376 THEORY OF HEAT. [CHAP. IX. 



Kt 



TYT, instead of t, in the integral (i) or in the integral (f). We 



\jJLJ 



shall now indicate some of the results which follow from these 

 equations. 



377. The function which serves as the exponent of the 

 number e* can only represent an absolute number, which follows 

 from the general principles of analysis, as we have proved ex 

 plicitly in Chapter II., section IX. If in this exponent we replace 



Tfj. 



the unknown t by 7^, we see that the dimensions of K } C, D and t, 

 (jU 



with reference to unit of length, being 1, 0, 3, and 0, the 



Kt 

 dimension of the denominator -^ is 2 the same as that of each 



term of the numerator, so that the whole dimension of the expo 

 nent is 0. Let us consider the case in which the value of t increases 

 more and more; and to simplify this examination let us employ 

 first the equation 



which represents the diffusion of heat in an infinite line. Suppose 

 the initial heat to be contained in a given portion of the line, 

 from x = htox = +g, and that we assign to a? a definite value X y 

 which fixes the position of a certain point m of that line. If the 



time t increase without limit, the terms -r-r and - - which 



4&amp;lt;t 4 



enter into the exponent will become smaller and smaller absolute 



_* 2 _ 2 _o? _ ft2 



numbers, so that in the product e & e *t e & we can omit 

 the two last factors which sensibly coincide with unity. We thus 

 find 



,, N 

 daf(a) 



This is the expression of the variable state of the line after a 

 very long time ; it applies to all parts of the line which are less 

 distant from the origin than the point m. The definite integral 



*2 



* In such quantities as e~ * . [A. F.] 



