ON OUR KNOWLEDGE OF THERMODYNAMICS. 79 



The Functional Determinants for Encounters and Collisions. 



31. The reasoning of Case II. of the last article cannot be regarded as 

 conclusive without further investigation if the forces of encounter become 

 impulsive as in the case of a collision. For Watson ' has pointed out that 

 the determinantal relation (1) 



9 jPi', • • • gnO _i^ 



^{Pu • • ' 9,.) 

 with the total time t constant is inapplicable, and, moreover, the S func- 

 tion used in proving it becomes discontinuous in the case of impacts. 



The difficulty can be overcome by regarding impulsive forces as the 

 limit of finite forces, and supposing the initial and final states separated 

 by a small constant interval of time during which the impact occurs ; 

 but it is certainly highly desirable to treat the problem separately. 



32. According to Watson, the objections will be avoided if, instead of 

 supposing the time constant, we assume the initial and final states to be 

 connected by a geometrical relation between the co-ordinates which can be 

 expressed in the form 



and that in that case the functional Jacobian 



d(Pu . . . g n-l) q,/ ,^^. 



a(K.- • .?„-i'r~^ * • • ^^''^ 



This relation is verified by Watson for the case of a projectile in his 

 letter in ' Nature.' But in his ' Kinetic Theory of Gases ' ^ he derives it 

 from the relation (1) with t constant, so that his method applies only to 

 finite forces. 



If, however, in the course of an encounter between molecules, impulsive 

 action takes place owing to the energy-function changing discontinuously 

 when the geometrical relation 



is satisfied, we will now show that the states before and after the impulse 

 are connected by the relation 



S( P/. • • • Pn, q\ ' ■ • g n-l')_ j.g.. /OQX 



o{p\, . . . Pn, qi ■ . • qn-\) qn 



provided that the principle of Conservation of Energy is satisfied. 



For if \ be the instantaneous increase of potential energy when q^ passes 

 through the value c, then the initial and final kinetic energies satisfy the 

 relation 



T-T'=\ (24) 



where X may be a function of ^i . . . qn-v 



Now since no impulsive action takes place through the \arIation of 

 the co-ordinates q^ . . , q^-x we have 



P\—P\, Pi=Pi, • • • Pn-\=P,^-\■> 

 and therefore 



dpx'=dpx, dp2=dp.2, . . . dp„_^'=dp„.i. 



' Nature, May 12, 1892, p. 29. 

 ' Kinetic Them-y of Gases, p. 37. 



