MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
169 
fr 
cos(2''a^ cos’"^ sin(r— 2)^+(m.+r— 1)^) = 
Let r=2, then we have 
Tt d.y" 
da 
IT 
n 
2 dScos”‘'^^Scos(m + l)$_^^^ll- i/l—c 
1— ccos^fl m dc 
2 
where (c) is of course less than unity ; an integral given by Abel. 
When 2 ’'c 4 is less than unity we can always integrate with respect to {z), but may 
obtain a single integral more simply by proceeding as follows : — 
{m + nr—\) (wz + wr— 2) ...(m + n(?’— 1) + 1) 
We have 
1.2.3...ZZ — 1 
TT 
QTO+«r— 1 
I dS COS'”+’”‘”'^ j(m+n(r-2)+l);fl . 
2 
consequently we find by summing a geometrical progression, 
f " .7.1 cos(ffl + y— l)a-2’'acos’'9cos(TO+l)9 l df^ 
j^^at/cos ‘'\i_2'-+‘Gcos’'9cos(r- 2)a + 2Vcos"’-flJ~2“+’-‘m da 
2 
When r=2 this result coincides with that last obtained. We may obtain a very 
general result by applying Fourier’s theorem to the series of Lagrange and Laplace 
as follows : — 
If u=f{y), and y=z-\-x(p{y), 
we have u=f{z)+{(p{z)f{z)]x-\-^{fzfz)^^-\-hc.^ 
.-. %=^{z)f\z)^^^{f{z)fz]x^^^{(p\z)f{z)]f^^-\-hc.-, 
fdci.dz’ 
" 297 ’ 
^00 ^00 
Now we generally have F(z)=i 1 cos a(;s— s')Fz 
J — caj 
^00 ^00 J 
whence 
and 
Hence substituting in the above series, we find 
dd 
297 
du_c°° r° 
^ioc(z 
:>y.d«dd 
27r 
Consequently we find the following definite integral : 
J J doidz(p(z')/'(z') cos a(^z — z'-i-x<p(z')'j = 2‘T^- 
