ON  THE  COLOURS  OF THICK  PLATES.                     165
and therefore
2V(7 ^+ c£~c ~cJ^=A?~~(J (c' ^+ c   V ......^(9)*
Again
___ a c_l         __        c_2 +1                   c_8
^1    ^ c '     ^2         ' c_x 4- *'     ^3        ^2 c_2' whence
-4- ™ = - -11 — t~ -^2.....-a- }• = — (--------), nearly.
c     c_s     c I      (c, + 0 cj      c Vc,    c_2/            ^J
Neglecting i altogether in the formulae (7), we get
1 _!_2/l_l\
o,    c^fi\c    _P)> and therefore
c     c_8     c    c_a     ^ (c \o     /)>
Let AJK + A^ = AE.    The formulas (6), (9), (10), and the corresponding formulae relating to A^R, give
The condition of distinctness, as has been already observed, is that AJS shall vanish independently of x and y, in which case the elementary systems of rings corresponding to the several elements of the dimmed surface will be superimposed on the screen. The coefficient of &^>2+y^2 will vanish when either of the following equations is satisfied :
In order that the coefficients of x and y may vanish independently of particular values of a and b' we must have
c' = c = _jo ........................... (IS),
which equations satisfy at the same time both of the equations (12), of which it would have been enough that one should have been satisfied. Hence the rings are formed most distinctly when the luminous point and the screen are both at a distance from the