16 Prof. B. Osgood Peirce on the 



b 



It is evident that when one end of a regular right prism 

 of 2n sides made of homogeneous material is kept at a constant 

 iemperature Y , and the other end at a constant temperature 

 Yi, while its other faces are kept as nearly as possible at some 

 constant temperature between Y and Y/, the temperatures on 

 the axis of the prism in its final state depend very largely on 

 the ratio of the length of the axis of the prism to that of a 

 diagonal of a cross section ; and that, if this ratio be small 

 enough, the temperature conditions to which the sides are 

 subjected are of slight importance. 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 measurably 

 different,, whether the curved surface is kept at 0° C. or 

 100° 0., 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. 



It is easy to see, moreover, that in the case of a prism or 

 cylinder of poorly conducting material, the bases of which are 

 kept at constant temperatures while its side faces are exposed 

 to variations of temperature, the introduction of such metal 

 sheets as we have sometimes used certainly affects to some 

 slight degree the temperatures in the rest of the prism. 



►Since it was extremely difficult to control the temperatures 

 on the side faces of our prisms, it was necessary to make sure 

 that the dimensions of the prisms were such that probable 

 temperature changes at these faces could not sensibly in- 

 fluence temperatures at points near the axis. We were able 

 to get some useful information, to be checked afterwards by 

 experiment, from a discussion of the solutions (obtained on 

 the very approximate supposition that the thermal conduc- 

 tivities of our materials did not change with the temperature) 

 of the tw r o simple problems in heat-conduction which follow. 



(1) The radius of the base of a homogeneous right cylinder 

 of revolution of length I is a. If in the final state the tem- 

 perature V (which is harmonic within the cylinder) has a 

 constant value Y on one (the lower) base, the constant value 

 Vj on the upper base, and the constant value V on the convex 

 surface; and if the axis of the cylinder be used as axis of z, 

 with origin at the centre of the lower base, V is given by the 

 equation 



Y=Y+22 



i- J » ( ?){(Vo-v)^(^)+re-?>^Cf)} 



