318 



34 



where the summations are to be extended over all positive and negative entire 

 values of r including zero. Now from (78) and (86) we find by means of elementary 

 calculations, omitting, here as well as in the following, for the sake of simplicity 

 the argument t£ of the Bessel coefficients, 



1^ - 1^ = 2 rs^ + ''^ ~ '^''"-^ + - ''^ 



-^'^é^'''^-^ ''^-^'-^ <93.) 



%T (^) - 1-"'^"'' ° (-.4; (f") (t) ) = h'^-^'"'^. 



Introducing these values in (92) we find after some simple reductions 



_ ZeFlP \ 



• {(1 -V £')((! +^') f3c'— 21 — -c' (5- 3£'2)) Jr^^ + (1 - £') ((1 - £') (3e' + 2) + re (5 - 3e'-)) Jr+i) • 



• cos 2 ;r (r^ 2 + 2 Wj) / 

 + 2 2 {(s'/^' + (1 + ^-') (2 + 5 rc'//2)) J^_, + (.'^,2 _ (1 _ (2 + 5 r£',.i^)) Jr+i} cos 2 ;r™, . 



The expressions for the coefficients become undefined for r = 0, but by directly 

 introducing r = in (92) we find that the coefficient to cos 2 r ( — 2(u-^-\- 2ü}^l \s equal 

 to ^'J^ , while the constant term in the second series becomes equal to ^- — ^ "j~ ^ . 

 Further it will be observed that in the second series the terms corresponding to 

 values of r, which are numerically equal but of opposite sign, may be taken 

 together, so that we finally get for 



.{(l+s')((l+c') (3£'-2)-r£'(5-3e'^'))Jr-i(r£) + (l-c') ((1-e') (3 e'+ 2) + re (5 - 3s-)) J.+i (re)) 



•cos2 7r(f^2\ü, H-2û;2)/ 



- - - J - + 2'^ ((2 + 5fj- £' - r)./r-i (rs) -f (- 2 + ôf,'' s"t)Jt+, (re)) cos 2nTco, /] 



3(-s^ + e- V) 

 s' 



where 



r-^ + n + n, s' = £ = i/r=-v^, y ^j^^t«' ^ = 



and where in the first series the summation is to be extended over all positive and 

 negative entire values of r except r = 0, and in the second series only over all 

 positive entire values of z except r = 0. 



By a calculation quite analogous to that for we may from (91), (88), (87) 

 and (77) deduce similar expressions for r' + iy^. Thus we find 



