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. into a«8 + «-8— » «-» + «-! — » «-! + — , w + 1, ai+|[4, a8+ai/£, Os + Oj^, ... . respectively, and moreover changes A^ into A +(26— 1)(1 + Je) + ^^-ii^ + «^i— » jIj into^i + (26— 1)^1 + 6(1 +^o)f4 + «^» — » ^.1 into A^i+ (26— 1)^.1 + eA^^ + 6(1 + -4o) — , jlji intojlg +(26 — l)A^ + eA^-\-eA^ — , iL., into ^_g + (26 — 1) ^_, + 6^.3|t£ + 6^«i— , 200 MacMahon : Third Memoir on a New Theory of Symmetric Functions. 131. Hence if any symmetric function be = ^ = '4/, the altered value will be which... by Taylor's theorem is symbolically exp (do + d^fi + d^i — ) 4> , where do =9«» ^-1 = 9a-i + «-l9a-. + «-«9a_, + » and the multiplication of operators is symbolic, and will also be ^\a, +(26-1)(1 +Aoy +eA^^ + eA,-^, ^1 +(2e-l)A +e{l +Ao)n + eA, 1- , which by Taylor's theorem is symbolically exp \e (jfQ + giii + g^i — ^ + {e—l)go\.^, where g^ = (1 + ^) 3^^ + Aidj,^ + A^d^^ + + -4-i9.i.. + ^-«9^. + » gi = (1 + -^o) 9.1, + A9.1. + ^9.1. + + ^-i9.io + -4-»9.i.x + ' g^, = (1 + ^) a^_^+ A,d^^ + j,a^, + .
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