34-0 THEORY OF HEAT. [CHAP. IX. 



350. If we make =0, we have ~ = I . JM which cor- 



Jo 1 + 2 



responds to the initial state. Hence the expression - I ^ - 



is equal to e- x . It must be remarked that the function F(x), 

 which represents the initial state, does not change its value accord 

 ing to hypothesis when x becomes negative. The heat communi 

 cated by the source before the initial state was formed, is 

 propagated equally to the right and the left of the point 0, which 

 directly receives it: it follows that the line whose equation is 



2 f^dqcoaqx . , f . i i ^ T-I 



y = I = 2&quot; 1S composed ot two symmetrical branches whicii 



are formed by repeating to right and left of the axis of y the part 

 of the logarithmic curve which is on the right of the axis of y, and 

 whose equation is y = e~ x . We see here a second example of a 

 discontinuous function expressed by a definite integral. This 



function - I ^ C S f^- is equivalent to e~ x when x is positive, but 

 it is e x when x is negative 1 . 



351. The problem of the propagation of heat in an infinite 

 bar, one end of which is subject to a constant temperature, is 

 reducible, as we shall see presently, to that of the diffusion of heat 

 in an infinite line; but it must be supposed that the initial heat, 

 instead of affecting equally the two contiguous halves of the solid, 

 is distributed in it in contrary manner; that is to say that repre 

 senting by F(x) the temperature of a point whose distance from 

 the middle of the line is x, the initial temperature of the opposite 

 point for which the distance is &, has for value F (x). 



This second problem differs very little from the preceding, and 

 might be solved by a similar method: but the solution may 

 also be derived from the analysis which has served to determine 

 for us the movement of heat in solids of finite dimensions. 



Suppose that a part ab of the infinite prismatic bar has been 

 heated in any manner, see fig. (16*), and that the opposite part 

 a/3 is in like state, but of contrary sign ; all the rest of the solid 

 having the initial temperature 0. Suppose also that the surround- 



1 Of. Biemann, Part. Diff. Glcich. 16, p. 34. [A. F.] 



