572 PHILOSOPHICAL TRANSACTIONS. [aNNO 1722. 



portions of the yielding substance, the opposition that substance makes to the 

 motion of the globes, will be the same in both, however different the velocities 

 be, with which they move. This I shall demonstrate as follows. 



Let A and b be two globes, equal in magnitude, but of different weights, 

 which are equally immersed into a yielding substance. Suppose the velocities 

 with which they move in their present situation, to be reciprocally in the sub- 

 duplicate ratio of the weights of the globes ; that is, let the ratio of the weight 

 of the globe a to the weight of the globe b, be duplicate of the ratio of the velo- 

 city of the globe b, to the velocity of the globe a. Since therefore the ratio 

 of the quantity of motion in the glob(j a, or of the force with which it moves, 

 to the quantity of motion in the globe b, or to the force with which that globe 

 moves, is compounded of the ratio of the weight of the globe a, to the weight 

 of the globe b, and of the ratio of the velocity of the globe a, to the velocity 

 of the other globe b, the force with which the globe a moves, is to the force 

 with which the globe b moves, as the velocity of this globe b, to the velocity 

 of the other globe a. But if the same opposition be made to the motion of the 

 globes when they bear upon equal portions of the yielding substance, the effect 

 of that opposition, while the globes enter farther into the substance by equal 

 spaces, will be proportional to the time in which the globes are moving those 

 spaces, or in which the opposition is made, if we consider those spaces while 

 nascent or in their first origin ; the effect therefore of this opposition will be 

 reciprocally proportional to the velocity of each globe; namely, the momenta- 

 neous loss of force in the globe a, will be to the momentaneous loss of force in 

 the globe b, as the velocity of the globe b, to the velocity of the globe a; and 

 the whole force of the globe a has been found to bear the same ratio to the 

 whole force of the globe b, consequently these globes, while they penetrate 

 equal spaces into the substance, lose parts of their force which bear the same 

 proportion to the whole; and therefore, if their velocities be at any time reci- 

 procally in the subduplicate ratio of their weights, so that the forces or degrees 

 of motion, with which they move, be reciprocally proportional to their velo- 

 cities, the forces with which they press into the yielding substance, at equal 

 indentures made in the substance, will continue in the same proportion; and 

 therefore, on the theory of resistance here supposed, when the whole force and 

 nmtion of both these globes is entirely lost, they will be plunged into the sub- 

 stance at equal depths. 



Now whereas in the experiment of Polenus, the globes, falling from heights 

 reciprocally proportional to their weights, strike the yielding substance with 

 velocities reciprocally in the subduplicate proportion of their weights, and the 

 effect is in all cases found to be what is here deduced from the theory of 



