170 
MR W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
Again, from Laplace’s theorem, we have 
j 1 dadz' c,o^ci{z—z' -\-x(p^(p^z')(p^(p^z'f%z'=2r-^, 
where u=f{y), y—(p^{z-\-x(p,y). 
These theorems of course suppose the series from whence they were derived to be 
convergent. 
As examples we may take the following. 
y=\-^xy\ 
Let 
then 
Also let 
then 
^00 ^oc 
1 I doidz' co'& ail — 
^-00 J— 00 
(4+\/ {ho~\/ 
y=\-{-x^y, 
J I cosa(\—z'-\-xz’‘')z‘dadz = Y^^^^ 
which we may modify thus ; by eliminating {x) 
j-J- = 
27r£3' 
2 — y 
Analogous methods apply to series involving Bernouilli’s numbers ; thus we have 
B. 
-^ = 1— 
s ^— 1 2 1.2 1 . 2 . 3. 4 
a?"*-)- &c. 
— ( — 1 \ 
V{2n+ 1) — 22»-i,r2“\^12«“22»^32'*^ j 
.1/ l\2n-l 
(log.^) 
1 1 
•22 »-i^ 2«P2 w 
1— S’ 
e’’^— 1 '2 ttJ^ \—z 
r T 1 r * sin (« loff.S’) dz tt +1 1 
Hence we have \ — ^ — — =~- — — — L. 
Jo 2—1 2 £ 2“’^— 1 2 « 
In this formula (a) must lie between 0 and 1, as it is necessary for the convergence 
of the above series that x should be less than 27r. 
I now enter upon the consideration of the processes I have before mentioned for 
reducing multiple integrals to single ones. We easily see the truth of the following 
equation : — 
1-f 
— . 1^ 2^ 1.3 i2 22 2^ 1 3 5 2^ 32 2® 
2 ' 2'2‘ ' ‘ 2'2'2‘ 
-j- &C. 
= 2 - 
1.2.1 ' 1.2. 3. 4. 1.2^1. 2. 3. 4. 5. 6. 1.2.3 
+ &C. — 1. 
