168 Prof. R. Clausius on the Second Axiom 



equations referring to heat, we denote by SL the work performed, 

 we can put 



8L=SU (20) 



When, on the other hand, the change in the motion is occa- 

 sioned by the fact that the force operating on the point has 

 been changed, the thing is not quite so simple, but requires 

 special consideration. 



6. As said above, the alteration of the force may be imagined, 

 mathematically, conditioned by a constant which occurs in the 

 ergal changing its value by an infinitely small quantity. With- 

 out going into this more closely, we will only make the following 

 assumption, which comes to essentially the same thing. The 

 ergal, which with the original motion was represented by the 

 function U, shall with the altered motion be represented by the 

 sum U + pY, in which V signifies any other function of the co- 

 ordinates, and (j, an infinitely small constant factor. 



In regard to the occurrence of the increase fiV, however, we 

 will preliminarily make the subsidiary assumption that the in- 

 crease does not take place suddenly at a certain moment, but 

 proceeds gradually, during an entire revolution, — the infinitely 

 small factor which stands before V increasing uniformly during 

 that time, so as just to reach the value jju at the end of the revo- 

 lution, and then preserving this value constant during the suc- 

 ceeding revolutions. Accordingly, during one element of the 



time dt, the factor will increase by ~~ } or, which is the same 



thing, during an element of the phase deb the factor will increase 

 by fi d(f>. 



In order now to determine the work-variation SL which cor- 

 responds to the entire transition from the one stationary motion 

 to the other, we must first give the work-variation for any selected 

 individual phase <fr v For this purpose let us consider the move- 

 able point from the moment when, in its revolution in the ori- 

 ginal path, it just passes the place which belongs to the phase 

 (f> v and let us follow it hence through two entire revolutions. 

 These comprise, 1 st, the remainder of the revolution commenced 

 in the original path ; 2nd, the revolution during which the alte- 

 ration in the ergal takes place; and, 3rd, the commencement of 

 the revolution in the new path as far as the phase <£ r The work 

 done during this time may be divided into two quantities, corre- 

 sponding to the original ergal U and the increase fjN. 



The first quantity is expressed very simply ; for if Uj denotes 

 the value of U in the original path belonging to the phase <p l} 

 and Uj + SU! the value belonging to the same phase in the new 

 path, then SUj is the first quantity of work. 



