| x | + | z | = φ ( 1 − | y | ), | x | − | z | = | 1 | ( 1 + | y | ). |
| a | c | b | a | c | φ | b |
It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculating plane of the curve (see § 38 below) is a tangent plane of the torse all along a generating line.
35. Transformation of Coordinates.—There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. We have, then, two sets of rectangular axes, Ox, Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown by the diagram—
| x | y | z | |
| x1 | α | β | γ |
| y1 | α | β′ | γ′ |
| z1 | α″ | β″ | γ″ |
that is, α, β, γ are the cosine-inclinations of Ox1 to Ox, Oy, Oz; α′, β′, γ′ those of Oy1, &c.
And this diagram gives also the linear expressions of the coordinates (x1, y1, z1) or (x, y, z) of either set in terms of those of the other set; we thus have
| x1 = α x + β y + γ z, | x = αx1 + α′y1 + α″z1, |
| y1 = α′x + β′y + γ′z, | y = βx1 + β′y1 + β″z1, |
| z1 = α″x + β″y + γ″z, | z = γx1 + γ′y1 + γ″z1, |
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation p = α′ξ + β′η + γ′ζ, adapted to the problem in hand.
But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically x² + y² + z² = x1² + y1² + z1², it appears that these satisfy the relations—