MR, S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
413 
The system of two spheres has twelve degrees of freedom. We require, there¬ 
fore, eleven linear functions of the velocities to be invariable. They are as follows, 
viz., four components of velocity in the common tangent plane, x, y, X, y, and seven 
others, viz., 
u + U = Y = u + U', 
cpu + Aw 1 = s 1 = c'pu + Aw'j, 
cqu + Bwo = s. 2 = cqu + Bo/ 3 , 
CTll —p Oci)g ^ Sg = CVU ~p Ooj o, 
cPU - An, = Sj = cPU' - An'i, 
cQU — Bn 3 = So = cQU' ~ Bob, 
cKU - Ca 3 = $ 3 = cKU' - Cfi' 8 . 
As before, we form the equations, 
u — u' + U — U' = 0 , 
• < 
Cp (u — u) + A (aq — CO : ) = 0, 
cq ( u — u) B (a>. 2 — oj' 2 ) = 0, 
cr (u — u') + C (w 3 — o/ 3 ) = 0, 
cP (U — U') - A (Xlj — n\) = 0, 
cQ (U — U') - B (n. 2 - xi' 3 ) = o, 
cB (U — CP) — C (fig — fl'g) = 0, 
and by the conservation of energy, 
(u -|- u ) (u — u') + (U -p U 7 ) (U — U') -p A (wj -p (oq — a)\) -p &c. = 0 
and equating the determinant to zero, 
u — U — c i^p>(^\ ~P qvto -p xo ) 3 -p Pfij -p Qn 3 -p Ilfig) = p, 
U — U — C (po) ^ -p q(0 3 -p T(x) 3 ~P Pfi j -P Qfi 3 ~P Pvfi g) ~ — p 
We can now substitute Y, s l5 s. 2 , s 3 , S l5 S 3 , S 3 , p for S 2 . . . in the general 
equation, and we obtain, as before, the result that the Maxwell-Boltzmann distri¬ 
bution, given existing, is not affected by collisions. 
10. Professor Burnside obtains the same eight equations as above given, and I 
acknowledge my obligation to him, but he originally deduced the result that the 
energy of rotation is twice the energy of translation, instead of equal to it, as, 
according to the theory, it should be. He has since seen reason to change his views 
with regard to this problem. 
