378 Lord Kelvin [April 27, 



§ 28. Premising that the mean kinetic energies with which the 

 Boltzmann-Maxwell doctrine is concerned are time- integrals of ener- 

 gies divided by totals of the times, we may conveniently divide the 

 whole class of problems, with reference to which the doctrine comes 

 into question, into two classes. 



Class I. : Those in which the velocities considered are either con- 

 stant or only vary suddenly — that is to 6ay, in infinitely small times — 

 or in times so short that they may be omitted from the time-integra- 

 tion. To this class belong : 



(a) The original Waterston-Maxwell case and the collisions of 

 ideal rigid bodies of any shape, according to the assumed law that the 

 translatory and rotatory motions lose no energy in the collisions. 



(b) The frictionless motion of one or more particles constrained 

 to remain on a surface of any shape, this surface being either closed 

 (commonly called finite though really endless), or being a finite area 

 of plane or curved surface, bounded like a billiard table, by a wall or 

 walls, from which impinging particles are reflected at angles equal to 

 the angles of incidence. 



(c) A closed surface, with non- vibratory particles moving within it 

 freely except during impacts of particles against one another or 

 against the bounding surface. 



(d) Cases such as (a), (b), or (c), with impacts against boundaries 

 and mutual impacts between particles, softened by the supposition of 

 finite forces during the impacts, with only the condition that the dura- 

 tions of the impacts are so short as to be practically negligible, in 

 comparison with the durations of free paths. 



Class II. : Cases in which the velocities of some of the particles 

 concerned sometimes vary gradually ; so gradually that the times 

 during which they vary must be included in the time-integration. To 

 this class belong examples such as (d) of Class I. with durations of 

 impacts not negligible in the time-integration. 



§ 29. Consider first Class I. (6) with a finite closed surface as the 

 field of motion and a single particle moving on it. If a particle is 

 given, moving in any direction through any point I of the field, it will 

 go on for ever along one determinate geodetic line. The question 

 that first occurs is, does the motion fulfil Maxwell's condition (see 

 § 18 above) ; that is to say, for this case, if we go along the geo- 

 detic line long enough, shall we pass infinitely nearly to any point Q, 

 whatever, including I, of the surface an infinitely great number of 

 times in all directions? This question cannot be answered in the 

 affirmative without reservation. For example, if the surface be exactly 

 an ellipsoid it must be answered in the negative, as is proved in the 

 following §§ 30, 31, 32. 



§ 30. Let A A', B B', C C, be the ends of the greatest, mean, 

 and least diameters of an ellipsoid. Let U x U 2 U 3 U 4 be the 

 umbilics in the arcs A C, C A', A' C, C A. A known theorem in 

 the geometry of the ellipsoid tells us, that every geodetic through 

 Uj passes through U 3 , and every geodetic through U 2 passes through 



