386 Prof. L. Natanson on the 



si ti oris equivalent to those indicated by these investigators 

 could be enunciated in the form of a simple and very general 

 formula ; we venture to think that the fundamental principle 

 which it embodies is worth attention. Besides, it seems to 

 afford the proper foundation for an attempt to arrive at some 

 deeper insight into the laws of dissipation of energy. 



Part I. 



§ 1. Introductory . — Conceive a system : it may be either 

 finite or infinitely small ; it may be an independent system, 

 or it may be only a part of some other system. Let the state 

 of the system, at time t, be determined by the values of 

 certain variable quantities, q () and of their first differential 

 coefficients with respect to the time, s i or dgjdt. We shall 

 suppose that the energy of the system consists of two parts, 

 the first of which, T, is a function of the q { and the s fj homo- 

 geneous of the second degree with respect to the s v and the 

 second, say U, is a function of the g. only. Let 3 denote the 

 absolute temperature of the system : $ may be an independent 

 variable, or otherwise it must be a definite function of the 

 variables. Suppose that the quantities q v s i received certain 

 arbitrarily chosen infinitesimal increments 8q v Ss i ; the energy 

 T will then become T + oT, and U will becoine U + SU. Let 

 then %P.Bq i be the work done on the system reversibly, 

 during the transformation, by extraneous forces, and let SQ 

 or ItRjbq. be the quantity of heat simultaneously absorbed by 

 the system from the exterior ; P { will then be the generalized 

 or Lagrangian extraneous " force " in the " direction " of 

 the variable q { , and R^ will be the " caloric coefficient/' as 

 it is called by M. Duhem, or the generalized "thermal 

 capacity " of the system with respect to the variable q.. 



With respect to the quantity SQ we now make the following 

 assumption, which we shall find is in accordance with fact. 

 Let us suppose that every variation Bq. t.ikes the special value 

 dq { or sflt ; then the values of the variables q { will become 

 9i + dq { ; the energies T and U will become T + dT, U + dU ; 

 the work done by external forces will be XF i dq i ; and the 

 quantity of heat absorbed will be dQ or 2R^fy f . If now the 

 variables be allowed to return to their primitive values q v T 

 and U will resume their former values T, U, the external 

 work — 2P.tf^ will be done, but the quantity of heat absorbed 

 will generally not be dQ but a different quantity, say djQ. 



