of Thermodynamics, 245 



p for the position,, relatively to the centre of the matter con- 

 tained in the space, of a point which is fixed so long as bf=0. 

 Because the motion is steady, each particle of matter which suc- 

 cessively arrives at the point p assumes the velocity, direction, 

 and curvature of motion proper to that point. Let v be that 

 velocity, and r the radius of that curvature ; then for a particle 

 of mass unity, in the act of traversing p, 



actual energy of mass 1 = — = kr, . . . (1) 



where r is a quantity upon whose uniformity throughout the 



space the steadiness of the motion depends, and k a function 



of {m,f,p); and 



v 2 2kr 

 centrifugal force of mass 1 = — = ; . . (2) 



2k 

 in which ?-, and consequently — , are functions of {m,f,p). 



Now let the change denoted by Bf take place, and let the 

 steadiness of the motion be restored : let Bn be the length of 

 a line drawn through the original position of the point p y so as 

 to be perpendicular to the path of the stream of particles which 



formerly traversed p ; and let rn be the angle made by Bn with 



r. Then Bn and rn are both functions of (m, /, p, Bf). Also 

 the work done, or energy converted, for a unit of mass at the 

 point p, while the path of the particles that traverse^ is shifted 

 through Bn, is as follows : 



A 



v 2 j, a 2kr.8n.cosm ~ . - , - ~ - 



— .on. cos rn= =t x iunction ot (m, j,p, of). (3) 



The energy converted during the change Bf throughout the 

 whole space considered, is the sum of the quantities of energy 

 converted for each unit of mass within the space. But t by 

 definition is uniform ; and the sum of a set of functions of p is 

 a function of/ and m\ therefore the whole energy converted is 



A 



2n l)7) O^ 7*71 



r . 2 . — '- ! = X function of (m, /, Bf) ; . (4) 



and because Bf is indefinitely small, the preceding expression is 

 equivalent to the following : 



energy converted =t . function (m, /) . 8f=r . SF(m, f). (5) 

 Let t be called absolute temperature, and this is the second law 

 of thermodynamics. It is to be observed that / may be, and 

 often is, a function of t. 

 September 6, 1865. 



