SECT. II.] FUNCTIONAL EXPRESSION. 2G3 



or 27T$ (x, t} = Idxfty {I + (2 sin x sin a. + 2 cos x cos a) e~ w 



+ (2 sin 2x sin 2a + 2 cos 2x cos 2a) e~ 22k * 



+ (2 sin 3# sin 3a + 2 cos 3# cos 3 a) e&quot; 3 ^ + &c.} 



= fda/(a) [1 + 22 cos i (a - a?) e **^. 



The sign 2 affects the number i, and indicates that the sum 

 must be taken from 4 = 1 to i = oo . We can also include the 

 first term under the sign 2, and we have 



a?, = cfa/(a) 2 cos / (a - a?) &amp;lt;r X 



We must then give to i all integral values from co to + oc ; 

 which is indicated by writing the limits oo and + oo next to the 

 sign 2, one of these values of i being 0. This is the most concise 

 expression of the solution. To develope the second member of the 

 equation, we- suppose 4 = 0, and then i= 1, 2, 3, &c., and double 

 each result except the first, which corresponds to i = 0. When 

 t is nothing, the function &amp;lt; (x, t) necessarily represents the initial 

 state in which the temperatures are equal to / (x), we have there 

 fore the identical equation, 



(B). 



We have attached to the signs I and 2 the limits between 



which the integral sum must be taken. This theorem holds 

 generally whatever be the form of the function / (x) in the in 

 terval from x = to x = 2?r ; the same is the case with that which 

 is expressed by the equations which give the development of F (x\ 

 Art. 235; and we shall see in the sequel that we can prove directly 

 the truth of equation (B) independently of the foregoing con 

 siderations. 



280. It is easy to see that the problem admits of no solution 

 different from that given by equation (E), Art. 277. The function 

 &amp;lt;/&amp;gt; (x, t) in fact completely satisfied the conditions of the problem, 



and from the nature of the differential equation -=- = k -, , no 



dt da? 



