332 THEORY OF HEAT. [CHAP. VIII. 



intermediate problems, such as the problem of the linear movement 

 of heat in a prism whose ends are maintained at fixed temperatures, 

 or exposed to the atmospheric air. The expression for the varied 

 movement of heat in a cube or rectangular prism which is cooling 

 in an aeriform medium might be generalised, and any initial 

 state whatever supposed. These investigations require no other 

 principles than those which have been explained in this work, 



A memoir was published by M. Fourier in the Memoir es de V Academic des 

 Sciences, Tome vii. Paris, 1827, pp. 605 624, entitled, Memoire sur la distinction des 

 racines imaginaires, et sur Vapplication des theoremes d analyse algebrique aux 

 equations transcendantes qui dependent de la theorie de la chaleur. It contains a 

 proof of two propositions in the theory of heat. If there be two solid bodies of 

 similar convex forms, such that corresponding elements have the same density, 

 specific capacity for heat, and conductivity, and the same initial distribution of 

 temperature, the condition of the two bodies will always be the same after times 

 which are as the squares of the dimensions, when, 1st, corresponding elements 

 of the surfaces are maintained at constant temperatures, or 2nd, when the tem 

 peratures of the exterior medium at corresponding points of the surface remain 

 constant. 



For the velocities of flow along lines of flow across the terminal areas *, s of 

 corresponding prismatic elements are as u-v : u -v , where (u, v), (i/, 1/) are tem 

 peratures at pairs of points at the same distance A on opposite sides of s and s ; 

 and if n : n is the ratio of the dimensions, u-v : u -v =n :n. If then, dt, dt be 

 corresponding times, the quantities of heat received by the prismatic elements are 

 as sk (u -v) dt : s k (u - i/) dtf, or as n^n dt : itf ndt . But the volumes being as 

 n 3 : n 3 , if the corresponding changes of temperature are always equal we must have 



n?n dt _ n 2 ndt dt__&amp;lt;n?_ 

 ri* :; ra 3 r &amp;lt;^&quot; ~^* 



In the second case we must suppose H : H =ri: n. [A. F.] 



