244

OF THE BINOMIAL THEOF.EM.

TRACT 12.

they do not constitute the whole of that term; for if they

, , ,, n n — 1 n — 2 n 11 — 1 , .

did. then would--- a 3, d-.- ab be = Omall

’ 12 3 12

values of n, or -j—A 2 + b = 0 : but it has been just shownabove, that 1 a 2 + b = 0 ; it would therefore follow that

would be

11 — 1

- , a circumstance which can only hap-

o 2

pen when n = — 1, instead of taking place for every valueof 11 . Some other quantity must therefore be joined withthese to make up the whole of the fourth term. And thissupplemental quantity can be no other than yC, becauseall the other following quantities are evidently plupollentthan a 3 or ab. It follows therefore, that c is equipollent"with a 3 , and therefore contains the 3d power of x. And thewhole value of c is

1 —n 11 — 2

1—11

1 — 2n

~AB — -

1—2 n

- AB=_ £br B * :

1 1 - n 1 — 9.ii

-x\

2 ' 3 “ ' 1 3 3 ii “ n c 2n 3 n

And the process is the same for all the other followingterms. Thus then we have proved the law of the wholeseries, both with respect to the coefficients of its terms, andto the powers of the letter .r.

Since the above account was first written, almost 30 yearsago, other demonstrations have been given by several inge-nious and learned writers; which may be seen in some of tlie-later volumes of the Philos. Trans, and elsewhere.