Theory of Machine
The maximum efficiency of spiral gears is (where θ = Shaft angle, and φ = Friction angle)

cos (θ - φ) + 1/ cos (θ + φ) + 1
sin (θ + φ) + 1/ cos (θ - φ) + 1
cos (θ - φ) + 1/ sin (θ + φ) + 1
cos (θ + φ) + 1/ cos (θ - φ) + 1

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

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