CONSTITUTION OF MATTER AND ANALTTICAL THEOEIES OF HEAT. 17 



Then 



t(x+^)-t{x) = e^'Cö = -|-, 



where X"^ X X + ^- 



6 ?r 7c 

 Therefore — = ^ is greater than ^ . since evidently 



^ ^x^'ix) 



—7p'{x') decreases as x' increases. Bat —x'ip'{x')^l since [x') ~< l • 



Therefore — >~1 as x approaches zero, (1) 

 X 



New, it is easily seen that as t approaches zero 

 But 



according to (1). 



And %'>~d. Therefore ^ > — ^ ^'^^i consequently, 



3ä 



Therefore I^it^ has the sign (—1)"". 



16. Since {■^(^\ltj) — rmt\ approaches zero as approaches zero and, con- 

 sequently, m approaches infinity, it follows from Art. 14 that 

 has the sign (—1)"'. 

 Therefore 1{Q lAQ + IM the sign (-1)™. 



Therefore it is proved that , as t approaches sero , V {x, f) mahes an indefi- 

 nitely large number of fluduaüons, from positive to negative values , with indefinitely 

 increasing amplitude. For the sake of brevity I will, hereafter, express such 

 behaviour by the symbolism 



V'{x, t) ±>^1. 



2. ^{x')r^l(^^)j. 



= (2Vrf+^J?cos|(z^)+y|, 



17. 



Abhandig. d. K. Ges. d. Wies, zn Göttlngen. Math.-Phys. Kl. N. F. Band 2,4. 



