Page images


[ocr errors]


[ocr errors]

In the case of a curve we have between the coordinates (x, y, z) a these are expressions for the current coordinates in terms of a twofold relation: two equations f(x, y, z) = 0, $(x, y, 3) = 0 give parameter p which is in fact the distance from the fixed point such a relation; i.e. the curve is here considered as the intersection (a,b,c). of two surfaces (but the curve is not always the complete intersection It is easy to see that, if the coordinates (x, y, s) are connected by of two surfaces, and there are hence difficulties); or, again, the co- any two linear equations, these equations

can always be brought ordinates may be given each of them as a function of a single variable into the foregoing form, and hence that the two linear equations parameter. The form y=$(x), x=(x), where two of the coordinates represent a line. are given in terms of the third, is a particular case of each of these Secondly, taking for greater simplicity the point Q to be coincident modes of representation.

with the origin, and a', B', 7. p to be constant, then p is the perpen29. The remarks under plane geometry as to descriptive and dicular distance of a plane from the origin, and 2', 8, 7 are the cosinemetrical propositions, and as to the non-metrical character of the inclinations of this distance to the axes (a1 +8+?=1). P is method of coordinates when used for the proof of a descriptive any point in this plane, and taking its coordinates to be (x, 7. ) then proposition, apply also to solid geometry; and they might be (5, 7 s) are = (x, y, z), and the foregoing equation pra's+86+y'n illustrated in like manner by the instance of the theorem of the radical becomes centre of four spheres. The proof is obtained from the consideration

a'x+s'y+y':D that S and S' being cach of them a function of the form x' +3* +22+ which is the equation of the plane in question. ax+by+c2+d, the difference S-S' is a mere linear function of the

I!, more generally, Q is not coincident with the origin, then, coordinates, and consequently that S-S'=o is the equation of the taking its coordinates to be (a, b, c), and writing A instead of P, the plane containing the circle of intersection of the two spheres S=0 and S'=0.

equation is 30. Meirical Theory:—The foundation in solid geometry of the and we thence have pop-laa'+b8°+cy), which is an expression

a'(x-a)+8'(y-b)ty' (2-0)=PI metrical theory is in fact the before-mentioned theorem that if a finite right line PQ be projected upon any other line OO' by lines for the perpendicular distance of the point (a, b, c) from the plane

perpendicular to. 00', then the length of the
P'Q' is equal to the length of PQ the coordinates can always be brought into the foregoing form,

It is obvious that any linear equation Ax+By+C:+D=o between
into the cosine of its inclination to P'Q'-or and hence that such an equation represents a plane.
(in the form in which it is now convenient
state the theorem) the perpendicular and the distance QP, = p; to be constant, say this is = d, then, as

Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), distance P'Q of two parallel planes is equal before, the values of E, ni $ are -, y—b, 3-6, and the equation to the inclined distance PQ into the cosine of the inclination. The principle of $ 16,

+y+4= p becomes that the algebraical sum of the projections of

the sides of any closed polygon on any line is which is the equation of the sphere, coordinates of the centre =(0,04).
zero, or that the two sets of sides of the and radius=d.
polygon which connect a vertex A and a A quadric equation wherein the terms of the second order are
vertex B have the same sum of projections **+ya+s, viz. an equation
on the line, in sign and magnitude, as we pass

from A to B, is applicable when the sides do
not all lie in one plane.

can always, it is clear, be brought into the foregoing form; and it 31. Consider the skew quadrilateral QMNP, thus appears that this is the equation of a sphere, coordinates of the sides QM, MN, NP being respectively the centre - A. - JB. – {C, and squared radius= }(A+B:+C)-D. parallel to the three rectangular axes Ox,

33. Cylinders, Cones, ruled Surfaces:- If the two equations of a Fig. 60.

Oy, Oz; let the lengths of these sides be straight line involve a parameter to which any value may be given, 8. n. 5, and that of the side QP be = p; and

we have a singly infinite system of lines. They cover a surface, and let the cosines of the inclinations (or say the cosine-inclinations) of the equation of the surface is obtained by eliminating the parameter e to the three axes be a, B. Vi then projecting successively on

between the two equations. the three sides and on QP we have

If the lines all pass through a given point, then the surface is a 5.7. $ = pa, pl. pr.

cone; and, in particular, if the lines are all parallel to a given line, and p=a+Bntrs.

then the surface is a cylinder. whence p++s, which is the relation between a distance e

Beginning with this last case, suppose the lines are parallel to and its projections 5, 7, § upon three rectangular axes. And from the line x=ms, y=nz, the equations of a line of the system are the same equations we obtain a' +82 + y2 = 1, which is a relation con

x=mz+a, y=nz+b,--where a, b are supposed to be functions of necting the cosine-inclinations of a line to three rectangular axes.

