Researches in the Calculus of Variations; Principally on the Theory of Discontinuos Solutions an Essay to Which the Adams Prize Was Awarded in the University of Cambridge in 1871

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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. 1871 edition. Excerpt: ...two different curves which touch at the common point. And the common point R must be such that here y is less than b; and in fact this is ensured by the nature of the cycloid. For as the cycloid is to touch the circle internally the radius of curvature of the cycloid must be less than the radius of the circle; that is twice TR must be less than OR; therefore TR is less than TO, which makes y less than b for the point R. It is obvious that 2 sin a is greater or less than unity according as the circular arc BA is less or greater than an arc of 60. 154. It may be useful to shew that it is possible to draw such a cycloid as we have supposed when 2 sin a is less than unity. Suppose we take r--b for the diameter of the generating circle of the cycloid, and put the vertex of the cycloid at B; this cycloid will fall without the circle at B, because 2r--2b, which is the radius of curvature of the cycloid at the vertex, is by supposition greater than r. On the other hand, if the diameter of the generating circle of the cycloid is indefinitely great, and the cycloid be made to pass through B, it will obviously fall entirely within the circle. Starting from the last case diminish the diameter of the generating circle of the cycloid continuously, making the cycloid always pass through B. Then we must arrive at the case in which the cycloid just touches the circle before cutting it. The point of contact will not be at A, for the tangent to the cycloid would then be vertical, while the tangent to the circle would not be vertical. The point of contact will not be at B, for there the tangent to the circle is horizontal, while the tangent to the cycloid would not be horizontal. Hence the contact must take place at some intermediate point, as we require. 155....