Statistical mechanics 



23 



Consequently we have 



P = + l 



and 



/ = ± 1 . 



Whether / has tlie value + I or — 1 is for us indifferent, as only the absolute 

 value of the determinant is of interest. We find, by the way, (through making the 

 Z-axis coincide with the symmetry-line, so that I = 0 = m and « = 1) that /= — 1. 



16. The remaining determinants are now easy to compute. We get 



jlB\A'\ sint)-' 



^r, 'YJ sin B 

 I 



Substituting these values in (24) we get 



J. Ä;'*(m^ + m^Y -f fc^ co* sin i? ^ sin ■&' lc^{m^ -\- m^) (ü'^ 



k^{mj^ + ■)n^)o)^ sin i)- sin B' k*(m^ -f "'s)^ 4" à'^ m'^ 



But 



& = y- JB= B'; h - h'\ 0) = 0)' 



so that 



|/| = 1. 



Q. E. D. 



17. It would have been possible to compute 



more simply in the following manner. 



Consider first generally the transformation from rectilinear — X, Y, Z — to 

 polarcoordinates — r, b, a, so that 



X = r sin h cos a 

 Y = r sin b sin n . 

 Z — r cos b 



The lacobiana of this transformation is 

 (32) /F: f'f) = r^sin6. 



r, b, a 



Let us now, for a moment, introduce a parameter p and put 



2) = p sin Q- cos ']j = p ^ 

 q = p sin d- sin (jj = p tj 

 r = p cos 8- = p C 