the variable parameter, or, what is the same thing, there is between Suppose we have through Q any other line QT, and let the cosine

them a relation f(a, b)=0: we have c= x-ms, b=y-ns, and the inclinations of this to the axes be a', s', 7 and 8 be its cosine.

result of the elimination of the parameter therefore is f(x-7, inclination to QP; also let p be the length of the projection of QP y-nz)=0, which is thus the general equation of the cylinder the upon QT; then projecting on QT we have

generating lines whereof are parallel to the line r=m2, y=. The p-a'E+B'nty's=pô.

equation of the section by the plane

3=0 is f(x, y) =0, and conversely

if the cylinder be determined by means of its curve of intersection And in the last equation substituting for 5, 7. $ their values pa,

with the plane s=0;.then, taking the equation of this curve to be PB, py we find

f(x, y) = 0, the equation of the cylinder is f(x-mz, y-n2) = 0. Thus, d=aa'+BB' tyr',

if the curve of intersection be the circle (x-a)?+68-B)!=y. we which is an expression for the mutual cosine-inclination of two have (x-me-a)" +(y-ns-B)*=qe as the equation of an oblique lines, the cosine-inclinations of which to the axes are a, B, 7 and cylinder on this base, and thus also (x-2)+-8)'=ge as the a', 8, 7 respectively: We have of course a:+A+y = I and equation of the right cylinder. a" +8+=1; and hence also

'If the lines all pass through a given point (a, b, c), then the equa1-88 = (a +82 +92)(a" +8"+y)-(aa'+BB'+ry'),

tions of a line are :-2=(2-c), y-0 = B(3-6), where a, 8 are By' - 6'7) + (ya'-'a)+(ab' -a'B)";

functions of the variable parameter, or, what is the same thing, so

there exists between them an equation f(a, b)=0; the elimination foot. These formulae are the foundation of spherical

trigonometry of the parameter gives, therefore, 1(E) =0; and this 32. Straight Lines, Planes and Spheres. — The foregoing formulae give at once the equations of these loci.

equation, or, what is the same thing, any homogeneous equation For first, taking Q to be a fixed point, coordinates (a, b, c), and (x-2, 3-6, 2-6)=0, or, taking s to be a rational and integral the cosine-inclinations (a, B. ~) to be constant, then P will be a function of the order n, şay (*) (x-3, -6, 2-6)* = 0, is the general point in the line through Q in the direction thus determined; or, equation of the cone having the point (a, b, c) for its vertex. Taking taking (x, y, z) for its coordinates, these will be the current co- the vertex to be at the origin, the equation is (*) (x, y, z) =0; and, ordinates of a point in the line. The values of £, n $ then are in particular, (*) (x, y, z) =o is the equation of a cone of the second 1-2, y-6, 8-c, and we thus have

order, or quadricone, having the origin for its vertex. x-ay-bs-ci

34. In the general case of a singly infinite system of lines, the *-o).

locus is a ruled surface (or regulus). Now, when a line is changing

its position in space, it may be looked upon as in a state of turning which (omitting the last equation, : = p) are the equations of the line about some point in itself, while that point is, as a rule, in a state of through the point (a, b, c), the cosine-inclinations to the axes being moving out of the plane in which the turning takes place. ! ina, B. y, and these quantities being connected by the relation stantaneously it is only in a state of turning, it is usual, though not Q+8+y=1. This equation may be omitted, and then a, B: n. strictly accurate, to say that it intersects its consecutive position. instead of being equal, will only be proportional, to the cosine. A regulus such that consecutive lines on it do not intersect, in this inclinations.

sense. is called a skew surface, or scroll: one on which they do is Using the last equation, and writing

called a developable surface or torse. x, y, z=tap, b+Bp.ctye,

Suppose, for instance, that the equations of a line (depending on


[ocr errors]


[ocr errors]
[ocr errors]


the variable parameter ) are 5+2=0(1+), (-7) : equations of one second degree are called quadric surfaces. quadric

Species Quadric -by then, eliminating 0, we have ---- .. or say if + 1, coefficienta in the general equation are limited to satisfy certain the equation of a quadric surface, afterwards called the hyperboloid particular one plane twice repeated, and (2) cones, including in marked that we have upon the surface a second singly infinite particular.cylinders; there is but one form of cone, but cylinders series of lines; the equations of a line of this second system (des may be elliptic, parabolic or hyperbolic. pending on the variable parameter o) are

K discussion of the general equation of the second degree shows

that the proper quadric surfaces are of five kinds, represented +-(1-3). (+3).

respectively, when referred to the most convenient axes of reference, It is easily shown that any line of the one system intersects every

by equations of the five types (a and b.positive): line of the other system.

(1) e +elliptic paraboloid. 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


hyperbolic paraboloid. a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculat- (3) en ++3=1, ellipsoid. ing plane of the curve (see $ 38 below) is a tangent plane of the torse all along a generating line.

