of the Mechanical Theory of Heat, 163 



We have now to find for this equation an explanation founded 

 on mechanical principles. For this purpose the above theorem 

 concerning the virial furnishes a clue, inasmuch as it indicates 

 the nature of the considerations which must be employed. But 

 it is not alone sufficient ; the investigation requires in addition 

 certain new and peculiar developments, which are to form the 

 subject of the present memoir. 



2. To begin with a case as simple as possible in relation to 

 the kind of motion, and thereby facilitate the view of the mode 

 of consideration which here comes into use, we will first suppose 

 a single material point, operated on by a force which may be 

 represented by an ergal — that is, the components of which, re- 

 ferred to three rectangular-coordinate directions, are expressed 

 by the partial differential coefficients of the three coordinates of 

 the point, taken negatively. Under the influence of this force, 

 the point will have a periodical motion in a closed path. 



Now let us imagine this motion to undergo an infinitely small 

 alteration, resulting in a new periodical motion. This conver- 

 sion of the motion can be occasioned in three ways : at any place 

 in the path, through a passing external influence, the velocity- 

 components -j-t -j-i and -7- may be infinitesimally altered, and 



then the point may again be left to the operation merely of the 

 original force ; or an infinitesimal alteration may occur in the 

 force operating on the point — for example, a change in the value 

 of a constant occurring in the ergal. The third cause of con- 

 version of the motion will not occur in our considerations on 

 heat, but is of interest for a comparison which we shall make 

 further on : it is the point being compelled to describe a path 

 somewhat deviating from the one chosen by itself — which is also 

 connected with an alteration of the force, because then to the ori- 

 ginal force is added the resistance which the new path-curve has 

 to perform. 



We will now investigate whether, in all these circumstances, 

 there exists a universally valid relation between the alterations 

 of the different magnitudes occurring in the motion. 



3. The alterations undergone by the coordinates of the point, 

 its velocity- components, the components of the force, &c. shall, 

 as differentials of those magnitudes, be denoted as usual by the 

 prefix d; so that, for example, dx will signify the variation in x 

 during the time dt. On the other hand, the alterations of those 

 magnitudes which result from a different motion taking the 

 place of the original one shall be called variations of the magni- 

 tudes, and be denoted by prefixing the letter 8 ; so that, e. g., 

 the difference between a value of x in the original motion and the 

 corresponding value in the altered motion will be signified by tx. 



M 2 



