Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/242

 to that rectangle. Then A will be the place from whence the other body fell. For compleating the rectangle DRSE, ince the area AbFD is to the area DFGE as VV to 2VI, and therefore as $$\scriptstyle \frac 12VI$$, that is, as half the whoe velocity to the increment of the velocity of the body falling by the unequable force; and in like manner the area PQRD to the area DRSE, as half the whole velocity to the increment of the velocity of the body falling by the uniform force; and ince thoe increments (by reaon of the equality of the nacent times) are as the generating forces, that is, as the ordinates DF, DR, and conequently as the nacent area's DFGE, DRSE; therefore ex æquo the whole areas ABFD, PQRD will be to one another as the halves of the whole velocities, and therefore, becaue the velocities are equal, they become equal alo.

Whence if any body be projected either upwards or downwards with a given velocity from any place D, and there be given the law of centripetal force acting on it, its velocity will be found in any other place as e, by erecting the ordinate eg, and taking that velocity to the velocity in the place D, as a right line whoe power is the rectangle PQRD, either increaed by the curvilinear area Dfge, if the place e is below the place D, or diminihed by the ame area DFge if it be higher, is to the right line whole power is the rectangle PQRD alone.

The time is alo known by erecting the ordinate em reciprocally proportional to the quare root of PQRD + or - DFge, and taking the time in which the body has decribed the line De, to the time in which another body has fallen with an