Theory of Machine
A disc is a spinning with an angular velocity ω rad/s about the axis of spin. The couple applied to the disc causing precession will be (where I = Mass moment of inertia of the disc, and ωP = Angular velocity of precession of the axis of spin)

Iω²
I ω ωP
(1/2).Iω²
(1/2). I ω ωP

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Theory of Machine
The displacement of the reciprocating roller follower, when it has contact with the straight flanks of the tangent cam, is given by (where r₁ = Minimum radius of the cam, r₂ = Radius of the roller follower, and θ = Angle turned by the cam from the beginning of the follower displacement)

(r₁ + r₂) (1 + cosθ)
(r₁ + r₂) [(1 - cosθ)/cosθ]
(r₁ - r₂) [(1 - cosθ)/cosθ]
(r₁ - r₂) (1 - cosθ)

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Theory of Machine
Two heavy rotating masses are connected by shafts of lengths l₁, l₂ and l₃ and the corresponding diameters are d₁, d₂ and d₃. This system is reduced to a torsionally equivalent system having uniform diameter d = d₁ of the shaft. The equivalent length of the shaft is

(l₁ + l₂ + l₃)/3
l = l₁ + l₂.(d₁/d₂)³ + l₂.(d₁/d₃)³
l₁ + l₂ + l₃
l = l₁ + l₂.(d₁/d₂)⁴ + l₃.(d₁/d₃)⁴

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Theory of Machine
For an isochronous Hartnell governor (where r₁ and r₂ = Maximum and minimum radius of rotation of balls respectively, S₁ and S₂ = Maximum and minimum force exerted on the sleeve respectively, and M = Mass on the sleeve)

S₁/S₂ = r₁/r₂
(m.g - S₁)/(m.g - S₂) = r₂/r₁
(m.g + S₁)/(m.g + S₂) = r₁/r₂
S₂/S₁ = r₁/r₂

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