536 



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



according as m is even or uneven. The rest of the determi- 

 nant is the same as in the first determinant for o>(n). 







(ii.) 







,(»)=<-)*-' 



1, 



1, 1, 



1, 1, 



1,... 





1-, 



0, 3-, 



0, -5', 



0,... 





o, 



1", o, 



0, 0, 



3',.. 





o, 



0, V, 



o, o, 



0,... 





o, 



0, 0, 



1", 0, 



0,... 





o, 



0, 0, 



0, 1-, 



0,... 



This determinant may be derived from the second deter- 

 minant for A r (n) by changing the sign of every element ±s r , 

 in which 5 is of the form 4m + 3. 



Expressions for E/(w), § 28. 





§28. For E/(V) we obtain the expressions :■ 





(*•) 





H/(n) =(-)»"' 



l', 2', &", 4T, b r , 



6-,... 





1, 0, 1, 0,-1, 



0,.,. 





0, 1, 0, 0, 0, 



1,... 





0, 0, 1, 0, 0, 



0,... 





0, 0, 0, 1, 0, 



0,... 





0, 0, 0, 0, 1, 



0,... 



Excepting the first row, this determinant may be derived 

 from the second determinant for E r (w) by putting r = 0. 



E/O) = (-)»-' 



(ii.) 



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



V, -2*-, -3 r , 0, -5 r , 6',.. 



0, V, 0, -2-, 0, -3',.. 



0, 0, 1, 0, 0,-2',.. 



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



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



The with element of the first row is or ( — 1)*0-D ? accord- 

 ing as m is even or uneven; the rest of the determinant is the 

 same as in the second determinant for a r {n). 



