α²+ β²+ γ²= 1,  α² +α′²+ α″²= 1,
α′²+ β′²+ γ′²= 1,  β²+ β′²+ β″²= 1,
α″²+ β″²+ γ″²= 1,  γ²+ γ′²+ γ″²= 1,
α′a″+ β′β″+ γ′γ″= 0,  βγ+β′γ′+ β″γ″= 0,
α″α+ β″β+ γ″γ= 0,  γα+ γ′α′+ γ″α″= 0,
αα′+ ββ′+ γγ′= 0,  αβ+α′β′+ α″β″= 0,

either set of six equations being implied in the other set.

It follows that the square of the determinant

α,β,γ
α′,β′,γ′
α″,β″,γ″

is = 1; and hence that the determinant itself is = ±1. The distinction of the two cases is an important one: if the determinant is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a rotation about O be brought to coincide with Ox, Oy, Oz respectively; if it is = −1, then they cannot. But in the latter case, by measuring x1, y1, z1 in the opposite directions we change the signs of all the coefficients and so make the determinant to be = + 1; hence the former case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found that we have the equality α = β′γ″ − β″γ′, and eight like ones, obtained from this by cyclical interchanges of the letters α, β, γ, and of unaccented, singly and doubly accented letters.

36. The nine cosine-inclinations above are, as has been seen, connected by six equations. It ought then to be possible to express them all in terms of three parameters. An elegant means of doing this has been given by Rodrigues, who has shown that the tabular expression of the formulae of transformation may be written

x y z
x1 1 + λ² − μ² − ν² 2(λμ − ν) 2(νλ + μ)
y1 2(λμ + ν) 1 − λ² + μ² − ν² 2(μν + λ)
z1 2(νλ − μ) 2(μν + λ) 1 − λ² − μ² + ν²
÷ (1 + λ² + μ² + ν²),

the meaning being that the coefficients in the transformation are fractions, with numerators expressed as in the table, and the common denominator.

37. The Species of Quadric Surfaces.—Surfaces represented by equations of the second degree are called quadric surfaces. Quadric surfaces are either proper or special. The special ones arise when the coefficients in the general equation are limited to satisfy certain special equations; they comprise (1) plane-pairs, including in particular one plane twice repeated, and (2) cones, including in particular cylinders; there is but one form of cone, but cylinders may be elliptic, parabolic or hyperbolic.

A discussion of the general equation of the second degree shows that the proper quadric surfaces are of five kinds, represented respectively, when referred to the most convenient axes of reference, by equations of the five types (a and b positive):