CEETAIN APPLICATIONS TO THE THEOEY OE DEFINITE INTEGEALS. 775 
Hence on substitution we have, since the number of values of y is 
— / 
f cos 
/ 2r7r / . 2r'i! 
X— (cos — +x/-lsin 
'TTl \ . — 1 
which agrees with the result in question. 
We now see why it is that, although in the investigation of the latter formula we 
had to take distinct account of the terms u and v in the fraction y ; in the final result 
they are recombined, and only present themselves implicitly as component parts of 
It is due to the fact that the equation determining the irrational factor of the original 
integral wants a second term, i. e. that^i = 0. 
If^ j = 1 , the term — 0 <p{x) 
Po 
log V in (11.) becomes — ©[^(^)]p, log w, or -- '2Q[(p{x)'\yr log 
whence the theorem assumes the following form, 
2:j>(^)j/dr=:r;=r©[?>(^’)]i/.{log (x-y.)H- logt^} = 2 ;z“e[<p(^-)] 2 / 4 og {u-y,.v). 
We may remark m concluding this section, that all general theorems lilte the above 
for the comparison of algebraical transcendents are difficult of application, from the 
necessity which they impose of developing logarithmic and irrational functions. This 
difficulty we avoid by employing directly the theorem of transformation exemplified in 
the earher problems of this section. The application of that theorem requires only the 
development of rational fractions, and this can always be effected by the operations of 
multiphcation and division. "When this form of procedure is adopted there remains, 
however, an integi’ation to be performed. Cu’cumstances must decide which of these 
methods is preferable, but generally I conceive it will be the latter one. I will only 
add, that the interest attaching to the entire subject of the comparison of algebraical 
transcendents appears to me to be chiefly of a speculative character. It is to be valued 
rather as affording evidence of the powers, and at the same time of the limitations of 
analysis, than as offering any prospect of increased command over the problems of 
physical science. Such at least seems to be the tenor of present indications. 
Application of the General Theorem of Transformation to the reduction of Functional 
Transcendents. 
29. Let us first apply the general theorem to the reduction of the expression 
where (p and are any rational functions of x, and f a general functional symbol, the 
simultaneous values of x in the several integrals being determined by an equation, 
'^=v, ( 1 -) 
in which w is a new variable. 
We have 
( 2 .) 
We must now seek the value of the expression 'S,f{v)(pdx, subject to the condition (1.), 
and then integrate with respect to v. 
