406 To prove this we consider any line p in space. 'l‘lie flat pencil in S, which lies in the plane drawn through p and the corresponding axial pencil in S._. dcteriniiie on p two )l'0- jective rows, and those points in these which coincide with their corresponding points lie on the surface. llut there cxist only two, or one, or no such points, unless every point coincides with its corresponding point. In the latter case the line lies altogether on the siirface. This proves also that a plane cuts the surface in a curve of the second order, as no line can have more than two points in common with it. To show that this is a curve of the same kind as those considered before, we have to show that it can be gencratcd by projective flat pencils. We prove first that this is tiiie for any plane through the centre of one of the pencils, and afterwards that every point on the surface may be taken as the centre of such pencil. Let then a, be a plane through S,. To the flat pencil in S, which it contains correspomls in S, a projective axial pencil with axis 41,, and this cuts a, in a second flat pencil. These two flat pencils in a, are projective, and, in general, neither concentric nor per- spective. They generate ll1(‘l‘(.'f0l‘0 a conic. But if the line a._. passes through S, the pencils will have S. as common ccntrc, and may therefore have two, or one, or no lincs united with their correspond- ing lines. The section of the surface by the plane a, will be accordingly a line-pair or a single line, or else the plane a, will have only the point S, in common with the surface. Every line I, through S, cuts the surface in two points, viz., first in S, and then at the point where it cuts its corresponding plane. If now the corresponding plane passes tlirough S,, as in the case just considered, then the two points where I, cuts the surface coin- cide at S,, and the line is called a tangent to the surface with S, as point of conta'l. Hence if I, be a tangent, it lies in that plane -2-, which corresponds to the line S28, as a line in the pencil S2. The section of this plane has just been considered. It follows that- All tangents to guazlric szr.2_-face at the can/rc of one of the rcciprocal pencils lie in a plane zchich is called the tangcnt plane to the smfacc at that point as point of contact. To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its ccntrc. The tangent plane to aqzaulric surface (it/zcr cuts the surface in two lines, or it has only a single linr, or also only a single point in common with the surface. In thcjirst case the point of contact is said to be hyperbolic, in the srcond parabolic, in the third cllip/ir. § 95. It remains to be proved that crery point S on the surface ni.i_~ be taken as centre of one of the pencils which generate the szirface. Let S be any point on the surface <1)’ generated by the reciprocal pencils S, and .S._.. 'e have to establish a I‘eeipi'0cal iorrespondcnce between the pciicils S and S,, so that the surface generated by them is identical with (D. To do this we draw two planes a, and 3, tliiough 3,, cutting the surface ‘I1 in two conics which we also denote by a, and 3,. These conics meet at S,, and at some other point T where the line of intersection of a, and 3, cats the surface. In the pencil S we diaw some plane 0' which passes through T, but not through S, or S3. It will cut the two conics first at '1‘, and therefoie each at some other point which we call A and ll re- spectively. These we join to S by lines a and b, and now establish the required correspondence between the '[lCll('.llS S, and S as follows:—— To S,'l' shall correspond the plane a’, to the plane a, the line 0, and to 3, the line b, hcnce to the flat pencil in a, the axial pencil a. These pencils are made projective by aid of the conic in a,. In the same manner the ll-ll,]l|‘l1L'll in 3, is made projective to the axial p.-ncil b by aid of the conic in 3,, corresponding elements being those which meet on the come. This deterniincs the corrcspoiidciicc, for we know for more than four rays in S, the corresponding planes in S. The two pencils S and S, thus made reciprocal generate a (jll.'1lll'l(‘, surface ‘D’, which passes through the point S and tliioiigli the two conics a, and 3,. The two surfaces <11 and ti)’ have therefore the points S and S, and the conics a, and 3, in coniinon. 'l‘o show that they are idcntical, we draw a plane through S and 5,, cutting each of the conics a, and 3, in two points, which will always be possible. This plane cuts fl) and <1)’ in two conics which have the point S and the points where it cuts a, and 3, in coinnion, that is five points in all. The conics therefore coincide. This proves that all those points P on <1)’ lie on <1) which have th: property that the plane S9,!’ cuts the conics a,, 3, in two points each. If the plane S.'__,1" has not this property, then we diaw a plane SS,P. This cuts each surface in a conic, and these conics have in comnion the points 8, 3,, one point on each of the coiiics a,, 3,, and one point on one of the coni.-s through S and S., which lie on both siii'facr's, hence five points. They are tlicrcfore coincident, and our llI('Ol'('lll is proved. § 96. The following piopositions follow 2-- ,l quad-ric .5-zu:f(n,'c h.a,s at rrrry point a tangrnl plane. 1;'rcry plnnc sr-ction of a I1!