








TOTAL INTENSITY OF INTERFERING LIGHT. 319 
eae) ATL yy) | de dy az dy dp dq 
= init otf fff Aa — (a = 

Nb a 
(a+b) 
Aab 
Now, when a vanishes, the whole of the integral 
ies es CED 
cos (ff? —a? +472 —y") dz dy dz dy’. 
ww) 
is ultimately comprised between limits for which 2’ is infinitely close to 2, and 
similarly with respect to vy’; so that ultimately 
1 (a +b) 
Nab 
within the limits for which the quantity under the integral sign does not vanish. 
Hence, passing to the limit, we get 
DeIl=% wf fax ay= & A, as before. 
cos (7? —a? +y/?—y?)=1 
Case III. Everything the same as in Case II., except that the phase of vibra- 
tion is retarded by p, where p is some function of x and y. 
This case is very general. It includes, as particular cases, those numbered I. 
and II. The experiment with Fresnev’s mirrors or a flat prism is also included 
as a particular case.* 
From what precedes, it is plain that we should have in this case 
28 
= limit ASL Ce 2" B24 eae 
A 6 
cos ee [2/?—2? +y/2—y"]—p'+p } dzdydz dy, 

where p’ is the same function of a’ and y’ that p is of z and y. The same reason- 
_ ing as before leads to the same result. 
I do not regard the preceding demonstration of a result which you were the 
first to announce, as of any physical interest after what you have yourself done. 
Still it may not seem wholly uninteresting, in an analytical point of view, to de- 
monstrate the proposition for any form of aperture. 
* Thus, in the case of the flat prism, if P, Q be the virtual images corresponding to the halves 
AB, BC, if we produce A B to D, we may suppose the light D 
which falls on B C, instead of coming from Q, to come from P, and 
to have been accelerated by the passage through the wedge DB Co 2 
of air instead of the same wedge of glass. 
