SECT. I.] TWO PROBLEMS. 335 



what is the state of the line after a given time. This problem 

 may be made more general, by supposing, 1st, that the initial 

 temperatures of the points included between a and b are unequal 

 and represented by the ordinates of any line whatever, which we 

 shall regard first as composed of two symmetrical parts (see fig. 16); 



Fig. 16. 



2nd, that part of the heat is dispersed through the surface of the 

 solid, which is a prism of very small thickness, and of infinite 

 length. 



-.JO* 6 second problem consists in determining the successive 

 states of a prismatic bar, infinite in length, one extremity of 

 which is submitted to a constant temperature. The solution of 

 these two problems depends on the integration of the equation 



dv _ K tfv HL 

 dt~CDdx z CDS V 



(Article 105), which expresses the linear movement of heat, v is 

 the temperature which the point at distance x from the origin 

 must have after the lapse of the time t ; K, H, C, D, L, S, denote 

 the internal and surface conducibilities, the specific capacity for 

 heat, the density, the contour of the perpendicular section, and 

 the area of this section. 



345. Consider in the first instance the case in which heat is 

 propagated freely in an infinite line, one part of which ab has 

 received any initial temperatures; all other points having the 

 initial temperature 0. If at each point of the bar we raise the 

 ordinate of a plane curve so as to represent the actual tempera 

 ture at that point, we see that after a certain value of the time t, 

 the state of the solid is expressed by the form of the curve. 

 Denote by v = F(x) the equation which corresponds to the given 

 initial state, and first, for the sake of making the investigation 



