12 Colorado College Studies. 



and having as values of n the numbers s and /, be multiplied 

 together, the product will be a series in the form 



71 n n — 1 



1 + -a- + -. x"- +...., 



1 1 2 



having for the value of n the number s + /. 



The actual multiplication, through the first few terms, may 



be written out as follows: 



S V s — 1 S 6 — 1 6—2 



1 + - a; + -. .1-^ + -. . x^ +. . . . 



1 12 12 3 



t t t—1 t t—l t—2 



1 H a- H . X' H . . a.^ +.... 



1 12 12 3 



X s s-l s 6—1 6-2 



1 r 2 ' 1 ' 2 ' 3 



t s t s s — 1 t 



— X -\ . — X- -j . . — .T^ +. 



1 11 12 1 



t t—l 6 t t—l 



— . x^ H . — . .r^ +. 



12 112 



t t-l t—2 

 1 " 2 ' 3 ' 



Here it is obvious that that which must be shown, in order 



to prove the theorem, is that the coefficients of the successive 



powers of x, found by adding the above columns, are identical 



with the coefficients of corresponding powers in the series 



.<? + f s-\-t .s-\-t—l 



I H X . x^ +. . . . 



1 1 2 



It is apparent that 1 is identical with 1, and 



s t s -r t 



\ with ; 



1 1 1 



and it may be proved by trial, (with successively greater diffi- 

 culty in the necessary reductions) that 



6 6 — 1 6 t t t—l 



12 11 12 



s + i s + /—I 

 is identical with . , etc., etc., as far as we please to 



