«HHM 
[101 
4 NOTES ON LAGRANGE S THEOREM, 
this is immediately transformed into 
l^ s />if) 
\ a !4> (p +1) (p + s+1) (SP./’««/) (S»/-p—*-■) 
- & (p + « + 0) (8pfr+*) (8? ./-P-s-e-ig/)}, 
in which last expression p + q = (p — 1). Of this, after separating the factor 8/, the 
general term is 
1 —8 a+1 f si - 
k[ a]“ / (b-a] i,_a [q] q 
(f>(p + l)(p + s + l) (8v- a fp +s ) (8?/-p-*- 0 ->) 
[pYlq-*}*-* <t ‘ p (p+ s + e) (S,fr+,) - 
equivalent to 
\ pp a ‘ +,/ - 8 bFM 5 W> (p+« + 0 (p+«+* +1) W w °) 
■s—a—e—l 
-<f)p(p+s +6) (8 p /p +s ) (Bvf-P-*- 9 - 1 )}, 
in which last expression p+q=k-a— 1. By repeating the reduction j times, the 
general term becomes 
1 1 
k (k — a —l)(k — a — ß — 2) ... [a] a [yÖ] ß ... 
1 
8 a+1 f. 8 ß+1 f... 
x S- 
2 {(—y~f cf>(p + a + ß ... +/) [p + s + a + ß ... +j’Y 
M № 
x[p + s + 0 + a + ß ...+j- IF* (8PfP +s+a+ ^-) 
where the sums a + /3 ... contain j terms, f being less than j or equal to it, and 2 
extends to all combinations of the quantities cl, /3... taken j and j' together (so that 
the summation contains terms). Also p + q = k — cl — /3 ... (j terms) — j, and the 
products k{k — a —\ ){k — cl — ¡3 — 2) ... and [a] a [/S]' 8 , ... 8 a+1 f. 8^ +1 f... contain each of 
them j terms. Suppose the reduction continued until k — cl — /3 ... (j terms) — j = 0, then 
the only values of p, q are p = 0, q = 0; and the general term of 
becomes 
1 
8 a+1 f. 8? +l f 
1 
k (k — ol — 1) (k — cl — ß — 2)... [a] a [#]0. 
x 2 {(—y~> cf)(a + ß ... +/) [s + a+ ß ... +j'Y [s+0+OL + ß...+j - 1]^'}. 
101] 
If 0 
whence fi 
becomes 
k(k - cl — 
and it is 
the equal 
The 
that is, 
where n 
Hen 
&c. ; and 
n, p, q.. 
The 
I propos 
develope 
will be