Introducing one more coefficient

Looking better to the expressions,

Agent2 = A x Agent 1
Alkali 1 = B - C x Agent 1
Alkali 2 = D - Alkali 1

I see that Alkali 1 in the last expression could include a multiplier coefficient, let as call it E. So, the generalized expressions should be written like this:

Agent2 = A x Agent 1
Alkali 1 = B - C x Agent 1
Alkali 2 = D - E x Alkali 1

The question now is to know why a general developer with two complentary agents should obey to these expressions. In the exceptional cases where the developing agents appear alone, they have an alkali combination that is better for it to work well. For instance, if you use just Metol, it works better with just sodium sulfite alone, that's to say, 0 g from sodium carbonate. If you use Hydroquinone alone it works better  with almost equal amounts of sodium sulfite and carbonate. Most Coffee developers use just sodium carbonate, maybe coffee works better with just sodium carbonate, and so on.

Let as see now how to find the coefficients for a particular developer. I will use here Reinhold G. Caffenol C-M recipe:

For 1 liter developer, we have:

54 g Sodium Carbonate

16 g Vitamine C

40 g Instant Coffee

Calculating the coefficients, A = Coffee/Vit. C = 2.5, assuming that Agent 1 = Vit. C and Agent 2 = Coffee.

If now we assume that Alkali 1 is Sodium Sulfite (0 g), B - C x 16 = 0 and Alkali 2 = D = 54. It is not possible to know the coefficient E, it is undetermined mathematically. It could be determined if we use another developer of the same family containing other amounts of sodium sulfite and carbonate.

So A and D are now fixed in 2,5 and 54. The equation B - C x 16 = 0 has a lot of solutions. If we fix however the maximum amount of Sulfite in 100 g/l, B = 100 and C = 6.25. E is not determined mathematically but assuming the approach that the sum of the amounts of alkalis is more or less constant, than E=1 (must not be).

We may then say that Reinhold's Caffenol C-M belongs to the family:

Vit. C, Coffee, S.S., S.C. = 2.5, 100, 6.25, 54, 1. 

Changing B, we have another different family that also includes this developer. Example, assuming that the maximum amount of sulfite is 120 = B, then C = 7.5

Vit. C, Coffee, S.S., S.C. = 2.5, 120, 7.5, 54, 1.