This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 Excerpt: ...coefficients being positive integers, and it also will be a general magic square. An ordinary magic square is a particular case of a general magic square, but the particular property of an ordinary magic square that it must involve different integers in the compartments prevents the formation of new squares by addition ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 Excerpt: ...coefficients being positive integers, and it also will be a general magic square. An ordinary magic square is a particular case of a general magic square, but the particular property of an ordinary magic square that it must involve different integers in the compartments prevents the formation of new squares by addition, so that within the group of such squares there is no syzygetic theory. 158 THE SYZYGETIC THEORY SECT. VIII 405. We proceed to investigate the syzygetic theory of general magic squares of the Third Order. Such squares do not exist in the case of the Second Order except in the trivial case a a a a but there are squares in which the row and column but not the diagonal properties are in evidence. For let such a square be a1 -2 This must clearly have the form a. a. a, a, where if oo be some integer a, +a = a5) and we find S'iZfaf = n.--., wherein we recall that the symbol fl denotes that in the ascending expansion of the function to which it is prefixed we are to retain only those terms which are free from the auxiliary X. The expression clearly has the value (1--XiXil-X2X2)' the denominator factors denoting the fundamental solutions Thus we have the fundamental squares I t) 0 1 0 1 10 and the most general square of the type 1 1 0 10 1 'a, a, ai n l + 0,2 I n-0 1 1 0 I Oj a, a trivial result. (the exponent of xw being unity shewing that the number associated with the rows and columns is unity) connected by the simple syzygy 100 010 001 010 +001 +100 001 100 010 100 010 001 = 001 +100 +010 010 001 100 each side being equal to 1 1 1 1 L J. Xj Xj) Xq X Xg tt-g Xj Xft Xa VU-q a 1 1 1 This is the complete syzygetic theory of these squares. CH. VII PROPERTIES OF BOTH DIAGONALS IN EVIDENCE and writing these AB=CD = E we have the s...
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