414 THEORY OF HEAT. [CHAP. IX. 



The two equations (a) and (b) give therefore for g and h the 

 two integrals 



\dy sin y z cos 2&# and \dy cos ?/ 2 cos 2% 



which we shall denote respectively by A and B. We may now 

 make 



sn cos &amp;gt;2 = 



y z = p z t, and Zby = pz ; or i 

 we have therefore 



fj&quot;t\dp cosp*t cos)2 = A, *Jt\dp si] 



The values 1 of g and /& are derived immediately from the known 

 result 



r + oo 



VTT = I dx e~ x *. 



J -00 



The last equation is in fact an identity, and consequently does 

 not cease to be so, when we substitute for # the quantity 



The substitution gives 



= r 1 \ dy e &quot;^ = f 1 \ dy 



Thus the real part of the second member of the last equation 

 is N/TT and the imaginary part nothing. Whence we conclude 



N/TT = -j= (\dy cos y*+jdy sin y*) , 



1 More readily from the known results given in 360, viz. 



fdusinu /^ , du . .. 



~~r~ = \/ o Let u = z &amp;gt; % 1= =dz &amp;gt; then 

 x/w v 2 Ju 



I e?2sins 2 =i \/ J. and I dzsinz*=2 I dzsiuz&quot;*= \/ J. 



Jo V 2 J-oo Jo V 2 



So for the cosine from p**^ /* [B.L.B.] 



/ w ^ 2 



