B в B FIG. 34. If, on the other hand, we take a point P in the plane of a conic, conjugates with regard to the foci. If therefore the two foci F, and we get to each line a through P one conjugate line which joins PF, be joined to P, these lines will be harmonic with regard to the to the pole of a. These pairs of conjugate lines through P form an involution in the pencil at P. The local rays of this involution are the tangents drawn from P to the conic. This gives the theorem reciprocal to the last, viz: A conic determines in every pencil in ils plane on involution, corresponding lines teing conjugale lines with regard to the conic. If the point is wilhout the conic the involution is hyperbolic, the tangents from the points being the focal rays. If the point lies on the conic the involution is parabolic, the tangeni at the point counting for coincident focal rays. If the poin: is within the conic the involution is elliptic, having no focal rays. It will further be seen that the involution determined by a conic on any line o is a section of the involution, which is determined by the conic at the pole P of p. $ 83, Foci.-The centre of a pencil in which the conic determines a circular involution is called a "focus" of the conic. In other words, a focus is such a point that every line through it is tangent and normal. As the latter are perpendicular, they will perpendicular to its conjugate line. The polar to a focus is called a bisect the angles between the other pair. Hencedirectrix of the conic. The lines joining any point on a conic 10 the two foct are equally From the definition it follows that every, focus lies on an axis, for inclined to the langent and normal at that point. the line joining a focus to the centre of the conic is a diameter to In case of the parabola this becomeswhich the conjugate lines are perpendicular; and every line joining The line joining any point on a parabola to the focus and the diameter two foci is an axis, for the perpendiculars to this line through the foci through the poini, are equally inclined to the langent and normal al are conjugate to it. These conjugate lines pass through the pole of that point. the line, the pole lies therefore at infinity, and the line is a diameter, From the definition of a focus it follows thathence by the last property an axis. The segment of a tangent between the directrix and the point of It follows that all foci lie on one axis, for no line joining a point contact is seen from the focus belonging to the directrix under a righe in one axis to a point in the other can be an axis. angle, because the lines joining the focus to the ends of this As the conic determines in the pencil which has its centre at a focus segment are conjugate with regard to the conic, and therefore a circular involution, no tangents can be drawn from the focus to perpendicular. the conic. Hence each focus lies within a conic; and a directrix does With equal ease the following theorem is proved: not cut the conic. The two lines which join the points of contact of two langerls each Further properties are found by the following considerations: to one focus, but not both to the same, are seen from the intersection of $ 84. Through a point P one line p can be drawn, which is with the tangenis under equal angles. regard to a given conic conjugate to a given line 2. viz. that line $ 86. Other focal properties of a conic are obtained by the following which joins the point P to the pole of the line q. If the line q is made considerations: to describe a pencil about a point, then the line will describe a Let F (fig. 35) bę a focus to a conic, f the corresponding directrix, pencil about Þ. These two pencils will be projective, for the line A and B the points of contact of two tangents meeting at T, and P passes through the pole of 9, and whilst g describes the pencil Q. the point where the its pole describes a projective row, and this row is perspective to line AB cuts the directhe pencil P. trix. Then TF will be We now take the point P on an axis of the conic, draw any line the polar of P (because A through it, and from the pole of P. draw a perpendicular a to popolars of F and I meet Let 2 cut the axis in l. Then, in the pencils of conjugate lines, at P). Hence TF and which have their centres at P and Q. the lines p and g are conjugate PF are conjugate lines lines at right angles to one another. Besides, to the axis as a ray through a focus, and in either pencil will correspond in the other the perpendicular to the therefore perpendicular. axis ($ 72). The conic generated by the intersection of corresponding They are further har. lines in the two pencils is therefore the circle on PQ as diameter, monic conjugates with so that every line in Pis perpendicular to its corresponding line regard to FÅ and FB in l. (84 64 and 13), so that Ad to every point P on an axis of a conic corresponds thus a point they bisect the angles Q. such that conjugate lines through P and Q are perpendicular. formed by these lines. We shall show that these point-pairs PQ form an involution. This by the way To do this let us move P along the axis, and with it the line p, proves keeping the latter parallel to itself. Then P describes a row, pa The segments between perspective pencil of parallels), and the pole of p a projective row. the point of intersection At the same time the line q describes a pencil of parallels perpendicular of two langents to a conic to p, and perspective to the row formed by the pole of P. The point and their points of conQ, therefore, where a cuts the axis, describes a row projective to the lact are seen from a focus row of points P. The two points P and Q describe thus two pro- under equal angles. B: jective rows on the axis; and not only does P as a point in the first If draw row correspond