Statistical meclianics 59 



Consequently 



fJ. = lÄ. ., A. ., (c(') '£(2) 



where 



8''y(M,) dj''-f(i\) d''''£{w,) 8'^'f(?<2) d''--p{v^) d''"i{w.^ 



<V./,-, wv.fe g,(;^/.-i a^,^/, gy^y^ 9,^^fe 



^ 8Mi'' 8?(;,'- e^^'^ dv^'"- "div^^ ^ T'aW^.'V.fe 



where 



1 — 97? (Ml' + î^l' + <f 1' + V + "2' + '«2') 



SO that 

 and 



(131) ^('171 A-i t272A-2 — ^!i7iA:i -^(272A-2 • 



The product /^/g is thus represented as an yl-series having the generating 

 factor cc. 



I 



Making the change of variables from u^, v^, etc. to U^, V^, etc., the product 

 /j /g becomes a function of the new variables TJ^, l\, etc. Using the generating 

 function 



_ _1_ V.^+ K-f W/) 



^ ^ ' a«(27r)3^ 



we now get 



(132*) f^f^^ Z Bi,j, ^iu'i A-i 12J2 A-2 • 



Our next task is to give the relations between the B:s and the A:s. 



Substituting the expression above for f^f^ into (127), and observing that I is 

 only dependent on U.^, V^, W^, we find that the integration respecting U^, F^, 

 may be immediately performed. 



It is, however, 



/ M ,■272 A-2 d dV,dW, = 0 

 as soon as any one of the indices ^\, or is different from zero, because of the 

 fact that the functions f{U^), œ(FJ, ^(M^) as well as all their differential coefficients 

 vanish for C/^ (or , or W^) = ± 00 . Consequently we have to consider only 

 terms in (132*) for which 



h=Ji = ^h = ^- 

 In order to express the B:s by the A:s we start from the formula 



(133) f{u^,v^,w^; u^, v^, tv^)^^!!^, F^, TF^ ; U^, V,, TFg) 



which we derive regarding the different variables. 



