Motion of heavy Bodies. 173 



on increasing, still the degrees by which it increases go oa 

 ishing. The same may be observed with respect to the degrees 

 by which the motion of a ship arrives at uniformity ; the action 

 of the wind upon the sails diminishes according as the ship ac- 

 quires motion, because it is withdrawn so much the more from 

 this action, according as it has more velocity. 



280. The principles necessary for determining the circum- 

 stances of this kind of motion are easily deduced from the prin- 

 ciples that we have laid down with regard to uniform motion, 

 and motion uniformly accelerated. 



(1.) In whatever manner motion is varied, if we consider it 

 with respect to instants infinitely small, we may suppose that the 

 velocity does not change during the lapse of one of these instants. 

 Now, when the motion is uniform, the velocity has for its express- 

 ion the space described during any time , divided by this time. 

 Accordingly, when the motion is uniform only for an instant the 

 velocity must have for its expression the infinitely small space 

 described during this instant divided by this instant. Hence, if 

 s represents the space described, in the case of a variable motion, 

 du 'ing any time , d s will represent the space uniformly describ- 

 ed during the instant d t ; we have, therefore, 



ds , 

 v = or d s = v d t, 



as the first fundamental equation of varied motion. 



281. In the equation v = g f, we have understood by g the 

 velocity which the accelerating force is capable of giving to a 

 body in a determinate time, as one second, by an action that is 

 supposed to continue constantly the same. In the equation 



d v = g d /, 



the same thing is to be understood. But we must observe that 

 the accelerating force being supposed to be variable, the quanti- 

 ty g which represents the velocity that the accelerating force 

 is capable of producing, if it were constant for one second, this 

 quantity g, I say, is different for all the different instants of the 

 motion. Indeed, it will be readily conceived, that when the 

 accelerating force becomes less, the velocity that it is capable 



