176 Mr. J. J. Waterstone's Exposition of 



to be exerted. Since perfect resiliency however occurs, the 

 force of impact will be reflected in contrary directions, and the 

 supplementary momentum will be exerted at p onfd in the 

 direction p v, and on fd in the direction jo to in both lines ; it 

 will therefore tend to augment the rotatory, and simultaneously 

 lessen the rectilineal motion. The actual ratio which these 

 quantities will have to each other, after the condition of the 

 medium is established, will be influenced by the following cir- 

 cumstances; 1st, It is equally probable that the point of con- 

 course p may be anywhere situated in the lines a b, df; the 

 extreme cases are, when it coincides with their extremities or 

 centres of gravity : in the former by the above formula the 

 rotatory motion is f ths of the whole ; in the latter it is ; the 

 mean quantity or y%ths will, therefore, show the ratio of the 

 whole quantity of rotatory momentum generated in the me- 

 dium by this cause singly, and y^ths that of the remainder, 

 by which the lines continue their rectilineal motions. 2ndly, 

 The rotatory motion by the diversified concourse of the 

 particles will be performed simultaneously 

 in planes perpendicular to each other. 

 For while a h revolves in the plane ar b ; 

 another particle may communicate a rota- 

 tory impulse in any other direction v a, 

 which by a well-known principle in dyna- 

 mics will cause the line ab to revolve at 

 the same time in the planes atb, arb, 

 perpendicular to each other. In this manner the whole quan- 

 tity of rotatory momentum effected by the first cause will be 

 nearly doubled, whilst the elastic recoil will tend further to 

 diminish the rectilineal motion of the particles. Before ascer- 

 taining exactly this ratio, the principles of chance require to 

 be employed in estimating the frequency of peculiar modes 

 of concurrence, and thus discovering the mean results of the 

 combined action of the whole medium. It is not intended, 

 however, to enter upon this investigation at present, as it is 

 unnecessary to prove the truth of what has been advanced, 

 that the primitive momentum is sepai-ated into two parts, 07ie 

 of which is employed in sustaini7ig a rectilineal and the other a 

 rotatory motion. 



II. Let a rigid plane be introduced into this medium : What 

 will be the corresponding change in its relative density ? Let 

 the line a b, having a rectilineal motion in the direction d e, 

 and a rotatory motion hg, impinge against the plane of which 

 A B is a section ; the whole momentum in the line will imme- 

 diately after the first collision be separated into 1st, a reflect- 

 ing impulse in the direction af; and 2ndly, the influence of 



the 



