) 
2as^f{<r) ds 
whence 
-^=h,hAn^^ (i+lj 
= hjt2h'rp( - 
Jo 
\)(l+/i?s)^(l+Als)^{l+A^sr 
2as s~^f{<T)ds 
+ fhS (1 + /4s)^(l + hls)H 1 + 
5 M 
MDCCCLVir 
