Statistical mechanics 65 



Suppose first k to be an odd number, then, according to our assumption, i -\- j 

 is an even number. Then we have 



cos = cos £ + ^3 cos 'èB + cos bB -\- ... 



sin >'B = sin B -\- B^ sin 3JS + B^ sin 55 + . . . 



and the product has the form 



C^sm2B -\- C^sm4B ... 



TZ 



which integrated between ± ^ gives the value zero. 



Suppose next li to be an even number and hence i^j an odd number and 

 consider the integral 



cos 'J. sln^'J^ dA 



0 



where now i j is odd 



1) i an even number: 



cos = + cos 2,A -|- cos 4zl + . . . 

 sin i A = sin A + sin o A B^ sin 5 J. . . . , 



the product is a sinus-series, the integral vanishes. 



2) i an o(!!(i number: 

 even here does the integral evidently vanish. 



We have hence found that integrals of the form : 



/ Uj W^<^ IdUdVdW 



vanish as soon as i -\- j -\-lc is an odd number. 

 But <ï>,y/f is a sum of terms of the form 



Ui' VJ' W' <E> 



where i' -\- j' + ^' is an odd number simultaneously with i j Ic 

 Consequently it follows that 



«..,=0 



as soon as i -\- j -\- k is an odd number. 

 Let us next consider the coefficient 



^200 ■ 



We have 



^200 -"-200 ^ 



so that 



Lnnds Universitets Årsskrift. N. F. Avd. 2. Bd 28. 



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