164 

 Hence 



R. J- M ERCER: STURM-LIOUVILLE SERIES OF NORMAL 



/ f 

 X (,) = cosh P * (1 + ^) + \ sinh I* (*'-* j> 



Supplying this value of 6, (s) in the formula 



ff> (s) = p sinh ps+h' cosh ps- \qA (*i) cosh p (*-*,) d* lt 



it is easily shown that 



//if a (p, s)\ 



0*. (s) = p sinh ps -I- cosh ps (h f ?i i + - ) ' 



.'o p / 



By employing the device of III., 6, it may be deduced from these results that 



and that 



</>' A (s) = p sinh p (TT s) cosh 



It follows from these formulae that 



/ f A a (P 



= p sinh p7r+cosh pnlh'+n. % j ^aSiH -- 



\ i 



hence 







Again we have 



v /. , a (p, s)\ , cosh p (TT-.S) sinh ps /, , l f 



^(s) </> A (s) = cosh ps cosh p (TT s) 1 + ^ 7 + - I J ?i ( 



\ p p .0 



cosh ps sinh p (TT s) /TT/_ j^ f' T \ 

 P \ 



= AcoshpTT+icosh P(TT 2s) + i ^(/t'+H' i o-j t/Sj 



P \ -JV ; 



gil-tj^l<-iy - , 



Since the integrals of the fourth and fifth terms between the limits and IT are 



both of the form 



at (p) cosh piT 



we see from this that 



Jo/ 



