Theory of Machine
Two pulleys of radii r₁ and r₂ and at distance x apart are connected by means of a cross belt drive. The length of the belt is

π (r₁ + r₂) + [(r₁ - r₂)²/x] + 2x
π (r₁ - r₂) + [(r₁ + r₂)²/x] + 2x
π (r₁ + r₂) + [(r₁ + r₂)²/x] + 2x
π (r₁ - r₂) + [(r₁ - r₂)²/x] + 2x

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

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Theory of Machine
The velocity of piston in a reciprocating steam engine is given by (where ω = Angular velocity of crank, r = Radius of crank pin circle, θ = Angle turned by crank from inner dead center, and n = Ratio of length of connecting rod to the radius of crank)

ωr [cos θ + (cos 2θ/n)]
ω²r [sin θ + (sin 2θ/n)]
ω²r [cos θ + (cos 2θ/n)]
ωr [sin θ + (sin 2θ/n)]

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