OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 737 



If, as in equations (82) to (86), we suppose the origin of co-ordinates to 

 be the centre of mass of the whole system, the axis of z to pass through 

 the particle m n , and the axes of x and y to be in the directions of the 

 principal axes of the section of the momental ellipsoid normal to z, then writing 



g=u-qz, t) = v+pz, t, = w ....................... (100), 



so that 77, are the velocity- components of m n relative to axes moving as 

 the system would do if it were then to become rigid, with the angular velocity 

 whose components are p, q, r, we may write 



The sum of the last three terms of this expression, with its sign taken positive, 

 represents the part of the internal motion of the system which is due to the 

 fact that the particle m n is moving with the relative velocity whose components 

 are 77, . 



"We may also define it as the work which would be done by the particle 

 m n against the internal forces of the system, if these forces were suddenly to 

 become such as to render the whole system rigid in an infinitely short time. 



Comparing this result with that obtained in equation (48), we see that 

 the law of distribution of the velocities of the particle m n is the same as 

 what it would be in a fixed vessel containing n 2 particles, provided that we 

 substitute for w a , if, V? the quantities (l fynz 2 )"^ 2 , (1 a/AZ 2 )" 1 ?? 2 , " respectively. 



Hence the mean square of the velocity in the direction of the line joining 

 the particle with the centre of mass is the same at all points of the system, 

 but the mean square of the velocity in other directions is less than this in 

 the ratio of 1 a/zz 2 to 1, where z is the perpendicular from the centre of mass 

 on the line of relative motion of the particle, and a is the moment of mobility 

 of the system about an axis through the centre of mass and normal to the 

 plane through that centre and the line of motion. 



When the product of the mass of the particle into the square of its 

 distance from the centre is so small that it may be neglected in comparison 

 with the moments of inertia of the system, then quantities like a/*z 2 and fytz 2 

 may be neglected in respect of unity, and we may assert that the mean square 

 of the relative velocity, for a particle of given mass, is the same in all directions 

 and at all points of the system; but that for different particles it varies 



VOL. II. 9S 



