and an Asymptotic Theorem in Bessel's Series. 195 

 To pass to the case of oblique incidence we write 

 x sin a + y cos a for y in cos k(Vt+y) and in <£(r, y )) 

 tfsina— ycosafor — ?/incos£(V£ — y)andin0(r, — y)\ ' ^ 

 Thus for example </>{r,y, a) being 



__ ^r+zsina+ycosa 



-zKT —\ cosj — + &(V£+a?sin«+#cos«— u ) >— ?- (4) 



</>(r, y, a) replaces </>(r, y) in (2). The regions B and C are 

 now separated by the line of the incident ray through in 

 place of OY' ; the regions A and C by the line of the ray 

 reflected from instead of OY. These are the only changes 

 needed. 



Lastly for an inc ident wave cos k{~Vt + lx-\-my + nz), let 

 *r= \ J x 2 + y\ ri= \/l-n 2 , and (7, m) = w'(sin a, cos a) ; then 

 </>(ot, y, n') being 



i^+zsina+ycos a 



2\/--j cosi j- + k(Vt + lx + my + nz — n'u) > 



du 



Tu ^ 



<j>{vr, y, n') replaces <f>(r, y) in (2j, while the opening terms are 



i cos k( Yt + lx+my + nz) - J cos k (Yt -\-lx- my + nz). 



These are the only changes needed, the separation of regions 

 being as in the last case. 



§ 2. The above constitutes a solution in terms of definite 

 integrals the evaluation of which depends on well known 

 series. To pass to the second solution in terms of Bessel's 

 functions we set out from these series, which are therefore 

 briefly quoted. If 



p cosudu . ) 



— =r{p)coap + Q(p)amp, j 



V« 



then 



i 



Jp sin u du 

 o ~—^~ =?W smp-Qfp) cosp | 



(6 a) 



2.2.2 



2.2 



2.2.2.2 



series convergent for all values of p. For p great, asym- 

 ptotic values are 



PAi7. 1%. S. 6. Vol. 36. No. 212. Aug. 1918. O 



fi 



