740 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
Now 
[a] b =a(a—l). .(«— 5+ 1) 
I>+1) 
— r(a-i+l)’ 
provided that a-\- 1 and a—h-\- 1 are positive. This law we can extend symbolically to 
expressions in which D appears, provided that, in the application of the symbolic forms 
thence arising, D shall admit of an interpretation which shall effectively make the sub- 
jects of the symbol T to be positive numerical magnitudes. Under this condition we 
have then 
iY— D + 
where 
Now 
\ n n J \n n J 
/n—l „ m . „ /D m \ 
s "'r(D+i) 
O(D) 
T(D— pn+ 1) 
= O(D)T(D)^ +i)0 , 
r(— d+-W5-”\ 
\ n n J \n n J 
|1(D + 1) 
T(D— pn + 1) 
r C-^ D+ ^- 1 0 r ^^) 
T(D y pn + i)6 ='¥(pn+iy pn+i)a 
r(i+i) 
\ n n J \ n J 
We see then that the conditions 
(n— l)«+m>0, i—m> 0 
(3) 
must be satisfied. For «=0 these conditions are inconsistent, and the proposed employ- 
ment of r therefore unlawful. For values of i greater than 0 the conditions will be 
found to amount to this, viz. that m must lie between the limits — (n — 1) and 1. We 
shall suppose m thus conditioned, and shall consider first the case in which z> 0. 
Here then we have, interpreting D by pn-\-i in T(D), but leaving it unchanged in 
®(D)> 
fe ie =: A n nj \n n) 
r(*+i) 
rcD+i) 
1. m\ (i — m\ 
r (— ’+»>(— ) 
— s>(pn+i)8 
— m\ e ’ 
(4) 
it being seen that if we similarly interpreted D in <E>(D) the conditions relative to T 
would be satisfied throughout. 
