Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/290

 body H arrives at L, to the emi-periphery HKM, the time in which the body H will come to M. And the velocity of the pendulous body in the place T is to its velocity in the lowet place R, that is, the velocity of the body H in the plane L to its velocity in the place G, or the momentary increment of the line HL to the momentary increment of the line HG. (the arcs HI, HK increaing with an equable flux) as the ordinate LI to the radius GK or as, $$\scriptstyle \sqrt {SR^2 - TR^2}$$ to SR. Hence ince in unequal ocillations there are decribed in equal times arcs proportional to the entire arcs of the ocillations; there are obtained from the times given, both the velocities and the arcs decribed in all the ocillations univerally. Which was firt required.

Let now any pendulous bodies ocillate in different cycloids decribed within different globes, whoe abolute forces are alo different; and if the abolute force of any globe QOS be called V, the accelerative force with which the pendulum is acted on in the circumference of this globe, when it begins to move directly towards its centre, will be as the ditance of the pendulous body from that centre and the abolute force of the globe conjunctly, that is, as CO x V. Therefore the lineola HT which is as this accelerative force CO x V will be decribed in a given time; and if there be erected the perpendicular TZ meeting the circumference in Z, the nacent arc HZ will denote that given time. But that nafcent arc HZ is in the ubduplicate ratio of the rectangle GHT, and therefore as $$\scriptstyle \sqrt {GH \times CO \times V}$$. Whence the time of an entire ocillation in the cycloid QRS (it being