778 PEOFESSOli BOOLE OX THE COMPAEISOX OP TEAXSCEXDEAT:S, ^TH 
which entails as a necessary consequence A—v, B=0, we find as the form of the equa- 
tion B=0, 
4. ^ i_ 
Now as the function within the brackets is essentially positive, the above equation can 
only be satisfied by making §'=0. But this indicates that all the roots are real. 
Besuming (5.), it is evident from what precedes, that if the lower limits of integration 
••i’s+n corresponding to v=]y, are arranged in the ascending order of magnitude, 
the upper limits q^, q ^ .. will also be ranged in the same order. Moreover and q^ 
will both be less than \ and q^ 'will lie between 7^^ and 7.^-, finally, and q„^^ will 
lie between and oo. Hence, then, the elements in the different integrals in the fii’st 
member of (5.) will be all different, the superior limit of each integral being less than 
the inferior limit of the integral which follows it. 
31. Let us now examine the case in which the integration relative to v in the second 
member of (5.) is from — oo to oo. 
Let ]) — — oo , and q — oo, we then have 
Pi ? Pi 2^^ "^I'-Pn + l 
2'l= ^2 , §'2 — ^25 !?3 ^3 • • + • 
Thus 5 'i=j ?25 ^i—P’i-’ln—Pn+ii 01’ tke upper limit of each integral coincides with the 
lower limit of the integral which follows it. 
It is more strict, however, to regard p and q as tending to the respective limits — oo 
and 00 . The first inferior limit, then tends to — 00 , and the last superior limit. 
fln+ii to OO, while the superior limit of each integral but the last tends upwai'd to the 
same limiting value to which the inferior limit of the integral following tends do-wn- 
ward. The different integrals close up into a single definite integral taken between the 
limits — 00 and 00 . Thus we have 
f ('•) 
«/ —00 — 00 ^ 
The reasoning is evidently independent of the natui’e of the function symbolized by/! 
That function may either be continuous or discontinuous. We thus arrive at the follow- 
ing theorem, in the expression of which we shall restore to \p its complete value, and 
shall replace the rational function (p by (p{x), and the symbol which is no longer neces- 
sary for distinction, by d. 
32. Theoeem. — If(p(x) denote a 7'ational function ofx, mid ifi he a general functional 
symbol, then 
i 
x- 
oc—K 
dx 
/•oo 
= t dvf{v)Q[p{x)'\ 
ty — 00 
V — x-\-- 
• + 
a, I 
• ( 1 -) 
x — K 
promded that a,, a,, .. a„ are real and jiositive, and .. real. 
