156 THEORY OF HEAT. [CHAP. III. 



more distant from the origin, the movement of heat is less com 

 plex : for if the distance x is sufficiently great, each term of 

 the series is very small with respect to that which precedes it, 

 so that the state of the heated plate is sensibly represented by 

 the first three terms, or by the first two, or by the first only, 

 for those parts of the plate which are more and more distant 

 from the origin. 



The curved surface whose vertical ordinate measures the 

 fixed temperature v, is formed by adding the ordinates of a 

 multitude of particular surfaces whose equations are 



^ = e* cos y, 7 ~ = - K 3 * cos 3# ^ =^&quot; 5 * cos 5 y t &c. 



The first of these coincides with the general surface when x 

 is infinite, and they have a common asymptotic sheet, 



If the difference v v l of their ordinates is considered to be 

 the ordinate of a curved surface, this surface will coincide, when x 

 is infinite, with that whose equation is ^irv 2 = e~ Zx cos 3y. All 

 the other terms of the series produce similar results. 



The same results would again be found if the section at the 

 origin, instead of being bounded as in the actual hypothesis by 

 a straight line parallel to the axis of y, had any figure whatever 

 formed of two symmetrical parts. It is evident therefore that 

 the particular values 



ae~ x cos y, le~ 3x cos 3y, ce~ 5x cos 5y, &c., 



have their origin in the physical problem itself, and have a 

 necessary relation to the phenomena of heat. Each of them 

 expresses a simple mode according to which heat is established 

 and propagated in a rectangular plate, whose infinite sides retain 

 a constant temperature. The general system of temperatures 

 is compounded always of a multitude of simple systems, and the 

 expression for their sum has nothing arbitrary but the coeffi 

 cients a, b, c, d, &c. 



192. Equation (a) may be employed to determine all the 

 circumstances of the permanent movement of heat in a rect 

 angular plate heated at its origin. If it be asked, for example, 

 what is the expenditure of the source of heat, that is to say, 