(4) *+- = 1, hyperboloid of one sheet.

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,

(5) ent---1,

=-1, hyperboloid of two sheets. Ox, Oy, Os, and Oxı, Oyı, Oz, the mutual cosine-inclinations being shown by the diagram

It is at once seen that these are distinct surfaces; and the equa. tions also show very readily the general form and mode of genera.

tion of the several surfaces.

In the elliptic paraboloid (fig. 61)
B' g'

the sections by the planes of zx and

ay are the parabolas g


Za'2 -26 that is, e, B. y are the cosine-inclinations of Oxi to Ox, Oy, Oz; having the common axes Oz; and a', 8, 7 those of Oyı, &c. And this diagram gives also the linear expressions of the co

the section by any plane 2=7 ordinates (x1, 9, 31) or (x, y, z) of either set in terms of those of the parallel to that of xy is the ellipse other set; we thus have x= x+Bytys, * = axi ta'yı ta'zi,

y=ža +20: Yu=4' x+B'yty's, y=Bxı+B'yı +B'zi,

so that the surface is generated by

Fig. 61. 31= "x+B'yty's, z=yxity'yi to'si,

a variable ellipse moving parallel to itself along the parabolas as which are obtained by projection, as above explained. Each of directrices. these equations is, in fact, nothing else than the before-mentioned In the hyperbolic paraboloid (figs. 62 and 63) the sections by the equation p=a's +8'rty'si adapted to the problem in hand.

But we have to consider the relations between the nine coefficients. planes of zx, sy are the parabolas s=28=- . having the opposite By what precedes, or by the consideration that we must have identically 3*+ya+s:=x;'+yi'+2, it appears that these satisfy axes Oz, Oz', and the section by a plane :=y parallel to that of the relationsa +B2 tr a'tan ta" -1, xy is the hyperbola vy=20-25

which has its transverse axis parallel +B2 tin

p+82 +8" - 1,
a? +B"? ty = 1,
woty? 7 -1,

to Ox' or Oy according as y is positive or negative. The surface is thus
a'a'+8'8"+yY=0, By+BY+B"y'=0,
a'a +B'B +ry -0,

ya+y'a'ty'a' = 0, aa' +BB ty = 0,

ab ta'b' ta's' = 0, either set of six equations being implied in the other set. It follows that the square of the determinant

B a'. Bi

B", is = t; and hence that the determinant itself is = #1. The distinction of the two cases is an important one: if the determinant is = +!, then the axes Oxi, Oy, O2 are such that they can by a

3 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 a1. 91, 2; in the opposite directions we change the signs of all the coefficients and so make the determinant to be - +1; hence

FIG. 62.

Fig. 63. the former case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found generated by a variable hyperbola moving parallel to itself along that we have the equality a = B'q'-B'y', and eight like ones, the parabolás as directrices. The form is best seen from fig. 63, obtained from this by cyclical interchanges of the letters a, B, 7,

which represents the secand of unaccented, singly and doubly accented letters.

tions by planes parallel to 36. The nine cosine-inclinations above are, as has been seen, the plane of xy, or say the connected by six equations. It ought then to be possible to express contour lines;

the conthem all in terms of three parameters. An elegant means of doing tinuous lines are the sec, this has been given by Rodrigues, who has shown that the tabular tions above the plane of expression of the formulae of transformation may be written xy, and the dotted lines

the sections below this

plane. The form is, in
x 1+1*-**-

2(v + m)

fact, that of a saddle.

In the ellipsoid (fig. 64) 2(^u +») 1-13 + ? - put 2(6+1)

the sections by the planes

of zx, sy, and xy are each
vu) 2(ur+X) 1-12-17 + m2

of them an ellipse, and the

section by any parallel +(1+12 + m +r),

plane is also an ellipse. Y the meaning being that the coefficients in the transformation are The surface may be con

Fig. 64. fractions, with numerators expressed as in the table, and the common sidered as generated by denominator.

an ellipse moving parallel to itself along two ellipses as directrices.



[ocr errors]


In the hyperboloid of one sheet (fig. 65), the sections by the planes absolute curvature is the limiting position of the point where the of zx, sy are the hyperbolas

principal normal at (x, y, z) is cut by the normal plane at a neighbour

ing point, as that point moves up to (x, y, z). - - = 1,

39. Differential Geometry of Surfaces. Let (x, y, z) be any chosen having a common conjugate axis Os'; the section by the plane of point on a surface f(x, y, z)=0. As a second point of the surface x, y, and that by any parallel plane, is an ellipse; and the surface moves up to (x, y, 3), its connector with (x, y, z) tends to a limiting may be considered as generated by a variable ellipse moving parallel position,

a tangent line to the surface at (x, y, z). All these tangent to itself along the two hyperbolas as directrices. If we imagine two

lines at (x, y, z), obtained by approaching (x, y, z) from different equal and parallel circular disks, their points connected by strings directions on a surface, lie in one plane of equal lengths, so that these are the generators of a right circular

