Mobius's Theorem on the Reversion of certain Series. 521 



then 



f(x) = F(» -FO 2 ) -F(^ 3 ) -FO 5 ) + F(> 6 ) - &c. ; 

 and if 



FW=/W + /(* 3 ) +/(* 5 ) + /0* 7 ) + /(*') + &°., 

 then 



/O) = FO) - FO 3 ) - FO 5 ) - FO 7 ) - FO 11 ) - &c. 



All the results given in this section are due to Mobius, 

 and in some respects they might be more properly called by 

 his name than the general theorem in § 3. 



Two-fold application of the Theorem, § 5. 

 § 5. It follows from § 3 that, if 



F(^) = /W + ^/(^) + ^/(^ 3 ) + ^/(^) +e 5 f(^)+ &c, 

 then 



/(*) = FO) -* 2 F<> 2 ) - e 3 F0 3 ) -««BV) + e 6 F<> 6 ) - &c. 

 Now suppose 



f (%)=:% + r) 2 a? + yz% 3 + t]iX^ + &c, 



where 972, 973, 7]^, ... are quantities of the same nature as 

 % e$i e±, • • ; viz. r) 2 , rj z , r) 5 , . . . are independent, and 57^= 

 ?7 m x 77^ for all values of m and n. 

 We thus have 



F(V)= # + ?7 2 a? 2 +97 3 ^ 3 + 7?^ 4 + &c. 

 + £ 2 2 + ^ 4 + *?3# 6 + ^ 8 + &c.) 



+ 63 (^ 3 + 7] 2 % 6 + ^ + V 12 + &C.) 



= 00) + 77 2 (/)0 2 ) + ^O 8 ) + %<^>0 4 ) + &C, 



where 



$0) ss a? 4- ^a? 2 + e 3 o? 3 + e^ -f &c. ; 

 and, applying Mobilises theorem, we find 



<f>( X ) = V(x)- V2 F(x?)- Vs F(w s )- Vi F{x 5 )+r) e F(a:°)- . . . 



Whenever, therefore, f(x) is of the form x -f- tj 2 x 2 + r) s x B + &c, 

 we obtain a formula in terms of the F's not only for /(#), 

 but also for another quantity c/>0) : in fact, since 



FO0 = /O) WO 2 ) W(^ 3 ) + ^/ 4 ) + &c, 

 and = $0) + y 2 <l>(% 2 ) + y z $(x z ) + 774$ (^ 4 ) + &c, 



we have 



/(*)=F(«)-^M-«,FM-e 4 FM + &c, 

 and <^) = F(*) -77 2 F(^)-77 3 F(^ 3 )-77 5 F0 5 )+ &c. 

 PM. 1%. S. 5. Vol. 18. No. 115. Z>ec. 1884, 2 M 