to Q, but also Q as a point in the first corresponds through A and B lines to P. The two rows therefore form an involution. The centre of parallel to TF, then the this involution, it is easily seen, is the centre of the conic. points As, B. where A focus of this involution has the properly that any two conjugate these cut the directrix lines through it are perpendicular; hence, il is a focus to ihe conic. will be harmonic conju Such involution exists on each axis. But only one of these can gates with regard to P have foci, because all foci lie on the same axis. The involution on and the point where FT one of the axes is elliptic, and appears ($ 80) therefore as the section cuts the directrix. The of two circular involutions in two pencils whose centres lie in the lines FT and FP bisect other axis. These centres are foci, hence the one axis contains two therefore also the angles P foci, the other axis.none; or every central conic has two foci which lie between FA, and FB. on one axis equidistant from the centre. From this it follows The axis which contains the foci is called the principal axis; in easily that the triangles case of an hyperbola it is the axis which cuts the curve, because the FAA and FBB, are foci lie within the conic. equiangular, and therefore similar, so that FA : AA. =FB : BB. In case of the parabola there is but one axis. The involution The triangles AA, A, and BB, B, formed by drawing perpendiculars on this axis has its centre at infinity. One focus is therefore at from A and B to the directrix are also similar, so that AA. : AA, infinity, the one focus only is finite. A parabola has only one -BB, : BB2. This, combined with the above proportion, gives focus. FA : AA - FB : BB. Hence the theorem: *85. If through any point P (fig. 34) on a conic the tangent PT The ratio of the distances of any point on a conic from a focus and and the normal PN (T.e. the perpendicular to the tangent through the corresponding directris is constant. the point of contact) be drawn, these will be conjugate lines with To determine this ratio we consider its value for a vertex on the regard to the conic, and at right angles to each other. They will principal axis. In an ellipse the focus lies between the two vertices therefore cut the principal axis in two points, which are conjugate on this axis, hence the focus is nearer to a vertex than to the correin the involution considered in $ 84; hence they are harmonic sponding directrix. Similarly, in an hyperbola a vertex is nearer B we next Fig. 35. to the directrix than to the focus. In a parabola the vertex lies RULED QUADRIC SURFACES halfway between directrix and locus. It follows in an ellipse the ratio between the distance of a point the same plane, in which case lines joining corresponding points $ 89. We have considered hitherto projective rows which lie in from the focus to that from the directrix is less than unity, in the envelop a conic. We shall now consider projective rows whose parabola it equals unity, and in the hyperbola it is greater than bases do not meet. In this case, corresponding points will be joined unity. It is here the same which focus we take, because the two foci like every surface generated by lines is called a ruled surface. This by lines which do not lie in a plane, but on some surface, which lie symmetrical to the axis of the conic. If now P is any point on surface clearly contains the bases of the two rows. the conic having the distances 9 and ra from the foci and the distances d, and do from the corresponding directrices, then ri/d.=ra/dz=e, obtain two axial pencils which are also projective, those planes If the points in either row be joined to the base of the other, we where e is constant. Hence also ad being corresponding which pass through corresponding points in the In the ellipse, which lies between the directrices, dites is constant, the axial pencils passing through them, then AA will be the line therefore also strz.. In the hyperbola on the other hand di-d, is of intersection of the corresponding planes a, a' and also the line constant, equal to the distance between the directrices, therefore joining corresponding points in the rows. in this case 71-7 is constant. If we cut the whole figure by a plane this will cut the axial pencils If we can the distances of a point on a conic from the focus its in two projective fat pencils , and the curve of the second order focal distances we have the theorem: generated by these will be the curve in which the plane cuts the * In an ellipse the sum of the focal distances is constant; and in an surface. Hence hyperbola the difference of the focal distances is constant. The locus of lines joining corresponding points in two projectie This constani sum or difference equals in both cases the length of rows which do not lie in the same plane is a surface which contains the the principal axis. bases of the rows, and which can also be generated by the lines of inter section of corresponding, planes in two projective axial pencils. This PENCIL OF CONICS surface is cut by every plane in a curve of the second order, kence either $ 87. Through four points A, B, C, D in a plane, of which no three in a conic or in a line-pair. No line which does not lie altogether on lie in a line, an infinite number of conics may be drawn, viz. through which is therefore said to be of the second order or is called a ruled the surface can have more than two points in common with the surface, conics is called a pencil of conics. Similarly, all conics touching four quadric surface. fixed lines form a system such that any fifth tangent determines one That no line which does not lie on the surface can cut the surface and only one conic. We have here the theorems: in more than two points is seen at once if a plane be drawn