54 Proceedings of the Royal Society of Edinburgh. [Sess. 
although there may be rings due to straining of the media at the surfaces 
of contact (vide Stokes’ Papers, vol. ii, p. 86). 
( b ) There is one case in which | A' | and | ft | will vanish, however, viz. 
when P = Q = 0 simultaneously and either R*0 or S*0. 
This case will exist when 
and when 
i.e. when 
cos 0 = 0, P = 0 ; 
r i r 3 =r 2 2 > Q = °; 
i.e. when 
Kdn 2 = (2k + 1) — , where k = 0, 1, 2, 3 . . 
a ° + tR^ k + 1 P' 
(i) When 0 = 0, i.e. for a o = 0, 
| A' |*0 | B' |*0. 
In this case we shall have a bright spot at the centre and interference 
bands given in radii by the equation 
2r 2 p 2 = R(2& + 1)A. 
(ii) If, however, a 0 *0, | A' | and | B' | are each zero when 4 r 2 a 0 = A, and 
in this case we shall have a dark spot at the centre and interference rings 
given by the equation 
r 2 p 2 = R/iA. 
(iii) When <x 0 *0 and 4r 2 a 0 *A, | A' | *0 and j B' | *0 at the centre, and 
we have a bright spot at the centre and interference rings given by the 
equation 
4r, (“o + © = (2 * +1)jL 
It is stated usually that Brewster verified the existence of a bright 
spot at the centre of Newton’s Rings when the refractive index of medium 
2 is intermediate between that of medium 1 and medium 3. This relation 
is not incompatible with the condition r^r^r^ or with the condition 
Tl r B = r 2 \ 
When r x r 3 * r 2 2 we have shown in section (a) that no interference 
rings are obtainable. 
When, however, r 1 r 3 = r 2 2 and 4a 0 r 2 *A, we shall have a system of 
interference rings with a bright spot at the centre, but in the case when 
rpr 3 = r 2 2 and 4 a 0 r 2 = A we shall have a system of interference rings with a 
dark spot at the centre. 
