Highly Magnified Images. By J. W. Gordon. 27 



be depicted in the plane of the page. Writing u for the divergence 

 angle of the beam A A rj we have at once — 



Distance of the middle point of 8 S from -q = D + sin u p. 



Length of perpendicular from ^ upon the cone = cos u p. 



D = tan a cos u p. 



Distance from 8 S to ^ = sec a cos u p. 



The point at which the perpendicular from yj x meets the surface of 

 the cone may conveniently receive a name, since it must be frequently 

 referred to in the following discussion. Since it is the point from 

 which the normal to the surface of the cone issues which passes through 

 ■q 1 in the focal plane, I propose to call it the normal point. 



In reckoning the value at the point rj l of the light radiated from 

 the small surface 8 S, there are seven matters to be taken into account, 

 namely — 



1. The area of 8 S. 



2. The projection along the axis 8 S . . . rj l of this area. 



3. The amplitude of the undulation in 8 S. 



4. A coefficient of condensation (or diffusion as the case may be) 

 representing the change in light density involved in passing from 

 SS to t) x . 



5. A coefficient of efficiency, depending upon the polyphasal character 

 of the diffracted beam. 



6. A coefficient of economy, expressing the fact that the original 

 impulse in which the diffracted rays take their rise is itself travelling 

 along the radiant surface, so that it is able to originate impulses which 

 reach the point ^ simultaneously from more points than one on the 

 edge of the beam. 



7. The phase in which the light arrives at i^. 



If we write \f/ for the amplitude of the light undulation at -q u and <f». 

 for its phase, we shall have — 



dxf, = {(1) X (2) X (3) X (4) x (5) X (6)} d D and <f> = (7) 



To facilitate the writing out of these values, let the following 

 symbols be adopted with reference to Fig. 23 : 



s = sin u ; c = cos u : n = ; N= J<n? 4- i 



cp 



r = the radius of the cone at the level of 8 S. 



Moreover, I propose to substitute for the integral ij/ a finite series, 

 having Af= kAD = kA, where k represents the above coefficient of 

 d D in the expression for d if/ suitably modified to meet the change in- 

 volved in the substitution of a short segment of one wave-length of the 

 edge of the beam for the infinitesimal increment of D. Then— 



1 . The area of 8 S = m . 8 r . A, m being a constant to be determined 

 by observation. 



2. The projection of (1) = (1) cos a = (1) — . 



8 R 



3. The amplitude = M r = - M ; if M = the amplitude in the- 



o r 



aperture A A. 