(-x)+ (1–2)+(5-2) =0. cylinder, and if we turn one of the disks about its centre through an

дх angle in its plane, the strings in their new positions will be one This plane is called the tangent plane at (x, y, z). One line through system of generators of a hyperboloid of one sheet, for which a=b; (x, y, z) is at right angles to the tangent plane. This is the normal and if we turn it through the same angle in opposite direction,

[ocr errors]
[ocr errors]
[ocr errors]

The tangent plane is cut by the surface in a curve, real or imaginary, with a node or double point at (x, y, z). Two of the tangent lines touch this curve at the node. They are called the “ chief tangents (Haupi-tangenten) at (x, y, z); they have closer contact with the surface than any other tangents.

In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a pode, it must therefore consist of two straight lines. Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboload of two sheets, and elliptic paraboloid.

A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chief-langent curves; on a quadric surface they are the above straight lines.

40. The tangents at a point of a surface which bisect the angles between the chief tangents are called the principal tangents at the point. They are at right angles, and together with the normal

constitute a convenient set of rectangular axes to which to refer the Fig. 65

FIG. 66.

surface when its properties near the point are under discussion. At a special point which is such that the chief tangents there run

to the circular points at infinity in the tangent plane, the principal we get in like manner the generators of the other system; there will be the same general configuration when a #6. The hyperbolic of the surface.

tangents are indeterminate; such a special point is called an umbilic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. pair cutting one another at sight angles through every point upon it,

There are two singly infinite systems of curves on a surface, a The figures should be studied to see how they can lie. In the hyperboloid of two sheets (fig. 66) the sections by the planes respective points of contact. These are called lines of curvature,

all tangents to which are principal tangents of the surface at their of ex and sy are the hyperbolas

because of a property next to be mentioned.

As a point Q moves in an arbitrary direction on a surface from

coincidence with a chosen point P, the normal at it, as a rule, at having a common transverse axis along z'Oz; the section by any

once fails to meet the normal at P; but, if it takes the direction of a plane z = y parallel to that of xy is the ellipse

line of curvature through P, this is instantaneously not the case.

We have thus on the normal two centres of curvature, and the x?

distances of these from the point on the surface are the two principal a

radii of curvature of the surface at that point; these are also the radii provided q? >c, and the surface, consisting of two distinct portions of curvature of the sections of the surface by planes through the or sheets, may be considered as generated by a variable ellipse normal and the two principal tangents respectively; or say they are moving parallel to itself along the hyperbolas as directrices. the radii of curvature of the normal sections through the two principal

38. Differential Geometry of Curves.-For convenience consider the tangents respectively. Take at the point the axis of - in the direction coordinates (x, y, z) of a point on a curve in space to be given as of the normal, and those of x and y in the directions of the principal functions of a variable parameter 0, which may in particular be one tangents respectively, then, if the radii of curvature be a, b (the signs of themselves. Use the notation x', x' for dx/do, dex/da?, and simi- being such that the coordinates of the two centres of curvature are larly as to y and s. Only a few formulae will be given. Call the 2=a and :=b respectively), the surface has in the neighbourhood current coordinates (5, 1, $),

of the point the form of the paraboloid The langenl at (x, y, z) is the line tended to as a limit by the connector of (x, y, z), and a neighbouring point of the curve when the latter moves up to the former: its equations are (8 - x)/x' = (1-y)/y' = (3-2)/2'.

and the chief-tangents are determined by the equation 0 = +2 The osculating plane at (x, y, s) is the plane tended to as a limit by The two centres of curvature may be on the same side of the point that through (x, y: :) and two neighbouring points of the curve as these, remaining distinct, both move up to (x, y, z): its one equation the paraboloid is elliptic, and the chief-tangents are imaginary;

or on opposite sides; in the former case a and b have the same sign, is (5-x)(y's"-y"z')+(n-y)(:'r" -z'x')+(5-2)(x'y'—x'y') = 0.

in the latter case a and b have opposite signs, the paraboloid is

hyperbolic, and the chief-tangents are real. The normal plane is the plane through (x, y, z) at right angles to the The normal sections of the surface and the paraboloid by the same tangent line, i.e. the plane

plane have the same radius of curvature; and it thence readily x'(&-x)+7(7-y)+:'(3-2) = 0.

follows that the radius of curvature of a normal section of the surface It cuts the osculating plane in a line called the principal normal. by a plane inclined at an angle & to that of zx is given by the equation Every line through (x, y, z) in the normal plane is a normal. The

1-coso_ sine normal perpendicular to the osculating planc is called the binormal.

