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 be given : — 
fJf'Waf 
x VWy 
/(«) - Av) 
= </•(«) - 0(0)- 
From 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 <£. 
dx 
x+l 
log 
log (g+1) 
, a + 1 
1 °S j+T 
log (a+l). 
-4v r ? (. 
p x - 1) dx = ~(eP a - 1). 
If 
a dp 
1 
lo g j- 
j — fj 4~ Ag A 4* -A-2^ + 
Y = l-A 0 -^(l + i)-y 2 (l + J + J)-^(l + J + l + i)-&c. 
If 1 + J + J + - • • . + he P u f i n if s lowest terms, the 
numerator is devisable by n, if n be prime. 
The sums of infinite series, such as 
— oo e -(^+i)6 00 (£c+l)e-(^+ 1 ) & 
^ O (» 2 a 2 + l)((a:+l) 2 a 2 + l) and ^ ° (A 2 + l)(i+l 2 a 2 + 1 ) ' 
Several of the above results are easily verified by the usual 
methods : some, however, seem not readily attackable. 
Another formula is 
J ~ f(a,x)dx F (x,y)dy =* d yJ^ y f(wW( x >y) d *- 
