388 Lord Kelvin [April 27, 



Maxwell doctrine, which makes the time-integrals of translatory 

 and rotatory energies equal. 



§ 43. In the solution for the biassed ball (masses 1 and 100), we 

 found great irregularities due to " ruus of luck " in the toss for plus 

 or minus, especially when there was a succession of five or six pluses 

 or five or six minuses. We therefore, after calculating a sequence of 

 200 flights with angles each determined by lottery, calculated a 

 second sequence of 200 flights with the equally probable set of angles 

 given by the same numbers with altered signs. The summation for 

 the whole 400 gave 555 '55 as the time-integral of the whole energy, 

 and an excess, 82 ■ 5, of the time-integral of the translatory, over the 

 time-integral of the rotatory energy. This is nearly 15 per cent. 

 "We cannot, however, feel great confidence in this result, because the 

 first set of 200 made the translatory energy less than the rotatory 

 energy by a small percentage (2*3) of the whole, while the second 

 200 gave an excess of translatory over rotatory amounting to 35 • 9 

 per cent, of the whole. 



§ 44. All our examples considered in detail or worked out, hitherto, 

 belong to Class I. of § 28. As a first example of Class II., con- 

 sider a case merging into the geodetic line on a closed surface S. 

 Instead of the point being constrained to remain on the surface, let 

 it be under the influence of a field of force, such that it is attracted 

 towards the surface with a finite force, if it is placed anywhere very 

 near the surface on either side of it, so that if the particle be placed 

 on S and projected perpendicularly to it, either inwards or outwards, 

 it will be brought back before it goes farther from the surface than 

 a distance h, small in comparison with the shortest radius of curvature 

 of any part of the surface. The Boltzmann-Maxwell doctrine asserts 

 that the time-integral of kinetic energy of component motion normal 

 to the surface, would be equal to half the kinetic energy of component 

 motion at right angles to the normal ; by normal being meant, a 

 straight line drawn from the actual position of the point at any time 

 perpendicular to the nearest part of the surface S. This, if true, 

 would be a very remarkable proposition. If h is infinitely small, we 

 have simply the mathematical condition of constraint to remain on 

 the surface, and the path of the particle is exactly a geodetic line. 

 If the force towards S is zero, when the distance on either side of S is 

 + h, we have the case of a particle placed between two guiding 

 surfaces with a very small distance, 2 h, between them. If S, and 

 therefore each of the guiding surfaces, is in every normal section 

 convex outwards, and if the particle is placed on the outer guide- 

 surface, and projected in any direction in it, with any velocity, great 

 or small, it will remain on that guide-surface for ever, and travel 

 along a geodetic line. If now it be deflected very slightly from 

 motion in that surface, so that it will strike against the inner guide- 

 surface, we may be quite ready to learn, that the energy of knocking 

 about between the two surfaces, will grow up from something very 

 small in the beginning, till, in the long run, its time-integral is 



