Two Proofs of the BmOMIAL THEOREM, hj the REV. 

 SAMUEL VINCEy A.M.F.R.S. Plumian Professor of 

 Astronomy and experimental Philosophy, in the University of 

 Cambridge 



Eead May, 1810 



When n is a whole positive number, it is proved bjr 

 common algebra, that 



(Ifo;) " = 1 + w* + n. -==ii?^ + - - -n. "-=i "idtlx'+ &c. 



Now if this be not true when w is a fraction, let the general 

 co-efficient be C + w. ~- "-^^ .r'. Then the quan- 

 tity C must vanish when w=l,2, oo. Now as C is 



expressed in terms of w and given co-efficients, it must alv^ays 

 be of the same form whatever n is, and, as it must vanish 

 Avhen w— I, 2, oo, it must be represented by ^^ ^ n ix n 2 



X n — 00—^^(7/ — an~\+ &c.) where r is infinite j this 



therefore must be the value of C. But when n is a fraction, 

 this value of C becomes infinite, which it cannot be, and as 

 no other value of C can enter in addition, but this, the gene- 



