for a System of Correlated Variables. 5 



shall be denoted by (j>(i\ . . . r n )dn . . . dr n , so that 



jj ...<j>(r 1 ...rjdr 1 ...dr n =l 



between the given limits; and I assume the function <f>(si . . . s M ) 

 to be finite and continuous for all values of s l ... s n within 

 the given limits. 



It may be the case that a certain function of s x ... s^ 

 say -^(s! ... s n ), is necessarily constant, and that the varia- 

 tions oi s v ... s n are subject to that condition. Or there may be 

 more than one such constant function. If so, the chance 

 <£($! . . . s n )ds l . . . ds n must be understood as subject to the 

 constancy of yjr, or other such functions as aforesaid. 



10. From this we can express the mean values of 5 : . . . s n , of 

 their squares, or binary products, or other functions, but it 

 will not be necessary to go beyond powers and products of 

 the second degree. Such mean values will be denoted as 

 usual by a bar. Thus 



Also 

 and 



•V 



6' S 



==jj . . . (/>(>! . . . sjs^ . . . ds ni &c. 1 



=jj . . . </>(>! . . . sjsfdii . . . ds n , &c, } • • (1) 



=JJ ...£(*!... s>A^> • • • d C &c - J 



11. The object of the first part of this paper is to prove 

 that, Si... s n being as above stated correlated inter se, then 

 for very great values of n, (j>(s 1 ... s n ) necessarily has the 

 form <£($! ... s n ) = Ae~ Q , in which A is a constant, and Q is a 

 homogeneous quadratic function of «i ... * n , involving both 

 their squares, and (as a consequence of the correlation) their 

 products, namely 



Q = i« lSl 2 + /3 uSl sz + K*/ + ■ ■ • + KV • • ( 2 ) 



and the coefficients a/3 are functions_of the mean values 

 s^...^, of the mean squares s{ 2 ...s } ?, and of the mean 



products ~s^s 2 ■ • • V<?> & c - 



This is a well known result for the case in which the 

 quadratic function Q is incapable of becoming negative. 

 It is necessarv, however, shortly to give the proof, which is 

 done in Parti. In Part II. I propose to show inter alia why 

 it is necessary for Q to be positive, and I shall then apply 

 the theorem to certain physical problems. 



