Statistical mechanics 



79 



Using the notation 



(188) 



we substitute in (187*) 



V - 



A T 



^000-^ 



^200 



It 



^020 = ^ « 



It 



A — 



002 ^ ^ ' 



and obtain for determining the quantities X, a, h, c the relations 



(X + 2v) rt — V h — V c = 0, 



(189) — v« + (X + 2v)fc— vc = 0, 



— V rt — V + (X + 2v) c = 0 . 



Putting: 



X = sv 



we get the » equation in s>-> : 



or 



(190) A= + 2, — 1, - 1 =0 



- 1, 5 + 2, - 1 

 -1, -1, 5 + 2 



A = s{s -f 3)2 = 0, 



6- = 0 



s = — 3 (double). 



To each root corresponds a system of coefficients a, b, c, of which one is 

 arbitrary. 



For s = 0 we get 



so that the roots are 



VCo = 0 



— va^ -f 2\ibQ — vCy = 0 



— va^ — v&Q -|- 2vCo = 0 

 which equations have the solution 



^0 ~ ^0 ~ ^0 



If the root s = — 3 is treated in the same manner, we get three identical 

 equations : 



0 = v(a + i + c). 



But s = — 3 is a double root. According to a known theorem in the theory 

 of linear differential equations with constant coefficients, the corresponding integrals 

 have the form: 



