Mathematical Essays

Mathematical Essays

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1819 edition. Excerpt: ...shall now conclude with a short examination of some new' properties of the function L:, 'which arise from this mode of integration. neral expression for the quadratic factor of 1' +.z,"" is 1--2.2' Cos. 1 +.1," by using a notation which we have employed in two other instances, we ' 29-1 m might assume (1--2.1" Cos. 11-l+ to represent the sum of all the terms produced by the successive substitution of 1, 2, 3, &c. for 0. But as we shall have to consider the variations _of this expression, both according to9 and to m, we shall endeavour to generalize the notation, so as to render it applicable to all cases that may occur, whatever may be the number of variables they may contain, or_ however far the sums of the variations may be extended. In this view, we observe that there are two things principally to be attended to. 1st, to denote clearly all the quantities that are subject to variation, and the order iniwhich the variations are to be made; and Qdly, to indicate the number of times to which each of the variations is to be continued. Now, to combine these two objects, we shall still retain the same symbols, only with these additions, that at the upper comer of the bracket to the right hand, we shall write all the variable quantities successively, in the order in which they are to be varied; and in the same place of the opposite bracket we shall set down the letters which indicate the number of the terms. Thus, if we suppose Z133; to be a function containing.z' and y, then " Z.r, _1/ F Will represent the sum of all the terms arising from n successive variations of.z'; m Zr, _1/ y will denote the sum of m variations of y; and "n'" Z.r, y FF'? will express the sum of all more

Product details

  • Paperback | 42 pages
  • 189 x 246 x 2mm | 95g
  • United States
  • English
  • black & white illustrations
  • 123677910X
  • 9781236779106