CONSTITUTION OF MATTER AND ANALYTICAL THEOEIES OF HEAT. 



11 



Since both tlie series on the right side of the above equation are uniformly 

 convergent in 



lim Q (x, t) = 0, 



X = 31 0 



lim Q {x, t) = 0. 

 X = — jr+0 



10, Thus the group of conditions tvhich is necessary and sufficient , in Order 

 that V{x, t) he the Solution of the prohlem, is the following : 



i. For every value x or, at least, for an everywhere dense aggregate 

 of its values , there exists a finite constant P such that , for any value of t, 

 however small, | F' i^) | < P, — ä < a; < 



11. Lim V(x, t) = f{x) if the limit exists, or, the limit does not exist, and 

 <=+o 



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

 aches zero. 



Behaviour of V {x, t) for t small. 



11. As V {x, t) is an odd function of x and F' (0, t) = 0, it would be suffi- 

 cient to consider the case 0 ■< x <%. 

 Let ® {ij, t) stand for 



Then 



l + 22cosm?/e , ^ > 0, 0<?/<23t. 

 1 



@' t) — — 22 sin my e 



Now, 



V' [x, i5) = — 2 sin * 

 1 



— 2jI msm mx cos mx e f [x ) dx 



Tt 1 'O 



= 2 / (x + x') e f{x') dx' 



^ 1 'o 



7f 1 'o 



Since the series 2 '^^ ^ * is uniformly convergent in y in the inter- 



1 



val (0, 2jc), it foUows that 



2 1»- sin m (ic' -}- a?) e and 2 sin ?)z (x' — x)e 



1 1 



2* 