through the line, for this will cut the surface in a conic. It follows also that The pairs of points in which The pairs of tangents which any line is cut by a system of can be drawn from a point together on the surface. a line which contains more than two points of the surface lies altoconics through four fixed points a system of conics touching four are in involution. $ 90. Through any point in space one line can always be drawn four-point, then any line will cut two opposite sides XC, BD in then through every point in either one line may be drawn cutting We prove the first theorem only. Let ABCD (fig. 36) be the cutting two given lines which do not themselves meet. If therefore three lines in space be given of which no two meet, the other two. If a line moves so that it always cuts three given lines of swhick no two meet, then it generates a ruled quadric surface. Let a, b, c be the given lines, and p.g.r... lines cutting them in the points A, A', A'...; B, B', B'...; C, C', C'... respectively; then the planes through a containing P: 9,7, and the planes through b containing the same lines, may be taken as corresponding planes in two axial pencils which are projective, because both pencils cut the line c in the same row, C, C', .:.; the surface can therefore be gener ated by projective axial pencils. F of the lines p, q,?... no two can meet, for otherwise the lines a, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of a. These lines are said to form a set of lines on the surface. If now three of the lines P. 9, 7 be taken, then every lined cutting them will have three points in common with the surface, and will therefore lie altogether on it. This gives rise to a second set of lines on the surface. From what has been said the theorem follows: A ruled quadric surface contains two sets of straight lines. Every Fig. 36. line of one set culs every line of the other, but no two lines of the same the points E, E', the pair AD, BC in points F, F', and any conic Any two lines of the same set may be taken as bases of two projective of the system in M, N, and we have AICD, MN)=B(CD, MN) rows, or of two projective pencils which generate the surface. They are If we cut these pencils by l we get cut by the lines of ihe other set in two projective rows. (EF, MN)=(F'E', MN) The plane at infinity like every other plane cuts the surface either (EF, MN)= (E'F', NM). in a conic proper or in a line-pair. In the first case the surface is But this is, according to $ 77 (7), the condition that M, N are called an Hyperboloid of one shcet, in the second an Hyperbolic Paraboloid. corresponding points in the involution determined by the point pairs E, E', F, F in which the line 1 cuts pairs of opposite sides of the The latter may be generated by a line cutting three lines of which four-point ABCD. This involution is independent of the particular. to a given plane. one lies at infinity, that is, cutting two lines and remaining parallel conic chosen. $ 88. There follow several important theorems: QUADRIC SURFACES Through four points two, one, or no conics may be drawn which touch $91. The conics, the cones of the second order, and the ruled any given line, according as the involution determined by the given quadric surfaces complete the figures which can be generated by four-point on the line has real, coincident or imaginary focí. projective rows or flat and axial pencils, that is, by those aggre Two, one, or no conics may be drawn which louch four given lines gates of elements which are of one dimension (88,5, 6). We shali and pass through a given point, according as the involution determined now consider the simpler figures which are generated by aggregates of by the given four-side at the point has real, coincident or imaginary two dimensions. The space at our disposal will not, however, allow focal rays. us to do more than indicate a few of the results. For the conic through four points which touches a given line has $ 92. We establish a correspondence between the lines and planes its point of contact at a focus of the involution determined by the in pencils in space, or reciprocally between the points and lines in four-point on the line. two or more planes, but consider principally pencils. As a special case we get, by taking the line at infinity: In two pencils we may either make planes correspond to planes Through four points of which none is al infinity either two or no and lines to lines, or else planes to lines and lines to planes. !! 'parabolas may be drawn. hereby the condition be satisfied that to a flat, or axial, pencil The problem of drawing a conic through four points and touching corresponds in the first case a projective flat, or axial, pencil, and in a given line is solved by determining the points of contact on the the second a projective axial, or flat, pencil, the pencils are said to be line, that is, by determining the foci of the involution in which the projective in the first case and reciprocal in the second. line cuts the sides of the four-point. The corresponding remark For instance, two pencils which join two points S, and S, to the holds for the problem of drawing the conics which touch four lines different points and lines in a given plane * are projective (and and pass through a given point. in perspective position), if those lines and planes be takea as M set meet. or corresponding which meet the plane r in the same point or in the In the first case the point of contact is said to be hyperbolic, in the same line. In this case every plane through both centres S, and S, second parabolic, in the third elliptic. of the two pencils will correspond to itself. If these pencils are $ 95. It remains to be proved that every point S on the surface brought into any other position they will be projective (but not may be