32 THE REV. F. H. JACKSON ON CERTAIN 
The identity (14) is a particular case of F(a 8 y8e1) and may be thrown into the form 
(x), = (2a), + SS (2a — 27) (iv P+ 1), . (16) 
in which 
(oe 
ewe-lae-2...e-n+1 
A more general form is 
1) : , a 
[zh=[22h+ Se + er ee Qr|,-f 20 —r+1],. . an 
! Ae ao 
[7h EES iiss ac [ce=n+1] lS -1 
These and other interesting theorems due to change of the sequence (p,p,.... ) 
must be left to another paper. 
We now proceed to obtain the coefficients B, B,, ete. 
Suppose that II,(a+(m)) is capable of expansion in the form 
I(a+(m))\= Boe Be iy Oe rc: 
Pi Pe! Ds! 
(m), denoting (p, +=. . » +P) at seeder eer Oet ais. tm) 
ps ” Pr PoPs °° * * Ps 
In (18) substitute (m)=0, then we have 
By = 1,(2) 
Similarly, if we substitute (m)—(1)=0 we obtain 
B, +B p= He+ ) 
Continue the process of substitution by putting successively 
(m)-(2)= 
(m) —(3)=0 
We obtain the following set of equations for determining the coefiicients B, B,, ete. 
II (a) = By 
(a+ (1))=B, +28, 
Py 
II(a + (2))=B, 4 Pit Pe Bi feed +PoPop, ‘ b ‘ 4s 
Pi P\P2 
(a+ (m))=B, +B, + ap, + by eyo 4 (np 
! ! 
Py Po: Pn! 
