ON THE TOTAL INTENSITY OF INTERFERING LIGHT. 229
the integration being extended over the whole aperture. If it should be necessary to suppose a change of phase to take place in the act of diffraction, such change may be included in the constant B. If, then, / be the intensity,
n2T /IT . Zirpx + qy 7 7\2 / ff 27rpx+qy7 7 \2 D2/ = sin— * , ™dxdy) + ( cos — ^ ^™ dxdy} ;
\J J A> 0 / \J J A* Q /
and if / be the total illumination,
-00 ,00
/= Idpdq.
J _oo J — oo
Now,
the limits of a?', y7 being the same as those of #, y. Hence,
= 1 1 1 /
cos TT~ (p^ — x + qy' — y) dec dy dx dy'.
In the present shape of the integral, we must reserve the integration with respect to p and q till the end; but if we introduce the factor e*0^*^, where the sign — or -f is supposed to be taken according as p or q is positive or negative, we shall evidently arrive at the same result as before, provided we suppose in the end a and /3 to vanish. When this factor is introduced, we may, if we please, integrate with respect to p and q first. We thus get
D2/= limit of
27T
j—- (px — x + qy' — y) dec dy dxf dy' dp dq. Now,
-oo /""
eTtt-P cos (kp -Q)dp= cos Q eTa^ cos kp dp
J -oo J — °°
1.00
4- sin Q I e*"P sin ij) dp
J -00
f°° . . 2a cos 0
= 2cosQ e-**co*kpdp= ,,.,-.