322 THEORY OF HEAT. [CHAP. VII. 



pressed; but it was useful to notice this conformity in a new 

 problem, which had not yet been submitted to analysis. 



332. Suppose the half side I of the. square which serves as the 

 base of the prism to be very long, and that we wish to ascertain the 

 law according to which the temperatures at the different points of 

 the axis decrease ; we must give to y and z mil values in the 

 general equation, and to I a very great value. Now the construc 



tion shews in this case that the first value of e is -^ , the second 



-x- , the third , &c. Let us make these substitutions in the general 

 2 2i 



equation, and replace n^ nj, n a l, nj, &c. by their values Q,-~-, 



A 2t 



f&amp;gt; t-r X IT 



--, ~ } and also substitute the fraction a for e&quot; 1 * ; we then find 



L L 



-&C. 



We see by this result that the temperature at different points 

 of the axis decreases rapidly according as their distance from the 

 origin increases. If then we placed on a support heated and 

 maintained at a permanent temperature, a prism of infinite height, 

 having as base a square whose half side I is very great; heat would 

 be propagated through the interior of the prism, and would be dis 

 sipated at the surface into the surrounding air which is supposed 

 to be at temperature 0. When the solid had arrived at a fixed 

 state, the points of the axis would have very unequal tempera- 

 tares, and at a height equal to half the side of the base the 

 temperature of the hottest point would be less than one fifth part 

 of the temperature of the base. 