bo A tangent, principal normal, and binormal are a convenient set of rectangular axes to use as those of reference, when the nature of a The section in question is that by a plane through the normal curve near a point on it is to be discussed.

and a line in the tangent plane inclined at an angle & to the principal Through (x, y, s) and three neighbouring points, all on the curve, tangent along the axis of x. To complete the theory, consider i he passes a single sphere; and as the three points all move up to (x, y, z) section by a plane having the same trace upon the tangent plane, continuing distinct, the sphere tends to a limiting size and position. but inclined to the normal at an angle : then it is shown without The limit tended to is the sphere of closest contact with the curve at difficulty (Meunier's theorem) that the radius of curvature of this x, y, z); its centre and radius are called the centre and radius of inclined section of the surface is =p cos 6. spherical curvature. It cuts the osculating plane in a circle, called the AUTHORITIES.—The above article is largely based on that by circle of absolute curvature; and the centre and radius of this circle Arthur Cayley in the 9th edition of this work. Of early and imare the centre and radius of absolute curvature. The centre of l portant recent publications on analytical geometry, special mention

[ocr errors]




[ocr errors]

is to be made of R. Descartes, Géométrie (Leyden, 1637): John, are called Complexes, Congruences, and Ruled Surfaces or Skews Wallis, Tractalus de sectionibus conicis nova methodo exposilis (1655, respectively. A Complex is thus a system of lines satisfying one Opera mathematica, i., Oxford, 1695); de l'Hospital, Traité analytique condition—that is, the coordinates are connected by a single relation; des sections coniques (Paris, 1720): Leonhard Euler, Introductio in and the degree of the complex is the degree of this equation supposing analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “ Appli- it to be algebraic. The lines of a complex of the nth degree which cation d'algèbre à la géométrie (Journ. Ecole Polytech., 1801); pass through any point lie on a cone of the nth degree, those which Julius Plücker, Analylisch-geometrische Entwickelungen, 3. Bde. lie in any plane envelop a curve of the nth class and there are n lines (Essen, 1828-1831); System der analytischen Geometrie (Berlin, of the complex in any plane pencil; the last statement combines 1835); G. Salmon, a Treatise on Conic Sections (Dublin, 1848; the former two, for it shows that the cone is of the nth degree and 6th ed., London, 1879): Ch. Briot and J. Bouquet, Leçons de géo- the curve is of the nth class. To find the lines common to four métrie analytique (Paris, 1851; 16th ed., 1897); M. Chasles, Traité complexes of degrees ni, m, 13, 14, we have to solve five equations, viz. de géométrie supérieure (Paris, 1892); Wilhelm Fiedler, Analytische the four complex equations together with the quadratic equation Geometrie der Kegelschnille nach G. Salmon frei bearbeitet (Leipzig, connecting the line coordinates, therefore the number of common Ste Aufl., 1887-1888); N. M. Ferrers, An Elementary Treatise on lines is 2nınınina. As an example of complexes we have the lines Trilinear Coordinates (London, 1861); Otto Hesse, Vorlesungen meeting a twisted curve of the nth degree, which form a complex aus der analytischen Geometrie (Leipzig, 1865, 1881); W. A. Whit- of the nth degree. worth, Trilinear Coordinales and other Methods of Modern Analytical A Congruence is the set of lines satisfying two conditions: thus Geometry (Cambridge, 1866): J Booth, A Treatise on Some New a finite number m of the lines pass through any point, and a finite Geometrical Methods (London, i., 1873; ii.. !877); A. Clebsch- number » lie in any plane; these numbers are called the degree F. Lindemann, Vorlesungen über Geometrie, Bd. I. (Leipzig. 1876, and class respectively, and the congruence is symbolically written 2te Aufl., 1891); R. Baltser, Analylische Geometrie (Leipzig, 1882); (m, n). Charlotte A. Scott, Modern Methods of Analytical Geometry (London, The simplest example of a congruence is the system of lines 1894); G. Salmon, A Trealise on the Analylical-Geometry of three constituted by all those that pass through m points and those that Dimensions (Dublin, 1862; 4th ed., 1882): Salmon-Ficdler, Analy- lie in n planes; through any other point there pass m of these lines, tische Geometrie des Raumes (Leipzig, 1863: 4te Auf., 1898): P: and in any other plane there lie n, therefore the congruence is of Frost, Solid Geometry (London, 3rd ed., 1886; Ist ed., Frost and degree m and class n. It has been shown by G. H. Halphen that the J. Wolstenholme). See also E. Pascal, Reperlorio di matematiche number of lines common to two congruences is mm' +nn', which may superiori, II. Geometria (Milan, 1900), and articles now appearing be verised by taking one of them to be of this simple type. The in the Encyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2. lines meeting two fixed lines form the general (1, 1) congruence;

