Elements of the Theory of Mechanics

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General Books, Mar 6, 2012 - 48 pages
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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. 1822 Excerpt: ... shew the angle at which the descending branch of the curve cuts the axis. 249. Coroll. 2. Beyond that point, the ordinate y becomes negative and increases indefinitely: not so the abscissa;r, which has a limit, that it does not pass; whence we may conclude that the descending branch of the curve has a vertical assymptote. For, when p is negative, we have P = _ p t/(l + ps) + log.--p + V( + pfi). therefore P =--p V( + p2)--log. p + 1/(12). And if p, besides being negative, be very great, we may assume 1/(1 + p) = p, and the logarithm may be neglected; because the logarithm is inconsiderable when compared with the number 'itself. When, therefore, p is negative and very great, P=--p3; and since the constant quantity C may also then be neglected, in comparison with p, we shall have--and Ph--Ch hp' Ph-Ch hp' Whence dx =----, and Jk = eonst. +---; which value hp hp does not increase indefinitely. But with respect to y, since dy =-, if--const. +-log. p; hp' h which value increases indefinitely along with p. 250. Proposition II. To construct the trajectory of a heavy projected obliquely in the air, when the angle of elevation is very small. The labour of constructing the curve by approximation becomes in this case more simple; for since the curve deviates very little from the axis, the abscissa x may be put for the arc s: whence eh'dx Integrating, so that x = 0 may give p = tan. /, there results dy, 1- ' = = And integrating afresh, so that x = 0 may give y = 0, 2 H h cos./ whence, for any abscissa x, the corresponding ordinate may be computed. 251. Coroll. Putting this value of y equal to nothing, the resulting equation, besides x = 0, will give another value of x, which will be the amplitude of the projection. If, likewise, the value of p be made equal to n...

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