The Messenger of Mathematics Volume 3

The Messenger of Mathematics Volume 3

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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. 1866 edition. Excerpt: ...viz. when v = 0, for then 7, = 0, so that Gj = (f'f, and is always positive; so that when f vanishes, for this particular system of 7's, the three last couples in the pair of progressions before and after such transit will be respectively and thus, contrary to what usually happens, there will, on the above hypothesis, be an extra gain of two double permanences. 14. Theorem No. 2 may also be stated as follows: If c0 + e1x + ctx +...+ cnxn =f(x), and v be any real quantity not intermediate between 0 and--n, and if c, 1.2 1.2.3 Co v(v+l)C y(y+l)(v + 3)Cs', say aa a a2, a3, ... be taken as the simple elements of/(x), and say A, A A A-, A as the quadratic elements of the same; and if we understand by the (cA) association the paired series A, A and if p-P(X) signifies the number of double permanences in the cA) association, corresponding tojf-t-X), then PP()-pP(X) = () + 2k, where h is zero or some positive integer, fi being supposed greater than X. It is easily seen from this that also vP()-vP(fi) = (p, X) + 2k'; and this gives us a second general theorem, and these two theorems will give different limits of (fi, X) unless k = k', i.e. unless P(p) = P(), for P(ji)-P(X) = 2 k-k'). We may observe that there is nothing corresponding to this in Fourier's theorem, since p (fi)--j (X) and v(X)--v (fi) give the same limits. 15. The existence of a theorem No. 3, including No. 2, and containing two arbitrary parameters, becomes apparent from the consideration that f(x) = Q will have the same finite roots as F(x) = 0, where F(x) = sx + x'f(x) + v, e, i) being two infinitesimals: this brings in two parameters, which, so far as this particular method of demonstration of their existence applies, would seem to more

Product details

  • Paperback | 54 pages
  • 189 x 246 x 3mm | 113g
  • United States
  • English
  • black & white illustrations
  • 1236979796
  • 9781236979797