and its Application to the Molecular Motions of Gases. 29 



For shortness' sake we will introduce single letters instead of 



pq'i and pq'i 2 , putting 



Cv=2^J v ™&Uv=P v 7jl (4) 



Equations (II.), (IIa.), (Ill a.) then change into 



8E=s!&+S — Be, (He.) 



i dc 



te=^?81ogtf+sS& l .... (He.) 



w-jsjfc+xj*. .... (in*.) 



8U=-^'81ogtt + S^&. . . (IIIc.) 



Since the occurrence of the quantity U is characteristic of 

 these equations, we will name the theorem which is expressed, 

 only in different forms, by each of them the theorem of the mean 

 ergal. 



§ 3. As already said, each of the equations (I.), (II.), and 

 (III.) can be analyzed into as many partial equations as there 

 are independent variations on the right-hand side. We can, for 

 example, write equation (IIIc.) in the following form: — 



dV. t dTK , , dV« , dV . , dXJ , , B 

 1 r hu Y , 1 j- Su 2 , 1 j-8u n 



■+^+t 8C > + kC \ (5) 



If now we assume that all the variations are independent of one 

 another, we can suppose the factors of each variation standing 

 on both sides equal to one another. Applying this, first, to the 

 variations 8u, we get n equations of the form 



dV 1 — ^1 



or, 



dur* PA ' v u; 



dU dV 1— , M 



^JB^ 88 ^ (r,) 



Therefore the differential coefficient of the mean ergal according to 

 the logarithm of a u is equal to the part in question of the vis viva. 

 Equations (I.) and (II.) can be treated in like manner; and 



