;{()») 



22 



The last expression contains only (p and y and is therefore a function of and 

 only, which allows of an expansion of the form 



(x+ iy)e2T/ _ 2'firi,r2e2-nn«'H-r^i«'»), (61) 



where the summation is to be extended over all positive and negative entire values 

 of Tj and ^rid where the coefficients B according to Fourier's theorem are equal to 



r, + L,)(M, + L,)JJ^ 



Br,,T. = ^ I j (Ml H-L,) COS + I sin I). 



• [(Mg + L,) cos^- + /Ä-sin|-} e + + i 



We will now transform this expression into an integral which is taken over (p and 

 j(, making use of the expression (56) for the functional determinant of the trans- 

 formation. At the same time we will introduce the abbreviations 



.. = 1/^. ',-V\. .,-|Æ3. ..-i/Hi. . = 14. m 



which allow us to express the quantities K, M,, Lj, M,, L.,, and in the form 



K — xlu:, Ml = xPul + il^), Li = 2xPc,i,,„ a, = i^t^,, \ 



In this way we get, denoting -\- 4- 1 by z, 



• (1 + <Tl COSç^- — COS;^)e-'<^' + '/»H''~''''^'Sinç/'-*(r. + V2);r- i-^zsinXd^d^ 



.prr 



= (- ir^, \ \ (ri.,e'ç''/-f rie-HA'=) (^^^ e + ^ " '^Z-^* (1 + h h-, cos ç^- + £,£.,3 cos^) • 



We see that the last expression becomes equal to the sum of a number of terms 

 each consisting of the product of two integrals of the type (58), where p is equal 

 to or 1. Making use of formula (34), we may write each of these integrals as a 

 sum of Bessel coefficients of the same argument and of different orders. By means 

 of elementary calculations and making use of (35), we get in this way for the JS's 

 the final expression 



xP \ I 



ß-i, *a = — V ) «23 '^-1 {'Oi)Jr,. ( r<7.) — £j L> Jr, + 1 (^■'^i) -I 1 (^<^2) f • '^5) 



This expression becomes indefinite for - = 0, but by introducing Ihis value of r 

 directly in (64), we easily find 



