70 Mr. Jarrett on Algebraic Notation. 



18. Theorem. S,„S„a„,^„ = S„.5'"'„.a„_„+i,„. 



For, Cs„a„„„ contains every term in which the sum of the in- 

 dices subscript amount to any positive integer from 2 to oo, that is, in 

 which it amounts to 7?i + 1 from m = l to ni= cc. 



Also S"'„a„_„+,,„ contains everi/ term in which the sum of the 

 indices subscript amounts to m+ 1 . 



CO CO 



.-. S„S^„a„_„+i,n contains exactly all the terms of S„5„a„,„. 



CO 00 00 



19. Theorem. <S'„,S„5,a„,,„,, = a. &'"„. 6%.a„,_„+ ^ „_,+ ,,,. 

 For ^^a„,„,, = £s\a,„^, , + i,,, by the last Article; 



Oo 00 00 J" ^ ^ 



00 



bv the same Article. 



In precisely the same manner we may show that 



CD CO OC CO 



S„ .S„^.S„.^... S„^ . (p (m, m^,m^, &c. m.) 

 = §!.S„,'".S„™i...S„;".-'.(|) (?n-?n, + l, m,-m,+ l, &c. »i._, -?«,+ !, «?.)• 



20. Prop. To arrange, according to powers of x, the series 



X CO CO CO 



a; . <p {m, r) . a;- ' . 5',., . 0, (r, r,) . x'.-' . 5,,^ . . . S^^ . <^, (r,_, , n) • x'.- ' . 



For this purpose we must first remove every quantity to the right of all 

 the symbols of summation (Art. 13.); in the next place, we must write 

 y. — r^ + l for r, r, - r, + 1 for r, , and so on ; observing to leave r, un- 

 altered ; and lastly, place over the symbols of summation, beginning at 

 the second, the quantities r, i\, Vo, &c. (Art. ig.). 



