(J48 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3. 



exerting the same effort, would produce more force in the same body when in 

 a space carried uniformly forwards, than if the space was at rest, or that springs 

 acting equally on two equal bodies in such a space, would produce unequal 

 changes in the forces of those bodies. 



Various problems concerning the collision of bodies are resolved in a more 

 general manner than usual. Mr. Bernouilli had determined the motions when 

 the elasticity is perfect, and one body strikes two equal bodies in directions that 

 form equal angles with its direction ; or when there are any number of bodies 

 impelled by it on one side in various directions, providing equal bodies be im- 

 pelled by it on the other side, in directions equally inclined to its own direction. 

 But the problem is resolved here without these limitations; some others of this 

 kind are subjoined, and this doctrine is applied for determining the motions of 

 bodies that act on each other while they descend by their gravity. 



The general principle derived from these inquiries is, that if there be no colli- 

 sion, or sudden communication of motion from one body to another, while 

 they descend together, and in any case, if the elasticity be perfect, the sum of 

 the products, when each body is multiplied by the square of the velocity ac- 

 quired by it, is the same as if all the bodies had descended freely from the same 

 > respective altitudes to their several places; only in collecting that sum, if any 

 body is made to ascend, the product of it multiplied by the square of its velo- 

 city is to be subducted; and if the bodies be supposed to ascend from their 

 places with the respective velocities acquired by them, then their common 

 centre of gravity will rise to the same level from which it descended. In other 

 cases, however, the ascent of the centre of gravity will be less than its descent, 

 but is never greater. 



After demonstrating the usual rule for finding the centre of oscillation, the 

 author treats of the motion of water issuing from a cylindric vessel. The 

 effect of the gravitation of the whole mass of water is considered as threefold. 

 It accelerates, for some time at least, the motion with which the water in the 

 vessel descends ; it generates the excess of the motion with which the water 

 issues at the orifice above the motion which it had in common with the rest of 

 the water; and it acts on the bottom of the vessel at the same time. Then 

 supposing the last two parts of the force to be in any invariable ratio to each 

 other, when the diameters of the base and orifice are given, he determines by 

 logarithms the velocity with which the water issues at the orifice; and shows 

 that this velocity will approach very nearly to its utmost limit in an exceedingly 

 small time. When the water is supposed to be supplied in a cylinder, so as to 

 stand always at the same altitude above the orifice, there is an analogy between 

 the acceleration of the motion of the water that issues at the orifice, and the 



