Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/22

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Of the Motion of Bodies that are reited in the duplicate ratio of their Velocities.

If a body is reited in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progreion, proceeding from les to greater terms: I ay that the velocities at the beginning of each of the times are in the ame geometrical progreion iinverely; and that the paces are equal, which are decribed in each of the times.

For ince the reitance of the medium is proportional to the quare of the velocity, and the decrement of the velocity is proportional to the reitance; if the time be divided into innumerable equal particles, the quares of the velocities at the beginning of each of the times will be proportional to the differences of the ame velocities. Let thoe particles of time be AK, KL, LM, &c. taken in the right line CD; and erect the perpendiculars AB, Kk, Ll, Mm, &c. meeting the Hyperbola BklmG, decribed with the centre C, and the rectangular aymptotes CD, CH, in B,k,l, m, &c., then AB will be to Kk, as