406 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 



Also, by Prop. V., if a: be the average velocity of one set of particles, and y 

 that of another, then the average value of the sum or difference of the velocities is 



from which it is easy to see that, if in each individual case 



u = ax + by + cz, 



where x, y, z are independent quantities distributed according to the law above 

 stated, then the average values of these quantities will be connected by the 

 equation 



PROP. XXII. Tivo perfectly elastic bodies of any form strike each other : 

 their motions before impact, and the line of impact, to find their motions 

 if/i'r impact. 



Let 37, and 37, be the centres of gravity of the two bodies. 3/,A',, 37,}",, 

 and 37,Z, the principal axes of the first ; and 37^T,, 

 37, y, and 37,Z, those of the second. Let 7 be the 

 point of impact, and 7^,772, the line of impact. 



Let the co-ordinates of 7 with respect to 37, be 

 .r u v,s,, and with respect to 37, let them be x^f, t . 



Let the direction-cosines of the line of impact 

 72,7/2, be /,w,n, with respect to 37,, and Ijnajn^ with 

 respect to 37,. 



Let 37, and 37, be the masses, and ^4,7^,C, and 

 inertia of the bodies about their principal axes. 



Let the velocities of the centres of gravity, resolved in the direction of 

 the principal axes of each body, be 



U lt V lt W lt and U t , V n W n before impact, 

 and V u V u W\, and V u V u W. after impact. 



Let the angular velocities round the same axes be 



p t . q a r u and p n q r n before impact, 

 and p' it q\, r\, and p' n q' K r' a after impact. 



the moments of 