(E. B. EL.) and the chords of a twisted cubic form the general type of a (1, 3)

congruence; Halphen's result shows that two twisted cubics have V. LINE GEOMETRY

in general ten common chords. As regards the analytical treatment, Line geometry is the name applied to those geometrical the difficulty is of the same nature as that arising in the theory of

curves in space, for a congruence is not in general the complete investigations in which the straight line replaces the point as intersection of two complexes. element. Just as ordinary geometry deals primarily with points A Ruled Surface, Regulus or Skew is a configuration of lines and systems of points, this theory deals in the first instance which satisfy three conditions, and therefore depend on only one with straight lines and systems of straight lines. In two dimen- parameter. Such lines all lie on a surface, for we cannot draw one sions there is no necessity for a special line geometry, inasmuch through an arbitrary point; only one line passes through a point of

the surface; the simplest example, that of a quadric surface, is as the straight line and the point are interchangeable by the really two skews on the same surface. principle of duality; but in three dimensions the straight line The degree of a ruled surface qua line geometry is the number of is its own reciprocal, and for the better discussion of systems which meets a given line is the degree of the surface qua point geo

its generating lines contained in a linear complex. Now the number of lines we require some new apparatus, e.g., a system of co- metry; and as the lines meeting a given line form a particular case ordinates applicat:le to straight lines rather than to points. of linear complex, it follows that the degree is the same from whichThe essential features of the subject are most easily elucidated cver point of view we regard it. The lines common to three comby analytical methods: we shall therefore begin with the notion plexes of degrecs, hin?", form a ruled surface of degree 2nınans;

but not every ruled surface is the complete intersection of three of line coordinates, and in order to emphasize the merits of the

complexes. system of coordinates ultimately adopted, we first notice a In the case of a complex of the first degree (or linear complex) system without these advantages, but often useful in special the lines through a fixed point lie in a plane called the polar plane investigations.

or nul-plane of that point, and those lying in a fixed plane


pass through a point called the nul-point or pole of the In ordinary Cartesian coordinates the two equations of a straight plane. If the nul-plane of A pass through B, then the

complex. line may be reduced to the form y=rx+s, 2= 4x+u, and , s, t, u nul-plane of B will pass through A; the nul-planes of all points on may be regarded as the four coordinates of the line. These co- one line li pass through another linel. The relation between l; and ordinates lack symmetry: moreover, in changing from one base of Iz is reciprocal; any line of the complex that meets one will also reference to another the transformation is not linear, so that the meet the other, and every line meeting both belongs to the complex. degree of an equation is deprived of real significance. For purposes They are called conjugate or polar lines with respect to the complex. of the general theory we employ homogencous coordinates; if | On these principles can be founded a theory of reciprocation with 219121w1 and 129:27w; are two points on the line, it is easily verified respect to a linear complex. that the six determinants of the array

This may be aptly illustrated by an elegant example due to A.

Voss. Since a twisted cubic can be made to satisfy twelve conditions, XzY22162

it might be supposed that a finite number could be drawn to touch are in the same ratios for all point-pairs on the line, and further, four given lines, but this is not the case. For, suppose one such can that when the point coordinates undergo a linear transformation containing the four lines is a curve of the third class, i.e. another

be drawn, then its reciprocal with respect to any linear complex so also do these six determinants. determinants for the coordinates of the line, and express them by the twisted cubic, touching the same four lines, which are unaltered symbols !, 1, m, 4, n. » where l=x162-xqw, * = 4122-y:zi, &c. in the process of reciprocation; as there is an infinite number of There is the further advantage that it cabicide and 6620,d be two complexes containing the four lines, there is an infinite number of planes through the line, the six determinants

cubics touching the four lines, and the problem is poristic.

The following are some geometrical constructions relating to the la bicidi

unique linear complex that can be drawn to contain five arbitrary

lincs: are in the same ratios as the foregoing, so that except as regards a To construct the nul-plane of any point O, we observe that the factor of proportionality we have = bıça-b2c1, l=cido-cod. &c. two lines which meet any four of the given five are conjugate lines The identical' relation lá + min tnv=o reduces the number of inde of the complex, and the line drawn through O to meet them is pendent constants in the six coordinates to four, for we are only therefore a ray, of the complex; similarly, by choosing another concerned with their mutual ratios; and the quadratic character four we can find another ray through 0: these rays lie in the nul. of this relation marks an essential diffcrence between point geometry. plane, and there is clearly a result involved that the five lines so and line geometry. The condition of intersection of two linės is obtained all lie in one plane. A reciprocal construction will enable n'+12+ mu't m'u tny' tn'v=0

us to find the nul-point of any plane. Proceeding now to the metrical

properties and the statical and dynamical applications, we remark where the accented letters refer to the second line. If the coordinates that there is just one line such that the nul-plane of any point on it are Cartesian and l, m, n are direction cosines, the quantity on the is perpendicular to it. This is called the central axis; if d be the left is the mutual moment of the two lines.

