Researches in the calculus of variations, principally on the theory of discontinuous solutions; an essay to which the Adams prize was awarded in the University of Cambridge in 1871

by Isaac Todhunter

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. 1871 edition. ... to a fixed point B, with the condition that the path is not to pass inside the given circular arc AB, which does not exceed a quadrant, B being the lowest point of the circle. A.fter the discussion of the problem enunciated in Art. 149 it will be sufficient to state the results of the present problem. The required curve consists of the arc of a cycloid having its cusp at A and touching the circle at some point R, and the portion BB of the circle. As the cycloid is outside the circle at B twice TR is greater than OB, and therefore TB is greater than OT. Hence for the point B we have y greater than b, and thus ¥---Yydx is necessarily positive for the part BB of the path, J 2ry since By is positive. If AB is an arc of 90 there is no cycloidal portion, and the required curve consists entirely of the circular quadrant. 156. The problems of Arts. 149 and 155 include as special cases a problem proposed by the late Dr Whewell in the Smith Prize Papers for 1846. His enunciation was as Prove that an arc of a circle from the lowest point which does not exceed 60 is a curve of quicker descent than any other curve which can be drawn within the same arc; and that the arc of 90 is a curve of quicker descent than any other curve which can be drawn without the same arc. It would be interesting to know how the distinguished philosopher treated the problem himself. It will be seen that in our investigation we obtain for the circular arc J 2ry this expression shews at once that the results enunciated are true with respect to such curves as differ infinitesimally from the given circular arc; but it would still remain to shew that the results are true for curves which differ to a finite extent from the given circular arc. The investigation which we have...

58 pages, Paperback

First published April 4, 2010

About the author

English mathematician known for his writings on the history of mathematics.