254 



THIRD REPORT — 1833. 



(r + r') IT 1 1 

 Tf" + -J- = sm X + -TT sin 3 z + — sin 5 jr + &c. (5.) 



2 



3 



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



4 4 



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

 to 3 It, &c., or from to — tt, from — it to —2it, &c.: its 

 values at those limits are zero. 



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

 his TMorie 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 



2 



is that of an indefinite straight line, 



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



■g-j and which passes through the origin of the coordinates : 

 whilst the equation 



y sz sinx zr- &m2x + -jr- sin 3 a; ;i- sin 4 .r + &c. 



» o ^ 



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 It from each other : the portion d C alone coincides with the 



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



Again, the line whose equation is y = ~^, is parallel to the 



1 

 i 



IS 



G' d, C 



* From page IG/" to ipo ; also 267 and 3')6. 