shortest distance, the angle between it and a ray of the complex, Since a line depends on four constants, there are three distinct types then d tan 8 = P, where p is a constant called the pitch or parameter. of configurations arising in line geometry--those containing a triply. Any system of forces can be reduced to a force R along a certain line, infinite, a doubly. infinite and a singly-infinite number of lines; they and a couple G perpendicular to that line; the lines of nul-moment


XI 12


for the system form a linear complex of which the given line is the the polar form 200, +2bb, +2cc, -211, -de, -,e vanishes. Comcentral axis and the quotient G/R is the pitch. Any motion of a paring this with the equation xiix+x3+x2–2x5o given rigid body can be reduced to a screw motion about a certain line, above, it appears that this sphere geometry and line geometry are i.e. to an angular velocity w about that line combined with a linear identical, for we may write a=xi, b=x2, <=15, r=825-1, d=15, velocity u along the line. perpendicular to the direction of its motion is its nul-plane with = 4*, but it is to be noticed that

a sphere is really replaced by two respect to a linear complex having this line for central axis, and the lines whose coordinates only differ in the sign of me, so that they are quotient ulw for pitch (cf. Sir R. S. Ball, Theory of Screws). polar lines with respect to the complex x,-0. Two spheres which

The following are some properties of a configuration of two linear touch correspond to two lines which intersect, or more accurately complexes:

to two pairs of lines ( R') and(9, 9'), of which the pairs (P. 9) and The lines common to the two-complexes also belong to an infinite (P.q') both intersect. By this means the problem of describing a number of linear complexes, of which two reduce to single straight sphere to touch four given spheres is reduced to that of drawing a lines. These two lines are conjugate lines with respect to each of pair of lines (1, 1), (of which 1 intersects one line of the four pairs the complexes, but they may coincide, and then some simple modifi- (op')., (99'). (rr'). (ss'), and l'intersects the remaining four). We cations are required. The locus of the central axis of this system may, however, ignore the accented letters in translating theoremas, of complexes is a surface of the third degree called the cylindroid, for a configuration of lines and its polar with respect to a linear which plays a leading part in the theory of screws as developed complex have the same projective properties. In Lie's transforma. synthetically by Ball. Since a linear complex has an invariant of tion a linear complex corresponds to the to:ality of spheres cutting a the second degree in its coefficients, it follows that two linear com given sphere at a given angle. A most remarkable result is that lines plexes have a lineo-linear invariant. This invariant is fundamental: of curvature in the sphere geometry become asymptotic lines in is the complexes be both straight lines, its vanishing is the condition the line geometry. of their intersection as given above; if only one of them be a straight Some of the principles of line geometry may be brought into line, its vanishing is the condition that this line should belong to the clearer light by admitting the ideas of space of four and five other complex. When it vanishes for any two complexes they dimensions. are said to be in involution or a polar; the nul-points P, Q of any Thus, regarding the coordinates of a line as homogeneous com plane then divide harmonically the points in which the plane meets ordinates in five dimensions, we may say that line goometry is the common conjugate lines, and each complex is its own reciprocal equivalent to geometry on a quadric surface in five dimensions. with respect to the other. As regards a configuration of these A' linear complex is represented by a hyperplane section; and if linear complexes, the common lines from one system of generators two such complexes are in involution, the corresponding hyperpla nes of a quadric, and the doubly infinite system of complexes containing are conjugate with respect to the fundamental quadric.. By prothe common lines, include an infinite number of straight lines which jecting this quadric stereugraphically into space of four dimensions form the other system of generators of the same quadric.

we obtain Klein's analogy. In the same way geometry in a linear If the equation of a lincar complex is Al+Bm+Cn+DA+Ev+complex is equivalent to geometry on a quadric in four dimensions; Fr=0, then for a line not belonging to the complex we may regard when two lines intersect the representative points are on the same

the expression on the left-hand side as a multiple of the generator of this quadric. Stercographic projection, therefore, General

moment of the line with respect to the complex, the word converts a curve in a linear complex, s.c. one whose tangents ali line co

moment being used in the statical sense; and we inser belong to the complex, into one whose tangents intersect a fixed

