Mobius's Theorem on the Reversion of certain Series. 533 



if the corresponding number in the first row is the simple 

 product of an even number of different primes, —1 if it 

 is the simple product of an uneven number of different 

 primes, and if it contains a squared factor. The third row is 

 formed by dividing the numbers in the first row by 2, and 

 entering, under those which are so divisible, 1, —1, or 0, 

 according as the quotient consists of the simple product of an 

 even or uneven number of different primes or contains a squared 

 factor. The fourth row is formed in the same manner, the 

 divisor being 3; and so on. The spaces left blank are to be 

 filled in with ciphers. The number unity is to be regarded as 

 the product of an even number of primes, i. e. corresponding 

 to the quotient unity, 1 is to be entered. 



In the second determinant the ciphers and the signs of the 

 elements are the same as in the first determinant ; but the 

 actual quotients themselves are entered, and each element in 

 the first row is replaced by unity. 



Both determinants are of the nth order. 



Expressions for o" r (Vi), § 23. 



§ 23. Proceeding in the same manner and using the general 

 formulas of § 6, we find : — 



0-) 



CTr (»)=(-) n - 1 1', 2', 3', 4', 5', 6',... 

 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,... 



this determinant differing from the first determinant for <r(n) 

 only in the first row, the elements of which are raised to the 

 power r; and also: — 



(ii.) 

 *>(«)=(-)»-' i, i, i, i, i, i, , 



r, -2', -3% 0,-5% &,. 



0,. 



1', 



o, 



~v, 



0,-3',.. 



o, 



o, 



1', 



o, 



0,-2',.. 



o, 



o, 



o, 



1', 



0, .0, .. 



o, 



o, 



o, 



o, 



1', 0, .. 



