318 PROFESSOR STOKES ON THE 
In the present shape of the integral, we must reserve the integration with 
respect to p and g till the end; but if we introduce the factor e+” +?7, where 
the sign — or + is supposed to be taken according as p or ¢ is positive or nega- 
tive, we shall evidently arrive at the same result as before, provided we suppose 
in the end a and @ to vanish. When this factor is introduced, we may, if we 
please, integrate with respect to p and q first. We thus get 
D? I = limit at fff ffferer cos 55 (p= +9y=9) du dy dx dy dp dq. 
ies} 2) 
Now, fe e**? cos (kp—Q) dp= cos af eT“? cos kp dp 
=o -—o 
an 
+ sin Qf et“? sinkp dp 
—o 
a 
=2c08Q f eu cos kp dp= = SOO. 
A similar formula holds good for g, whence 
p? I= limit Laer . eB w= 8) Ve ; c rv=0)"| dxdydz dy’. 
or 

Let now 
Qa (¢—2) _ PONG 
<a Hau, whence dz’= = du, 
and the limits of w are ultimately —» and +o, since a ultimately vanishes. 
Hence 



2adxv br du 
limit of + (FE Give apey ae io Lap = 
A similar formula holds good for vy’, and we have, therefore, 
D?I= en fy dzdy=0 2A, 
if A be the whole area of the aperture or apertures. 
Now I ought to be equal to A, and, therefore, 
D=bX. 
Case II. Aperture in front of a screen. 
The formula for the illumination is given in Airy’s Tract, Art. 73. We have 
as before, 
pet = limit of ff fff ferr*t cos 29 { (v— 22)" 
of 2 eR i te JG +} 
(« are +(y sry" -(y-2 Ee didydx dy dpdq 
=limit of (ff ff ferer*62 cos { TTD [yt a8 4 y2—y"I 

