Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/245

 Draw IC meeting KE in N, and on IK let fall the perpendicular NT; and let the interval DE or IN, between the circumferences of the circles be very mall; and imagine the bodies in D and I to have equal velocities, Then becaue the ditances CD and CI are equal, the centripetal forces in D and I will be alo equal. Let thoe forces be expres'd by the equal lineolæ DE and IN; and let the force IN (by cor 2. of the laws of motion) be reolved into two others, NT and IT. Then the force NT acting in the direction of the line NT perpendicular to the path ITK of the body, will not at all affect or change the velocity of the body in that path, but only draw it aide from a rectilinear coure, and make it deflect perpetually from the tangent of the orbit, and proceed in the curvilinear path ITKk. That whole force therefore will be pent in producing this effect; but the other force IT; acting in the direction of the coure of the body, will be all employed in accelerating it; and in the leat given time will produce an acceleration proportional to it itelf. Therefore the accelerations of the bodies in D and I produced in equal times, are as the lines DE, IT; (if we take the firt ratios of the nacent lines DE, IN, IK, IT, NT); and in unequal times as thoe lines and the times conjunctly. But the times in which DE and IK are decribed, are, by reaon of the equal velocities (in D and I) as the paces decribed DE and IK, and therefore the accelerations in the coure of the bodies through the lines DE and IK are as DE and IT; and DE and IK conjunctly; that is, as the quare of DE to the rectangle IT into IK. But the rectangle IT x IK is equal to the quare of IN; that