R= ) £,exp Jil, (cosa- COSa5 +y, (cosB-cos8p ) 
+ s + =il\e = 
z, (cosy cosy, ) p, (cosy, 1) Po ,(cosy, , 1)] (2) 
By simple geometry, it can be shown that 
P, = \/D°-2D(x, cosaty, cos8t+z,cosy)+tr,? (2a) 
t t t t t 
where 7..° = x,*+y,2+z.2, and that 
t hein! 
D-(x,cosaty, cos8tz cosy) 
U U U (2b) 
cost. = 
Ve 0; 
If compensation is made for a plane wave front arriving 
from a direction (cosa, COs, , COSy,) and if the actual 
source is a finite distance D in a direction defined by 
(cosa, cos8, cosy), it can be shown that the directivity 
function is 
WV 
Re Ve, exp Jhlx., (cosa- cosa, ry, (cos8- cosa ) 
-_ U 
Gal 
+2, (cosy-cosyo +p, (cosy ,-1)] (3) 
Now if D> =, in equation (3), the directivity function be- 
comes in the limit that for a plane wave front: 
MV 
R= » B exp Jhl x, (cosa-cosay yy, (cos8-cos8o ) 
t=1 
tz, (cosy- cosy, )] 
because p .(cos,-1)50 as D>». This can be shown by use of 
L'H6pital'S rule’ 
