20 Mr. S. H. Burbuiy on the Law of Probability 



mim 2 ...m n is the product of n factors, which are the n roots 

 of equation (B) ; also that the sum of the coaxial minors of D 

 each of t 2 constituents, and each divided by the proper factors 

 m, t in number, is equal to the sum of the products of the 

 n factors of A n taken t together, whatever number t may be less 

 than n. For A„_i is by the theory of equations the sum of 

 the products of the n roots taken n — 1 together. And it is 

 also the sum of the coaxial minors of D each of (?i — l) 2 

 constituents divided as aforesaid, &c. It is possible to 

 construct a determinant which shall possess this property, 

 but its demonstration would be too long for the present 

 paper. 



Since Q=\??E, and E is positive, Q has the same sign 

 as X ; and since we are limited to positive values of Q, we are 

 limited to positive values of X. That is, equation (B; can have 

 for our purpose no negative roots. But D is proportional to 

 the product of the n roots of (B). Therefore we are limited 

 to positive values of D. 



35. Every maximum value of e~^ determines a state of 

 stable equilibrium for the system, that is, if Ui...u„ are 

 changing rapidly, a state of stationary motion. There may 

 be many such states corresponding to different real roots 

 of (B). We may call the state corresponding to any particular 

 real root X, the state X. If X x and X 2 be two real roots, and 

 X 2 >X l5 then the state Xi is more probable than the state X 2 in 

 the ratio e (Aa— *i) mE . If X 2 — X x is not nearly zero, and n very 

 great, then the state X x is more probable than the state X 2 in 

 a very high ratio, and the more so as nE increases. It 

 follows that very great values of X are in a very high degree 

 improbable, and may be neglected. The state corresponding 

 to the least root of B is the most probable of all the states of. 

 stable equilibrium — I define it to be the normal state. 



36. I think the method thus investigated is applicable to 

 determine the normal state of any material system whose 

 parts mutually influence each other, and therefore become 

 correlated. And is not this the case with almost all material 

 systems in nature ? A rare gas is perhaps the only known 

 system to which the assumption of no correlation has been 

 or can be legitimately applied. Further, a system of mutually 

 acting, and therefore correlated, parts is a living system. 

 On the other hand, if Q be reduced to a sum of squares, as 

 Boltzmann's H theorem professes to prove, it would be, if 

 left to itself, a dead system, for which no further change is 

 possible. 



