186 DR. J. MERCER: STTJRM-LIOUV1LLE SERIES OF NORMAL 



Now at all points of the closed interval (0, ir) 



^ tanli ps < 1 ; 

 and, whatever value belonging to the interval s may have, 



oo8hp*..8inhp(*-j) 

 - ' 



cosh ps 



= i 

 * 



cosh 



^ tauh ps+% 



siuh p(s 2.?! 



cosh 



if ,, is a point of the closed interval (0, s). Hence, if * be the greatest value of 

 | in (0, ir), we see from (9) that 



which may be written 



It follows that &(*) is -limited for values of p that are greater than a certain 

 positive number, and of * that lie iu the closed interval (0, 77). 



G. It will be convenient in what follows to use a (p, s) as a shorthand symbol for 

 the phrase " a function of p and s which is limited for values of p that are greater 

 than a certain positive number, and of s that lie in the closed interval (0, ir)." 



With this convention it follows from the result obtained in the preceding 

 paragraph, that (9) may be written 



whence, in virtue of (8), we obtain the formula 



A () = cosh 



(11) 



In order to obtain an asymptotic formula for ^ x (s), we turn back to the equation (7). 

 Differentiating with respect to s, we obtain 



[' 



0\(s) = p sinh ps+k' cosh ps q$^ (s^) cosh p(ss t ) dsi. 



Jo 



Using (11) this becomes 



^A (") = p sinh ps+h' cosh ps- q l cosh ps t cosh p (s-.s'i) ds v 



1 r 



pJo^ 1 ^"' l ' "^ i) \- \ ) 



Since 





cosh ps t . cosh p (a i_) 

 cosh ps 



