330 THEORY OF HEAT. [CHAP. VIII. 



small dimensions; -^having a very small value, the same is the 

 case with e, we have therefore -^ = e 2 , hence the fraction 



-3-Jfe - 



e &amp;lt;*&amp;lt;* is equal to e cva . 



Thus the ultimate temperatures which we observe are expressed in 



_!^ TP . ,, . na cos na h 



the form Ae CDa. If now in the equation : - =1 -j^a, we 



sin na K. 



suppose the second member to differ very little from unity, we find 



^ n * a i ^ A - -W - -- 



-^= -^-, hence the fraction e is e cva. 

 JK. o 



We conclude from this that if the radius of the sphere is very- 

 small, the final velocities of cooling are the same in that solid and 

 in the circumscribed cube, and that each is in inverse ratio of the 

 radius ; that is to say, if the temperature of a cube whose half side 

 is a passes from the value A to the value B in the time t, a sphere 

 whose semi-diameter is a will also pass from the temperature A 

 to the temperature B in the same time. If the quantity a were 

 changed for each body so as to become a, the time required for 

 the passage from A to B would have another value t , and the 

 ratio of the times t and t would be that of the half sides a and a. 

 The same would not be the case when the radius a is very great : 

 for 6 is then equal to JTT, and the values of na are the quantities 

 TT, 27T, 3-7T, 4?r, &c. 



We may then easily find, in this case, the values of the frac 

 tions e & , e ^ 2 ; they are e~^ and e~~&quot;* . 



From this we may derive two remarkable consequences: 1st, when 

 two cubes are of great dimensions, and a and a are their half- 

 sides ; if the first occupies a time t in passing from the temperature 

 A to the temperature B, and the second the time t for the same 

 interval ; the times t and t will be proportional to the squares a 2 

 and a z of the half-sides. We found a similar result for spheres of 

 great dimensions. 2nd, If the length a of the half-side of a cube 

 is considerable, and a sphere has the same magnitude a for radius, 

 and during the time t the temperature of the cube falls from A to 

 B } a different time t will elapse whilst the temperature of the 



