Law of Distribution of Energy. 155 



dfifo ) 

 vanish. In order to find ■ J we integrate according to 



each of the other variables between the limits ± go . The 

 result will be proportional to e"< A *i 2+Bz " Ir) , where A does not 



concern us, and , oc B, or the condition that j shall 

 ' dx r ' dx r 



A 

 vanish is that B = 0. Also B= -r— — , where A is the deter- 



]rrl 



rninant of r rows S + 2a 1 2a 2 . . . . and A lrH a coaxial minor. 



(15) We have, then, to find a relation between the coefficients 

 a ly bi2, &c, which will make A lr vanish compared with A lr <rl 

 without making all the coefficients b vanish which connect 

 the variables in any one of the original associations with those 

 in any other. That can be done in many ways. It will be 

 sufficient to give one example. Let us suppose all the a's 

 equal to each other, and all the 5's of the form b p p~[ equal 

 to one another, and all other 6's zero. 



Further, let 6 12 =5 23 = &c. =2a0. 



Then the determinant assumes the form 



A = 



2a b . . 



b 2a b . 



. b 2a b 



. . b 2a 



and if f r be its value for r rows, the law of formation is 



/,=/_, -* 2 /,_ 2 (A) 



Since every/ is to be positive, f r _ l > f r . Ultimately as r 



increases the ratio J ^~ becomes constant. Let its constant 

 value be X. Then equation A gives 



or Vl-46 2 



2 -i- 



of which the positive sign will be taken. This result shows 



that if/ , or A, be always positive, 6 2 cannot be greater than £ 9 



r+ 1 

 or 6 greater than J. Also if = J, we easily find A = 9r (2a) r , 



and therefore . r— 1 /Q ._ 2 



&lrr] — 2r—2 V^ a j 



