of the Mechanical Theory of Heat from the First. 25 



tion), but requires a special demonstration, and which is 

 equally important with the other, because it is only the combi- 

 nation of the two that can form a complete basis for the mecha- 

 nical theory of heat." 



It is evident that Clausius's statement directly contradicts 

 the assertion of Rankine. The latter maintains that the Second 

 Proposition can be deduced from the First ; while Clausius says 

 that the Second is not contained in the First. 



The purpose of the present paper is to examine this important 

 question, and to prove that, without any further hypothesis, 

 the Second Proposition can be deduced direct from the First. 



Every material body can be regarded as an aggregate of an 

 immense number of material points, which, under the influ- 

 ence of internal and external forces, move according to certain 

 unknown laws. We will not bind ourselves to any hypothesis 

 concerning the nature of this internal motion ; our sole and 

 single assumption shall be that the particles are not at rest, 

 but in some kind of motion. 



To characterize this motion, or, in other words, to determine 

 the thermic condition of a body, it is known that two indepen- 

 dent variables are necessary. Let them be denoted by f and 

 7) ; then the state of the body remains unaltered so long as 

 neither f nor 77 changes ; but as soon as either f or 77 or both 

 together undergo a change, the state of the body in question 

 changes at the same time. 



As long as the thermic condition of a body remains un- 

 changed, so long do, according to my conception, the total 

 living force (T) by itself, and the total potential energy (U) by 

 itself , remain constant. 



If, therefore, the state of the body is given by the two inde- 

 pendent variables £ and 77, then, as long as Sf=0 and £77 = 0, 

 in like manner 



ST=0 (1) 



and 



SU=0. . (2) 



Let the mass of any material point whatever of the body be 

 m ; further, at the time £ = let x, y, z be its rectangular co- 

 ordinates, a/, y' , z' the components of the velocity, and x" , y", 

 z" the acceleration-components. Already in the first instant 

 — say, after the time St, the position, velocity, and acceleration of 

 the point will be different, viz. x + 8x..., x' + Baf. . . , x" + hx" . 

 But, from equation (1), must 



ST^mix'Bx'+y'Sy' + z'Sz')^; .... (1a) 



