168 THEORY OF HEAT. [CHAP. III. 



found, by replacing the trigonometrical quantities by imaginary 



exponentials. We thus obtain the values of -3- and -r- which 



ace ay 



we have just stated. 



The problem which we have now dealt with is the first which 

 we have solved in the theory of heat, or rather in that part of 

 the theory which requires the employment of analysis. It 

 furnishes very easy numerical applications, whether we make 

 use of the trigonometrical tables or convergent series, and it 

 represents exactly all the circumstances of the movement of 

 heat. We pass on now to more general considerations. 



SECTION VI. 



Development of an arbitrary function in trigonometric series. 



207. The problem of the propagation of heat in a rect- 



d 2 v d 2 v 

 angular solid has led to the equation -y-g + -=- = ; and if it 



be supposed that all the points of one of the faces of the solid 

 have a common temperature, the coefficients a, b, c, d } etc. cf 

 the series 



a cos x + b cos 3x + c cos 5# 4- d cos 7x + ... &c., 



must be determined so that the value of this function may be 

 equal to a constant whenever the arc x is included between JTT 

 and + JTT. The value of these coefficients has just been assigned; 

 but herein we have dealt with a single case only of a more general 

 ; problem, which consists in developing any function whatever in 

 an infinite series of sines or cosines of multiple arcs. This 

 problem is connected with the theory of partial differential 

 equations, and has been attacked since the origin of that analysis. 

 It was necessary to solve it, in order to integrate suitably the 

 equations of the propagation of heat; we proceed to explain 

 the solution. 



We shall examine, in the first place, the case in which it is 

 required, to reduce into a series of sines of multiple arcs, a 

 function whose development contains only odd powers of the 



