74 BEPORT — 1894. 



24. To prove this, Boltzmann has given (loc, cit.) a highly artificial 

 and laborious verification of the Jacobian relation 



HA y', e' ) (13J 



where 6) is the angle the direction of motion makes with the axis of x, and 6' 

 its value after a time t', which Boltzmann takes to be a small interval, M. 

 As Boltzmann 's proof is not easy to follow, it may be interesting to obtain 

 the same result much more simply and without imposing restrictions on 

 the magnitude of the time-interval t' by considering the Jacobian 



^_ d{x',y',u',v' )_di x',y\x',y' )^ 

 d{x, y, u, v)—d{x, y, i, ij)' 



We have, keeping the initial time constant, 



d^_ d{x',y',x',i/') ^ d{x',y',x',y' ) ^ d { x',y',x',y') ^ d {x' , y' , x' , \j') ^ 

 dt' 9 {x, y, X, y) d {x, y, x, y) 8 {x, y, x, y) d {x, y, x, y)' 



The first two Jacobian s evidently vanish. And by the equations of 

 motion 



Hence x', y' are functions of x', y', and the quantities in the numerators of 

 the last two Jacobians are not all independent : therefore these vanish. 

 Therefore integrating with respect to t' : 



A^ constant := 1, (its initial value) . . . (14) 

 If 



q^^ic^-^-v-, tan d=:v/u 



we have by the transformation from Cartesian to polar co-ordinates 



.^d{ x',y',iq '\6') 

 d{x, y, \q\ «)• 



In virtue of the equations of energy 



SJW=1 = ^S^t) 



8E dE 



._d{x',y',B',^)_d{x',y',e') 



'd{x, y, d, E) d{x, y, 6) 



whence (13) 



d{x, y, ti) 

 as was to be proved. 



25. And assuming the law of distribution 



/(E) dx dy du dv (15) 



we have evidently 



("CO Too (ca Coo 



f{\u'^+^v'^ + Y)n?dudv=\ f{\u'^+\v'^ + Y)vHvdu, 



J— ooj— CO J— coj— c» 



