The Messenger of Mathematics Volume 19

The Messenger of Mathematics Volume 19

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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. 1890 edition. Excerpt: ...for v from this equation in the expressions for /c'--re, 'r'--'r, we can express the potential energy in terms of u, w, and a, and the virtual moments of the acceleration can be expressed in terms of the same quantities. The use of Lagrange's method then leads to the general equations of motion. We shall apply this to the case of a circular helix. Calling V the potential energy of the wire, we have, for the variation of this, The part of the virtual moments due to acceleration is mf(' i3u] 530 + ibdw) ds. and, after substituting from the last equation in the first two eliminate A, B, G, we get the determinantal equation 3 7 I 2 2 1 from which we can see that there are two kinds of waves of given length which travel with different velocities. When the helix is of small pitch the natures of the two are very different. For, if we neglect 'r, the equations separate into the two sets Consequently there will be one wave which will appear as a progressive contraction of the radius of the helix, and whose velocity will be given by L Ir' 'n' I 11 m /c n and another wave which will appear as a crowding of the successive turns of the helix, and whose velocity will be Vibrations of a circular ring. The equations we have written down for the case 'r=0 are clearly the equations of vibration of a circular ring. In order to adapt the above formulae to this case it is only necessary to notice that here n must be of the form r/e where r is an integer, and ax= 1 where a. is the radius. Thus, for vibrations in the plane of the ring, we have and for vibrations normal to the plane of the ring, and involving torsion, The first of these results was given by Hoppe in Crelle, Bd. ea, 1871. ON THE EXHAUSTION OF NEUMANN' more

Product details

  • Paperback | 34 pages
  • 189 x 246 x 2mm | 82g
  • United States
  • English
  • black & white illustrations
  • 1236738349
  • 9781236738349