/'tl//‘II’ .s-mfm-c is a conic or a I inc-1 air. GEOM ETRY Ercry linc 'u-hich has three points in common with a qumlric siu_'/‘ace lies on the suifacc. Ercry conic zrhich has fire points in common with a quadric surfuic tics on the su7_'/‘ace. Through t-11:0 conics zrhich lie in (li_t/"cr_rntpluncs, but lmrc t1copuin.’>‘ i n common, and lhro ugh. one c.rtcrnat point always one qumlric .5-urfaac may be drawn. § 97. Ercry plane 'u'hich culsa. qumlric surface in a line pair is «I tangent plane. For every line in this plane through the centre of the line-pair(the point of intersection of the two lines) cuts the surface in two coincident points and is tlicrcfore a tangent to the surface, the centre of the I inc-pair being the point (3/‘contact. If a qunulric surface contains a. linr, lhcn crcry plane through this line cuts the surface in a line-pair (ur in tire chi"/zciiluit linv. 5). For this plane cannot cut the surface in a conic. llcnce If a quatlric surface contains one line p t/u-n it contains an 'in_/initc number of lines, and through. rrcry point Q on I/at siII_'/7'0‘, one line q can be a'rau'n 'u-hich cuts p. For the plane llirough thu- point Q and the line 12 cuts the surface in a linc-p.iir which niust pass throiigh Q and of which p is one line. No two such lines (1 on the surface can mcct. For as both meet p their plane would contain p and tlicrcfore cut the suifacc in .i tiianglc. 1-.'rcry line which cuts thrcc liars q 'u'ill be on the .su7_'facc; for it has three points in coinnion with it. Ilcncc the quatlric su7_'/‘aces 'u'hich contain lines arc the same as thc ruled quatlric surfaces considcrccl in §§ 89-93, but with one ini- portant exception. consideration the possibility of a plane having only one line (two coincident lines) in eoninion with a quadric SIIITIICC. § 98. To investigate this case we suppose first that there is one point A on the surface through which two diffciciit lines a, b can la- drawn, which lie altogether on the surface. If 1’ is any other point on the surface which lies neither on (I nor b, then the plane tlirough 1’ and a will cut the suifacc in a second line a’ which passes through 1’ and which cuts a. .5'iiiiilarl_' tlicic is a line b’ through l’ which cuts b. 'I‘la-sc two lines a’ and b’ may coincide, but then they must coincide with l‘-. Ifthis happens for one point P, it happens for (('l'_' other point 0. For if two dilferciit lines could be drawn tlirough Q, then by the saiiic reasoning the line PQ would be altogctlicr on the siiifau-c, hence two lines would be drawn through 1’ against the assiiniptioii. From this follows :5- 1_'/‘thcrc is one point on a quralric sin;/‘ace llzrough zrliiclz onc, l-ut only one, line can bc (Iran-n. on the siuyiicc, thcn tliroilglz cz-cry pmnl one line can be (lrazcn, and all these lines meet in a point. .'['lu' surface is a cone of the second 07'(lt')'. If through one point on a quallric surfacc, two, and onlylu‘o, Ii/us can bc tlrazcn on the surface, t/zen. through. crrry point two lines may be (lraicn, and the surface is a ruled qumlric su:_'/‘n(‘c. If through. one point on a quail/-ic surface 'no line on the sin;/"ac can be drawn, (hen the surface contains no luics. Using the definitions at the end of 3' 95, we may also say :— Un a quatlric surface the points are all hyperbolic, oral l parabolic, or all elliptic. As an example of a quadrie surface with clliptical points, we mention the sphere which may be gciieratcd by two i-eeipi-or-al pencils, where to each line in one corresponds the plane 1-cr- pendieular to it in the other. §99. Poles and Polar I’lancs.—The theory of poles and polars with regai'd to a conic is easily cxtciidcd to (lll.’lill‘l-'2 siirfav-cs. Let 1’ be a point in space not on the siirhicc, which '(- suppose not to he a cone. On every line tlirough 1’ which cuts the siii-face in two points we determine the harinoniv coiijugatc Q of 1‘ with regard to the points of iiiteiscctioii. 'l'lii'ough one of these liiics we draw two planes a and 3. The locus of the points Q in a is a line a, the polar of P with iega1'd to the conic in which a cuts the surface. Similarly the locus of points Q in [3 is a line I». This cuts (1, because the line of intersection of a and 3 contains but one point Q. The locus of all points Q tlicrcfore is a plane. This plane is C(tll"tl the polar plane of the point l’, zrith rrganl lo the quadric sm;/‘ace. If 1’ lies o-n the surface 'u'r talcr the l((lI_t[I1l/ plant: of l’ as its polar. The. following propositions hold :— 1. I;'z'u'y point has a polar plane, which is constructed by draw- ing the polars of the point with regard to the conics in which two planes tlirough the point cut the siii-face. 2. If Q is a point in the polar of l’, then I’ is (I paint in thcpular of (3, because this is true with regard to the conic in which a plane through l'Q cuts the surface. 3. Every plane is the polar plane of one point, 11'/4 [ch is Cllllftl the pull’ of the plane. The pole to a plane is found by constructiiig the polar planes of three points in the plane. Their iiitciscctioii will be the pole. 4. The points in 'which the polar plane 43/‘l’ cuts the .su/_'/ircc are points of contact of tangrnls ¢lraa'n_]ro:n l’ to the surface, as is easily seen. lb-iicc :- [l’ROJECTl'E.
In the last investigation we have left out of