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By H. F. Baker

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4xy + 47-" + 4*. 1 ind AM A,, wherein, A= ; and so If it is understood that A then, as x, y, ' z are supposed subject to the relation symmetrical, on. now D= . -r it is Au A,, 9 5- b 9 9 + 77^- + o 9 n Q= 5-, 9^ 9 3 j. 5- 9a; 9 9 + SV + T5s 9 9z . , can be easily verified that ^ = Q, For, of these, the first Pr=-, PT=Q?. 0) = A n (2^ + equation, multiplying by ijf, is 12xy - 4*). B - X, A,, (2X 4 y + Sxy - 4*) it ART. 12] from total single integrals. 43 the determinant gives = A + i\, A-B + 2^A23 - 2yA ] + 2y [JX, A,, - (4z + X.

Rip r . Riemann applications of theta functions to the surface we shall suppose the matrices a and h (and r) to be those arising in connexion with the algebraic integrals (p. 13). We 10. return now to the Riemann surface, and consider upon it the function of (x) expressed by 8 (," '"-e,, n/' m -e. ^), (A 3 ), (A t ), so rendering (x). ' 1 m v^ U vf- e, t , the right and left' sides of (A 3 ) a similar statement r m holds for (A t ). The function (w e) is an integral function of t'/ and therefore on the Riemann surface, capable, that is, of v/< '", analytical taken by vf> e l at ; ' 1 , representation about any place of the surface by a series of integral powers of the parameter for that place, there being no negative powers hence, the ; number of places (x) where the function vanishes to the first order, if any, or the sum of the orders with which it vanishes, is given by taking the integral J_ fd 2-rriJ round the closed curves (AjA^A^A^ contour the two sides of (A t ) 1 ), (A 3 A t Ai~ give no contribution ; l A ~ t 1 )- Of the former the two sides of (A,) give theta functions.

8 Ms (z) u*~" = T/. - ((z, x) - (z, a)] ~ . 7. Before passing on it seems necessary to make a few introductory remarks relative to a notation which will be found of great use in the sequel. m A rows and n columns, rectangular arrangement of mn elements, in be added to, or subtracted from, another such array or matrix, of the same number of rows and columns, the meaning being that the (r, s)th may element of the resulting array, namely the element in the r-th row and s-th column, is the sum, or difference, in the respective cases, of the corresponding elements a Ts Or the matrix of type (m, n), that is, of rows and n columns, with general element a r

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