4 PROCEEDINCJS OP THE AMERICAN ACADEMY. 



section ; and, if this ratio be small enough, the temperature conditions to 

 which the sides are subjected are of slight importance. For instance, 

 the temperatures at points on the axis of a relatively thin disk, one face 

 of which is kept at 0" C and the other at 100° C, are not measur- 

 ably different, whether the curved surface is kept at 0° Cor 100° C, 

 from the temperatures at corresponding points on the axis of an infinite 

 disk of the same thickness, the faces of which are kept at 0° C. and 

 100° C. respectively. 



On the other hand, if the temperature gradient on the side faces could 

 be made to follow the proper law, — or even if, for moderate values of 

 Fo — I'l, it could be kept constant, — the temperatures on the axis of the 

 prism would be much the same, whether the prism were slender or stout. 



In view of the extreme difllculty of controlling, or even of measuring 

 with accuracy, the temperatures on the side faces of a prism, it seemed 

 to us desirable to determine beforehand, as accurately as we could from 

 theoretical considerations, under each of a number of different assump- 

 tions with respect to the side temperatures, how short a prism of given 

 cross section must be in order that the temperatures on its axis, in the 

 case mentioned above, might be sensibly the same as if its cross section 

 were infinite in area. 



We shall find it convenient to write down at the beginning of our dis- 

 cussion some of the common equations* of the theory of heat conduction 

 in the forms which we shall need to use later on. If 6 represents 

 the temperature at -the time t at any point, J*, in an isotropic solid, the 

 rate of flow of heat at this time, at J\ in any direction, is usually assumed 

 to be the product of a scalar point function, /, and the negative of the 

 space derivative, taken at 7* in the given direction, of a certain function 

 of the temperature, /■ (6). If, therefore, u, v, and w are the components, 

 parallel to three mutually perpendicular co-ordinate axes, of the vector, 

 //, which represents the flow within the solid, 



„_ . 9/(6) _ , 9 6 



v = -/.f(6).l^^, w = -.'./' {6).^^^. (1) 



* Fourier, Theorie Analytique de la Chaleur. Poisson, Theorie Matlie'niatique 

 de la Chaleur. Lame, Lei.ons sur la Theorie Analytique de la Chaleur. Kelvin, 

 Article " Heat " in the Encj'clopanlia Britanniea. Kelland, Brit. Assoc. TJcp , 1811. 

 Preston, Theory of Heat, llieniann, Partielle Differentialgleichungen. 