taken as centre of one of the pencils which generate the perspective). surlace. Let S. be any point on the surface generated by the The correspondence between two projective pencils is uniquely reciprocal pencils S, and Sa. We have to establish a reciprocal determined, if to four rays (or planes) in the one the corresponding correspondence between the pencils S and Si, so that the surface rays (or planes) in the olher are given, provided that no three rays of generated by them is identical with R. To do this we draw two either sel lie in a plane. planes a. and ßi through Sı, cutting the surface o in two conics Let a, b, c, d be four rays in the one, a', 6', ', d' the corresponding which we also denote by a, and Bi. These conics meet at Si, and rays in the other pencil. We shall show that we can find for every at some other point T where the line of intersection of as and Bu ray e in the first a single corresponding ray e' in the second. To cuts the surface. the axial pencil a (b, c, d ... ) formed by the planes which join a to In the pencil S we draw some plane o which passes through T, b,c,d ..., respectively corresponds the axial pencil a' (6',c', d'.::), but not through S, or Sz. It will cut the two conics first at T, and and this correspondence is determined. Hence, the plane a' e' which therefore each at some other point which we call A and B respeccorresponds to the plane de is determined. Similarly, the plane tively. These we join to S by lines a and b, and now establish the b'e' may be found and both together determine the ray e'. required correspondence between the pencils S, and S as follows: Similarly the correspondence between two reciprocal pencils is To S.T shall correspond the plane o, to the plane as the line a, and determined if for four rays in the one the corresponding planes in to Be the line b, hence to the flat pencil in a, the axial pencil a. the other are given. These pencils are made projective by.aid of the conic in ai. $ 93. We may now combine In the same manner the fat pencil in Bi is made projective to the 1. Two reciprocal pencils. axial pencil b by aid of the conic in B1, corresponding elements being Each ray cuts its corresponding plane in a point, the locus those which meet on the conic. This determines the correspondence, of these points is a quadric surface. for we know for more than four rays in S, the corresponding planes 2. Two projective pencils. in S. The two pencils S and S, thus made reciprocal generate a Each plane cuts its corresponding plane in a line, but a quadric surface d', which passes through the point S and through ray as a rule does not cut its corresponding ray. The the two conics Qy and Bi. we draw a plane through S and Sz, cutting each of the conics a, and 3. Three projective pencils. Bi in two points, which will always be possible. This plane cuts The locus of intersection of corresponding planes is a and $' in two conics which have the point S and the points where cubic surface. it cuts aj and B, in common, that is five points in all.' The conics Of these we consider only the first two cases. therefore coincide. $94. If two pencils are reciprocal, then to a plane in either corre- This proves that all those points P on o' lie on which have the sponds a line in the other, to a flat pencil an axial pencil, and so on. property that the plane SSP cuts the conics a., B, in two points Every line cuts its corresponding plane in a point. If S, and S, be cach. If the plane SS,P has not this property, then we draw a plane the centres of the two pencils, and P be a point where a line ay in the SS,P. This cuts each surface in a conic, and these conics have in first cuts its corresponding plane as, then the line ba in the pencil S. common the points S, S., one point on each of the conics a., B., and which passes through P will meet ils corresponding plane B, in P. For one point on one of the conics through S and S, which lic on both 6, is a line in the plane ag. The corresponding plane B, must therefore surfaces, hence five points. They are therefore coincident, and our pass through the line 01, hence through P. theorem is proved. The points in which the lines in S, cut the planes corresponding $96. The following propositions follow:to them in Sy are therefore the same as the points in which the lines A quadric surface has at every point a tangent plane. in S, cut the planes corresponding to them in S. Every plane section of a quadric surface is a conic or a line pair The locus of these points is a surface which is cut by a plane in a Every line which has thrce points in common with a quadric surface conic or in a line-pair and by a line in not more than two points unless lies on the surface. il lies allogether on the surface. The surface ilself is therefore called a Every conic which has five points in common w a quadric surface quadric surface, or a surface of the second order. lies on the surface. To prove this we consider any line p in space. Through two conics which lie in different planes, but have two points The fat pencil in S, which lies in the plane drawn through pin common, and through one external point always one quadric surface and the corresponding axial pencil in Sy determine on p two pro may be drawn. jective rows, and those points in these which coincide with their 697. Every plane which culs a quadric surface in a line-pair is a corresponding points lie on the surface. But there exist only two, tangent plane. For every line in this plane through the centre of or one, or no such points, unless every point coincides with its the line-pair (the point of intersection of the two lines) cuts the corresponding point. In the latter case the line lies altogether on surface in two coincident points and is therefore a tangent to the the surface. surface, the centre