20 THE REV. F. H. JACKSON ON 
So that from the whole series we obtain 
{[m] - [x -v]}{[mq] -[-2 -v— 1} Aye — 
soul [m, — 1]A,a"™—4 
+ {[m,]-— [2 -v]}{[m.] -[- 2 -—v—-1]} Ae! - P oe [m.][m_—-1JA,er™-7 , (48) 
Choose mM, = My - 
bo bo 
M41 = M, + 27 
~ 
Also choose [m,] [m,—1]=0 so as to get rid of the term 
= m,] [m, — LJAx™—?) 
Then, in order that the expression may be of the form f(x)—/(#”), the coefficients 
A, A, Ag... . ete. must be chosen so as to satisfy the relation 
1 / 
pe [mi - Aw = {[m,]-[n—-v]}{[m,]-[-2-v-1]}A, 
since [m,][m,-1] = 0 either m = 0 
or mM, = | 
for the value m = 0 
Myo, = Ir 
and 
Ao A. ei ‘2 ulm =v —2r+2][m+v+2r—1]y 
rae. 7 ares [27] [2r =] 
NW” [n-v](n+v+1] p 
y = Afi = ee Palast ee (44) 
Sea [2] [1] 
ta) = (eo vl [mtv +1] aye f, -_ [may 2) [etr+3] \ tae 
Af (2) p 2n+4 | pie : (1] [2] te cee as f ( ) 
the general term of the series y being 
(aR pron [m-v—-Ir+2]..... {u—v]-[m+v+l]l]..... [w+v+2r-1),0n (46) 
Vig v) [2r]! 
When 7 is a positive integer and n—yv is even, the series (44) is C. P.(xA) as is 
evident if we consider that there are = terms in the series, and so by substituting 
n—v t 
no v for 7 we reverse the series. The general term becomes 
A pF) (n—ntresary aro 20 + 2)[27 +4]. . [nm - v]- [m+v+ 1}[x +ut+ Bion . . [2n- Qr- 1] 
I] [2]... - [w=v= 27] 
Ss [w—v][n-—v—- 2] eA cee [EA atin eee 
P eS Ses ee io 
% [2n — 27]! 5 
[m -- 7°]! [n —v — 27}! [7]! (2),,(2),- 
gly —2r] (47) 
[w—v—2r] 
