How the Davis steering gear mechanism work?

The Davis steering is an exact steering gear mechanism.the slotted links AM and BH are attached to the front wheel axle, which turns on pivots A and B respectively . the road CD is constrain to move in the direction of its length, by the sliding member at P and Q. these constrain are connected to the slotted links AM and BH by a sliding and a turning member at each end. the steering is effected by moving CD to right or to left of normal position. CD show the position of CD for turning to left.

let

a=vertical distance between AB and CD.

b=Wheel base.

c= Distance between the pivots A and B of the front axle.

d=Horizontal distance between AC and BD.

x=distance move by AC to AC=CC=DD, and

α=Angle of inclination of the link AC and BD , to vertical.

From triangle AAC

tan(α+φ)=AC/AA=d+x/a

From triangle AAc

tanα=AC /AA=d/a

From triangle BBD

tan(α-Θ)=BD=d-x/a

we know that tan(α+φ)=tanα+tanφ/1-tanα*tanφ

=d+a*tanφ/a-d*tanφ

(d+x)(a-d*tanφ)=a(d+a*tanφ)

tanφ=a*x/(a^2+d^2+d*x)

similarly , from tan(α-Θ)=d-x/a

We get tanΘ=a*x/(a^2+d^2-d*x)

we know that for correct steering ,

cotφ-cotΘ=c/d

1/tanφ-1/tanΘ=c/d

(a^2+d^2+d*x)/a*x-(a^2+d^2-d*x)/a*x=c/b

2d*x/a*x=c/d

2tanα=c/d (d/a=tanα)

Note: through the gear is theoretically correct, but due the presence of more sliding members, the wear will be increased which produce slackness between the sliding surface, thus eliminating the original accuracy.

Hence Davis steering gear is not in common use.