SECT. IX.] UNITS AND DIMENSIONS. 129 



TS- 



we perceive that it becomes : for the flow of heat varies 



m 



directly as the area of the surface, and inversely as the distance 

 between two infinite planes (Art. 72). As to the coefficient c 



which represents the product CD, it also depends on the unit of 





 length and becomes 3 ; hence equation (E) must undergo no 



change when we write mx instead of x, and at the same time 



- , = , 3 , instead of K, h, c - the number m disappears after 

 m m~ m 



these substitutions : thus the dimension of x with respect to the 

 unit of length is 1, that of K is 1, that of h is 2, and that of c 

 is .3. If we attribute to each quantity its own exponent of di 

 mension, the equation will be homogeneous, since every term will 

 have the same total exponent. Numbers such as $, which repre 

 sent surfaces or solids, are of two dimensions in the first case, 

 and of three dimensions in the second. Angles, sines, and other 

 trigonometrical functions, logarithms or exponents of powers, are, 

 according to the principles of analysis, absolute numbers which do 

 not change with the unit of length ; their dimensions must there 

 fore be taken equal to 0, which is the dimension of all abstract 

 numbers. 



If the unit of time, which was at first 1, becomes -, the number 



n 



t will become nt, and the numbers x and v will not change. The 



coefficients K, h, c will become , - , c. Thus the dimensions 



n n 



of x, t, v with respect to the unit of time are 0, 1, 0, and those of 

 K t h, c are - 1, - 1, 0. 



If the unit of temperature be changed, so that the temperature 

 1 becomes that which corresponds to an effect other than the 

 boiling of water ; and if that effect requires a less temperature, 

 which is to that of boiling water in the ratio of 1 to the number p- 

 v will become vp, x and t will keep their values, and the coeffi 

 cients K. h, c will become , - . - . 



P P P 



The following table indicates the dimensions of the three 



undetermined quantities and the three constants, with respect 

 to each kind of unit. 



F. H. 9 



