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-p
)
= st(2+p)p2
ru(1-2p+p
)
= 2stp
+ stp
ru - 2rup
+ rup
= 2stp
+ stp
(ru - st)p
+ (-2ru - 2st)p
+ 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
ax
+ bx + c
The quadratic equation formula to solve for x is:
In our case the unknown is not x, the unknown is p.
ax
+ 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 p
to solve for p. Otherwise we can
just use the table, rather than the quadratic.
How did Fisher develop the product formula:
