CEETAIN APPLICATIONS TO THE THEOET OF DEFINITE INTEGEALS. 
777 
of the functional symbol f, we may so determine the form of as to cause the several 
integrals included under the sign 2 in the first member to close up, if the expression is 
permitted, into a single definite integral whose actual value will then be given by that 
of a far more simple integral in the second member. 
Let the limits of v in the second member be p and q, and let the transforming equation 
'<P = V 
giye pi, P2. .pn+i for the values of x when and q^, q^.-qn+i for its corresponding 
values when v=q. Then we have 
. . . (5.) 
%)P\ %JP 2 ^Pn-\-\ ^P 
Now let us give to the form 
^ ^2 , ^ 
where a^, ..X„ are real, and a„ real and positive. The transforming equation 
is then 
.r ^^2 -L_=v; (6.) 
X — K^ X — K^ X — An ^ 
whence representing still the first member by -p', 
1 [ I ^2 ..-I 
dx {x — Xj)^ — ^2)^ — \j)^’ 
and as this expression is always positive, it follows, — 1st, that -vp regarded as a function 
of X is never a maximum or minimum ; 2 ndly, that whenever -py varies continuously while 
X increases, it varies by way of increase. 
From these properties, and from the form of the equation (6.), it readily follows that 
if Xi, >^, . .X„ are arranged in the order of increasing magnitude, then whatever real value 
V may have, the roots of (6.) wiU be real and will be disposed in the following manner, 
\iz. one root less than one root between and Xj.., one between X„_i and X„, and 
finally one between X„ and 00 . To prove this in detail, let x vary from — co to X„ then 
\p, as is evident Lorn its form, varies from — cx) to co , and it varies continuously by way 
of increase so as never to resume a former value. Once therefore in its course it will 
be equal to v. Wherefore one root of (6.) lies between — 00 and Xj. Supposing x to 
continue to increase, the value of -p/ suddenly changes when x passes over the value X, 
from CO to — CO , and as x varies from Xj to X2, -p^ again varies continuously, and by way 
of increase from — 00 to 00 , and therefore again becomes equal to v once in its course ; 
wherefore a value of x lies between Xj and X2. In like manner there is a value of x 
between and X3.. X„_, and X„. Finally, as x varies from X„ to 00 , \p once more passes 
over the value v. Whence the proposition is manifest. 
The reality of all the roots of (6.) may also be readily shown in the following manner. 
Let x=p-^q\/ — 1 . Then substituting in (6.), and reducing that equation to the form 
A+B/y/ — l=t;, 
5 I 
MDCCCLVII. 
