Part 6

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,

( s₀ ) σ sin ( ---- ) = ----- ( R ) R

in which

___________________ σ = √(x² + y² + z_²)

or,

( σ ) ( σ ) s₀ = R . arc sin( --- ) = R . arc tang( --- ) (5.) ( R ) ( t )

In this, s₀ 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 )

R²σ² χ² + ϒ² + ζ² = r² = --------- t²

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 ) s₀ = 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 s₀ < ½Rπ, one for which s₀ > ½Rπ. The extension in the direction of r is then

ds₀ R² ----- = ------------- dr R² + r²

In order to obtain corresponding expressions for pseudospherical space, let R and t be imaginary; that is, R = ℛi, and t = τi. Equation 6 gives then

s₀ r tang ------ = ± ------ iℛ iℛ

from which, eliminating the imaginary form, we get

ℛ + r s₀ = ½ℛ log. nat. --------- ℛ - r

Here s₀ has real values only as long as r = R; for r = ℛ the distance s₀ in pseudospherical space is infinite. The image in plane space is, on the contrary, contained in the sphere of radius R, and every point of this sphere forms only one point of the infinite pseudospherical space. The extension in the direction of r is

ds₀ ℛ² ----- = ----------- dr ℛ² - r²

For linear elements, on the contrary, whose direction is at right angles to r, and for which t is unchanged, we have in both cases

____________________ √dx² + dy² + dz² t τ σ ---------------------- = ----- = ---- = ----- ___________________ R ℛ r √dχ² + dϒ² + d_ζ²

_________________ √x² + y² + z² = ------------------- ____________ √χ² + ϒ² + ζ²