of a Periodical Functional Equation. 835 



hand of it, and a is the prime numerator of - . The number 

 of f~ s, or terms, must therefore be q. If now we put ge- 

 nerally A ( — $a r x embracing the log. in <p, and differentiate 

 (6) with respect to v, we shall find 



l- q _l'+2-j 2-5, — 1'+3— -? 3-g -l' 

 = A o f+x+fte.{Aj4,x+ftx.{A a f+x...f4,x.{A g _ l 4,x(7) 

 where the several orders of ftyx are factors to all the expres- 

 sion following on the right hand, and the acute accent in each 

 case over the — 1 signifies the differential function of the in- 

 verse function of f, the other indices denoting simply the di- 

 rect or inverse functions of /"as they happen to be positive or 



rf+q ' df p 



negative ; so that generally f x implies the -^— function of 



ax 



fqx. 



Otherwise thus : 



Put for tya r x bfx + v\ia r x, and the equation proposed 

 6tands y\p.r= b$ x + vtya r x (8) 



which becomes tyx=f~ 1 {bfx+vf~ l {b$a r x+vf~ l {b<pa 2r x 



+ . . . vf~ l \b<pa iq ~ 1)r x + v^x (9) 



by successively substituting for the last number its value de- 

 rived from (8). In this expression (9) the same import is 

 given to each / _ as in (6). Differentiating with respect to 

 b and v; putting afterwards b = and u=l ; letting A = <pa rt x; 



and comprehending in <p the quotient — -, we shall find 



l- g _i'-j_2- 9 1-q — V+3— » 3-<j 



\+f **+/ ^.[A,*/ *,x+f ^x.[A 2 +f * x 



-i' 



+ . . ./ i>x.[A _ x +i,x = (10) 



the orders and accented fs denoting the same as in (7). 



It may here be observed, that these expressions (7), (10) are 

 not necessarily confined to periodic functions. In case, how- 

 ever, ax is not a periodic, we must put every where \J/« yr .r for 

 \J/x. When ax is a periodic and of the «th order,y.r must 

 obviously be a periodic too, of the order — . Should in 



this instance fx not be a periodic, the arbitrary function will 

 receive a certain limitation, which being inconsistent with the 

 perfectly unlimited form in which it is introduced, and which 



it 



