and its Application to the Molecular Motions oj Gases. 31 



(recording to c^ the variable ergal represented by q 1? q 2 , . . . q n , c v 

 c>, §c. and take the mean value of this differential coefficient. 



In'a corresponding manner, from equations (I.) and (II.) are 

 obtained partial equations of the form : — 



dE aW ,_ , 



%^==P, ..... (8.) 



in which E is to be considered as a function of e v e , . . . e n > c v c 2 , 

 &c, and U — T as a function of i v i 2 , . . . i n} c l} c 2) &c. 



§ 4. As an example for the application of the theorem of the 

 mean ergal, in my former memoir I treated of the motion of two 

 material points under the influence of their reciprocal attraction. 

 Here we will first consider another case, which, on account of its 

 great simplicity, is well suited to make the thing evident. 



Given a material point with the mass m, its position determined 

 by rectangular coordinates which we will designate by x v oc 2 , 

 and x 3 . On this point a force acts, the components of which, 

 taken along the directions of coordinates, X p X 2 , and X 3 are 

 proportional to an odd positive power of the coordinates, so that, 

 if n is a positive even number, we can put 



v .n— 1 n .n—l . /v>«— 1 



X,— .^-, X,=.-«V' X 3=-»V'- • (9) 



where c lt c 2) c 3 represent three positive constants ; and accord- 

 ingly we can form for the ergal the equation 



'-©*♦ (?)"*©■■■ •■ <>») 



As in this case eaoh component of the force depends only 

 on the coordinate which belongs to it, and not on the other 

 two coordinates, we can consider the motion in each coor- 

 dinate-direction separately. The part of the vis viva which 



m rdx\ 



refers to a single coordinate-direction is — I — J or, written dif- 



ferently, ^-^' 2 ; and the" portion of the ergal referring to the 

 same coordinates we will designate by H, so that we can put 



=-©"• M 



Accordingly equation (I.), if referred only to the motion in the 



