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By Manin Yu.I.

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O), replacing L by L and (Ls > (it is recalled that [L°s, Z]=0) . We find on setting « Ar _ 1 =0, « w = l: AT-2 r JV *-0 La-0 P-1 This implies the two identities AT-2 J" N ff-l ft-0 La-0 p-l La-0 9-JV 1 J J (the second follows from the fact that the order of the commutator on the right is < — 1 ) The commutator in the first identity is equal to [(! Wv^{s)-{-^)v_^S)u^]^-r. 10: B-l We wish to represent the formula obtained in the form (cf. Sec. 8, Chap. I with appropriate B. kl . In order to compare (7) and ( 9 ) , we make i n (7) and (8) the change of i n d i c e s : a — p — ~{ = k, f4-8 = y, P —1—8 = / .

R e l a t i o n s (3) can be c o n s i d e r e d a system of e q u a t i o n s f o r which has t r i a n g u l a r form and can t h e r e f o r e be s o l v e d by i n d u c t i o n . ,v_N+l(s—\), The c l o s e d formulas (4) a r e most e a s i l y o b t a i n e d by w r i t i n g (3) i n o p e r a t o r form tofe&^sQ + doiyv^is-l), where T is the operator for increasing the index by one: T (v^) (5) =v(*l+1 . or-i)<-^i> and the fact that (6) is the inversion of (5) is obtained by induction on /.

U n+2 =l. In place of the columns and Y = (Y0, Y\, • • •)' with formal variables as coordinates; the action of d* on them is interpreted as the conversion of Xt, Yk i i ( the formal variables d Xi = X i^, d'Yk=Y l>), — 6u into independently of one another and of a<" . Each element of- the left and right sides of (14) will then be a formal infinite sum of monomials in a, X, Y and their derivatives which is trilinear in u, X, Y ; we simply verify that the coefficients of each such monomial at corresponding places on the left and right coincide.

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