104) Mr. Jolm Herapatli on Funciiofial Equaiions. 



But rj/ a ^~ X, ^l; cc v{/"~ X, . . . are manifestly the first, second, 



. . . orders of the functional root of a certain periodic of the 



7Zth order. Let this periodic be /3'\r = x, and the equation 

 will become 



.r=/[vl/-\r,^^-,/3=^a',..../3'^-\r] (4) 



Now because 



|3 .r = 4/ a 4/" JT, /3- cr = v|; «' ;}/" ^', /3^ A' = \I; a^ \l/~ ^, .... (5) 



it is evident that the arbitrary function or functions, which might 

 exist in the determination of \J/ generally, from any one alone 

 of these equations, must disappear when vp is considered to be 

 determined, as it should be, from all the equations (5) simul- 

 taneousl}^ Consequently if /3 has no arbitrary function, v(/ has 

 not ; and vice versa, if ^ has no arbitrary function, neither has 

 |3. Again, because of the invariable relation which the above 

 equations (5) establish between ^I' ^"i^l (3, considering the 

 form of u invariable, |3 determines the form of ^, and 4> the 

 form of /3. Let us now suppose a, instead of having a con- 

 stant form, has, and likewise ^, every possible form. This 

 will give to /3 its utmost generality, and at the same time ren- 

 der it independent of v}/. With such views (4) becomes 



x=f,{^x,^\r,^'x,...^''-'x] (6) 



Moreover, because /3 cannot be otherwise than a periodic of the 

 7ith order, its generality, if there be any difference, should be 

 greater when its form is determined from the simple condition 



0^ X = X, than when from this condition coupled with the 



properties of yj . But when /3 is determined from $"' x = x 

 I have demonstrated, (Annals for December 1824, p. 424,) that 

 its complete form has but one arbitrary function. Therefore 

 \J/ in (2), whose form is determined by that of |3, has only one 

 arbiti'ary function. And this must be equally true for each and 

 allthe/i — 1 contemporaneous equations (3), which together 



with (2) involve x under every change from «" x to a"~ '^. 

 Hence t/ie complete solution of any periodical functional equa- 

 tion ofthefrst order, as (2), a?id likexvise the comjjlete soltition 

 of a7iy periodical equation of any order, as (6), has but one ar- 

 bitrary function, containing nnder it every possible transforma- 

 tion X can have in the n simnltaneoiis eqtiations. 



It will perhaps have been observed (Philosophical Magazine 

 for September 1824, p. 199) that I once held a different opi- 

 nion respecting the number of arbitrary functions in the com- 

 plete solution ; but, as I then hinted, my ideas were not suffi- 

 ciently matured to speak confidently ; and I was led away by 



analogical 



