EQUATION AND ITS TRANSFORMATIONS. 
787 
Now, if n is a positive uneven integer, we have, by art. 16, U, -j-r/jV^R/ ; and 
M+gqN is equal to U 1 +< 7 1 V 1 when a is written for x, and a put = 2 ; so that, if n is a 
positive uneven integer, 
x u 1 e ^ r ' ! 
dx=\Y(\n) 
1 + 
n — 1 
?& —1 
(2a) 
(n — l)(n— 3) 
(n — l)(n—2) 
(n — l)(n — 3)(n — 5) (2a) 3 
(n-l)(n-2)(n-3) 3! i “ 
the series terminating at the term preceding the first term containing a zero factor in 
the numerator. 
Transforming the integral by assuming x=~„ we find that, if n is a negative uneven 
X 
integer, 
x 
\1l 1 rj X~ , 
-•dx=\T (-£»)*" 11 + — i (2«) 
n +1 
{n + l)(n + 3) (2«) 2 
(n + l)(n + 2) 
91 
+ &c. \e 
the series terminating when the first zero factor appears. 
Thus, generally, 
x n l e~ 
tlx 
u(i»)q+h_l(2“)- 
(n-l)(n- 3) (2a 2 ) (%-!)(»-3)^-5) (2a) 3 
(n — l)(n — 2) 21 ' (n-l)(n—2)(n-3) 3! 
-&c. |( 
,-2a 
+lr(_l n )a^ l+^ +1 (2a) + ( ^ 1)( - +3) (gfO!, (n + l)(n + 3)(n_+5) ^ , 2a ( } 
-hs 1 ! 2 n ) ^^ n + i^ a )^( n+ i)( n+ 2) 2! + («+l)(% + 2)(% + 3) 3! +^ c -r t • Yh 
if n is not equal to an integer : but if n — a positive uneven integer, the first series 
continued up to the first term containing a zero factor in the numerator is the value 
of the integral, the second series being ignored altogether; and if n = a negative 
uneven integer, the second series continued up to the first term containing a zero 
factor in the numerator, is the value of the integral, the first series being ignored 
altogether. The rule may therefore be stated as follows : if neither series terminates 
then (8) represents the value of the integral, but if one of the series terminates, the 
finite series represents the value of the integral, the other being ignored; a series 
being supposed to terminate at the term preceding the first term that contains a zero 
factor in the numerator. 
The apparent change of form is curious, but the reason for it has fully appeared in 
§ I., arts. 3-6. In the ‘British Association Report’ for 1S72 (Transactions of the 
5 I 2 
