Law of Distribution of Energy. 145 



It is then shown that 



X=l if 0=0, 

 X<1 if 0=£O; 



and therefore in forming X 1 *", N being very great, we may- 

 neglect powers of 6 above 6' 2 , and so 



XN 2 



and 



F(c)=C old e ^ 2 '-, 



which is proportional to 



i 2 - 

 € ; 



and the constant C is found from the condition that 



i 



¥(c)dc=l. 



(2) That proposition has been extended as follows. Let 

 #i, x 2 . . . x n be n mutually dependent magnitudes, and let 

 the chance that they shall simultaneously have the values 

 denoted by x\ . . . x 1 + dxi, &c, be. 



f(xi . . . x)dx 1 . . . dx 



»> 



where f{x\ . . . a? J is a function which vanishes for infinite 

 values c of the variables x x . . . a? w , and which is not altered 

 when they all change sign together. Let us call the simul- 

 taneous occurrence of x x . . . a? an association. The limits 

 for each a? shall be + co . Then we have 



jj • • • /(#i • • • xjdxi . . . dx n = 1, 



JJ . . . f{x x . . . x n ) Xidxi . . . dx n = &c» 



Let \\ . . .f(x x . . . x^x^dxi . . . dx n = x x 2 y 



jj . . ./(#! . . . xjxix^dxi . . . dx n = x x x 2 &c. 



Let there now be N such associations, N being very large, 

 and let each be independent of all the others. The variables 

 in the first association shall for distinction be called x n , 

 x i2 . . . x ln , those in the second association x 2i , x 22 . . • # 2 n> an( ^ 

 so on to x m , x m . . . x Nn . 



Phil. Mag. S. 5. Vol. 37. No. 224. Jan. 1894. . L 



