of a Periodical I'uintional Equation. 337 



f n x = x: it is required to find the form of any other order, 

 f x; r, ti, t being any numbers whole or fractional. 



We of course here suppose that the form rf f* x beino- 

 known, common algebra will give us the formof/^ r / _2r » 

 f~ 3r 



Assume / r x = r|/a r v(/ _ x, in which a is the particular form 

 ot tyxm our complete solution (1); and which is therefore al- 

 ways known for every order. For x put ^>x, and we have 

 /' <bx — fyx r x . Determine the form of i>x from (7) or (10) 

 by substituting f for/ and making $x in the expression 

 employed = jr. Then we shall have 



f x = $/ a. \J/ ~ x ' 



where a ' and ty are known forms. 



As an example, \etfx = b-x, and let it be required to find 

 the form off- x. 



Now, we easily see that/ * x must be a function of the 

 fourth order. The difficulty, however, does not lie in finding 

 a function of this or of any other order, but in finding such 

 a one that its second function shall have the particular form 

 assigned tof. By putting in (1) /•= 1, we have 



b-x = i,(-^- l x) 

 and, smce f fx= — 1, by (5) 



b — \J/.r = tyx 4. p( —x) — fx. 



Therefore *x = »-'(-*>+* = £+* aml ^ 1 r = M by 

 potting <px = x. 



Again : by ( J ), when /• = 1 and v = \ and because ctx = — .r, 

 «*.r = — i/ 1—x-; 



and therefore 



- Y 2 



An arbitrary function might here have been easily intro- 

 duced ; since, instead of making fx = x, we might have made 

 it equal to <px, any function whatever of x, that becomes — Qx 

 by dunging x into — r. Any odd power of x is an obvious 

 case. Such an arbitrary function would disappear in the given 

 particular form fx. 



We are now therefore in possession of a general and direct 

 rule for extracting any functionid root whatever of a periodic, 

 and that too in finite terms. 



Vol. 61. No. <n 9. Nov. 1824. U u As 



