[101 
notes on lagrange’s theorem. 
5 
' s )(^ _1 Sf) 
-*- 1 ) 
r 
the factor Sf, the 
S—0—1 
) 
)+S' 
r 
—s—a—e—l^ 
p+«) 
luction j times, the 
* + /3... +jy 
-p—8—0—j—a—p • • • ^ J ; 
equal to it, and 2 
j' together (so that 
terms) — j, and the 
.. contain each of 
terms) —j — 0, then 
a + ß ... +j — lp ■>’}. 
101] 
If 0 = 0, the general term reduces itself to 
1 1 
S a+1 f. S ß+1 f...f~K 
fc(jfe-a-l)(jfe-a-£-2)... [a]*[/3f... 
(s + cc + /3 ... +/) <f>(cc + /3 ... +/) [s + a + p ... + j - lp --1 }; 
whence finally, if (f>p = , the general term of 
S 
becomes 
(P + s ) [p] p № 
1 
K^*) 
k (k - a — 1) (k — a — ß — 2) ... * [a]“ [ߥ... 
8“«/ 52 {(-y-r [ s + a + ß ... + j- 
and it is readily shown that the sum contained in this formula vanishes, which proves 
the equation in question. 
IV. 
The demonstration of the equation (11) is much simpler. We have 
S*- (/-’8/) = 2 S-> (/-«-8/). (8 ! -"/‘)|, 
that is, 
where n extends from n = 1 to n = t. Similarly 
[t-n- lp“ 1 
= n S i [p-ip-'- W*> (8‘-“-V-“)}. 
t — n ^ \[t — n —p — l] 5-1 
gi—11—p f t ~ n 
&c. 
p 
[q- ip“ 1 
(S q fp) (§*-«-i>-5y *-»-*>) t, 
Hence, substituting successively, and putting t — n—p — q = r, &c., 
&f s = [ w p [yjp (^+ r ) [q-1]*- 1 (W"*) ^ ? / 9+r )} » 
&c.; and the last of these corresponding to a zero value of the last of the quantities 
n, p, q... is evidently the required equation (11). 
V. 
The formula (18) in my paper on Lagrange’s theorem (before referred to) is incorrect. 
I propose at present, after giving the proper form of the formula in question, to 
develope the result of the substitution indicated at the conclusion of the paper. It 
will be convenient to call to mind the general theorem, that when any number