Theory of Machine
In a reciprocating steam engine, when the crank has turned from inner dead centre through an angle θ, the angular velocity of the connecting rod is given by

ω cosθ/(n² - cos²θ)1/2
ω sinθ/(n² - sin²θ)1/2
ω cosθ/(n² - sin²θ)1/2
ω sinθ/(n² - cos²θ)1/2

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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)

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

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