Mr Pocklington, On the Kinetic Theory of Matter. 287 



happens, for example, in the case of a sphere rolling on a cylinder 

 under any forces, if we choose two coordinates of the centre of the 

 sphere, the longitudinal and transverse velocities and the normal 

 angular velocity as defining the phase, and we can prove the usual 

 theorem about the distribution of energy between the degrees of 

 freedom in the case of a set of such systems. In general, however, 

 the Jacobian is not unity, and we may expect to find the distri- 

 bution of energy obeying different laws. It will be useful to 

 examine what happens in some particular cases, and these may 

 conveniently be chosen to be such as can be approximated to by 

 using actual bodies. 



7. For a first case, let a body rest on a horizontal plane by 

 two smooth contacts and a massless wheel, so that the problem is 

 that of the plane motion of a body one point of which is con- 

 strained to move in the direction of a line fixed in the body. Let 

 this line pass through the centre of gravity, and let the distance 

 of the point of contact of the wheel from the centre of gravity 

 be a. Let the position of the body be specified by x, y, the 

 coordinates of the centre of gravity, and the angle that the line 

 joining the centre of gravity to the point of contact of the wheel 

 makes with the axis of x. 



The Lagrangian function is 



and the condition is 



i;sin — y cos — ad = (1). 



Hence the equations of motion are 



x + X, sin = 0, 



y — X cos = 0, 



¥0 -\a = 0, 



or 



n'rr -I- b 2 ft H1T1 ft == Ci) 



(2). 



ax + W'6 sin = 0) 



ay — W6cos0 — Oj 

 On differentiating (1) and eliminating x, y by (2), we have 



x cos + y sin = -r (3). 



a 



On differentiating this, and eliminating x, y by (2) and x, y 

 by (1), we have 



d _ ■ a*' p 



dt{0) F + a 2 



19—2 



