SECT. IV.] STEADY MOVEMENT IN A PRI-M. 97 



same quantity, dispersed into the air, is, by the principle of the 

 communication of heat, equal to %7rXhvJt ; we must therefore 



have at the surface the definite equation K-j- =hv. The 



nature of these equations is explained at greater length, either 

 in the articles which refer to the sphere, or in those wherein the 

 general equations have been given for a body of any form what 

 ever. The function t? which represents the movement of heat in 

 an infinite cylinder must therefore satisfy, 1st, the general equa- 



- dv K (tfv 1 dv\ , . . 



tion ~r ~^T} [TJ ~*~ ~ J~) wnicn ^PP^es whatever x and t may 



be; 2nd, the definite equation -^ v -f -j- = 0, which is true, whatever 



the variable t may be, when x X; 3rd, the definite equation 

 v = F(x). The last condition must be satisfied by all values 

 of r, when t is made equal to 0, whatever the variable x may 

 be. The arbitrary function F (x) is supposed to be known ; it 

 corresponds to the initial state. 



SECTION IV. 



Equations of the uniform movement of heat in a solid prism 

 of infinite length. 



121. A prismatic bar is immersed at one extremity in a 

 constant source of heat which maintains that extremity at the 

 temperature A ; the rest of the bar, whose length is infinite, 

 continues to be exposed to a uniform current of atmospheric air 

 maintained at temperature 0; it is required to determine the 

 highest temperature which a given point of the bar can acquire. 



The problem differs from that of Article 73, since we now W 

 take into consideration all the dimensions of the solid, which is 

 necessary in order to obtain an exact solution. 



We are led, indeed, to suppose that in a bar of very small 

 thickness all points of the same section would acquire sensibly 

 equal temperatures ; but some uncertainty may rest on the 

 results of this hypothesis. It is therefore preferable to solve the 

 problem rigorously, and then to examine, by analysis, up to what 

 point, and in what cases, we are justified in considering the 

 temperatures of different points of the same section to be equal. 

 F. H. 7 



