Mr. Jarrett on Algebraic Notation. 69 



14. The symbol <S'„, . S'„ a„, „ will correctly represent the sum 

 of a series consisting of r terms, of which the wj"" term is the 

 series S'„a„_„; and the same notation may be extended to any 

 number of symbols of summation. 



15. Theorem. {S\ a„. .) {S^„ J„) = S--^, a„, . S'„ b„ . 



For, {S'-„, a,„) (S'„ 6„) = a, . S'„ b„ + a^ S\ 6„ + &c. + o, . S'„ b„ , 

 by actual multiplication, 



16. Theorem. If r is independent of n, and s of w, then 

 shall 



For, S'-^.S'„-a„„„ = S'^ {«„, , + «,„_„ + &c. + a„,„ + &c. + a^, J 



= S^^ «,„. I + S^„ a„„ , + &c . + i^ a„. „ + &c. + S^„, . a„, . 

 = Sl'„ . 5'',„ a„, „ . 



17. Prop. To arrange 5'"„.. S",. a„,^. j;-^-' according to powers 



of X. 



+ S%a,^,.x'-' + &C. + S\.a„_,.x'-\ 



Taking successively the 1st, 2d, and 3d, &c. terms of these different 

 series, we get 



'S"„.'S'»,a„,,.x'--' = (a,^, +a.,^,+a^^^ + &c. + o,,,,) 



+ X (02,2 + "a, 2 + «3,2 + &c. + a„,,) 

 + a;'K3 + «4,3+«6,3 + &c. + a,,3)4-&c.+;r''--.«„,„) 

 = 5'V.a„, +x.S;—a,^,^, + .^^ 5/-^a,^„,, + &c. + x"-' .a„,„ 



