for the Theoretical Proof of Avogadro's Law. 315 



Hence n 



P=8ttW^4 \ I J^f{v',t)F(Y',t)rT'8*&dVdTdSdO.(14) 



Jo o'o Jo Jo 



Maxwell* has already given an equation which in our 

 notation is 



Ai/ 2 V'V = « 3 V 2 t (15) 



He has, it is true, indicated a proof of this equation, but 

 has not clearly shown its truth. 



I have proved this proposition more fully, and obtained a 

 similar one for polyatomic molecules ; on which account I 

 have also assumed it in my treatise on the thermal equilibrium 

 of gases upon which external forces act. The most general 

 proposition, in which all similar propositions are included, is 

 shown in Maxwell's paper " On Boltzmann's Theorem." t 

 But since all the necessary formulas are now at our disposal, 

 I will here verify equation (15) by direct calculation of the 

 functional determinants. 



If we first of all introduce, instead of V and T, the variables 

 r and G, we obtain Y 2 dYrdT=r 2 drydQ, as is geometrically 

 evident. So also 



Y /2 dY'T f dT' = r 2 dry'dG'. 



Since r is not altered by impact, we introduce, instead of dv, 

 dG, dO, the differentials of the three variables, 



, 2 2 , 4Mvr , t x , 4MW 



y = (m + M) v 2 + 2Mvrg + Mr 2 , 

 z=vyco. 

 The calculation of the functional determinants gives 



da dy dz = ^^ [(m + M.)v {2gsa - ys 2 o + ya 2 o) + 2Mrs<r] dv dGc dO. 



Further, from the spherical triangle RKB/ (fig. 1), we have 



co : a>'== sin W£lK : sin RI2K ; 

 and from the triangles Pflv and PO?/, 



sinROK : y =v : OP, 



sinR'&Kryr^XlP; 



* Phil. Mag. [4] vol. xxxv. (March 1868). 



t Cambridge Phil. Trans, vol. xii. part 3, p. 547 (1879). Wied. Beibl. 

 v. p. 403 (1881). 