that when the coordinates are replaced by lincar functions conic: when this conic is the imaginary circle at infinity the curve of themselves the new coordinates are multiples of the moments is what Lie calls a minimal curve. Curves in a linear complex have of the line with respect to six fixed complexes. The essential features been extensively studied. The osculating plane at any point of such of this coordinate system are the same as those of the original onc, a curve is the nul-plane of the point with respect to the complex, viz. there are six coordinates connected by a quadratic equation, and points of superosculation always coincide in pairs at the points but this relation has in general a different form. By suitable choice of contact of stationary tangents. When a point of such a curve is of the six fundamental complexes, as they may be called, this con given, the osculating plane is determined, hence all the curves through necting relation may be brought into other simple forms of which a given point with the same tangent have the same torsion. we mention two: (i.) When the six are mutually in involution it can The lines through a given point that belong to a complex of the be reduced to x,? +xz+x: +x;}+x3? +Xg2 = 0; (ii.). When the first nth degrec lie on a cone of the nth degree if this cone has a double four are in involution and the other two are the lines common to line the point is said to be a singular point. Similarly, the first it is x'+x; +*3*+*-2XyX5 = 0. These generalized a plane is said to be singular when the envelope of the

Noo-locer coordinates might be explained without reference to actual magni: lines in it has a double tangent. It is very remarkable como tude, just as homogeneous point coordinates can be; the essential that the same sursace is the locus of the singular points

pkeres. remark is that the equation of any coordinate to zero represents a and the envelope of the singular planes: this surlace is called the linear complex, a point of view which includes our original system, singular surface and both its degree and class are in general 2n(-1). for the equation of a coordinate to zero represents all the lines which is equal to four for the quadratic complex, meeting an edge of the fundamental tetrahedron.

The singular lines of a complex F=o are the lines common to F The system of coordinates referred to six complexes mutually and the complex in involution was introduced by Felix Klein, and in many cases is

8F 8F , 8F $F, 8F $F more useful than that derived directly from point coordinates; e.g.

Io + im otin

= 0. in the discussion of quadratic complexes: by means of it Klein has developed an analogy between line geometry and the geometry of As already mentioned, at each line l of a complex there is an infinite spheres as treated by G. Darboux and others. In fact, in that number of tangent linear complexes, and they all contain the lines geometry a point is represented by five coordinates, connected by a adjacent to l. If now l be a singular line, these complexes all reduce relation of the same type as the one just mentioned when the five to straight lines which form a plane pencil containing the line I. fundamental spheres are mutually

at right angles and the equation Suppose the vertex of the pencil is A, its plane a, and one of its lines of a sphere is of the first degree. 'Extending this to four dimensions : then being a complex line near l, meets & or more accurately of space, we obtain an exact analogue of line geometry, in which the mutual moment of ', and is of the second order of small quan (i.) a point corresponds to a line; (i.) a linear complex to a hyper- tities. If P be a point onl, a line through P quite nearl in the plane sphere; (iii.) two linear complexes in involution to two orthogonal will meet and is therefore a line of the complex; hence the hyperspheres; (iv.) a linear complex and two conjugate lines to

complex-cones of all points on l touch e and the complex-curves a hypersphere and two inverse points. Many results may be obtained

of all planes through I touch I at A. It follows that I is a double by this principle, and more still are suggested by trying to extend line of the complex-cone of A, and a double tangent of the complexthe properties of circles to spheres in three and four dimensions.

curve of a. Conversely, a double line of a cone or curve is a singular Thus the elementary theorem, that, given four lines, the circles line, and a singular líne clearly touches the curves of all planes circumscribed to the four triangles formed by them are concurrent, through it in the same point. Suppose now that the consecutive may be extended to six hyperplanes in four dimensions; and then line ?' is also a singular line, A' being the allied singular point, a we can derive a result in line geometry by translating the inverse the singular plane and any line of the pencil (A', :') so that is of this theorem. Again, just as there is an infinite number of spheres a tangent line at l' to the complex: the mutual moments of the closer nature, so there is an infinite number of linear complexes I meets the lines i and e' in two points very near A. This being true touching a non-linear complex at a given line, and three of these for all singular planes, near i the point of contact of a with its have contact of a closer nature (cf. Klein, Malk. Ann. v.).

envelope is in A, i.e. the locus of singular points is the same as the Sophus Lie has pointed out a different analogy with sphere envelope of singular planes. Further, when a line touches a complex geometry. Suppose, in fact, that the equation of a sphere of radius it touches the singular surface, for it belongs to a plane pencil like

(Aa), and thus in Klein's analogy the analogue of a locus of a hyperx + y ++2ax+2by+2cz+d=0,

surface being a bitangent line of the complex is also a bitangent lino

of the singular surface. The theory of cosingular complexes is thus so that y=0? +62 +c: -d; then introducing the quantity e to make brought into line with that of conlocal surfaces in four dimensions, this equation homogeneous, we may regard the sphere as given by and guided by these principles the existence of cosingular quadratic the six coordinates a, b, c, d, e, r connected by the equation a' t. complexes can easily be established, the analysis required being bo+c-gi--de =0, and it is easy to see that two spheres touch if I almost the same as that invented for confocal cyclides by Darboux

[ocr errors]
« PreviousContinue »