Mobius's Theorem on the Reversion of certain Series. 525 

 whence we find 



A/ (re) -2«'A/Q + 2aWg)-2a^A/(j- c ) + ... 



=0 or 1, according as n is even or uneven ; 



A/ W -2A/(|) + SA/(^)-2A/( 5 ^) + . . ..«, 



where A, B, C, . . . have the same meaning as in the preceding 

 section. 



Taking, as an example, n=6, the formulae give 



A/ (6) - 3*A/ (2) - 2'A/ (3) + 6*A/ (1) = 0, 



A/(6)-A/(2) = 6- 

 viz. 



2 r + 6 r -3 r . 2 J --2 r (l/ + 30 + 6 r =0, 



2 J, -f6 r — 2 r =6 r . 



If ?'=0, A' r (n) denotes the number of divisors of n whose 

 conjugates are uneven. 



Formulae involving E r (n), § 10. 

 § 10. Let e 2 = and e p = ( — l^P-Vp*, where p is any 

 uneven prime, and let f(x) = -z ; then 



J (#) = r ■ — q 3 + q s — .. 7 + &C. 



=x + E r (2> 2 + E r (3> 3 + E r (4> 4 + &c, 



where E r (V)* denotes the excess of the sum of the rth. powers 

 of those divisors of n which are of the form 4m + 1 over the 

 sum of the rth. powers of those divisors which are of the form 

 4m + 3. 



Thus we have 



/(#) = # + «? + x d + #* + &c, 



<j>(x) = x— 3 r o? 4- 5V — 7V+&C, 



F(» = x + E r (2> 2 + E r (3> 3 + E r (4)^ 4 + &c, 



e 2n =0, e 2n+1 = (-l) n (2n + \) r , Vn=h 



* As there is no risk of confusion between this function E r (w) and the 

 symbolic operator E w of §§ 1-3 and 15-20, 1 have thought it unnecessary 

 to change a notation which I have used in other papers. 



