A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 535 In this paper also the notion of chiastic homography is introduced. The characteristic feature of chiastic homography is, that every three pairs of corre- spondents a, y; ß, i; y, C, fulfil the relation The homography of impulsive and instantaneous systems is chiastic, and the relation has other physical applications. The substance of this paper has been reproduced in Chaps, xx. and xxi. of the present volume. Appell (B.)__Sur le Cylindroide. Revue de mathématiques speciales, 5th year, 1895, pp. 129, 130. It had been shown in the Theory of Screws, 1870, that the projections of any point on the generators of a cylindroid form an ellipse. Appell lias here shown conversely that if the projections of a point on the generators of a conoxdal surface lie on a plane curve, then the conoid can be no other than a cylindroid. Roubadi (O-)> PP- 181—183 of the same volume, gives some further geometrical investigations about the cylindroid. We may now enunciate a theorem still more general, that if the projections of every point on the generators of a ruled surface other than a cylinder are to form a plane curve then that curve must be an ellipse and the ruled surface must be the cylindroid (see p. 20). Ball (R. S.)— Further Development of the Relations between Impulsive Screws and Instantaneous Screws. Eleventh Memoir. Transactions of the Royal Irish Academy, Vol. xxxi., pp. 99-144 (1896). It is shown that when y is the impulsive screw and a the instantaneous screw, the kinetic energy of the mass M twisting about a with a twist velocity a is J/å3 — COS (a?/) The twist velocity acquired by a given impulse is proportional to COS (ay) Pa There is a second general relation, besides that proved in the last Memoir, between two pairs of impulsive screws y, %, and their corresponding instantaneous screws a, ß for a free rigid body. This relation is as follows —; cos (ßy) + —^Tßt\ cos (“£) = cos (ay) cos (pf) The following theorem is also proved. If two cylindroids be given there is, in general, one, and only one, possible correlation of the screws on the two surfaces, such that a rigid body could be constructed for which the screws on one cylindroid would be the impulsive screws, and their correspondents on the other cylindroid the instantaneous screws. Kotelnikof (A. P.)—Screws and Complex Numbers Address delivered 5th May 1896. Printed (in Russian) by order of the Physical Mathematical Socie y in the University of Kazan. After an introduction relating to the place of the Theory of Screws in