288 Mr Pocklington, On the Kinetic Theory of Matter. 



Putting a new variable for log 6, this becomes an equation of 

 the second order, the integral 1 of which is 



6 = sech b (t — e), 



a 



where b, e are the constants of integration. 



Hence 6, after possibly increasing at first, tends ultimately to 

 the limit zero, and the kinetic energy finally passes wholly into 

 energy of translation. 



We can prove from (3) and (1) that the body is then moving 

 along the line joining the point of contact of the wheel with the 

 centre of gravity, and in that sense. However the body is started, 

 if we except an unstable steady motion, it ultimately moves as 

 just described. We shall say that the motion tends to a final 

 state, in which 6 = 0. 



The motion is of course reversible. If reversed when 6 has 

 diminished nearly to zero, 6 increases, at first slowly, then more 

 rapidly, reaches a maximum, and then decreases to the limit zero. 

 The motion still tends to the same final state. The bearing of 

 the results of this paragraph on the kinetic theory of matter is 

 discussed in § 12. 



8. It will be interesting to consider the Jacobian 



J=d(x, y, 6, x, y)/d (x , y , O , x , y ). 



It can be evaluated by means of the integral equations of 

 motion by first finding the Jacobian relative to the constants of 

 integration, but we prefer to use a method of somewhat more 

 general application. 



Differentiating, we find dJIdt equal to a sum of five Jacobians, 

 each of which differs from J by having an additional dot in the 

 numerator. Dividing by J, and using the formula for change of 

 independent variables, 



ldJ ^ d(x, y, 6,x,y) ^ 

 J dt d (x, y, 6, x, y) 



_ dx dy dd dx dy 

 dx dy dd dx dy ' 



where the differentiations are partial, and the independent vari- 



1 This equation can be integrated again and, if {k 2 + a 2 )/a 2 is the square of an 

 integer, the values of x and y can be found in finite terms by integrating the values 

 of x and y found from (1) and (3). The path of the point of contact of the wheel 

 is somewhat simpler than that of the centre of gravity. Its intrinsic equation is 



fc 2 + a 2 , ad> 



s — log sec - 



sJk 2 



+ a* 



