If the particle is constrained to move on some given surface, any two independent specified functions of its rectangular coordinates x,y,z, may be taken as its coordinates q₁ and q₂, provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing q₁ and q₂ equal to their values in terms of x,y, and z the last-named coordinates may be uniquely obtained as explicit functions of q₁ and q₂.

If the particle is constrained to move in a given path, any specified function of x,y,z may be taken as its coordinate q₁, provided that by the two rectangular equations of its path and the equation formed by writing q₁ equal to its value in terms of x,y,z the last-named coordinates may be uniquely obtained as explicit functions of q₁.