of Edinburgh , Session 1877-78. 
271 
2. On Methods in Definite Integrals. By Professor Tait. 
(Abstract.} 
This paper deals with various formulae of definite integration 
which are, in general, put into forms in which they enable us with 
great ease to sum a number of infinite series. As a simple example 
of such a formula the following may he given : — 
fjmdaf 
x 
/(<*) - fiv) 
= <K a ) - <M 0 )- 
Prom this it is easy to deduce innumerable results, of which the 
annexed are some of the more immediate. They are written just as 
they are presented by the formula : when different forms are given 
to / and to cf>. 
dx 
x+l l0 S 
log (a + 1) 
l0£ 
a + 1 
x+l 
log (a+ 1). 
ZTj f ‘\ (•" - dx = j+ a - + 
If 
a djp 
1 
l0£ 
l ~ q + + A x (9 + A 2 0 2 + 
l-6> 
-I. 
a dx a — x 
— log. 
x ° x 
If 1+J + J + . . . . + n _ he put in its lowest terms, the 
numerator is devisable by n, if n he prime. 
The sums of infinite series, such as 
•j™, oo € — (#+ 1)6 00 (x + l)€ — ^+ 1 ) & 
^ o (a: 2 a 2 +l)((a;+l) 2 a 2 + l) and ^ ° (#+1 )(i+1V+1) ' 
Several of the above results are easily verified by the usual 
methods : some, however, seem not readily attackable. 
Another formula is 
ri a r* x r * « 
J o f(a,x)dxj o F (x,y)dy = J q dy J f(a,x)Y(x : y)dx. 
