Manchester Memoirs, Vol. Ixii, (191 7) No. 1 9 



On this assumption we replace I. and II. by 



I. The rate of change of the momentum of the particle of co- 

 ordinates x, y, z, in the direction of the axis of x 



S£ + 8£ + 8T_SV A + ^ 



8x 8y 8z 8y 8z ' 

 and 



II. The particle possesses an internal angular momentum, of 



which the rate of change about the axis of „r=^ 1 . 



The ^-elements of stress will no> doubt generally be null, 

 but I have retained them, as it is possible that they may play 

 a piart in a theory of fracture or of permanent set. 



The question of the mathematical expressions for the 

 momentum and rate of change of momentum of the particle 

 depends upon the amount and character of the degree of free- 

 dom which the particle is to enjoy. 



If the particle is to be unrestrained in its freedom to move, 

 and is only influenced by the forces arising from the field iof 

 stress in which it finds itself, we may suppose that the position 

 of the particle at any time is a function of the three quantities 

 which determined its position at some epoch, and of the time 

 elapsed since that epoch. If we take the mass of the particle 

 ito 'ble invariable, the expression for the rate of change of 

 momentum is known. Under this head comes the case of fluid 

 motion, and the propagation of a small disturbance, but not any 

 case of molar motion 'in which any finite portion of the material 

 suffers a change of position approximately comparable with a 

 rigid motion of that portion. 



Passing to the other extreme, we may take axes in motion 

 such that the origin has the velocity u , v , w„ and the axes 

 have angular velocities w x , w y , <o s about their own positions 

 in space. If then the velocity of each particle in the .directions 

 of the lines in space occupied by the axes is given by 



Ug — WzV + UJyZ 

 Vg — W X Z + U)-.X 



w - u) y x + u) x y 



where u . . . w x . . . are functions of / only, we can deduce 

 the rate of change of the particle's momentum. On integration 

 over the. whole of a body we might deduce the whole of Rigid 

 Dynamics. In the motion of a rigid body fthe parts of the body 

 are subject to stresses, which are not elastic stresses; these 

 stresses have no ^-element, and they are no- doubt quite 

 jdefinite, but they are not defined by the rates of change of 

 momentum of the constituent particles. Between the two ex- 

 treme cases we 'have mentioned, there exists a wide range of 



