242 Prof. J. H. Jeans on Temperature-Radiation 

 Thus we have 



N ! ??~ N Co" A 



a 1 ! a 2 ! ... <z» 



! * 



Using Sterling's approximation for the values of N !, a x \,&c. 

 this gives 



log/(f„ & -. f»)=C- 2 (*+*) log^' . (18) 



where C denotes the constant 



— log n ^— log (27rN) 4- ?i log ft) + log A. 



Write equation (17) in the form 



a s = a + $ s , (19) 



where 



a — — , IS = &) 1 to sin - — -, 



then, since S s maybe supposed small compared with a , we have 

 (^l)logf = ( fl o + i + S s )lo g (l + |) 



_ -^0 1 qj 2 



as far as terms of order (S s /a ) 2 « On summing, we obtain 



na. 



2a. -1 s Z n 



s=l -^ ^O s=l 



2 a — 1 ho 2 

 = Z^T--9~(a +& + ••• Cn). 



Replacing — .°— 2 - by ^ — in this, equation (18) becomes 

 4a " -<2 



iog/(6, E* ...f»)=c-«(fr+f 2 2 + ... £, 2 ), 



ftft) 2 XI 



where # = i — = i — , 



±a 4:V 



and so we obtain as the law of distribution 



/(&, 6...&)«i*...*.=A«-'<«i , +^ , + - 5»VfWf 2 ... <$. 



.... (20) 



