Development Of The Product Method

The formula for the product method for repulsion linkage is:

We could solve for p using the quadratic formula. First let us cross-multiply:

ru(1-p²)² = st(2+p²)p²

ru(1-2p²+p⁴) = 2stp² + stp⁴

ru - 2rup² + rup⁴ = 2stp² + stp⁴

(ru - st)p⁴ + (-2ru - 2st)p² + ru = 0

The coefficient for p⁴ is (ru-st); for p² it is (-2ru - 2st); and the coefficient for p⁰ is ru. For an equation in the form

ax² + bx + c

The quadratic equation formula to solve for x is:

In our case the unknown is not x, the unknown is p².

ax² + bx + c goes to (ru - st)x² + (-2ru - 2st)x + ru
where a = (ru - st); b = (-2ru - 2st); c = ru.

Now:

then substituting:

We have used the quadratic formula to solve for p and then taken the square root of p² to solve for p. Otherwise we can just use the table, rather than the quadratic.

How did Fisher develop the product formula: