254 



THIRD REPORT — 1833. 



(;• + r*) . TT 



-7j- sin 3 r + -p- sin 5 T + &c. (5.) 



which may be easily shown to be equal to -j- and j- altera 



nately, in the passage of x from to it, from v to 2 if, from 2 it 

 to 3 IT, &c., or from to — tt, from — ir to — Sir, &c.: its 

 values at those limits are zero. 



The series (2.) and (5.) have been investigated by Fourier, in 

 his Thior'ie de la Chaleur *, by a very elaborate analysis, which 

 fails, however, in showing the dependence of these series upon 

 each other and upon the principles involved in the deduction 

 of the fundamental series : and they present, as we shall now 

 proceed to show, very curious and instructive examples of dis- 

 continuous functions. 



The equation y ■= -^is that of an indefinite straight line, 

 Q A P, making an angle with the axis of x, whose tangent is 



-^, and which passes through the origin of the coordinates : 



whilst the equation 



y -St smx — -^ sia2 X + -3- sin 3 jc — j sin 4 « + &c. 



is that of a series of terminated straight lines, d' c, dC,T> C, 

 &;c., passing through points a, A, A', &c., which are distant 

 2 tt from each other : the portion d C alone coincides with the 



primitive line, whose equation is y = -3-. 



It 



Again, the line whose equation is y = -rr-, is parallel to the 



IB 



G' d, c 



* From page IG? to IPO ; also 267 and 350. 



