684 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
Q is its image. The optical distance between the conjugate points A and 
B is the same for all paths — e.g. for ARS B and ALMB, and the 
optical distance from P to Q is the same as from A to B. Consequently, 
the optical distance P R S Q is the same as A R S B, i e. /x . A P . sin a = 
/x' . B Q . sin /?, where /x, y! are the refractive indices near A and B 
respectively, a and /3 the divergence angles R AL, SBM for a given 
l ay, and A P, B Q denote the corresponding linear magnitudes of the two 
images. 
The author then proceeds to the actual calculation of the images to 
be expected upon Fresnel’s principles in the various cases. The origin 
of co-ordinates (£ = 0, rj — 0 ) is the geometrical image of the radi- 
ant point. Representing the vibration incident upon the lens by 
cos (2 7r Y t / X), where Y is the velocity of light, the vibration at any 
point £, y] in the focal plane is 
V<-/ + 
^j^X dxdy, 
in which / denotes the focal length, and the integration for x and y is to 
be extended over the aperture of the lens. 
In the case of a rectangular aperture of width a , b parallel to x and y 
respectively, the expression giving the diffraction pattern along the axis 
sinw 
£ can be simplified to the form , where u is equated to tt $ a / Xf. 
Values of the amplitude 
sin u 
and the intensity 
sim u 
for different 
values of — 
7 T 
are given 
in a table. The illumination first vanishes 
when u = 7r, i.e. when £ = X / af. 
The author has shown, in a previous paper, * that a self-luminous 
point or line at u — — tt is barely separated from one at u = 0. He 
now considers the case under three different conditions as to phase : — 
(i.) when the phases are the same ; (ii.) when the phases are opposite ; 
and (iii.) when the phase-difference is a quarter-period. In the first case 
the resultant amplitude is represented by 
sin 
in (ii.) by 
and in (iii.) by 
u 
U -f- 7T 
sin u 
sin (u -f 7r) # 
u 
U + 7T ’ 
'jsin 2 u 
j sin 2 (w + 7r)| 
of 
A table gives the values of these three functions for different values 
— . The graphs of the three functions in fig. 106 show that in i. 
* Phil. Mag., viii. p. 2GG. 
