504 PROFESSOR BURNSIDE ON THE PARTITION OF ENERGY IN THE 



that i//, ty are the angles made by ab, AB with arcs joining a, A to some fixed 

 point. Then, to satisfy the third assumption, 



ff dsdSdxf,d¥ 



must be taken between the limits and 2v for xff and Sk, and over the whole 

 sphere for each of the surface-integrals. 



Finally, as regards the magnitudes of the angular velocities, the integrations 

 are 



J~f. e-^^'+^-^^+^-^^+^^co^Q^^fi^cOgrfQa , 



and the limits for a) lt Q 1} &c, are ± oo. It might appear that, having taken all 

 possible positions for the principal axes, the limits for o^, Q 1} &c, should be 

 and oo ; but a little consideration will make it clear that these latter limits 

 would be correct only for bodies symmetrical with regard to three mutually 

 rectangular planes through the centre of inertia. 



In the process of finding T let the integrations with respect to (a u £l lf &c, 

 be first performed. The only part of T which will contribute anything after 

 integration is 



2(p* w \ + g V 2 + rW 3 + P 2 ^ + Q 2 Q 2 . 2 + R 2 fl 2 3 ) -(k + K ){u - U) 2 

 [2+c\k + K)Y 



Performing on this expression the integrations indicated, and dividing by the 

 corresponding part of the denominator of T, the result is 



2c 2 



^^- ( ^)^Mk J ^h^Ki. J ^) . 



[ 2+c 2(& + K)] 2 



The value, at the same step in the process, of the average increase in w r energy 

 at a collision is similarly found to be 



(pHP8) (((^_i ; )4,{ (gHQi)(g L_ i l ; ) +( ,, +E8)( A__^ )} ) 



[2+<tf*+K)]" 



and from this, by an interchange of symbols, the values of the corresponding 

 quantities for the other two rotations may be written down. 

 In the four expressions so far obtained, write 



IC 1 /,' L> K$ 



