126 PRACTICAL MATHEMATICS 



fJtl 



Now -j- = a { ke~ u sin (pt c) + pe- kt cos (pt c) } 



CvE 

 fJlJ 



and -j- = when ke~ u sin (pt c) + pe*** cos (pt c) = 

 ctt 



k sin (pt c) = p cos (pt c) 



T) 



tan (pt c) = - 



K 

 *n 



If a = tan" 1 * then tan (pt c) = tan a 



w 



and pt c = a, TU + a, 27C + a, . . . MTC + a 



Hence -^ = when 2 has the values 

 at 



a + c a+c 7c a+c 2rc a+c nrc 



p p P P P' P P 



Some of these values of t will give maximum values of y, while 

 the other values of t will give minimum values of y. 



,..,, , , , . a+c me 



Taking the general value of t, t= -- 1 -- 



Then pt c = WTC + a 



and sin (pt c) = sin (rnz + a) 



When n is an even integer, sin (pt c) is positive, and hence 

 maximum values of y occur when t has the values 



a+c a+c 27C a+c 4:r 

 - , - -] -- , -- 1 -- , etc. 



p P P P P 



When n is an odd integer, sin (pt c) is negative, and hence 

 minimum values of y occur when t has the values 



a+c 7i a+c STC a+c STT 

 + > + > + ^t etc. 

 p p p p p p 



T^ ... a+c , 



Putting - = b. 



Then when t = b y = ae~ bk sin a = M 



when t = b H -- y = ae~~ \ + F/ sin a 



_ 



= Me" ^ 

 when Z = b + - ?/ = ae~ k \ b+ ~p' sin a 



-lirfc 



= Me P 



