338 Mr. J. Herapath on a Periodical Functional Equation. 

 As another example of the application of the preceding 



theorems, take the equation \J/ ; '« r x = 4> q x, the condition be- 

 ing a - x = x, where p, q, r, n are any numbers. A particular 

 case of this equation has been considered by Mr. Babbage in 

 the 68 th Prob. of his and Messrs. Herschel and Peacock's 

 " Examples," &c. 



Take the \|/ _; ' function on each side, and ty q ~ p x = oi x. 



r , vji_ 



Whence fyx = a^x, and \J/ r L = u n x=x. tyx is there- 

 fore a periodic of the order (q—p).l, the q—p function having 



the particular form a r x. An arbitrary function will be com- 



r 



prehended in a g~S p x, which will disappear in a. x. 



The truth of this solution may be otherwise shown thus ; 

 take the \J/ ; ' function on both sides, and change x into a r x, the 

 proposed equation will become 



Another similar process gives \J/ 3; ' « 3r .r = \J/ 3!7 .r, and generally 



np n q «£ "g 



\f/ r a x = 4> .r, or 4» # = \|/ r x. 



Whence \J/ .r = x. 



Mr. Babbage, to whom the world owes so much for his dis- 

 coveries in this calculus, has given a different solution, when 

 r=l. He finds by an indirect process that tyx=$~ f<px' 

 will satisfy the conditions of the question whenyis any periodic 

 of the q —p order, and <p any symmetrical function of x, a. x, 



n-l 



o?x, . . . . « x ; n, p, q being whole numbers. His solution 

 comprehends the form of a not in the particular solution J", 

 which is indeed left perfectly arbitrary, but in a certain limi- 

 tation to the arbitrary function itself. This method likewise 



gives every value of \(/ x = x without any exception ; but 



q-r 

 if my views are correct, \f/ x should have but one value, and 



that =ux, and free from an arbitrary function. 



I have some other observations to make on this subject and 

 the nature of periodic functions, which brevity obliges me to 

 defer to another opportunity : but I may here observe, that 

 the functional theory properly considered, results from a much 



more 



