TRANSACTIONS OF SECTION A. 623 



in -which 



R = Ai^i* + Bi2^i^, + &c. -i>\e^+ . . . +r,Ai)V'^, 

 in which A, Bj„, &c., are known. 



And performing the integrations according to ^j . . . ^,„ we obtain as result the 

 expression 



in which the coefficients are known quantities wlien the functions /a, ft,, &c., and 

 the X's, ju's, &c., are known. 



It thus appears that the chance of the linear functions 



Xirtj + Xj«., + . . . 



f^A + FA+ • • • 



lying respectively within the limits 



}\ . . . i\ + d)\, 



r.^ . . . r.^ + dr^, Sec, 



is a quadratic function of {rj—i\), (?•; — r.,), &c. The exponential form always 

 occurs in the result, whatever he the forms of the functions /„, fi, &c. Only the 

 coefficients of the quadratic function are determined by the forms of these func- 

 tions. 



Paet II. 



9. Up to this point we have assumed that any one of our sets of correlated 

 variables, e.g. a, . . . a,,, or, as we may call it, any one ' association,' is independent 

 of the variables in any other. 



It is now proposed to treat the several associations «, . . . a„, Sj . . . i,„ ifcc, as 

 representing the state of the same material system, defined by n variables, at succes- 

 sive points of time. That is, if .t, . . . .t„ be the variables defining the system, 

 fcda^ . . . dan is the chance that when t = they shall lie within the limits 



.x'l = </, and .i\ = flj + da^, &c. 

 .r„ = a.j and a-^ = a., + da.j, &c. 



Similarly /brfJ, . . . db,, is the chance that when t = T they shall lie within the 

 limit .i\ = b^ and .r, = b^ + db^, &c., and so on. Then generally the variables b^ . . . b„ 

 are not independent of flfj . . , an, because given that when t = 0, a\ = a^, &c., that 

 aflects the chance that when t = T, .Tj shall =6,, &c. 



We cannot therefore at once apply our investigation of Part I. to the series of 

 functions /a, _/i, &c. 



10. It is now shown that in cases where the condition of complete independ- 

 ence is not satisfied, another condition, called the modified condition of independence 

 may be satisfied. That is, namely, that although the 6's are not independent of 

 the a's for such values of the time r, yet it may be the case that when r= or >T 



where T is a definite time, the Vs, are independent of the a's, or -~ vanishes. It is 



da 

 shovra that if that modified condition be satisfied for every pair of associations, or 

 states of the system, separated by an interval of time not less than T, we may 

 legitimately apply the method of Part I. to the whole series of associations, N" in 



T 

 number, at intervals of time ht or ^, exactly as if they were mutually inde- 

 pendent. If, that is, we form n linear functions of the type 



Xi(7j +Xo(rti +S«()+ . . . 

 /Mja^ + \i^c^(a„ + ha.^ + . . . 



and find the chance that they shall respectively lie within the limits r, , . . r^ + dr , 

 &c., by the method of Part I. we obtain correct results so far as the exponential 

 form is concerned. The result is of the form f-^di\ , . . drn, \iiih R a quadratic 

 function oi{i\ — i\), ('•j-'j); &C' 



