# A Treatise of Mechanics, Tr., and Elucidated with Notes, by H.H. Harte

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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. 1842 edition. Excerpt: ... = x, 1--cos a = /3; and -lso, at same time, d0= dx V2x-xi The formula (2) will become 9 V\$x--aVl--x and, in order to deduce from these the duration of t, a semioscillation, we must integrate from x--/3, (which answers to 0 = a, ) to x = 0, (which answers to 6 = 0.)(/) Now, developing by the formula for expanding a binomial, we obtain in this series, the general term is In--1 /xn L6 2ra 27' 1.3.5 2.4. and it always converges, because x is constantly less than 2. If, therefore, the order of integration be reversed, which we are permitted to do, by changing at the same time the sign of dt, and if we make -j3 x"dx a At the two limits a; = 0 and a: = /3, we have Vx---2= 0; and, therefore, by taking the definite integrals we shall have, by means of this last equation, (2n-l)fl If in this formula we make n = 1, - = 2, n = 3, &c., successively, we obtain from it Ai/SAo, A2= \$/3a, = /pao, A3 = 4Aa= 2.4.6 3a' &c.; consequently, we shall have, generally, 1.3.5 2m--1 n 2.4.6 2- and as to the value of A0, it is, when taken between x--o and x = (S, f/3 dx Aq =-=. IT. JOs/px-x By substituting these values of A, A, A., &c, in that of T, there will result for the required value of T, and which necessarily converges, since 3 is always less than unity. If the fourth power of a be neglected, we shall obtain /3 = o2; hence this series will be reduced to its two first terms, and the value of T will coincide with that of the preceding number. 186. Let us now proceed to consider the motion of a simple pendulum in a resisting medium. If the preceding notations be retained, the force of gravity resolved in the direction of the tangent Mt will be g. sin 9, because the angle which this line makes with the vertical Mn...