130 



THEORY OF HEAT. 



[CH. II. SECT. IX. 



162. If we retained the coefficients C and D, whose product 

 has been represented by c, we should have to consider the unit of 

 weight, and we should find that the exponent of dimension, with 

 respect to the unit of length, is 3 for the density D, and 

 for G. 



On applying the preceding rule to the different equations and 

 their transformations, it will be found that they are homogeneous 

 with respect to each kind of unit, and that the dimension of every 

 angular or exponential quantity is nothing. If this were not the 

 case, some error must have been committed in the analysis, or 

 abridged expressions must have been introduced. 



If, for example, we take equation (6) of Art. 105, 



dv _ K d*v hi 

 dt ~~GD ~da?~ CDS V 



we find that, with respect to the unit of length, the dimension of 

 each of the three terms is ; it is 1 for the unit of temperature, 

 and 1 for the unit of time. 



/ 2/2 



In the equation v = Ae~ x & of Art. 76, the linear dimen 

 sion of each term is 0, and it is evident that the dimension of the 



exponent x A/ ^~ is always nothing, whatever be the units of 

 length, time, or temperature. 



