CONSTITUTION OF MATTEß AND ANALTTICAL THEOEIES OF HEAT. 



9 



5, It should be noted that T{x, f) need not satisfy any conditions other 

 than those necessary to make the conditions (A), (B) and (C) intelligible and 

 these conditions themselves, For example, T' (x. t) need only be finite and inte- 

 grable in t ; thus it may not be possible to obtain an interval of time, however 

 small, in wbicb there are not an infinite number of instants at which the rate 

 of flow of heat does not exist. Also T' {x, t) may be meaningless for an aggre- 

 gate of values of t, of zero content ; in particular T' {x, 0) may not exist. 



Solutions of Fourier' s Type. 



6. Let the initial temperature be f{x), a finite, integrable, and even function 

 of X. I will find the necessary and sufficient conditions, that f{x) must satisfy, 

 in Order that the temperature at any subsequent time be given by V{x, t), where 



V(x, t) = \a^-\- 2 OOS luxe , 

 1 



being given by 



2 . 



= — / f ix') cos mx' dx' . 

 n 'o 



It should be noted that F(ic, t) is defined only for ^ > 0. 



7. Substitute in (A) and (B) 



T{x, t) = V{x, t), 

 T{x, 0) = fix). 



Then these conditions become 



0 



I V{x', t) - f(x') I dx' = f V (x, t') dt', -jKx<r.; (A') 



lim V{x, f) = f{x) if the limit exists, or, the limit does not exist, and then 

 < = +o 



f(x) is contained in the aggregate of values assumed by V{x, t) as t appro- 

 aches zero. (B') 



Consider (A'). The series for V{x', t) is uniformly convergent in x' and is 

 consequently integrable term by term ; also it is easily seen that 



/ j fi.^') - i «0 1 f^^' ^- 2 sin mx. 



Therefore the left side of (A') is equal to 



y, — sinma;e — >] — sm mx. (1) 

 1 w im ^ 



Abhandig. d. K. Ges. d. Wiss. zu Göttingen. Math.-Phys. KI. N. F. Band 2,4. 2 



