SECT. II.] TRIGONOMETRIC SERIES. 1 37 



termediate molecule communicates to that which follows it more 

 heat than it receives from that which precedes it. Thus it 

 follows that the excess of the heat which it acquires in the direc 

 tion of x, is exactly compensated by that whicn&quot; it loses in the 



direction of ?/. as the equation -^- 2 + -y- 2 =0 denotes. Thus 



ax dy 



then the route followed by the heat which escapes from the 

 source A becomes known. It is propagated in the direction 

 of x, and at the same time it is decomposed into two parts, 

 one of which is directed towards one of the edges, whilst the 

 other part continues to separate from the origin, to be decomposed 

 like the preceding, and so on to infinity. The surface which 

 we are considering is generated by the trigonometric curve which 

 corresponds to the base A, moved with its plane at right angles to 

 the axis of x along that axis, each one of its ordinates de 

 creasing indefinitely in proportion to successive powers of the 

 same fraction. 



Analogous inferences might be drawn, if the fixed tempera 

 tures of the base A were expressed by the term 



b cos 3y or c cos 5y, &c. ; 



and in this manner an exact idea might be formed of the move 

 ment of heat in the most general case ; for it will be seen by 

 the sequel that the movement is always compounded of a multi 

 tude of elementary movements, each of which is accomplished 

 as if it alone existed. 



SECTION II. 



First example of the use of trigonometric series in the theory 



of heat. 



171. Take now the equation 



1 = a cos y + b cos oy + c cos oy + d cos 7y + &c., 



in which the coefficients a, b, c, d, &c. are to be determined. 

 In order that this equation may exist, the constants must neces- 



