and an Asymptotic Theorem in Bessel's Series. 197 

 Thus in view of to = ~ ~ t~ we g e ^ 



Kl7a) 



<K r >y)-<Kr, -y) 



= s/2 cos (J +*V*)[j 4 (*r) sin | + J f sin y + ...] 



-a/2 sin(^ +*vA[j f siny + J, sin — + ...1 



and 



=v /2 cos (^ 4. A V«V Jx COS ^ + J|- COS y + . . \ 



+ <y 2 sin ( j -\-kYtji J 3 cos — +J, cos — + ...) # 



Recalling the statement in connexion with (11), thej: ormulse 



q 



(15) represent the left-hand members from = to — and 



3t ^ 



from — to 2ir the left-hand members with sign changed. 



Thus in (17 a) the series represents the part of formula (2) 

 which contains (/>, with the signs attached for the different 

 regions ; and (17 b) gives the corresponding forms for the 

 2nd solution. Thus (9 a) and (9 b) represent the original 

 solution. 



§4. For oblique incidence r + x sin a. + y cos « takes the 

 place of r(l + sin #), and so in passing from cf)(r f y) to <£(r, y } a), 



(d is changed to — ~ j ; and in like manner co' is changed to 



~ — h j. Thus we get in the region A 



= ^/2 cos (~ + &Vn I J^ (&/■) sin = (cos " — sin -A 



T . 50 / 5a ! ±&mm 5 a \ -, 



+J i- sm ^( cos T~ sm y; + •••] 



j>(176) 



•2 sin (j + *V*) [ J3 sin ^ (cos 3 | 



+ sm T j 



}>(18a) 



, T . 10/ let , . 7a\ "I 

 -f J^sin-^-fcosy + sm-^-)... I 



