80 



NEWTON S PRINCIPIA. 



u 2, and v = */2 . Z ; also d t = - - = 7= - 



Therefore t = 



x 

 - = \/2 . ^, or the time is as 



the area . In these expressions, therefore, to find Z and 

 we have to substitute the values of X and Z in terms 

 of x, and integrate. 



It is hardly necessary to add, that if, instead of the 

 velocity and the time being sought (Z and ), these are 

 given, and the place reached by the body be sought, we 

 find it by the same construction ; and ascertaining what 

 value of x gives the value of Z, the square root of the 

 area. But it may be well to note here, that if O M be 

 the curve, whose ordinate P M or y = X, the centripetal 



force at P in. terms of A P or x, or the gravitation of 

 any particle of a homogeneous fluid towards S at the point 

 P ; then the column of that fluid whose altitude is A P 

 will press at P, as the area A P M O, or as v 2 , the square 

 of the velocity acquired by a body falling through A P. 



iv. The next object of research is to generalise the pre 

 ceding investigations of trajectories from given forces, and 

 of motion in given trajectories, applying the inquiry to all 

 kinds of centripetal force, and all trajectories, instead of 

 confining it to the conic sections, and to a force inversely 



