which is an expression for the mutual cosine-inclination of two lines, the cosine-inclinations of which to the axes are α, β, γ and α′, β′, γ′ respectively. We have of course α² + β² + γ² = 1 and α′² + β′² + γ′² = 1; and hence also
1 − δ² = (α² + β² + γ²)(α′² + β′² + γ′²) − (αα′ + ββ′ + γγ′)²,
= (βγ′ − β′γ)² + (γα′ − γ′α)² + (αβ′ − α′β)²;
so that the sine of the inclination can only be expressed as a square root. These formulae are the foundation of spherical trigonometry.
32. Straight Lines, Planes and Spheres.—The foregoing formulae give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c), and the cosine-inclinations (α, β, γ) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current coordinates of a point in the line. The values of ξ, η, ζ then are x − a, y − b, z − c, and we thus have
| x − a | = | y − b | = | z − c | (= ρ), |
| α | β | γ |
which (omitting the last equation, = ρ) are the equations of the line through the point (a, b, c), the cosine-inclinations to the axes being α, β, γ, and these quantities being connected by the relation α² + β² + γ² = 1. This equation may be omitted, and then α, β, γ, instead of being equal, will only be proportional, to the cosine-inclinations.
Using the last equation, and writing
x, y, z = a + αρ, b + βρ, c + γρ,
these are expressions for the current coordinates in terms of a parameter ρ, which is in fact the distance from the fixed point (a, b, c).