SECT. VI.] MODIFICATION OF THE SERIES. 205 



of the interval which includes the arc which represents F(x}\ 

 the function becomes F ( j, which we may denote by /(#). 



The limits x = TT and x = TT become = TT. = TT ; we 



r r 



have therefore, after the substitution, 



&amp;lt;p? 



X [ . 7T# , 27T.T f ,, . 277-tf , 



-f cos TT - I f(x) cos dx-\- cos I / (x) cos c&e f etc. 



x f ., N . TTX j . 27r# /* ,. . . 2?nr , 

 + sin TT - I /(a?) sin dx -f sm \f(x) sm d# + etc. 



All the integrals must be taken like the first from x = r to 

 x = +r. If the same substitution be made in the equations (v) 

 and (yu,), we have 



cos -- I f(x) cos dx 



2?ra; /*/./ 27ra; 

 + cos- - \f(x) cos -- 



1 /., x . 7T5? F ~ f 



2 r /W = sm \ f( x 



^ J 



In the first equation (P) the integrals might be taken from 

 from x = to x = 2r, and representing by x the whole interval 2r, 

 we should have * 



1 It has been shewn by Mr J. O Kinealy that if the values of the arbitrary 

 f unction /(x) be imagined to recur for every range of x over successive intervals X, 

 we have the symbolical equation 



and the roots of the auxiliary equation being 



t ^J ^ , 7 = 0, 1, 2, 3... cc, [Turn over. 



