128 THEORY OF HEAT. [CHAP. II. 



the volume ; but it would be preferable to employ the coefficient c 

 which we have just denned ; magnitudes measured by the unit 

 of weight would not then enter into the analytical expressions : 

 we should have to consider only, 1st, the linear dimension x, the 

 temperature v, and the time t\ 2nd, the coefficients c, h, and K. 

 The three first quantities are undetermined, and the three others 

 are, for each substance, constant elements which experiment 

 determines. As to the unit of surface and the unit of volume, 

 they are not absolute, but depend on the unit of length. 



160. It must now be remarked that every undetermined 

 magnitude or constant has one dimension proper to itself, and 

 that the terms of one and the same equation could not be com 

 pared, if they had not the same exponent of dimension. We have 

 introduced this consideration into the theory of heat, in order to 

 make our definitions more exact, and to serve to verify the 

 analysis; it is derived from primary notions on quantities; for 

 which reason, in geometry and mechanics, it is the equivalent 

 of the fundamental lemmas which the Greeks have left us with 

 out proof. 



161. In the analytical theory of heat, every equation 

 expresses a necessary relation between the existing magnitudes 

 x, t, v, c, h, K. This relation depends in no respect on the choice 

 of the unit of length, which from its very nature is contingent, 

 that is to say, if we took a different unit to measure the linear 

 dimensions, the equation (E} would still be the same. Suppose 

 then the unit of length to be changed, and its second value to be 

 equal to the first divided by m. Any quantity whatever x which 

 in the equation (E) represents a certain line ab, and which, con 

 sequently, denotes a certain number of times the unit of length, 

 becomes inx, corresponding to the same length ab ; the value t 

 of the time, and the value v of the temperature will not be 

 changed ; the same is not the case with the specific elements 



h, K, c\ the first, h, becomes , ; for it expresses the quantity of 



i(Ylt 



heat which escapes, during the unit of time, from the unit of sur 

 face at the temperature 1. If we examine attentively the nature 

 of the coefficient K, as we have defined it in Articles 68 and 135, 



