422 Mr. Herapath on the Solution o/'vj/' x = x. [Dec. 



the 4*" 1 x, ^~ a x, .... values of which are evidently equal to 

 the functional roots of the former. 



All the preceding properties are analogous to the properties of 

 the roots of unity ; but it would not be correct to conclude from 

 hence that the parallel is strictly true in all cases. For instance, 

 all the roots of unity, except one or two at most, are imaginary. 

 In the case however of a periodic function, every root is real, if 

 the order be real. ' 



It is not one of the least remarkable features of our solutions 

 that every order of the function positive, negative, &c. is given 

 by the same formula, without the necessity of algebraic resolu- 

 tion ; — a circumstance that has not been accomplished in any 

 other solution with which I am acquainted. Several other con- 

 sequences of high importance also readily flow from our last very 

 simple solution, which I presume it would be in vain to expect 

 from any other. Thus the extraction of any functional root of 

 a periodic function a x of a given form and order ; and the reso- 

 lution of the equation \|/" x = x, so that any order \p v x of it may 

 have one given form only, are questions easily resolved from the 

 preceding solution, and obviously much too important in their 

 consequences to need the notice of any particular instances, 

 especially as I have treated on the subject in another place. I 

 shall, therefore, merely indulge myself in the solution of a very 

 common form, but more general functional equation of the kind 

 than I have yet seen solved. Let the equation be 



4, x « 4- « r x ; (27) 



where «"x = x, and 11, r, are any numbers. By our (25) we 

 have 



«' x = /3 sin J?I^ + sin- 1 jS- x^ (28) 



(3 being the particular form of the arbitrary function which gives 

 the solution in the rth order, the form a' x and k l being any one 

 invariable integer. Let us assume x = 11. and ax — v : + l . Then 

 a r x = 

 and 



But the condition a" x = x gives u : + „ = it,, and therefore 

 x = u. = B sin (30) 



n 

 11 r a ■ *'&**■ 



as well as « x = u. + r = p sin . 



and 2sa= ^ sin " ' j3 ~ * x. 

 Substitute this value for 2 z A in (29), and we have 

 }x = <psin ^~ r sin"' (3~ l x^ (31) 



