SECT. I.] GENERAL SOLUTION. 343 



the sum of the series is simply an integral, which must be taken 

 with respect to q from q = to q = oo . Hence 



v - e~ M \dqe-W* sin qx \dxF(x)smqx ......... (a), 



the integral with respect to x must be taken from x = to x = oo. 

 We may also write 



TTl) f 30 f 



_ Q-U \ dqe-Wt sm q x I 

 * Jo Jo 



7TV f 30 f 30 



~^ Q ~ u \ d^F(^]\ dq e- 

 * Jo Jo 



sm 



or 



Equation (a) contains the general solution of the problem; 

 and, substituting for F(x] any function whatever, subject or not 

 to a continuous law, we shall always be able to express the value 

 of the temperature in terms of x and t : only it must be remarked 

 that the function F(x) corresponds to a line formed of two equal 

 and alternate parts 1 . 



354. If the initial heat is distributed in the prism in such a 

 manner that the line FFFF (fig. 17), which represents the initial 



Fig. 17. 



state, is formed of two equal ares situated right and left of 

 the fixed point 0, the variable movement of the heat is expressed 

 by the equation 



TTV f 30 f 00 



-_ = e~ u I -d&F(a) I dq e~W cos qx cos ga. 



Fig. 18. 



If the line ffff (fig. 18), which represents the initial state, is 

 i That is to say, F(x)=-F(-x}. [A.F.] 



