CHAP. VIII.] CUBE AND SPHERE COMPARED. 329 



6, is between and - TT, e 2 is between TT and , e 3 between 2?r and 



- TT, the roots e 2 , 6 g , e 4 , &c. approach more and more nearly to the 



inferior limits TT, 2-Tr, 3-7T, &c., and end by coinciding with them 

 when the index i is very great. The double arcs 2e l5 2e 2 , 2e 3 , &c., 

 are included between and TT, between 2?r and 3?r, between 4?r 

 and OTT ; for which reason the sines of these arcs are all positive : 



. . sin 2e, .. , sin 2e p . . . 



the quantities 1 H - - , 1 H ^ - 2 , &c., are positive and included 



16 1 ^ 2 



between 1 and 2. It follows from this that all the terms which 

 enter into the value of ^ V are positive. 



340. We propose now to compare the velocity of cooling in 

 the cube, with that which we have found for a spherical mass. 

 We have seen that for either of these bodies, the system of tem 

 peratures converges to a permanent state which is sensibly attained 

 after a certain time ; the temperatures at the different points of 

 the cube then diminish all together preserving the same ratios, 

 and the temperatures of one of these points decrease as the terms 

 of a geometric progression whose ratio is not the same in the two 

 bodies. It follows from the two solutions that the ratio for the 



. 3 3 Je 



sphere is e~ n and for the cube e 2 . The quantity n is given by 

 the equation 



cos na h 



na - - = 1 ^,&amp;lt;7, 



sm na K 



a being the semi-diameter of the sphere, and the quantity e is given 

 by the equation e tan e = -^a, a being the half side of the cube. 



This arranged, let us consider two different cases; that in 

 which the radius of the sphere and the half side of the cube are 

 each equal to a, a very small quantity ; and that in which the 

 value of a is very great. Suppose then that the two bodies are of 



