408 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF OASES. 



Substituting the values of the accented quantities in equation (65) by means 

 of equations (63) and (64), and transposing terms in It, we find 



2 {UJ, - UJ, 



(68) , 



the other terms being related to y and z as these are to x. From this equation 

 we may find the value of R ; and by substituting this in equations (63), (64), 

 we may obtain the values of all the velocities after impact. 



We may, for example, find the value of U\ from the equation 



(69). 



TT* I 1 ' * (y.n.-s.m.)* (y.n. - 2,,)' } M, 

 J *\ff+Jl* ~AT ~T *J /, 



= u [ !'- 1 l * i (yi n i- z > m >)\(y* n >- z > m y | jfc c ] 3/i 



1 \ Jtf, iVf, 4, 4 t ' '/ /, 



+ 2UJ,- 2p, (y.n, - Z.TO,) + 2p 3 (y,n t - z.w,) - &c. 



PROP. XXIII. To ^wcZ <Ae relations between the average velocities of trans- 

 lation and rotation after many collisions among many bodies. 



Taking equation (69), which applies to an individual collision, we see that 

 T, is expressed as a linear function of U u U n p t) p u &c., all of which are 

 quantities of which the values are distributed among the different particles 

 according to the law of Prop. IV. It follows from Prop. V., that if we square 

 every term of the equation, we shall have a new equation between the average 

 itdues of the different quantities. It is plain that, as soon as the required 

 relations have been established, they will remain the same after collision, so that 

 we may put U?= U? in the equation of averages. The equation between the 

 average values may then be written 



(. .If. U* - M. Uf) + ( Jf, L? - A jfl (y ^ ' + ( Jf. U! - Aff) (y ' n * - + &c. = 0. 



Now since there are collisions in every possible way, so that the values of 

 /, m, n, &c. and x, y, z, &c. are infinitely varied, this equation cannot subsist 

 unless 



MM = M,U; = A, P ? = A,p? = Ac. 



The final state, therefore, of any number of systems of moving particles of 

 any form is that in which the average vis viva of translation along each of the 



