106 MAJOR P. A. MAOMAHON: MEMOIR ON THE 



Art. 29. We may now obtain properties of the sub-lattice functions 

 For, in the relation 



put I = -l-m-n and observe that (-s) = -oT'(s). 

 We find 



/_ \ 



and | NM | , | NM ! , . . . | NM ! , is unaltered to a factor ^ 

 Hence the identity 



Herein putting 



/ = we find L, = x''* aat L , 



1=1 Iv^.aj^-'- 



1 = 2 L^ a =x^- n ^ n 



1 t 







exhibiting an elegant property of the functions. 

 From the relation 



&iNM|,|NM| 1 ...|NM|, 

 we find, giving I the values 0, 1, 2, ... in succession, 





