90 Mr. Jarrett on Algebraic Notation. 



Hence, if the supposed law holds for r eliminations, it will hold 

 for r + 1 : but it does hold for 4 ; therefore it will hold for 5, 6, &c. 

 and hence it is generally true. 



And thus finally : 



VL ,^^,,!m ia-}Y , g (a- - 1)'+'' 

 n \t_ \t + r 



t 



'■+1 n ) 



Ex. 5. Taylor's Theorem : E^.u = S,„D^"'-' rf;-' : u. 



For, whatever the form of u may be, it can be reduced into a series 

 of monomials, into none of which x enters as an exponent. We may 

 therefore assume 



u = S';„a„x' (1); 



£.,.«« = 5''-„o„.:p+Dxr. 



= Sr,a„.l.K .,— !— ..r«„-"-'I)^"'-' 



, |m— 1 



m — 1 I 



by the binomial theorem 



» 1 



= S',S„,a„.\a„ .,— ^— .x\""'*\D3!\"'-' (Art. 13.) 



= s!. Sr„ a„ . [a^. -^ . .r"n-"- ' . S¥l"'- ' (Art. 15.) 



= S,„ i^^ . S^,a„ .[a^x«.-"-' (Art. 13.) 



