CONSTITUTION OF MATTER AND ANALYTICAL THEOEIES OF HEAT. 



33 



lim {&, X, t) == 0, 



lim Q^{6, x,t) = 0, ^ ^ 



X — Tt 0 



where t) is the quantity of heat which flows across any area <?, placed 



parallel to the faces and distant x from the origin, in the interval (0, t). New, 

 as has been already indicated in Art. 30, Q.^{o) cannot be approximated to for 

 every area 6. Therefore the third condition , in all its generality , cannot be 

 approximated to. 



The equations 



-.t 



lim I f Y'{x,t')dt'\ = 0, 



. = -.^ (C.) 



lim \fY'{x,t')dt'\ = 0, 

 X = 7t — 0 ' 0 



are equivalent to (3) when 6 is restricted to be circular and of unit area, the 



quantity neglected being Thus, (CJ is an approximation to the fact that 



the quantity of heat which flows across any circle of unit area, situated any- 

 where on any face of the slab, is nil. Now from this fact, of course, follows 

 the fact that the quantity of heat which flows across any area 6, situated on 

 any face of the slab, is nil; but it is seen without difficulty that the possi- 

 biKty of an approximation to the former does not necessarily involve the possi- 

 bility of an approximation to the latter. With this understanding, (Cj may be 

 regarded as the third approximate condition of the phenomenon. 



Auxiliary^) Functions of Fourier's Tjrpe. 



32. Let the initial auxiliary function Y{x, 0) be f{x), a finite , integrable, 

 and even function of x ; further , let f {x) , f" {x) not only exist but also be 

 finite and continuous. It will be sufficient for the purposes of this essay to 

 consider only such functions f {x) in the present theory. I proceed now to find 

 what necessary and sufficient conditions f{x) must satisfy in order that the 

 auxiliary function at any subsequent time be given by V{x, t) where, as in the 

 last part, V{x, t) Stands for 



1) It is scarcely necessary to mention that tlie function Y{x, t) which exactly satisfies (Aj), 

 (Bj), (Cj) is not the same as the real auxiliary function. The real auxiliary function satisfies (Aj), 

 (Bj), (C,) only approximately ; thus the exact Solution of these conditions can be only an approxi- 

 mation to the real auxiliary function, and may be calied the pro-auxiliary function. However, in 

 this and subsequent articles, I will use the briefer name „auxiliary function" as there is no chance 

 of its being confused with the real auxiliary function. 

 Abliandlg. d. K. Ges. d. Wiss. zu Göttingen. Math.-Phys. Kl. N. F. Band 2,4. 5 



