SECT. III.] VALUES OF HIGHEST TEMPERATURES. 389 



To ascertain the highest temperature V, we remark that the 

 exponent of e~ l in equation (a) is ht + -jy- Now equation (&) 



# 2 1 x z x 2 1 1 



gives fa = jf- - ~ ; hence ht + 77- ; = ny-, - ~ , and putting for - its 



&quot;rA-C 2 db/JC Zfff 2 I 



/p2 l\ 



known value, we have ht + TT~ , = \/ T + 7 ^ 2 ; substituting this ex- 



j^rCv y T /(J 



ponent of e&quot; 1 in equation (a), we have 



and replacing */#& by its known value, we find, as the expression 

 of the maximum V, 



4/i 1 _1 



X* 



The equations (c) and (d) contain the solution of the problem ; 



TTJ jr 



let us replace h and k by their values TT/T^ an d ^7^ ; let us also 



-I Q 



write 5 g instead of -=- , representing by g the semi-thickness of the 



prism whose base is a square. We have to determine Fand 6, 

 the equations 



w e - 



I*B ,l 



V^^+i 



These equations are applicable to the movement of heat in a 

 thin bar, whose length is very great. We suppose the middle of 

 this prism to have been affected by a certain quantity of heat bf 

 which is propagated to the ends, and scattered through the convex 

 surface. V denotes the maximum of temperature for the point 

 whose distance from the primitive source is a?; is the time 

 which has elapsed since the beginning of the diffusion up to the 

 instant at which the highest temperature V occurs. The coeffi- 



