FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 163 



2. The solution u satisfies the boundary condition (2) by its definition. It is 

 easily seen from (4) that it will also satisfy the boundary condition (3) if 



tan WT = ;ZF' ( 7 > 



where 



ftr / JI/ \ H'/l' f* / H' \ 



a,M, ( cos rs, sin TS, } </ and P' = g^M, ( sin TS, + cos TS, ) e/a,. 

 . u \ r I T Jo \ T I 



Using the formula (5), we see that 

 P = /t'+H' o, cos* TS, </s,+ ^Ll t p' = _ \ n l 8 in TS, . cos TS, </s,+ ?H/. 



Jo T Jo T 



Hence the equation (7) may be written 



tan ITT = - (V+H'- f", cos 2 TS, </*,) + ^ . 



T \ Jo T J 



It is easily seen from this that the large positive roots of (7) are of the form 



- n I 



3| CtOj I / \ 



+-?-> (8) 



n \ TT in' 



where n is a positive integer. 



It will l>e clear that a positive integer n may be chosen great enough to ensure that, 

 for n ^ n, the numtars T B are roots of (7) which increase continually with n. Thus, 

 since (7) is the condition that there should be solutions of (1) satisfying B', 



T J t T . +i T ' t ... 



are corresponding singular values arranged in increasing order of magnitude. It 

 follows that, for values of n which are not less than a certain positive integer, 



T.' = ^-, (9) 



where m is a positive or negative integer, or zero. The paragraphs which immediately 

 follow will be devoted to the determination of m. 



3. Referring to 5, 6 of the previous section, it will be seen that, by using (9) 

 and (10) we obtain 



r i \ . A ' , i If cosh ps, sinh p (ss,) , , a(p, s) 



* (s) = 1 + tanh ps o, - -2* as,-)- M* -' . 



p p Jo cosh ps p" 



Since 



2 cosh ps, sinh p (s s,) = sinh ps + sinh p (s 2s,), 

 and 



f' sinh p (s-2s,) = 

 Jo 



cosh ps p 



we have 



C, () = 1 + - tanh ps (h'-l \' qi d*} + 

 P \ -o 



Y 2 



