. relative to Stationary Motions. 237 



deavoured to apply the equation to the science of heat. The 

 subject, however, appears to me, even from a purely mechanical 

 point of view, to be of so great importance that I have taken 

 the trouble to pursue it further in this direction, and to give the 

 equation as general a form as possible, by which also its appli- 

 cation to special cases is of course facilitated and gains in cer- 

 tainty. The result of this investigation I take leave to commu- 

 nicate in the following. 



1. It will serve our purpose first to briefly cite the equation 

 in the same form as hitherto in order to connect with it our 

 further considerations. 



Given a movable material point of mass m, which, under 

 the influence of a force that has a force-function or, according to 

 another nomenclature, ergal, moves in a closed path. Let the 

 ergal be denoted by U, the velocity of the point by v } and its 

 period by i. Of the quantities which are variable during the 

 motion the mean value shall be taken, which shall be signified 

 by a horizontal stroke above the symbol representing the va- 

 riable. 



Besides the originally given motion of the point, let us further 

 consider one deviating infinitely little from it. The deviation may 

 be occasioned by the point having begun its motion from another 

 place, or having had at the commencement other components of 

 velocity than with the original motion. Besides, the ergal may 

 have undergone a change. The latter we will imagine expressed 

 by this — that in the function U, in addition to the space-coor- 

 dinates, one or more quantities c l9 c 2 , &c. occur, which are con- 

 stant during each motion, but may change at the transition from 

 the one motion to the other. 



If now, for every quantity that comes into consideration, we 

 regard the difference of the two values which it has in the ori- 

 ginal and in the deviating motion as the variation of the quan- 

 tity, and indicate it by 8, and for abbreviation collect the terms 

 which relate to the quantities c v c 2 , &c. under the sign of sum- 

 mation, the equation in question reads : — 



8U-2^& = 5g-5 + wl ^g logf . (1) 



2. In order to generalize this equation, it might be assumed 

 that instead of one material point several are given, all moving 

 in closed paths. If all their periods were equal, and changed 

 in the same ratio on the one motion passing into the other, the 

 extension of the equation to such a case would be of itself intel- 

 ligible at once; but if the periods are different and change in 

 different proportions, special considerations are needed for this 

 extension. 



