EQUATION AND ITS TRANSFORMATIONS. 
761 
p(p + 1 ) 1 . + + _1 
2 aaO 2.4 a*a 
&c., 
p being a positive integer, then we might suppose that the corresponding particular 
integral, when p was not an integer, would be obtained by continuing the series, which 
does not then terminate, to infinity. This infinite series, when p is not an integer, 
still satisfies the differential equation, but is divergent; and the true integral is 
obtained by commencing the series at the other end and continuing it to infinity 
backwards. In general, when we have a series which terminates of itself for a parti¬ 
cular form of p, we may derive from it two infinite series, when p has not this form, 
by commencing it at either end. One of these will be an ascending series and the 
other a descending series; and we can thus, as it were, pass from the one to the other 
through the intervention of the finite series. 
The fifth section contains the evaluations of the definite integrals 
cos bx 
lo (<# + &)* 
^r-dx, 
\ n J 
where m denotes any real quantity and n any positive quantity. These integrals have 
been evaluated when m is of the form 0 ^ and when n is a positive integer; but, so 
far as I know, the general formulae given are new. It is known that these integrals 
satisfy differential equations of the forms (4) and (1) respectively, so that their values 
are necessarily connected with the solutions of these equations considered in §§ I. and 
III. The results are curious, as they exhibit changes of form similar to those referred 
to in describing the contents of § I., and which are due to the same cause—viz. the 
recommencement of the terminating series after the zero terms. 
When n is unrestricted it is shown that we have 
a- 
xplx — 
i 
+kn< 2 “>- 
(A-1 )(»■ —3) (2a) 2 . 
\n-l)(n- 2) 2! 
+IU 
0 + 1)0+ 3) (2 ay 
0 +1)0+2) “IT 
&c. 
but when n is a positive integer the first series is to be continued till it terminates, and 
the second is to be ignored ; and if n is a negative integer the second is to be con¬ 
tinued till it terminates, and the first is to be ignored. The well-known value of the 
integral when n— l = an even integer =2 i, viz. 
x~'e 
^dx— v -~a l 1 1 
J o 
MDCCCLXXXI. 
i(i+ 1) 1 , (i—l)i(i + l)(i + 2) 
2 2a ' 2.4 " 
5 F 
+ &c. [e 
