Applied Mechanics and Graphic Statics
The ratio of the ranges on the inclined plane with motion upward and with motion downward for a given velocity, angle of projection will be

cos (α - β)/cos (α + β)
sin (α - β)/sin (α + β)
tan (α - β)/tan (α + β)
sin (α + β)/sin (α - β)

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Applied Mechanics and Graphic Statics
Varingon’s theorem of moment’s states

Algebraic sum of the moments of two forces about any point, is equal to the moment of their resultant about that point
Arithmetical sum of the moments of the forces about any point in their plane, is equal to the moment of their resultant about that point
Algebraic sum of the moments of the forces about any point in their plane, is equal to the moment of their resultant about that point
Arithmetical sum of the moments of two forces about any point, is equal to the moments of their resultant about that point

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