30 THE REV. F. H. JACKSON ON CERTAIN 
1428 CatlBBtla, || ywatle.... se a tees B+™-l mw, Bp 
‘i Lge 12.2?-y-y + 3 phase 82) B Cee ae myyt3....y-lt+n E (3) 
2, 12 
I pea a Maen MEISE a eds FU oe, . a 
eae eee +9" if © 
aB aatl? BB+)? 4 
l tet (ok ee a . : D (5) 
It is well known that if II,(a+m) denote the product of s factors 
(a, +m)(a,+m)(ag+m) ... » (a,+m) m integral 
an identity of the following kind can be established, 
I(a+m) = B+ Byn+ Bym(m—-1)+ ... . +Byn(m-1)... (m-st1) 
where B, B, . . . . B, are constants, that is, are independent of m. 
We proceed to establish a generalisation of this, on which all subsequent work will 
depend. 
Let I1,(a2+(m)) denote the product of s factors 
(a, +p, + pot »- +. +Pm(agtpyt+Pot .-- +Pm) eee s (a,+p,+Po+ . ++ +Pm) 
then I,(a+(m)) is ae equal to 
B,+B, +B, (m)((m Sh. eae # p mim) C0) Saar ((m) — (s—1)) (6) 
a Dep ges es 
which may be more conveniently written 
pies UL Prien 20 vavall aed F +Bi™s SG 
Py "PiP2 Ps! 
The coefficients B, B,, etc. are independent of m and are given by 
->(- V@rarapete-n) ’ £ : (8) 
Before proceeding to obtain these coefficients it will be well to explain the notation 
clearly. 
(2 —1) is not the same as (m)—(1), for (m—1) denotes p,+pytp3+ .- - +Pm-15 
while (m) —(1) denotes p»+p3+p,+ . ~~. +Dm | 
(m),=(py+Po+ --. ee att Piet wee en) <a (Dr... Ee 
a a fey a ; 
(3)! = (nr, +P. +P3)(Pe SAL . = m + Pot Pi)(Po soa 
(2)! = (p, + P2)Po {2}! = (pg + Po)Po 
(ya: uy le==epe : : : ~ (10) 
In the expression for B,° four elements p, , p., Pz, p, appear and 
(4)! = (p+ Po+ Pz t+P4)(Po+P3+Ps)(P3+Ps)Pa {4}! = (py +P34+Po+P1)(P3 + Po +P1)(Po+ Py)P1 
(3)! = (py + P.+ Ps)(Po+ Ps)Ps {3}! = (p+ 3+ P2)(P3 + P2)(Po) 
(2)! = (p, + Pe)Pe {2}! = (py +Ds)Ps 
(1)! = p, {l}i=p, . ; : : > aay 
