CHAP. VII.] CASE OF A THIN BAR. 319 



e, or e lf is \J j- ; by inspection of the figure we know the values of 

 the other roots, so that the quantities e lt e 2 , e 8 , e 4 , e 6 , &c. are the 

 following A/ -j- , TT, 27r, STT, 4-Tr, &c. The values of n v n v n 3 , n^ n y &c. 

 are, therefore, 



!_ /h 7T 27T 3?T 



v^v & J i ~i 



whence we conclude, as was said above, that if I is a very small 

 quantity, the first value n is incomparably greater than all the 

 others, and that we must omit from the general value of v all the 

 terms which follow the first. If now we substitute in the first 

 term the value found for n, remarking that the arcs nl and 2nl are 

 equal to their sines, we have 



hl\ x /? 



the factor A/ -j- which enters under the symbol cosine being very 



small, it follows that the temperature varies very little, for 

 different points of the same section, when the half thickness I is 

 very small. This result is so to speak self-evident, but it is useful 

 to remark how it is explained by analysis. The general solution 

 reduces in fact to a single term, by reason of the thinness of the 

 bar, and we have on replacing by unity the cosines of very small 



A* 



arcs v = e~ x * kl , an equation which expresses the stationary tempe 



ratures in the case in question. 



We found the same equation formerly in Article 76 ; it is 

 obtained here by an entirely different analysis. 



331. The foregoing solution indicates the character of the 

 movement of heat in the interior of the solid. It is easy to see 

 that when the prism has acquired at all its points the stationary 

 temperatures which we are considering, a constant flow of heat 

 passes through each section perpendicular to the axis towards the 

 end which was not heated. To determine the quantity of flow 

 which corresponds to an abscissa x, we must consider that the 

 quantity which flows during unit of time, across one element of 