of the line-pair being the point of contact. This proves also that a plane cuts the surface in a curve of the If a quadric surface contains a line, then every plane through this second order, as no line can have more than two points in common line cuts the surface in a line-pair (or in two coincident lines). For with it. To show that this is a curve of the same kind as those this plane cannot cut the surface in a conic. Hence considered before, we have to show that it can be generated by If a quadric surface contains one line p then it contains an infinite projective fat pencils. We prove first that this is true for any number of lines, and through every point Q on the surface, one line plane through the centre of one of the pencils, and afterwards that can be drawn which cuts p.. For the plane through the point Q every point on the surface may be taken as the centre of such pencil and the line p cuts the surface in a linc-pair which must pass through Let then a. be a plane through S. To the flat pencil in s, which Q and of which,p is one line. it contains corresponds in S, a projective axial pencil with axis No two such lines q on the surface can meet. For as both mect p and this cuts a in a second flat pencil.. These two flat pencils their plane would contain p and therefore cut the surface in a in an are projective, and, in general, neither concentric nor per- triangle. spective. They generate therefore a conic. But if the line az passes Every line which cuts three lines q will be on the surface; for it through Ş the pencils will have S, as common centre, and may has three points in common with it. therefore have two, or one, or no lincs united with their corresponding llence the quadric surfaces which contain lines are the same as the lines. The section of the surface by the plane a, will be accordingly ruled quadric surfaces considered in $8 89-93, but with one important a line-pair or a single line, or else the plane as will have only the exception. In the last investigation we have left out of considerapoint S, in common with the surface. tion the possibility of a plane having only one line scwo coincident Every line h through S cuts the suriace in two points, viz. first lines) in common with a quadric surface. in S, and then at the point where it cuts its corresponding plane. $ 98. To investigate this case we suppose first that there is one If now the corresponding plane passes through S., as in the case point A on the surface through which two different lines a, b can be just considered, then the two points where li cuts the surface coincide drawn, which lie altogether on the surface. at Si, and the line is called a langent to the surface with S, as point If P is any other point on the surface which lies neither on a nor of contact. Hence if b, be a tangent, it lies in that planer which , then the plane through P and a will cut the surface in a second corresponds to the line SS, as a line in the pencil S. The section line a' which passes through P and which cuts a. Similarly there of this plane has just been considered. It follows that is a line b' through P which cuts b. These two lines a' and ' may All langents to quadric surface al the centre of one of the reciprocal coincide, but then they must coincide with PA. pencils lie in a plane which is called the tangent plane to the surface If this happens for one point P, it happens for every other point at that point as point of contact. Q. For if two different lines could be drawn through O, then by the To the line joining the centres of the two pencils as a line in one same reasoning the line PQ would be altogether on the surface, corresponds in the other the langent plane at its centre. hence two lines would be drawn through P against the assumption. The langent plane to a quatric surface either cuts the surface in From this follows:two lines, or it has only a single line, or else only a single point in If there is one point on a quadric surface through which one, bul only common with the surface. one, line can be drawn on the surfur, then through every point one line seen. can be drawn, and all these lines meel in a point. The surface is a cone projective pencils meet form a congruence. We shall see this con. of the second order. gruence consists of all lines which cut a twisted cubic twice, or of If through one point on a quadric surface, two, and only two, lines all secants to a twisted cubic. can be drawn on the surface, then through every point two lines may § 102. Let l be the line Sise as a line in the pencil Sa. To it be drawn, and the surface is a ruled quadric surface. corresponds a line l in S... Al cach of the centres iwo corresponding If through one point on a quadric Surface no line on the surface can lines meet. The two axial pencils with ly and ly as axes are probe drawn, then the surface contains no lines. jective, and, as their axes meet at Sa, the intersections of correUsing the definitions at the end of $ 95, we may also say: sponding planes form a cone of the second order ($58), with Sq as On a quadric surface the points are all hyperbolic, or all parabolic, centre. If and 72 be corresponding planes, then their intersection or all elliptic. will be a line pe which passes through Sp. Corresponding to it in As an example of a quadric surface with elliptical points, we s, will be a line Pi which lies in the plane , and which therefore mention the sphere which may be generated by two reciprocal meets pz at some point P. Conversely, if y be any line in S, which pencils, where to each line in one corresponds the plane perpendicular meets its corresponding line P. at a point P, then to the plane ho to it in the other. will correspond the plane lipe, that is, the plane S, Sp. These $99. Poles and Polar Planes.