Derivation of the difference equation 3B:

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Starting with 3A we have:

(W₁₁p² + W₁₂pq)/W - p =

[W₁₁p² + W₁₂pq - Wp] /W =

[W₁₁p² + W₁₂pq - W₁₁p³ - W₁₂*2p²q - W₂₂pq²]/W.

Doing some regrouping this becomes:


[W₁₁ (p² - p³) + W₁₂(pq - 2p ²q) - W₂₂pq²]/W

note the following:

(p² - p³) = pq*p

and

(pq - 2p²q) =

pq(1 - 2p) = pq(q - p) since 1 = p + q

Then by substitution and by regrouping we find that:

D p = pq [p(W₁₁ - W₁₂) - q(W₁₂ - W₂₂ )]/W

 as asserted earlier .