174 . UNDULATORY THEOEY OF LIGHT. 



and of the intercept itself. When w has any considerable magnitude, this 

 tactor rapidly reduces the value of — — ; indicating a similarly rapid diminu- 



tion in the fluctuations of value of the integral. On the other hand, when to 

 is very .'mall, this factor may be regarded as a constant, and assumed, without 

 sensible error, as equal to unity. 



The distance of these fringes from the boundary of the shadow may be de- 

 termined as follows. Suppose the screen S to be at a, and let the straight line 

 Itc/C be the boundary of the geometrical shadow. Draw aN perpendicular to 

 ItA. Call f/N, y/, and AN, x. By construction, Ij« — BA=.JA. Put EA=r, 

 BA:=.v, and B«=2'- Then, 



2 2 



2{s-\-x) 2s 



disregarding inappreciable terms of the root, and omitting x in the denominator 

 where its efiect on the value of the fraction is insensible. Also, in the circle 



.„2 



whose centre is R, x is the versed sine of A«, and is sensibly equal to ^ 



2r. 



Whcncc- 



2r 2s 2rs " ' ?• + *' 



And putting o, as before, for the distance BC, we have, 



r=-^-'^=^ ; and finally S=J ^^^. [10.] 



From this expression, which is the equation of an hyperbola, it appears that, 

 if the screen B move toward A, the /orus of all the points in space correspond- 

 ing to B will be. a hyperbolic curve, of which R and a arc the vertices. A sim- 

 ilar inference may be drawn from considering that, in all positions of the 

 screen B, Ba — BA is constant and equal to ^A ; or BA — Ba= — J>^, whence 

 RA-f-BA — Ba, or RB — Ba=RA — ^l, which is also constant. But this 

 is the property of a hyperbola whose Jori (not the vertices) are R and 

 a, and whose major axis is r — ^X. This latter result is the strictly coi-rect 

 one. The discrepancy between it and the former is owing to the omission of 

 minute terms in obtaining that result. Put the major axis equal A, and the 

 minor axis equal B. Ilien, by the law of the hyperbola, 



B2=?•2-A2=^=r2-(/•-J;.)'='•^-5^•^• or Ji=V7x, 

 disregarding the minute negative term. 



The equation of the curve, if we employ the exact values of the axes, will 

 be— 



which, when the minute terms are dismissed, simplifies itself to the expression 

 found before. 



The semi-axis major of this curve being ^r — ^X, it appears that the curve 

 itself passes behind the obstructing edge at the distance of ^X, at which distance 

 a wave reflected from the obstacle would meet a wave advancing with a differ- 

 ence of ^X. Whether this theoretic indication is actually realized, it would 

 perhaps be difficult experimentally to determine. Such a reflected wave, con- 

 sidering only the difference of path, would be out of harmony with the advancing 

 wave ; but, considering that its molecular movements would be reversed by 

 reflection from the dense ether of the obstacle, the harmony would be restored. 



i'^or the distance of the second fringe from the shadow, the expression already 

 found for the first answers perfectly, if we prefix to the quantity under the 



