Mobius's Theorem on the Reversion of certain Series. 537 



Expressions for <j>(n), § 29. 



§29. From Gauss's formula (§ 18) and the formula of 

 § 20, we find 



0(i)= i 



0(2) + 0(1) = 2 



0(3) +0(1) = 3 



0(4) +0(2) + 0(l)=4 



0(5) +0(1)=5 



0(6) + 0(3) + 0(2) +0(1)= 6 



and 



0(1)= i 



0(2) -20(1)= -1 



0(3) -30(1)= -1 



0(4) -20(2) = 



0(5) -5#1) = -1 



0(6) _20(3)-30(2) + 60(l)= 1 



The first system of equations gives : — 



#(») = (-)*"' 1, 2, 3, 4, 5, 6,... 



1, 1, 1, 1, 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,... 



where in the third row every alternate element is unity, in 

 the fourth row every third element is unity, and so on. 

 The second system of equations gives: — 



0W=(-)"- 1 



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

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

 0, 1, 0,-2, 0,-3,. 

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



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

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



Phil. Mag. S. 5. Vol. 18. No. 115. Dec. 1884. 



2N 



