Let p = ɑa, q = βb, r = ɣc,
Then we may write (ii.) as—
αaR + βbG + ɣcV = zZ—(iii.).
Now either ɑ, β, or ɣ must be smaller than the other two. As an example, if ɑ be the smallest, we multiply (i.) by ɑ when we get—
ɑaR + ɑbG + ɑcV= ɑwW—(iv.)
Subtracting (iv.) from (iii.) and we get—
(β-ɑ)bG + (ɣ-ɑ)cV = zZ - ɑwW.
Now it has already been stated that between V and G there is some ray which gives the same sensation of colour, mixed with a very small quantity of white light, as the above mixture of V and G—let us call it X and its luminosity x [x being evidently equal to (β-ɑ)b + (ɣ-ɑ)c], and μ the luminosity of the small quantity of white added.
We then get zZ = xX + (μ + ɑ) W.
Here we have the colour Z in terms of a single ray, and of white light.
This same holds good when in (ii.) ɣ is smaller than ɑ and β; but it does not do so should it happen that β is the smallest, for there is no part of the spectrum which contains simple colours giving the same sensation to the eye as mixtures of red and blue. There is, however, a very simple way in which the registration of such a colour (which it must be remarked must be of a purple tone) can be effected. It can be fixed by its complementary. To do this we must add to (ii.) a certain amount of R and V, which will make the whole white. Thus, suppose in (iii.) ɑ to be larger than ɣ and ɣ than β, then we must add ϕbG + θcV and we have
ɑaR + (β + ϕ)bG + (ɣ + θ)cV = nW = Z + ϕbG + θcV;
but (β + ϕ), and (ɣ + θ) each equal ɑ ∴ n = ɑw.
∴ Z + ϕbG + θcV= ɑwW.