Highly Magnified Images. By J. W. Gordon. 29 



,-s ■ IN — n ) N sin n 



v J IN f(N — «.)ir »0 



where # is a small angle that cannot be mathematically determined, but 

 must be ascertained by observation. 



6. The Economy factor. If every point upon the surface c t . . e 2 

 received its light from a separate ray in the principal beam, the efficiency 

 would be fully represented by the expression just found for (5). But we 

 are integrating over a single element of the light cone, and the original 

 impulse is travelling to a certain extent in the same direction as the 

 diffracted light which it gives off. Thus the light source is, so to speak, 

 economised, and a light source of shorter length than A suffices to yield all 

 the diffracted light which we have derived from D x . . D 2 . Thus since the 

 phase range in the surface c x . . e 2 is only (1 — sin a) 2 7r, it is evident 

 that the effective length of edge is (1 — sin a) A. Therefore 



(6) = 1 — sin a = — — — . 



v ' N 



We may now collect these various results into one expression, as. 

 follows : 



. , m . d r . A d R M , . . 



A^= ^ •V'' (ne + ,) 



v • l(N - ri)ir [ N N - n sin n 6 



X Sm \ X "1 (N -»»)„■• X -•" ' 



= m . d E . A . M . • . \ nc + s . I sin . &-*>* . **"!L 



It may be noted here that the expression m d R A M denotes the 

 radiation upon the focal point from a small surface A S A (equal in area 

 to the topmost segment of the radiant wedge) in the wave-front which 

 passes the aperture A . . A. It may, therefore, be fitly taken for the unit 

 of radiation for the given system, and expressed by the symbol M x . 



Accordingly the last expression may be written — 



„.- /nc-\-s 1 . (N — n)-7r sin n0\ . ^. 



A^ = s.M 1 ^ ^—.-.snP - N ; .- Q ) . (10) 



The successive terms of this series are to be taken upon the principle 

 of assigning one term to every segment of one wave-length measured 

 from the normal point along the edge of the beam facing the point j/ 1s 

 and therefore the values of n must be so chosen that— 



1 3 5 



J) 1 = n x c p = A ; 1) 2 = n 2 c p = '-X; D 3 = w 3 cp = ' A, etc. 



AAA 



It is plain that if n be taken very small N = 1 nearly, and, therefore, 



Is - ■ n 

 sin — == — it tends to = as n approaches 0. Also if n becomes very 



X ji 



large, N — n tends to become = 0, so that in that case also sin — - 



approximates to 0. 



