DYNAMICS. 55 



a. If the body begin to move from A with a certain 

 velocity, then the difference between the square of 

 its velocity at C and at A is equal to twice the area 

 ACED. 



b. In like manner, if the body be projected upward 

 with a given velocity, the difference between the 

 square of the initial velocity, and the square of its 

 velocity at any point C, will be double of the area 

 BCEF ; and if the motion of the body be destroy- 

 ed when it has ascended to a certain point A, the 

 square of the initial velocity will be double of the 

 area BADF. 



101. If a body be accelerated or retarded by 

 any number of forces acting in succession, and each 

 continuing its action while tbe body moves over a 

 certain distance ; the sum that is made up by mul- 

 tiplying each of these forces into the distance over 

 which it acts, and adding those products together, 

 will be proportional to the difference of the squares 

 of the velocities of the body at the beginning and 

 end of the action of those forces. 



102. Hence if the body is accelerated from rest, 

 the sum collected, as in the last proposition, will 

 be as the square of the velocity acquired ; and if 

 it be retarded or resisted till it come to rest, the sum 

 will be as the square of the initial velocity. 



Thus if F, F', F", F'", &c. are the forces that act in 

 succession, s, 6*', s", s'", the distances or spaces over 



which 



