44-0 THEORY OF HEAT. [CHAP. IX. 



which is equal to 2irf(x) t Arts. 415 and 350. Whence we con 

 clude the previous equation (A). 



When the variable x is exactly equal to TT or + TT, the con 

 struction shews what is the value of the second member of the 

 equation (A), [|/(-7r) or ^/(TT)]. 



If the limits of integrations are not - TT and + TT, but other I 

 numbers a and b, each of which is included between TT and 

 + TT, we see by the same figure what the values of x are, for which 

 the second member of equation (A) is nothing. 



If we imagine that between the limits of integration certain 

 values of /(a) become infinite, the construction indicates in what 

 sense the general proposition must be understood. But we do 

 not here consider cases of this kind, since they do not belong 

 to physical problems. 



If instead of restricting the limits TT and + TT, we give 

 greater extent to the integral, selecting more distant limits a 

 and b , we know from the same figure that the second member 

 of equation (A) is formed of several terms and makes the result 

 of integration finite, whatever the function /(#) may be. 



We find similar results if we write 2?r y instead of r, the 



limits of integration being X and + X. 



It must now be considered that the results at which we 

 have arrived would also hold for an infinity of different functions 

 of sin jr. It is sufficient for these functions to receive values 

 alternately positive and negative, so that the area may become 

 nothing, when j increases without limit. We may also vary 



the factor . -, as well as the limits of integration, and we 

 versm r 



may suppose the interval to become infinite. Expressions of 

 this kind are very general, and susceptible of very different forms. 

 We cannot delay over these developments, but it was necessary 

 to exhibit the employment of geometrical constructions ; for 

 they solve without any doubt questions which may arise on the 

 extreme values, and on singular values; they would not have 

 served to discover these theorems, but they prove them and guide 

 all their applications. 



