788 PEOFESSOE BOOLE OJf THE C03IPAEIS0X OF TEAXSCEXDEXTS, -WITH 
3rdly. From the value of any known definite integral, we com, hy th.e general theorem, 
deduce either the values of other definite integrals tadcen between the limits — cc and cc . 
or relations among the values of those integrals taken between other limits. 
To accomplish the first object, we have only to transfonn the given integral into one 
whose limits are — oo and oo , and then apply directly the general theorem. The method 
requires no illustration. 
To accomplish the second object, we must express the function under the sign of 
integration, not as a continuous function taken between the given limits, but as a dis- 
continuous function taken between the limits — oo and oo , the character of its discon- 
tinuity being such, that for all values within the given limits of integration it shall 
assume the form specified, and for all values without the given limits shall vanish. 
Thus let I f{v)dv be the definite integral whose value is given. We may extend the 
Jp 
integration from — oo to oo , provided that we regard /(i;) as vanishing when v falls 
without the limits and g. We shall thus have 
provided that — — ) vanishes whenever x — ^ falls ’without 
^ Aj x~Xnj a?— A] x—\n 
the limits p and q. Let the roots of the equation 
X 
O^n 
taken in ascending order of magnitude, be^i, ^^d the roots of 
X 
X — Aj 
Un 
x—K’^' 
dr 
taken in the same order, be ^i, ^' 2 ••$'«+!• We suppose X„ Aj, ..X„ also to be in ascending 
order of magnitude. Thus (art. 30) and q^ lie between — 00 and X^,p .2 and q., between 
Aj and Aa, and so on. Also £1 is greater than p^, q^ than p. 2 , &c. Hence we see that 
will only fall within the limits p and q when x falls either between 
2 )i and g',, or between and q. 2 . See. Thus we have in fact 
and still more generally, <p(x) being a rational function of x, 
.T — Aj x — \„ 
which is a reproduction of (5.), art. 30. I deem it, however, an important fact, that in 
the comparison of functional transcendents, formulae involving the sign of summation 
