MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES. 503 



Hence the dynamical equations may be written (the mass of a sphere being 

 taken as unity) 



TJ'+u'=TJ+u 



U' — u' — c[PQ\+pw\ + Qil' 2 + qco' 2 + 11Q' S + roo' n ] 

 = w - U + c[PQ 1 +po*i + Q£l 2 + qw 2 + R0 3 + ro) 3 ] 



A(Q\-n{) = cP(U-U') 

 B(£2' 2 -fi 2 ) = cQ(U-U') 

 C(£2' 3 -ft 3 ) = cE(U-U') 



A^'j — wj) = cp{ti' — u) 

 B(o)' 2 — o) 2 ) = cq(u — u) 

 (/(fc/3 — « 3 ) = cr{a' — u) . 



The elimination of the angular velocities after impact from these equations 

 leads to the following equations for XT' and u'\ — 



U'(l + c2K) - u'(l + c%) = u(l- c 2 k) - U(l - C 2K) + 2cm 

 U' +it =u +U, 



where again for brevity 



A i- B -t- c ^ 



^2 2 2 



A + B + C =l 

 po*! + qw 2 + rco 3 + Pfi x + QQ 2 + Ef 2 3 = vs . 



Hence the increase of energy of translation due to the impact ( = T say) 



= \ (U'2 + M'2_U2-«2) 



= 2c (* ~ U + cw X 2rar _ c ( k + K X M _ U )) . 

 (2 + C 2(^ + K)) 2 ' 



In the special state of the system the mean value of this quantity, as also of 

 the increases in the energies of rotation about the three principal axes, must 

 vanish ; and the three independent results so obtained should give k lf k 2 , k s in 

 terms of h. 



In determining the mean value of any quantity connected with a collision of 

 the spheres here considered, the fourth assumption made above permits the 

 integrations involving the speeds and directions of motion of the colliding 

 spheres and the direction of the line of centres to be performed just as in 

 § 21 of Prof. Tait's paper above referred to. The additional integrations to 

 be performed in this case will be obtained as follows : — Suppose lines drawn 

 from the centre of the unit sphere (in the figure of the paragraph referred to) 

 parallel to the A- and B-axes of each sphere meet the unit sphere in a, A, b, B ; 

 that ds, d& are elements of the surface of the sphere, surrounding a, A ; and 



