In like manner we may form the table

OO+EE

OE + EO

B+ A + B + A

B+ A + A′ = B

B' + A' + B' + A = B′ + A + A′ = B′

Hence, if the Nicol N be turned round, these overlaps will retain their tints; while if the analyser P, be turned, their tints will vary, although always remaining complementary to one another.

There remains the other pair of overlaps, viz. :—

Each of these is deprived of the pair of complementaries A, A, B, B'; and therefore each, as it would seem, ought to appear white of low illumination, i.e. grey. This effect is, however, partially masked by the fact that the dark bands are not sharply defined like the Fraunhofer lines, but have a core of minimum or zero illumination, and are shaded off gradually on either side until a short distance from the core the colours appear in their full intensity. Suppose, for instance, that B' and A' were bright tints, the tints resulting from their suppression would be bright. On the other hand, the complementary tints A and B would be generally dim, and the image B + A bright, and the overlap B + A + B + A' would have as its predominating tint that of B + A. And similarly in

other cases.

There are two cases worth remarking in detail, viz., first that in which

B = A', B' = A

i.e. when the same tints are extinguished by the combination Q P and by Q, P1. This may be verified by either using two similar quartz plates Q, Q1, or by so turning the prism P1, that the combination Q, P, used alone shall give the same complementary tints as Q P when used alone. In this case the images have for their formulæ the following:

1

in other words, O O and EO will show similar tints, and EO, E E complementary. A similar result will ensue if B A B′ = A'.

=

Again, even when neither of the foregoing conditions are fulfilled, we may still, owing to the breadth of the interference bands, have such an effect produced that sensibly to the eye

which imply that the images OO and OE may have the same tint; but that E O and E E need not on that account be complementary. They will differ in tint in this, that E E, having lost the same tints as EO, will have lost also the tint A, and will have received besides the addition of two measures of the tint A'.

Effect of Combination of two Colours.-A similar train of reasoning might be applied to the triple over

L

laps. But the main interest of these parts of the figure consists in this, that each of the triple overlaps is complementary to the fourth single image; since the re-combination of all four must reproduce white light. Hence the tint of each triple overlap is the same to the eye as the mixture of the two tints suppressed in the remaining image. And since by suitably turning the Nicol N or the prism P1, or both, we can give any required position to the two bands of extinction, we have the means of exhibiting to the eye the results of the mixture of tints due to any two bands at pleasure.

Effect of Combinations of three Colours.-A further step may be made in the combination of colours by using a third quartz Q, and a third double-image prism P2, which will give rise to eight images. And if cc' be the complementaries extinguished by the combina、ion Q2 P2, the formulæ for the eight images may be thus written :—

The total number of combinations of tint given by

the compartments of the complete figure will be

The most interesting features of the figure consists in this, that the subjoined pairs are complementary to one another, viz. :—

000
C+B+A

ΕΟΟ

C'+B+A

EEO
C+B'+A

EEE

C+B+A'

2

EOE
C'+ B'+A'

ООЕ
C+B'+A'

OEE
C'+B+A'

ОЕО
C'+B'+A

And if the prisms P, P1, P2 are so arranged that the separations due to them respectively are directed parallel to the sides of an equilateral triangle, the images will be disposed thus:

The complementary pairs can then be read off, two horizontally and two vertically, by taking alternate pairs, one in each of the two vertical, and two in the one horizontal row. And each image will then represent the mixture of the three tints suppressed in the complementary image.

Low-tint Colours.-A slight modification of the arrangements above described furnishes an illustration of the conclusions stated by Helmholtz, viz. that the low-tint colours (couleurs dégradées), such as russet, brown, olive-green, peacock-blue, &c., are the result of relatively low illumination. He mentioned that he obtained these effects by diminishing the intensity of the light in the colours to be examined, and by at the same time maintaining a brilliantly illuminated patch in an adjoining part of the field of view. If, therefore, we use the combination N, Q, P, P1 (i.e. if we remove the second quartz plate), we can, by turning the prism round, diminish to any required extent the intensity of the light in one pair of the complementary images, and at the same time increase that in the other pair. This is equivalent to the conditions of Helmholtz's experiments; and the tints in question will be found to be produced.