164 Prof. It. Clausius on the Second Axiom 



In reference to the latter, however, a special remark must be 

 made, which is of importance for the following. If the altered 

 motion is to be compared with the initial one in such a manner 

 as to show how the values of x in the one differ from the corre- 

 sponding values of x in the other motion, we must first settle 

 which values of x shall be regarded as corresponding to each 

 other. For this purpose, any two points infinitely near each 

 other in the two paths may first be taken as corresponding 

 points. Starting from these, in order to obtain the remaining 

 corresponding points we take as a measure a magnitude which 

 changes in the course of the motions, and settle that those points 

 in the two paths which belong to equal values of the measuring 

 magnitude are corresponding points. As measuring magnitude, 

 however, one must be chosen which for an entire revolution has 

 equal values in both paths ; for through an entire revolution the 

 moving point always arrives again at the chosen initial point in 

 each of the two paths, and these we have already taken as cor- 

 responding points. 



We will now determine the measuring magnitude in the fol- 

 lowing manner. Let i be the time of a revolution with the 

 original motion, and t the variable time which the moveable 

 point requires in order to pass from the initial position to another 

 one; then we will put 



/=i.<£ (3) 



For the altered motion, let the time of a revolution be denoted 

 by i'j and the variable time, reckoned from the point's leaving its 

 initial position, by i' ; then we put 



If, now, </> has equal values in both expressions, t and t ] are cor- 

 responding times. The corresponding times being in this man- 

 ner determined, the corresponding points of the two paths and, 

 accordingly, the corresponding values of x, y } z, &c. follow of 

 themselves. 



The magnitude (f> we will call the phase of the motion. During 

 one revolution the phase increases one unit. With further in- 

 crease, the phases which differ by a whole number of units may 

 be regarded as equal, in the same sense as angles which differ by 

 multiples of 2-7T. 



Subtracting the first of the two preceding equations from the 

 second, there results 



t>-t=(i J —i)<l>. 



The difference t' — t is the variation of t, and the difference 

 i 1 — i the variation of i. Denoting these by St and Si, we can 

 write 



St-Si.cj) } (4) 



