276 , THEORY OF HEAT. [CHAP. V. 



It follows from this that in order to obtain the value of the 

 coefficient a lt in equation (e), we must write 



2 \x sin UjX F(x) dx a^\X -^~ sin Zn^X] , 



the integral being taken from x = to so = X. Similarly we have 

 2 \x sin n z x F(x) dx=aAX^ si 



sn 



In the same manner all the following coefficients may be deter 

 mined. It is easy to see that the definite integral 2 Ix sin nx F (x) dx 



always has a determinate value, whatever the arbitrary function 

 F (x) may be. If the function F(x) be represented by the 

 variable ordinate of a line traced in any manner, the function 

 xF(x) sin nx corresponds to the ordinate of a second line which 

 can easily be constructed by means of the first. The area bounded 

 by the latter line between the abscissae x and xX determines 

 the coefficient a it i being the index of the order of the root n. 



The arbitrary function F(x) enters each coefficient under the 

 sign of integration, and gives to the value of v all the generality 

 which the problem requires; thus we arrive at the following 

 equation 



sin n^xlx sin n % x F (x} dx 



J - 



sin n z x Ix sin n z x F (x) dx 



- J - e-** + &c. 



This is the form which must be given to the general integral 

 of the equation 



in order that it may represent the movement of heat in a solid 

 sphere. In fact, all the conditions of the problem are obeyed. 



