18 Mr. S. H. Burbury on the Law of Probability 



functions of inter alia a quantity v, which, like E, is " langsam 

 veranderlich." And that they possess the property that as v 

 diminishes every b 2 increases cceteris paribus. That is, the 

 b coefficients may, as mentioned in art. 8, he functions of the 

 instantaneous distance between the variables to which they 

 relate, e. </., b pq may be a function of p, the distance which at 

 the instant separates u p and n g < or the things to which u p and 

 iiq relate, from each other. If then the distances p diminish 

 by the diminution of v, which will generally be the case if v 



denote the volume of the system, the condition that every — r 



. db 2 . . dv 



is negative will be satisfied, if every — is negative. 



32. Since the two functions ^)(rj ..r n ) and Ce~®, or let us 

 say <f> and F, are equal to one another throughout a certain 

 range of values of v, for all of which D is positive, it follows 



that at every point or value of v within that range ~ = - ■.- , 

 and all the derived coefficients of $, as -^| 5 &c. ; are respect- 

 ively equal to the corresponding derived coefficients of F. 

 Therefore, by Taylor's theorem, <£ = F for all values of v for 

 which that theorem can be legitimately applied, with initial v 

 within the given range. If, however, when a certain value of v, 

 say v = V, is reached, a discontinuous change takes place in v, 

 Taylor's theorem will in general at that point fail, and the 

 equation </> = F will cease to be true. 



33. Since e~ Q is maximum, 



|^a f^a &c=0 



au x du 2 



and since E is constant, 



?n 1 « 1 ()w 1 -j- m 2 u 2 ~d u 2 + &c. = 0. 

 Whence, if \ be the indeterminate multiplier, 



dQ , dQ _ x 



du L du 2 



or in the notation of (24) 



«i^i 4 b 12 u 2 + b 13 n 3 + &c. = Xniiiiy " 



b 12 u ± + a 2 u 2 + b 2S u 3 + &c.=Xm 2 u 2 I . . . (A) 



&c. 



