It is easily found by means of the methods used for three dimensions that the shortest lines are given by equations of the form _ax_ + _by_ + _cz_ + _ft_ = 0 } } (3.) _αx_ + _βy_ + _γz_ + _φt_ = 0 } in which _a_, _b_, _c_, _f_, as well as _α_, _β_, _γ_, _φ_, are constants. The length of the shortest arc, _s_, between the points (_x_, _y_, _z_, _t_), and (_ξ_, _η_, _ζ_, _τ_) follows, as in the sphere, from the equation cos_s_ (_xξ_ + _yη_ + _zζ_ + _tτ_) ------ = --------------------------- (4.) _R_ _R_² One of the co-ordinates may be eliminated from the values given in 2 to 4, by means of equation 1, and the expressions then apply to space of three dimensions. If we take the distances from the points _ξ_ = _η_ = _ζ_ = 0 from which equation 1 gives _τ_ = _R_, then, ( _s_₀ ) _σ_ sin ( ---- ) = ----- ( _R_ ) _R_ in which ____________________ _σ_ = √(_x_² + _y_² + _z_²) or, ( _σ_ ) ( _σ_ ) _s_₀ = _R_ . arc sin( --- ) = _R_ . arc tang( --- ) (5.) ( _R_ ) ( _t_ ) In this, _s_₀ is the distance of the point _x_, _y_, _z_, measured from the centre of the co-ordinates. If now we suppose the point _x_, _y_, _z_, of spherical space, to be projected in a point of plane space whose co-ordinates are respectively ( _Rx_ ) ( _Ry_ ) ( _Rz_ ) χ = ( ---- ) ϒ = ( ---- ) ζ = ( ---- ) ( _t_ ) ( _t_ ) ( _t_ ) _R_²_σ_² χ² + ϒ² + ζ² = _r_² = --------- _t_² then in the plane space the equations 3, which belong to the straightest lines of spherical space, are equations of the straight line. Hence the shortest lines of spherical space are represented in the system of χ, ϒ, ζ, by straight lines. For very small values of _x_, _y_, _z_, _t_ = _R_, and χ = _x_, ϒ = _y_, ζ = _z_ Immediately about the centre of the co-ordinates, the measurements of both spaces coincide. On the other hand, we have for the distances from the centre ( _r_ ) _s_₀ = _R_ . arc tang( ± ---- ) (6.) ( _R_ ) In this, _r_ may be infinite; but every point of plane space must be the projection of two points of the sphere, one for which _s_₀ < ½_R_π, one for which _s_₀ > ½_R_π. The extension in the direction of _r_ is then _ds_₀ _R_² ----- = ------------- _dr_ _R_² + _r_² In order to obtain corresponding expressions for pseudospherical space, let _R_ and _t_ be imaginary; that is, _R_ = ℛ_i_, and _t_ = τ_i_. Equation 6 gives then _s_₀ _r_ tang ------ = ± ------ _i_ℛ _i_ℛ from which, eliminating the imaginary form, we get ℛ + _r_ _s_₀ = ½ℛ log. nat. --------- ℛ - _r_