CONSTITUTION OF MATTER AND ANALYTICAL THEORIES OF HEAT. 



19 



V (x, t) (Sjt r'+^ cos I? (^) + 



for those values of t for which the cosine is not zero. 

 Therefore F' {x, t) ±>- 1. 



3. 



— d 

 18. I = fx'^'^^ cos I {x') I e c?ic' = f+f . 



0 d 0 



Proceeding as in tlie case of of the last article, it is easily seen that 



each of the integrals J and _/ is numerically less than 4P((Z). 

 d 0 



Therefore |1| < 8| P(tZ)| and, consequently, |i|;^[P(d)| as ^ approaches zero. 

 Case I : {x') >~ i- • 



As shown in the beginning of Art. 15, d is given by the equation 



X' jd) _ 



and since 



^(^')>-^, -^'(^')^^- 



Therefore — X{x') ^x'^'^^ and, consequently, V^— X{x')^ ^l(x'). 

 Therefore 



X' jx') ^ 1 . 

 X{x') ^ x'' 



hence ~ ^^^j consequently, d ^ \/^. 

 Now, 



hence — P(d) -< ci^"''* e 

 Since [/1;^-P((?), it foUows that 



I F' (ic, Ol -»^^ ^""^^"^'^^ ^^^1 consequently, -^1. 



3* 



