EQUATION AND ITS TRANSFORMATIONS. 
791 
Thus, if n is an even integer, we have 
j v n + (n — 1 )av n 
, (n-2)(n-3) 
9 i 
+ a i ' 1 \dv, 
which agrees with Boole’s formula. But if n is not an even integer, the expression 
in square brackets when expanded contains an infinite number of terms, and putting, 
as before, 
P„=( x n (f)(x)dx, 
J o 
the general formula is 
Q„ — V H +(n— l)aP n _ s + ^- ^ 3) a 2 B n _ 4 ,+ (ft ^^ )(n a 3 P w _ c +&c. ad inf. 
+ a n+l P_ n _ 3 d-(w+3)a n+2 P_ n _ 4 + ■— + a M+3 P_ TO _ 6 + &c . ad inf. 
24. This formula involves infinite terms unless <f> is such a function that the 
integrals P n _ 2 , P B _ 4 , . . . P_ m _ 3 , P _„_ 6 . . . are all finite. This condition is not fulfilled 
when (f\(oc) = e~ x \ for \fx n e~ z °dx is infinite when n— or < —1, so that we do not obtain 
by means of the formula a demonstration of the equation (8). If however we replace 
l™x n e~ z "'dx by T ( ---- j in all the terms, whether the integral be really infinite or not, 
we do in fact, as we should expect, obtain (8). For, putting <£( x) — e~ x \ substituting 
gamma-functions for the integrals, and writing n—1 in place of n, the formula gives 
j x’ 1 l e ** *‘dx=\\T(\n) +(n— 2)aT(^n— 1) +— ——— cdT(fn — 2) +&c. \e ~ a 
— — + y ,” + ^ a 2 r(— \n—2 )~t~&c. \e 2a 
i r (l n )J 1 .^lg/2a)4- (% - 3)(W ~- ) (2ft) 2 , (n-4) fr-5)(w-6) (2 a? 1 
2 (2 n )1 1 ^ n _ 2 ( Za /^( n -2)(n-4) 2<.(n-2)(n-4)(n-6) 31 + & j 
+w-k i+^|(2a)+fc±|&±i) m +&c . 
(n + 2)(% + 4) 2! (n + 2)(n + 4)(n + 6) 3! 
The coefficients are readily identified with those in (S), for evidently 
(n + r + l)(n-\-r + 2) . . . (n + 2r) _ (n-\-l)(n + 3) . . . (n + 2r—Y) 
(n + 2)(n + 4) . . . ( n-\-2r) (n + l)(n + 2) . . . (u + r) 
This process, regarded as a method of obtaining the formula (8), is of course unsound, 
