36 On the Collision of Perfectly Elastic Bodies of any Form. 



same after collision, so that we may put U 1 ' 2 =U l 2 in the equa- 

 tion of averages. The equation between the average values may 

 then be written 



(M.U.'-M^^g+tM.U.'-A.p,') ^" 1 ^"" ' 

 + (M.U,*- A,j»«) fo" 2 -*^' + fa. =0. 



A 2 



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

 values of /, m, n, &c. and SB, y } z, &c. arc infinitely varied, this 

 equation cannot subsist unless 



M 1 U 1 a =M 2 U 2 2 =A 1 jo l 2 =A 2 ^ 2 2 =&c. 



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 three axes is the same in all the 

 systems, and equal to the average vis viva of rotation about each 

 of the three principal axes of each particle. 



Adding the vires vivce with respect to the other axes, we find 

 that the whole vis viva of translation is equal to that of rotation 

 in each system of particles, and is also the same for different 

 systems, as was proved in Prop. VI. 



This result (which is true, however nearly the bodies approach 

 the spherical form, provided the motion of rotation is at all 

 affected by the collisions) seems decisive against the unqualified 

 acceptation of the hypothesis that gases are such systems of hard 

 elastic particles. For the ascertained fact that y, the ratio of the 

 specific heat at constant pressure to that at constant volume, is 

 equal to 1*408, requires that the ratio of .the whole vis viva to 

 the vis viva of translation should be 



whereas, according to our hypothesis, /3=2. 



We have now followed the mathematical theory of the col- 

 lisions of hard elastic particles through various cases, in which 

 there seems to be an analogy with the phenomena of gases. We 

 have deduced, as others have done already, the relations of pres- 

 sure, temperature, and density of a single gas. We have also 

 proved that when two different gases act freely on each other 

 (that is, when at the same temperature), the mass of the single 

 particles of each is inversely proportional to the square of the 

 molecular velocity; and therefore, at equal temperature and 

 pressure, the number of particles in unit of volume is the same. 



We then offered an explanation of the internal friction of 