—The theory of poles and polars planes intersect in P2, so that p. is a line on the quadric cone generated with regard to a conic is easily extended to quadric surfaces. by the axial pencils l, and la Hence: Let P be a point in space not on the surface, which we suppose All lines in one pencil which meet their corresponding lines in the not to be a cone. On every line through P which cuts the surface other form a cone of the second order which has its centre at the coutre in two points we determine the harmonic conjugate Q of P with of the first pencil, and passes through the centre of the second. regard to the points of intersection. Through one of these lines we From this follows that the points in which corresponding rays draw two planes a and B. The locus of the points Q in a is a line a, meet lie on two cones of the second order which have the ray joining the polar of P with regard to the conic in which a cuts the surface. their centres in common, and form therefore, together with the line Similarly the locus of points Q in B is a line 6. This cuts a, because S.S, or hi, the intersection of these cones. Any plane cuts each of the the line of intersection of a and B contains but one point Q. The cones in a conic. These two conics have necessarily that point in locus of all points Q therefore is a plane. This plane is called the common in which it cuts the line h, and therefore besides either polar plane of the point P, with regard to the quadric surface. If P one or three other points. It follows that the curve is of the tkird lies on the surface we take the tangent plane of Pas ils polar. order as a plane may cut it in three, but not in more than three, The following propositions hold: points. Hence:1. Every point has a polar plane, which is constructed by drawing The locus of points in which corresponding lines on two projectie the polars of the point with regard to the conics in which two planes pencils meet is a curve of the third order or a twisted cubic "k, kick through the point cut the surface. passes through the centres of the pencils, and which appears as the 2. if Q is a point in the polar of P, then P is o point in the polar intersection of two cones of the second order, which have one line is of Q, because this is true with regard to the conic in which a plane common. through PQ cuts the surface. A line belonging to the congruence determined by the pencils is a 3. Every plane is the polar plane of one point, which is called the secant of the cubic; it has two, or one, or no points in common with Pole of the plane. this cubic, and is called accordingly a secant proper, a langent, or $ The pole to a plane is found by constructing the polar planes of secant improper of the cubic. A secant improper may be considered, three points in the plane. Their intersection will be the pole. to use the language of coordinate geometry, as a secant with 4. The points in which the polar plane of P culs the surface are imaginary points of intersection. points of contact of langents drawn from P to the surface, as is easily $ 103. If a, and az be any two corresponding lines in the two Hence : pencils, then corresponding planes in the axial pencils having and 5. The langents drawn from a point P to a quadric surface form a a, as axes generate a ruled quadric surface. if P be any point on cone of the second order, for the polar plane of P cuts it in a coníc. the cubick, and if Pu, Pz be the corresponding rays in S, and S, which 6. If the pole describes a line a, its polar plane will turn about meet at P, then to the plane di Pi in S, corresponds as Do in Sy. These another line a', as follows from 2. These lines a and a' are said to be therefore meet in a line through P. conjugate with regard to the surface. This may be stated thus:100. The pole of the line at infinity is called the centre of the Those secants of the cubic which cut a ray an, drauen througktike surface. If it lies at the infinity, the plane at infinity is a tangent centre S, of one pencil, form a ruled quadric surface which passes ikrough plane, and the surface is called a paraboloid. both centres, and which contains the twisted cubic k. Of such surfaces The polar plane to any point oi infinity passes through the centre, an infinite number exists. Every ray through S, or S, which is net e and is called a diametrical plane. secant determines one of them. A line through the centre is called a diameter. It is bisected at the If, however, the rays a, and az are secants meeting at A, then the centre, The line conjugate to it lies al infinily. ruled quadric surface becomes a cone of the second order, having If a point moves along a diameter its polar plane turns about the A as centre., Or all lines of the congquence which pass through a persi conjugaie line at infinity; that is, il moves parallel to itself, ils centre on the twisted cubic k form a cone of the second order. In other words, moving on the first line. the projection of a twisted cubic from any point in the curve on to The middle points of parallel chords lie in a plane, viz. in the polar any plane is a conic. plane of the point at infinity through which the chords are drawn. It a, is not a secant, but made to pass through any point Q in The centres of parallel sections lie in a diameter which is a line space, the ruled quadric surface determined by d, will pass through conjugale to the line al infinity in which the planes meet. 0. There will therefore be one line of the congruence passing through O, and only one. For if two such lines pass through , then the lines TWISTED CUBICS SQ and SiQ will be corresponding lines; hence will be a point on $ 101. If two pencils with centres S, and S, are made projective, the cubic k, and an infinite number of secants will pass through it. then to a ray in one corresponds a ray in the other, to