Mobius's Theorem on the Reversion of certain Series. 531 



instead of n ; and that if (n) be a quantity defined by the 

 equation 



we find 



0,(i) +p(f) +4>M+<f> (i). . . +*.(*) 



Second formula involving <p(n) } §§ 20, 21. 

 § 20. From the formula proved in § 16 it follows that 



(1-2E,)(1-3E,)(1-5E,). . -C*M/K) 



= (l-E 2 )(l-E 3 )(l-E 5 ).../(*), 



and by equating the coefficients of /(# n ) in this equation we 

 obtain the formula 



(,) -S^Q + ta^ytaoc^ + . . . 



= or(-l)* 



according as n is divisible by a squared factor or is the pro- 

 duct of N simple prime factors. Thus, for example, putting 

 7i = 10, 20 and 30, the formula gives 



0(10) -20(5) -50(2) + 100(1) = 1, 



0(20) -20(10) -50(4) + 100(2) =0, 



0(30) -20(15) -30(10) -50(6) + 60(5) + 100(3) + 150(2) 



-300(1) = -1, 

 viz. 



4-8- 5 + 10 = 1, 



8-8-10 + 10 = 0, 



8-16-12-10 + 24 + 20 + 15-30= -1. 

 § 21. In a similar manner we find that 



* GO - Ut$ + *HS) - **«**(£) + • • • 



= or ( — 1) N , according as ?i is divisible by a squared 

 factor or is the product of N simple prime factors. 



Expressions for o" r (w), Ar(w), fyc. as Determinants, §§ 22-28. 



Expressions for <r(n), § 22. 



§ 22. From the formulae of §§ 6-11 we may readily deduce 

 expressions for cr r (n) &c. as determinants of n rows. 



