542 



Mr. stokes, on THE CRITICAL VALUES OF 



Let then the limit of nj„ be found. It will appear under the form 



C„ + Cj(- ly + SC cos 7iy (17). 



Comparing this expression with (l6), we shall have 



/(0)=^Co, /(a) = -^C,; 

 and for each term of the series denoted by S we shall have 



TT 2 



In particular, if /(r) is continuous, and if the limit ol nj„ is Lj or L^ according as n is odd or 

 even, we shall have 



: IfiO) + /(a) ] , /., = - {/(O) - f{a) I ; 



whence 



/(O) = - (L„ + i,), /(a) = - (L„ - 4). (18). 



4 4- 



If /(■'') were discontinuous for an infinite number of values of .ii lying between and a, it is 

 conceivable that the infinite series coming under the sign S might be divergent, or if convergent 

 might have a sum from which n might wholly or partially disappear, in which case the limit of 

 nA„ might not come out under the form (17). It was for this reason among others, that in Art. I, 

 I excluded such functions from consideration. 



^^ /('*■) 'be expressible algebraically between the limits ,r = and x = a, or if it admit of 

 different algebraical expressions within different portions into which that interval may be divided, 

 A„ will be an algebraical function of w, and the limit ol nA„ may be found by the ordinary methods. 

 Under the term algebraical function, 1 here include transcendental functions, using the term alge- 

 braical function in opposition to what has been sometimes called an empirical function, or a general 

 function, that is, a function in the sense in which the ordinate of a curve traced libera manu is a 

 function of the abscissa. Of course, in applying the theorem in this article to general functions, it 

 must be taken as a postulate that the limit o{ nA„ can be found, and put under the form (17). 



The theorem in question has been proved by means of equation (9), in which it is supposed 

 that ,f'(v) does not become infinite within the limits of integration. The theorem is however true 

 independently of this restriction. To prove it we have only got to integrate by parts once instead 



n 



of twice, and we thus get for the quantity which replaces — the integral 



:/"/'(■^') 



COS doc , 



a 



which by the principle of fluctuation* vanishes when n becomes infinite. There is however this 

 difference between the two cases. When the series (15) has been cleared of the part for which the 



• I borrow this term from a paper by Sir William R. Hamilton 

 On Flnclualxiuj Functions. Transactions of the Royal Irish 

 Academy, Vol. xix. p. 2(>-|. Had I been earlier acquainted with 

 this paper, and that of M. Dirichlet already referred to, I would 

 probably have adopted the second of the methods mentioned in the 

 introduction for establishing equation (2) for any function, or 



rather, would have begun with Art. 7. taking that equation as 

 established. I have retained Arts. (2)_(fi), first, because I 

 thought the reader would enter more readily into the spirit of the 

 paper if these articles were retained, and secondly, because I 

 thought that Section jii, which is adapted to this mode of viewing 

 the subject, might be found useful. 



