756 PEOFESSOE BOOLE OFT THE COMPAEISOX OF TEAXSCEXHENTS, TTITH 
property of 0 with respect to the term within the brackets, we see that (6.) is resolvable 
ultimately into a series of terms falling under the two typical forms 
All terms of the first form obviously vanish, since they can have no negative indices. 
It only remains, therefore, to consider terms of the second form. 
First, let i be equal to or greater than n. Then putting x—a—z, and x=.a-\-z, the 
1 • ■ 
coefficient of - in the development of the function — 
, to which 
[x-a) 
n is reduced. 
in ascending powers of z will be 
1.2..(n— 1) 
("•) 
1 37 * 
and the coefficient of - in the development of the function — in descending powers 
of X will be 
n(n+ l)..i 
1.2..{i-n+l) 
( 8 .) 
These expressions are equal, as may be shown by equating them and clearing of factions. 
Hence, in this case, 
Secondly, if ^=w~l, each of the above expressions (7.) and (8.) reducing to unity, the 
equation (9.) is still true. 
Lastly, if i is less than n — 1, neither will any term containing ^ present itself in the 
ascending development, nor any term containing - in the descending development, so 
that the equation (9.) remains true in this case also. 
Wherefore the theorem is proved generally. See Note A. 
General Theorem of Transformation. 
12. The foregoing properties of the symbol 0 have an important bearing upoli the 
general theorem for the transformation of integrals under the sign 2, to the demonstra- 
tion of which we shall now proceed. 
Theorem.— - i/^E=0 he an equation connecting the vanahle x ivith another set of variables 
a,, ag, . . a,., the function E being rational and entire with respect to x, and if F be any func- 
tion of-K and q/a,, Ua, .. a,, which is rational with respect to x, then., provided that F does 
not become infinite when E=0, we have 
2F£?^=©[F]^, 
where ^ indicates complete differentiation with respect to the variables a„ a.^, . . a,., and the 
