CERTAIN APPLICATIONS TO THE THEORY OP DEFINITE INTEGRALS. 
80 ;; 
Therefore 
* 4 , 7 , 
'3 da~ 
d(T as 
3 - -Jo (1 +A?s)*(l + /ilsy^(l 
Let 71 be the positive root of the equation 
1 , 
(18.) 
a^s 
bh 
l+4iS ‘ 1 + ^2® ' l+^aS 
and let the density be uniform ; theny’(ff)==l or 0, according as s is less or greater than r,. 
Before and after the break, therefore, -=^^=0. At the break we have, by the reasoning 
dfi^\ 
of a pre\ious section, --j^ds= — 1. We must therefore substitute this value in (18.), 
and in the rest of the expression under the integral sign change s into t;. Observing 
dc I)^ 
that this substitution converts -j- into * — 72^+ 7^+7 and that the integral 
being reduced to a single finite element we may reject the integral sign, we have 
dY 
3 da 
This result is due, I believe, to Mr. Cayley, but was originally obtained by an entirely 
dififerent analysis. 
It only remains to add, that when the index of differentiation is fractional we must 
revert to the first expression in (5.), Art. 42, and effect the integrations separately. The 
integration with respect to n may always be performed. The possibility of the two 
others will depend upon the nature of the problem under consideration. Thus writing 
the expression in the form 
V 
h-^h^..h 
II 
da.ds.s^~^f{a) 
dv.v 2cos|(a— (r)^;+(^^■— 
I'(0 Jo Jo (1 + . (1 + A: 
it is easily shown that, according as a is greater or less than c. 
I 
dv.d 2 cos |^(a— 1)^ [■ = 
r(i-!+l)sin(i-|+l)7 
or 0, 
(a-<ry-Y+‘ 
r(^-i)(«-Tr>' 
by a known property of the function F. The latter form gives 
or 0, 
r(.)r(|-i 
da.ds.s*~^f{a) 
(1 -f /iis)'®(l +/( 2 s)^..(l -\-hnS)^[a cr) 2 
This transformation, or one in effect equivalent to it, is due to Mr. Cayley *, who has 
applied it to obtain the value of a remarkable definite integral which occurs in the 
mathematical theory of electricity. 
* Cambridge and Dublin Mathematical Journal, vol. ii. p. 219, 
