198 

 and 



On a Di fraction Problem. 



+ 



= \/'2 cos (J + kVt) |^J\ cos - ( cos ~ + sin | j 



! T 50/ 5cc . 5u\ ~] 

 \ + J5 cos — (cos — + sin — 1 + . . . 



v2sm It-+™ Jgcos-^tcos-^ — sm-^ J 



K18&) 



, T 7 ^ 



70/ 7« 

 cos- 



sin 



7a 



)♦■■•] 



J 



In dealing with the three-dimensional wave and <£(ot, y, n'} 

 the right-hand member has Wot as argument for the 

 BessePs functions, and also Yt + nz takes the place of V*. 

 The connected forms shown in (8) are for the 3-dimensional 

 case 



^(OT,y,nO = 2-7^[{P(W/j) cos kn'p + Q(kn'p) sinWp} 



X cos -J t ^+k(Vt + lx + my+nz)\ + ...1 



= £y= [P ( Wp) cos ^j + *(V* 4 >w - rim) | 



-Q(Wp)sin |j+^(V^ + ^-n'^) J 1 



= - — ■=. iP(Wp)cos Wot + Q(W/o) sin Wot i 



2Y/7T U 



Xcos I ^+k(Yt + nz)\ + .;*], 



(8 ft) 



where 



/o = OT + A'sin « + ?/cos a and so n r p = ri-uT-\-lx + my. 



In the plane oblique case write n = 0, n' = l, OT=r in (8 6). 



§ 5. The asymptotic values of the series containing Bessel's 

 functions are assigned by their equivalents in terms of 

 P, Q, or (/>. An asymptotic value attaches to positions for 

 which p [r+y in the simplest case) is sufficiently great ; it 

 is not essential that y should be finite, but a finite angular 

 space must be excluded on both sides of the critical line, ft 

 say without further precision. 



