196 THEORY OF HEAT. [CHAP. III. 



of sines of multiple arcs, we employ the general equation (D). 

 The general term /&amp;lt; (x) sin ix dx is composed of three different 



2 

 parts, and we have, after the reductions, -^sin ia for the coefficient 



of sin ix, when i is an odd number ; but the coefficient vanishes 

 when i is an even number. Thus we arrive at the equation 



-7T(j)(x) = 2\ sin a. sin x + ^ sin 3a sin 3# 4- ^ sin 5a sin 5x 



Zi (^ O O 



+ 5=2 sin 7a sin 7# 4- &c. [ (X). 1 



If we supposed a = JTT, the trapezium would coincide with an 

 isosceles triangle, and we should have, as above, for the equa 

 tion of the contour of this triangle, 



~ 7r&amp;lt;f&amp;gt; (as) = 2 (sin a? ^ sin 3# + ^ sin 5% ^ sin 7# + &c. k 2 

 2 \ d / j 



a series which is always convergent whatever be the value of x. 

 In general, the trigonometric series at which we have arrived, 

 in developing different functions are always convergent, but it 

 has not appeared to us necessary to demonstrate this here ; for the 

 terms which compose these series are only the coefficients of terms 

 of series which give the values of the temperature ; and these 

 coefficients are affected by certain exponential quantities which 

 decrease very rapidly, so that the final series are very convergent. 

 With regard to those in which only the sines and cosines of 

 multiple arcs enter, it is equally easy to prove that they are 

 convergent, although they represent the ordinates of discontinuous 

 lines. This does not result solely from the fact that the values 

 of the terms diminish continually ; for this condition is not 

 sufficient to establish the convergence of a series. It is necessary 

 that the values at which we arrive on increasing continually the 

 number of terms, should approach more and more a fixed limit, 



1 The accuracy of this and other series given by Fourier is maintained by 

 Sir W. Thomson in the article quoted in the note, p. 194. 



2 Expressed in cosines between the limits and TT, 



ITT&amp;lt;P ()=__{ cos.2a; + - cos Gx + ^- cos Wx + &c. ) . 



o \ O O / 



Cf. De Morgan s Diff. and Int. Calc., p. 622. [A. F.] 



