Mathematical Questions and Solutions Volume 19
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Mathematical Questions and Solutions Volume 19

Mathematical Questions and Solutions Volume 19


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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. 1873 Excerpt: ...C is' x2+mxy + cny2--ax--cny = 0 (1); hence, when this passes through the centroid, we have aem--2?n = 2o2 (2). Now the coordinates x, y of the centre of (1) are = n(cm-2a) = 2cV b (2) (3) _ ma--2cn _ 2a2.... --'--in c(m---in)' hence, eliminating m and n from (2), (3), (4), we get (ac--2ay--2cr)'-= iacxy, which is the equation of an ellipse touching the triangle at the point of bisection of each side; it is therefore the maximum inscribed ellipse, and its area is--of the triangle. 3- 3876. (Proposed by J. J. Sylvester, F.R.S.)--Show that there are 3184 positions in a cubic curve such that at each of them curves of the 50th order may be drawn having 90-point contact with the cubic. Solution by Professor Clifford. The gross number of points where a curve of order n can have 3x-point contact with a cubic is 9h2. The problem is in fact the same as that of the division of the periods of an elliptic function by 3-, and as there are two periods, there are 9-2 solutions. (clebsch, Anwendung der Abelschen Functionen in der Gcometrie, Crelle, baii.) But in the case when n is a composite number, all the curves whose order is a divisor of -, and which have complete contact with the cubic, are included in the result. Thus each inflexional tangent, taken n times over, constitutes a curve of order - having 3--point contact with the cubic; and the nine inflexional tangents are thus always included in the 9-2 solutions. To obtain the number of proper solutions, then, we must subtract all these improper ones. When n = 30, the result is Here it is to be observed that the curves of order 5 are twice subtracted, with the curves of order 15 and 10; so that they have to be added in again. The same remark applies to the orders 3 and 2. ...


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Product Details
  • ISBN-13: 9781236271334
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 34
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1236271335
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 82 gr


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