Dynamical Theory of Heat and Light. 19 



reference to which the doctrine conies into question, into two 

 classes. 



Class L Those in which the velocities considered are either 

 constant or only vary suddenly — that is to say, in infinitely 

 small times — or in times so short that they may be omitted 

 from the time-integration. 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 enerov 

 in the collisions. 



(b) The frictionless motion of one or more particles con- 

 strained 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 anodes 

 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), (/»), 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 durations 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 1. with durations of impacts not negligible in the 

 time-integration. 



§ 29. Consider first Class I. (/>) 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 deter- 

 minate 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 geodetic line lono- 

 enough, shall we pass infinitely nearly to any point Q what- 

 ever, 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. 



2 



