30 Lord Kelvin on the 



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

 we found great irregularities due to "runs 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 

 tran slatory, 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 rot itory energy bv a 

 small percentage (23) of the whole, while the second 200 

 gave an excess of translatory over rotatory amounting to 

 359 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., consider 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 /<, 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 per- 

 pendicular 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 ±A, we have the case of a 

 particle placed between two guiding surfaces with a very 

 small distance, 2h, between them. ]f 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 



