302 On Boltzmann's Theorem on the average Distribution 



We have a special case if some of the r's are identical with 

 some of the «'s — if, for example, we retain the ^-coordinates 

 and transform only x and y into polar coordinates. Suppose 

 that t , i = «i, v 2 = ii2, • • • i'k=Uk, but that Vt+i...v m are given 

 functions of u i} n 2 , . . . u m ; then the functional determinant is 

 simplified to 



dv 1 dv a ...dv m =d^du 2 ...du m , 2 + ^±- 1 ...$2t. . (2) 



- dUk+i dum 



We may now apply this general formula to the former one. 

 Instead of u x . . . u m let us put the 2n + l Hamiltonian inde- 

 pendents q x . . . q nj q\ . . . (/„ E ; for i\ . . . v k let us put q 1 . . . y„; 

 but for Oi + , . . . v m let us put p x . . .^>, f T. Then equation (2) 

 becomes 



cZ</! . . . dq n diii . . . dp n dr = 



dft. ..*,.,//,. ..*/.<ffiS ± ^...^^. • • (3) 



Let us now in the generally-applicable formula. (2) intro- 

 duce other special values, viz. for u x . . . «,„ the Hamiltonian 

 independents again, but substitute q\ . . . q' n for v 1 ... v k , whilst 

 for v k+ i . . . v m we substitute the variables p\ ...j/ n r\ which 

 indeed, according to Hamilton's method, are also functions 

 of the independents introduced by him ; consequently equa- 

 tion (2) is applicable to this case just as much as to the 

 former. Equation (2) becomes by this substitution, 



dq\ . . . dq' n dp \ . . . dp' n dr = 



«-«.+~+**±%...%* • ■.« 



The reader is advised to write down the functional determi- 

 nants of equations (3) and (4) at length, and then for each 

 member of the functional determinant of equation (4) to sub- 

 stitute the value which it would have according to equa- 

 tion (1). We shall then have, except for sign and for an 

 exchange of horizontal and vertical lines, exactly the functional 

 determinant of equation (3). The two functional determinants 

 have therefore the same numerical value ; and since we are 

 here concerned simply with this, and in the equations (3) and 

 (4) the products of the differentials of the right sides are iden- 

 tical, it follows from these equations that 



dq 1 . . . dq n dp 1 . . . dp n dT=dq\ . . . d r q„ dp' x . . . dj/ n dr. 



Dividing each side by dr. we obtain 



dq x . . . dq» dp t . . . dp n tsdq / 1 . . ■ J</n dp\ . . . dp' n , . (5) 



