FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 169 



* are both of the form a(n, s) in (0,ir). By an application of the rule of integration 

 by parts, it is easily seen that 



y8[ (n, s) cos 2ns ds = -J , I &, (n, s) sin '2ns ds = a ' '. 

 Jo n Jo n 



Thus we have 



r* i r* rt a ( n \ 



2 cos ns = IT + - a*, q l sin nsi cos ns l ds t H ^ , 

 Jo n Jo Jo n 



which, taken in conjunction with (17), leads to 



TT 1 f* f* 1 a (n) 



w, as = - + - a 3 q l sm n, cos rw, cfo, H V^. 

 4 nJo Jo n 



Substituting the positive square root of this value of I ' ds, and the value of m 



Jo 



obtained above, in the right-hand member of (16), we eventually obtain the formula* 



M 18 ) 



H L \ '* '* / \ "'* '" /_ 



where 



7T 



i (n. s) = I ds a </i sin 2ns, ds t , 



^ ' O !, I * 



4TT J J J, 



j (n, *) = s </i cos 2)1*! </! (ir *) </i cos 2/is, c/.s, . 



2TT |_ Ji Jo 



'Jo -', 

 and 



Putting in evidence the argument of q as in III., 6, the reader will see that 

 /+, (s) is the normal function which, for /* = X. +1 , satisfies the equation 



and the boundary conditions 



~ - H' = at s = 0, d ~ + h'u = at s = w. 

 ds ds 



It follows that AI (H, s) should remain unaltered when we interchange h' and H', and 

 at the same time substitute it s for s, </(TT s,) for q (*,) ; also that A(.s), Ag(u, s) 

 should merely change sign. We have expressed A (*), Aj (n, s), A. 3 (n,s) in forms 

 designed to show that they possess these properties. 



* It should be observed that there is an ambiguity of sign in the determination of each normal 

 function (vide III., 3). By substituting the positive value of -y u n -ds in (16) we obtain the 



asymptotic formula for that determination of ^,, + i(^) which is positive for s = 0. Substituting the 

 negative value of the square root in (16) we obtain the formula for the other determination. This would 

 have served our purpose just as well as (18). 



VOL. CCXI. A. Z 



