Page:EB1911 - Volume 11.djvu/739

the variable parameter θ) are x/a + y/c = θ (1 + y/b), x/a − z/c = 1/θ(1 − y/b); then, eliminating θ we have x²/a² − z²/c² = 1 − y²/b², or say, x²/a² + y²/b² − z²/c² = 1, the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter φ) are

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—

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):

(1)	z = x2/2a + y2/2b, elliptic paraboloid.
(2)	z = x2/2a − y2/2b, hyperbolic paraboloid.
(3)	x2/a2 + y2/b2 + z2/c2 = 1, ellipsoid.
(4)	x2/a2 + y2/b2 − z2/c2 = 1, hyperboloid of one sheet.
(5)	x2/a2 + y2/b2 − z2/c2 = −1, hyperboloid of two sheets.

Fig. 61.

It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.

In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolas

z =	x2	,  z =	y2	,
	2a		2b

having the common axes Oz; and the section by any plane z = γ parallel to that of xy is the ellipse

γ =	x2	+	y2	;
	2a		2b

so that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices.

Fig. 62.	Fig. 63.

Fig. 64.

In the hyperbolic paraboloid (figs. 62 and 63) the sections by the planes of zx, zy are the parabolas z = x²/2a, z = − y²/2b, having the opposite axes Oz, Oz′, and the section by a plane z = γ parallel to that of xy is the hyperbola γ = x²/2a − y²/2b, which has its transverse axis parallel to Ox or Oy according as γ is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.

In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.