Mr. Jarrett on Algebraic Notation. 71 



We shall thus find 



S,(l> {m,r).x--\S^^(j>^ (r, r,) . a;^-^ 5?, . . . -C,"^, (»-,-i, O •^"•-' 

 = Sl. S,;. . . S,;.-' .(p{m,r-r^ + l).ct>, {r-r, + l,r,-r^+l) . .. 



0S-1 (rs_2-r,_, + l, r,_, -r^+l) .<^s (r,_,- ?■, + 1, rj .a;'-' 

 = S,, a"-'. 5,,'-0 (;k, r-r, + l) . 5,;.^i (r-ri + 1, r, -r,+ l) . S,; 



(Art. 13.) 



21. The form in which this last result appears, naturally 

 suggests a notation by which, indices being applied to brackets, 

 many very complicated expressions may be reduced to a very 



simple form. Let then {a„ be used as an abbreviation of the 



expression {ai. [a^. [a^- {...{a„, where a™ is any function of w, 



1 2 3 4 n 



and the result of the last Article will become 



S,.(p{m,r).x^-\\Sr^.(i),{r,^i,r,) XT'" = 5',a;'-'. 5,;0 (»i,r-ri + 1) . 



{ S^^,-x . <p,_, (r,_2 - r,_, + 1, »■,_, - r,+ 1) . 0s (r,_, -r, + 1, r,). 



s — 1 



22. Prop. To arrange according to indices subscript of b, 

 the series 



S„S\a„,„.b„, S^S^.a„,„.b^„,,, and S,„S'\a^,„.b,„. 



CO CO CO <» 



S„S\a„,„.b„ = bi.S„,a„,t + bo.S,na„+i,2 + tg-'S'ma^+s.s + &<=• 



CO 00 _ 



