194                     ON THE  COLOURS  OF  THICK PLATES.
that the two systems of rings intersected each other, I found in fact that whatever colour appeared in that part of a lycopodium ring which lay outside the interference system was predominant in the latter system throughout the remaining part of a circle described round the image. When the flame was placed in the axis, an abnormal inequality in the brilliancy of the rings of the interference system became very apparent. This inequality was easily seen to correspond to the alternations of intensity in the lycopodium system.
86. Let us now turn to the general case, in which the luminous point and the eye are supposed to have any positions, either in the axis or not far out of it.
The equations of the lines PL*, PE are
~ A ^
f-x     cj-y        h
Let the small angle L^PE be projected on the planes of zx and zy, and let a, ft be the projections, measured positively towards x, y, and from PL> towards PE. The preceding equations give
_x — a_:]    x—f c_:!            ~h    '
which becomes, when a_:! and c_:J are expressed in terms of a and c,
/2     1     l\       a    f                    _/0,,v
----   U. _+ _4.£ .................. (36).
p    c    lij       c    li
If a', j8' be the projections of the angle of diffraction (where E_s denotes the image of the eye,) we may find a', /3' from a, /3 by interchanging a, b, c, and /, g, h, and changing the sign, If now we change the signs of the resulting expressions, in order to allow for reflexion at the back, and so compare the circumstances of the two diffractions, we shall obtain the very same expressions as at first, since (36) and the corresponding expression which gives /3 remain unchanged when a, &, c and /, g, h are interchanged. Hence in the general case, as well as in the particular case first considered, the two diffractions take place under the same circumstances, and therefore the interference rings are not affected by any irregularities which may attend the mode