Mathematical Questions and Solutions Volume 23 (Paperback)

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. 1875 Excerpt: ...precisely the same manner as above, it may be shewn that if two material particles, constrained to move without friction in acircle, under the action of a central force, repulsive or attractive, emanating from a point in the plane of the circle and varying directly as the distance, have-a common absolute velocity of description under the action of the force; their chord of connexion will envelope another circle, coaxal with the first and with the circle of evanescence of their common absolute velocity under the action of the force; see answer to Question 4576, on p. 38 of this volume of the Reprint. III. Solution by R. W. Gobnese. By vis viva, the velocity at any distance r from the centre of forces is given by -2 = p (a-r2), oc distance from radical axis of this circle and the circle of evanescence of velocity of description. The remainder of the solution is that to my Question 4576, which I erroneously altered from a question in Tait and Steele's Dynamies, to which an unintelligible solution was appended. It is very interesting as introducing a property of elliptic integrals in its analytical solution; viz., that if--= constant, J. (1--e cos ey then 1 = a cos (P + a) + b cosi(/3--o), where a and b are constants connected by the relation 2ab = e(l-a2-42). I. Solution by Professor Townsend. It was shown by Professor Casey, in his able memoir " On Bicircular Quartics" (Transactions of the Royal Irish Academy, 1869), that every circle Z, intersecting orthogonally a focal circle E, and having its centre X on the corresponding focal conic F, of a bicircular quartic U, has double contact with the quartic, at the two points P and Q which are inverse at once to the circle E and to the right line T touching the conic F at the point X. And it wa...