CONSTITUTION OP MATTER AND ANALYTICAL THEORIES OP HEAT. 15 



Therefore, since / /"'"^'cos \rp (2 \/t y)\ e~'^\hj can be made as small as we 



0 



please, by choosing s sufficiently small, and | i/'(2£ V^) — ?^(2 VÖ} ^ 1, it follows 

 that 



i,c^i\/rf+^ cos U(2\/r)} 



for those values of t for which cos {tp(2\/T)] is not zero. 

 15. Consider I^. Put t{x') = r;. Thea 



^2 = j _ cos 7} dfj, 



X s 



where P(x') stands for 77-^ — • 



^ ^ ip [x ) 



Put , = X{x') and let ä be the value of x' for which —P(a;') attains 



p [X ) 



its maximam. 



Then d is given by the equation 



X'{d) _ _^ 

 X(d.) ^ 2t ' 



1 



Now, 



X- 



■i^'(x')\ 



since 



z(^)>-^(.x')=-l. 



Therefore 



Therefore l\ — X{x')] r^I(x') and, consequently, 



X'{x') _1_ 



dl — 

 Therefore ^ ^^"^r consequently, d r\j \Jt. 



Let C be chosen to be sufficiently large; then, 



Cx2\jt 



and —P(x') constantly decreases as x' increases from Cx2Sß' io 6. 

 Let ip{d) = ; then it is evident that 