a plane a Hence : plane, to a flat or axial pencil a projective flat or axial pencil, and Through every point in space nol on the twisted cubic one and only one secant to the cubic can be drawn. There is a double infinite number of lines in a pencil. We shall $ 104. The fact that all the secants through a point on the cubic see that a single infinite number of lines in one pencil meets its form a quadric cone shows that the centres of the projective pencils corresponding ray, and that the points of intersection form a curve generating the cubic are not distinguished from any other points on in space. the cubic. If we take any two points S, S' on the cubic, and draw of the double infinite number of planes in the pencils each will the secants through each of them, we obtain two quadric cones, meet its corresponding plane. This gives a system of a double which have the line SS' in common, and which intersect besides infinite number of lines in space. We know (Ś 5) that there is a along the cubic. If we make these two pencils having S and quadruple infinite number of lines in space. From among these we centres projective by taking four rays on the one cone as cortemay select those which satisfy one or more given conditions. The sponding to the four rays on the other which meet the first on the systems of lines thus obtained were first systematically investigated cubic, the correspondence is determined. These two pencils will and classified by Plücker, in his Geometrie des Raumes. He uses the generate a cubic, and the two cones of secants having $ and S'as following names:- centres will be identical with the above cones, for each has five A treble infinite number of lines, that is, all lines which satisfy one rays in common with one of the first, viz. the line SS' and the four condition, are said to form a complex of lines; e.g. all lines cutting lines determined for the correspondence; therefore these two cones a given line, or all lines touching a surface. intersect in the original cubic. This gives the theorem. A double infinite number of lines, that is, all lines which satisfy On a twisted cubic any two points may be taken as centres of protwo conditions, or which are common to two complexes, are said to jective pencils which generale the cubic, corresponding planes being form a congruence of lines; e.g. all lines in a plane, or all lines Those which meet on the same secani. cutting two curves, or all lines cutting a given curve twice. of the two projective pencils at S and S' we may keep the first A single infinile number of lines, that is, all lines which satisfy, fixed, and move the centre of the other along the curve. The pencils three conditions, or which belong to three complexes, form a ruled will hereby remain projective, and a plane a in S will be cut by its surface; e.g. one set of lines on a ruled quadric surface, or develop corresponding plane a' always in the same secant a. Whilst S' able surfaces which are formed by the tangents to a curve. moves along the curve the plane a' will turn about e, describing an It follows that all lines in which corresponding planes in two axial pencil. So on. B..B 1 IV RD FIG. 39. AUTHORITIES.-In this article we have given a purely geometrical Conversely any two points As, Az in a line perpendicular to the theory of conics, cones of the second order, quadric surfaces, &c. In axis will be the projections of some point in space when the plane doing so we have followed, to a great extent, Reye's Geometrie der 72 is turned about the axis till it is perpendicular to the plane me Lage, and to this excellent work those readers are referred who wish because in this position the two perpendiculars to the planes for a more exhaustive treatment of the subject. Other works and a through the points A, and Aj will be in a plane and therefore especially valuable as showing the development of the subject are: meet at some point A.' Monge, Géométrie descriptive: Carnot, Géométrie de position Representation of Points.-We have thus the following method (1803), containing a theory of transversals; Poncelet's great work of representing in a single plane the position of points in space Trailé des propriétés projectives des figures (1822); Möbins, Bary- we take in the plane a line xy as the axis, and then any pair of points centrischer Calcul (1826); Steiner, Abhängigkeit geometrischer A., Az in the plane on a line perpendicular to the axis represent a Gestallen (1832), containing the first full discussion of the projective point A in space. If the line AA, cuts the axis at Ao, and if at A. relations between rows, pencils, &c.; Von Staudt, Geometrie der a perpendicular be erected to the plane, then the point A will be in Lage. (1847) and Beiträge zur Geomelrie der Lage (1856-1860), in it at a height AjA=AA, above the plane. This gives the position which a system of geometry is built up from the beginning without of the point A relative to the plane #. In the same way, if in a any reference to number, so that ultimately a number itself gets perpendicular to ng through A, a point A be taken such that AgA = a geometrical definition, and in which imaginary, elements are A Al, then this will give the point A relative to the plane 2. systematically introduced into pure geometry; Chasles, A perçu $ 2. The two planes #1, #2 in their original position divide space historique (1837), in which the author gives a brilliant account of into four parts. These are called the four quadrants. We suppose the progress of modern geometrical methods, pointing out the that the plane w is turned as indicated in advantages of the different purely geometrical methods as compared fig. 37, so that the point P.comes to Q and with the analytical ones, but without taking as much account of R to S, then the quadrant in which