CHAP. VIII.] REMARKS. 331 



sphere is falling from A to JB, and the times t and t are in the 

 ratio of 4 to 3. 



Thus the cube and the inscribed sphere cool equally quickly 

 when their dimension is small ; and in this case the duration of 

 the cooling is for each body proportional to its thickness. If the 

 dimension of the cube and the inscribed sphere is great, the final 

 duration of the cooling is not the same for the two solids. This 

 duration is greater for the cube than for the sphere, in the ratio of 

 4 to 3, and for each of the two bodies severally the duration of the 

 cooling increases as the square of the diameter. 



341. We have supposed the body to be cooling slowly in at 

 mospheric air whose temperature is constant. We might submit 

 the surface to any other condition, and imagine, for example, that 

 all its points preserve, by virtue of some external cause, the fixed 

 temperature 0. The quantities n, p, q, which enter into the value 

 of v under the symbol cosine, must in this case be such that cos nx 

 becomes nothing when x has its complete value a, and that the 

 same is the case with cos py and cos qz. If 2a the side of the 

 cube is represented by TT, 2?r being the length of the circumference 

 whose radius is 1 ; we can express a particular value of v by the 

 following equation, which satisfies at the same time the general 

 equation of movement of heat, and the state of the surface, 



.. 



v = e cb cos x . cos y . cos z. 



This function is nothing, whatever be the time t t when x or y or z 



receive their extreme values + - or - : but the expression for the 



2i 2* 



temperature cannot have this simple form until after a consider 

 able time has elapsed, unless the given initial state is itself 

 represented by cos x cos y cos z. This is what we have supposed 

 in Art. 100, Sect. Yin. Chap. I. The foregoing analysis proves the 

 truth of the equation employed in the Article we have j ust cited. 



Up to this point we have discussed the fundamental problems 

 in the theory of heat, and have considered the action of that 

 element in the principal bodies. Problems of such kind and order 

 have been chosen, that each presents a new difficulty of a higher 

 degree. We have designedly omitted a numerous variety of 



