Mobius's Theorem on the Reversion of certain Series. 535 



Expressions for A/(w) ; § 26. 

 § 26. The corresponding expressions for A/(n) are: — 



(i.) 



1', 2', 3', 4', 5', 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,... 



A/0)=(-)»-' 



Excepting only the first line, the elements of this determi- 

 nant may be derived from those of the second determinant for 

 A r (n) by putting r=0. 



(ii.) 



A/0)=(-)-' 1, 0, 1, 0, 1, 0,... 

 1, -2',-3 r , 0,-5', 6',.. 

 0, 1', 0,-2% 0,-3',... 



o, o, r, o, o,-2-,... 

 o, o, o, r, o, o,... 

 o, o, o, o, r, o,... 



This determinant differs from the second determinant for 

 <r r (n) only in the first row, which consists of 1 and 

 alternately. 



Expressions for E,.(n), § 27. 

 § 27. For E (n) we obtain the expressions: — 



CO 



E, (»)=(-)»-' 



1', 0, 



-3', 0, 



5', 0,... 





1,-1, 



-1, 0, 



-1, !,••• 





0, 1, 



0,-1, 



0,-1,... 





0, 0, 



1, o, 



0,-1,... 





0, 0, 



0, 1, 



0, 0,... 





0, 0, 



0, 0, 



1, 0,... 



The with element of the first row is or ( — l)i( m -Vm r 