the the German as of the French authors; Id., Rapport sur les progrès point A lies is called the first, and we say de la géométrie (1870), a continuation of the Aperçu; Id., Trailé de that in the first quadrant a point lies above géométrie supérieure (1852); Cremona, Introduzione ad una teoria the horizontal and in front of the vertical 11 geometrica delle curve piane (1862) and its continuation Preliminari plane. Now we go round the axis in the C, B. di una teoria geometrica delle superficie (German translations by sense in which the plane r is turned and o Curtze). As more elementary books, we mention: Cremona, come in succession to the second, third Elements of Projective Geometry, translated from the Italian by and fourth quadrant. In the second a C. Leudesdorf (2nd ed., 1894); J. W. Russell, Pure Geomelry (2nd ed., point lies above the plane of the plan and 1905). (O. H.) behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its proThis branch of geometry is concerned with the methods for jection is taken. Fig: 38 shows how these are represented when representing solids and other figures in three dimensions by the plane ra is turned down. We see that drawings in one plane. The most important method is that above the axis; 'in the second if plan and elevation both lie above; in A point lies in the first quadrant if the plan lies below, the elevation which was invented by Monge towards the end of the 18th the third if the plan lies above, the elevation below; in the fourth if plan century. It is based on parallel projections to a plane by rays and elevation both lie below the axis. perpendicular to the plane. Such a projection is called ortho- If a point lies in the horizontal plane, its elevation lies in the axis graphic (see PROJECTION, $ 18). If the plane is horizontal the and the plan coincides with the point itself. If a point lies in the projection is called the plan of the figure, and if the plane is with the point itself. If a point lies in the axis, both its plan and vertical plane, its plan lies in the axis and the elevation coincides vertical the elevation. In Monge's method a figure is represented elevation lie in the axis and coincide with it. by its plan and elevation. It is therefore often called drawing of each of these propositions, which will easily be seen to be true, in plan and elevation, and sometimes simply orthographic the converse holds also. projection. 63. Representation of a Plane. -As we are thus enabled to represent $ 1. We suppose then that we have two planes, one horizontal, points in a plane, we can represent any finite figure by representing the other vertical, and these we call the planes of plan and of eleva; in this way, for the projections of its points completely cover the its separate points. It is, however, not possible to represent a plane tion respectively, or the horizontal and the vertical plane, and planes - and is, and no plane would appear different from any other. denote them by the letters and 72. Their line of intersection is But called the axis, and will be denoted by ry. any plane a cuts each of the planes #1, #, in a line. These are If the surface of the drawing, paper is taken as the plane of the point where the latter cuts the plane a. called the traces of the plane. They cut each other in the axis at the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing on the axis, and, conversely, any two lines which meet on the axis A plane is determined by its two traces, which are two lines that meet paper we turn it about the axis till it coincides with the horizontal determine a plane. plane. This process of turning one plane down till it coincides with If the plane is parallel to the axis its traces are parallel to the axis. another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when planes of projection at infinity and will be parallel to it. Thus a of these one may be at infinity; then the plane will cut one of the The whole arrangement will be better understood by referring to plane parallel to the horizontal plane of the plan has only one finite fig. 37. A point A in space is there projected by the perpendicular trace, viz, that with the plane ol elevation. If the plane passes through the axis both ils traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by me As it is perpendicular to *, it may be taken as the plane of elevation, its line of intersection y with a being the axis, and be turned down to coincide with n. This is represented in fig. 40. OC is the axis 'wy whilst OX and OB are the traces of the third plane. They lie in one line y. The plane is rabatted about to the hori. FIG. 38. zontal plane. A plane a through the axis xy will then show in it AA, and AA, to the planes m and az so that A, and Az are the a trace as. In fig. 40 the lines OC , horizontal and vertical projections of A. and OP will thus be the traces If we remember that a line is perpendicular to a plane that is of a plane through the axis xy, perpendicular to every line in the plane is only it is perpendicular which makes an angle POQ with to any two intersecting lines in the plane, we see that the axis which the horizontal plane. is perpendicular both to AA, and to AA, is also perpendicular to We can also find the trace A, A. and to A,A, because these four lines are all in the same plane. which any other plane makes Hence, if the plane a, be turned about the axis till it coincides with with 13. In rabatting the plane .... the plane a, then A2A, will be the continuation of A, A. This to its trace OB with the plane 17 will come to the position OD. position of the planes is represented in fig. 38, in which the line AA: Hence a plane B having the traces CA and CB will have with the is perpendicular to the axis x. third plane the trace Bs, or AD if OD=OB D FIG. 37 FIG. 40. |