534 



Mr. J. W. L. Glaisher on Applications of 



This determinant may be derived from the second determi- 

 nant for a(n) by replacing every element ±m by ±m r . 



Expression for v(n), § 24. 

 § 24. Putting r=0 in either of the determinants of the 

 preceding section, we find: — 



v(n) = (-)»- 1 



1, 1, 1, 1, 1, 1,... 

 1,-1,-1, 0,-1, 1,... 

 0, 1, 0,-1, 0,-1,... 

 0, 0, 1, 0, 0,-1,... 

 0, 0, 0, 1, 0, 0,... 

 0, 0, 0, 0, 1, 0,... 



Expressions for A r (re), § 25. 



§ 25. The formulae of § 8 lead to the following determinant- 

 expressions for A r (ra): — 



(i-) 

 1', 0, 3", 0, 5-, 0,... 



<M»)=(-)* 



1,-1, 



1, 0,-1, 



1, 0, 



0,-1,... 



o, 



0, 0, 1, 0, 0,-1,... 



0, 0, 0, 1, 0, 0,... 



0, 0, 0, 0, 1, 0,... 



This determinant may be derived from the first determinant 

 for ov(n) by replacing the elements which involve even 

 numerals in the first row by zeros. 







(ii.) 





A,0) = (-)"-' 



1, 



1, 1, 



1, 1, 1,.. 





l r , 



0, -3', 



0,-fr, 0,.. 





0, 



1'-, 0, 



0, 0,-3-,.. 





o, 



0, V, 



0, 0, 0,.. 





o, 



0, 0, 



1', 0, 0,.'. 





o, 



0, 0, 



0, 1', 0,.. 



This determinant may be derived from the second deter- 

 minant for <T r (n) by replacing every element involving even 

 numerals by zero. 



