198 UNIVERSITY OF VIRGINL\ PUBLICATIONS 



l|/( + x-0)+/(-7rH-0)|-|/(-7r + 0j-/( + 7r + 0)}^JJ'-^^/<. (22) 



But the series is a periodic function with period 27r. Therefore this 

 last value (22) is the same value which the series takes when x converges 

 to + -r + 0. Hence at the point tt the series has a shear whose upper and 

 lower ends are respect ivelj^ 



I j/(7r)+/(-7r)|±l|/(-7r)-/( + 7r)} 1.1796. (23) 



The series S can be made to take uniquely anj' number of this shear 

 as a limit by properly assigning c in the interval ( (0, tt) ). 



The restriction that c S ir may be removed, for if c = m-ir + h where h 

 lies between and x, the integral from to c can be decomposed into the 

 partial intervals. 



Jo Jtt Jim-l)w Jmir 



in each of which sin t/t keeps its sign imchanged and therefore to each 

 of which the first law of the mean is, as before, applicable. The integrals 



£ 



sin t ,, 



at 



t 



are alternately positive and negative and each numerically less than the 

 preceding. Hence as before whatever be the assigned positive number c 

 the value of (16) is given bj' (18). Since the function of c 



X' 



'- dx 



is sjTiimetrical wioh respect to the ordinate axis, c may be anj' real number 

 whatever. There are two positive values of c one less the other greater 

 than TT which give the same value to S, as also there are two corresponding 

 symmetrical negative values. In particular there is a value of c between 

 Itt and TT corresponduig to c = <» such that when x ( = ) tt — the value 

 of (S is / (tt — 0) , or the continuation of the uniformly convergent series 

 for X Ktt/' 



* In connexion with this subject reference is made to a paper by Professor Bocher, 

 Annals of Mathematics, VII, Second Series, p. 81, §9, 1906. Introduction to the Theory 

 of Fourier's Series. Also Philosophical Magazine, Series V, Vol. 45, p. 85, 1898. 

 The Harmonic Analyser of Michelson and Stratton. 



