caken at random in the plane will, as a rule, not satisfy the equation, 1 CRP tells us that, with the signs appropriate to their directions but infinitely many points, and in most cases infinitely many real attached to CR and RP, ones, have coordinates which do satisfy it, and these points are RP=CR tan a, i.e. MP-DC=(OM-OD) tan e, exactly those which lie upon some locus of one dimension, a straight and this gives that line or more frequently a curve, which is said to be represented by the cquation. Take, for instance, the equation y=mx, where m y-k=tan a (x-h), is a given constant. It is satisfied by the coordinates of every point an equation of the first degree satisfied by x and y. No point not P, which is such that, in fig. 48, the distance MP, with its proper sign, on the line satisfies the same equation; for the line from C to any is m times the distance OM, with its proper sign, i.e. by the co- point off the line would make with CR some angle B different from a ordinates of every point in the straight line through which we and the point in question would satisfy an equation y-k=tan B(x –h). arrive at by making a line, originally coincident with x'Ox, revolve which is inconsistent with the above equation. about O in the direction opposite to that of the hands of a watch The equation of the line may also be written y=mx+b, where through an angle of which m is the tangent, and by those of no other m=tan a, and b=k-h tan a. Here b is the value obtained for y points. That line is the locus whic it represents. Take, more from the equation when o is put for x, i.e. it is the numerical measure, generally, the equation y=(x), where $(x) is any given non-ambigu- with proper sign, of OB, the intercept made by the line on the axis ous function of x. Choosing any point M on x’Ox in fig. 1, and of y, measured from the origin. For different straight lines, m and b giving to x the value of the numerical measure of OM, the equation may have any constant values we like. determines a single corresponding y, and so determines a single Now the general equation of the first degree Ax+By+C=o may point P on the line through M parallel to y'Oy:. This is one point be written y=-ộr-8, unless Bro, in which case it represents a extreme left to the extreme right of the line x'Ox, regarded as line parallel to the axis of y; and - A/B, -C/B are values which extended both ways as far as we like, i.e. let x take all real values can be given to m and b, so that every equation of the first degree from – o to . With every value goes a point P, as above, on represents a straight line. It is important to notice that the general the parallel to y'Oy through the corresponding M; and we thus find equation, which in appearance contains three constants A, B, C, in that there is a path from the extreme left to the extreme right of effect depends on two only, the ratios of two of them to the third. the figure, all points P along which are distinguished from other In virtue of this last remark, we see that two distinct conditions points by the exceptional property of satisfying the equation by suffice to determine a straight line. For instance, it is easy from the their coordinates. This path is a locus; and the equation y=$(x) above to see that represents it. More generally still, take an equation /(x, y)=0 which involves both x and y under a functional form. Any particular +=1 value given to x in it produces from it an equation for the determination of a value or values of y, which go with that value of x in specify is the equation of a straight line determined by the two conditions ing a point or points («, »), of which the coordinates satisfy the that it makes intercepts DA, OB on the iwo axes, of which a and 6 equation f(x, y)=0. Here again, as x takes all values, the point or are the numerical measures with proper signs. note that in fig. 50 a points describe a path or paths, which constitute a locus represented is negative. Again, by the equation. 'Except when y enters to the first degree only in 9-»-*=(*–*). f(x, y), it is not to be expected that all the values of y, determined as going with a chosen value of x, will be necessarily real; indeed i.e. it is not uncommon for all to be imaginary for some ranges of values (vi - yu)x -(x1 - x2)y+xiy-xayı = 0, of x. The locus may largely consist of continua of imaginary represents the line determined by the data that it passes through points; but the real parts of it constitute a rea! curve or real curves. two given points (X., Yı) and (x, y). To prove this find m in the Note that we have to allow x to admit of all imaginary, as well as cquation y-y1=m(x-x) of a line through (x, y), from the conof all real, values, in order to obtain all imaginary parts of the dition that (x2, Yz) lies on the line. locus. In this paragraph the coordinates have been assumed rectangular. A locus or curve may be algebraically specified in another way; Had they been oblique, the doctrine of similar triangles would have viz. we may be given two equations x =j(0), y=F(0), which express given the same results, except that in the forms of equation y-k= the coordinates of any point of it as two functions of the same m(x - h), y=mx+b, we should not have had m= tan a. variable parameter 0 to which all values are open. As 8 takes all 9. The Circle. - It is easy to write down the equation of a given values in turn, the point (x, y) traverses the curve. circle. Let (k, k) be its given centre C, and p the numerical measure It is a good exercise to trace a number of curves, taken as defined of its given radius. Take P (x, y) any point on its circumference, by the equations which represent them. This, in simple cases, can and construct the triangle CRP, in fig. 50 as above. The fact that be done approximately by plotting the values of y given by the this is right-angled tells us that equation of a curve as going with a considerable number of values CR?+RP-CP, os x, and connecting the various points (x, y) thus obtained. But and this at once gives the equation methods exist for diminishing, the labour of this tentative process. (x-h)? +(-k): = p?. Another problem, which will be more attended to here, is that of A point not upon the circumference of the particular circle is at some determining the equations of curves of known interest, taken as distance from (h, k) different from p, and satisfies an equation defined by geometrical properties. It is not a matter for surprise inconsistent with this one; which accordingly represents the cirthat the curves which have been most and longest studied geometrically are among those represented by equations of the simplest cumference, or, as we say, the circle. The equation is of the form character. 8. The Straight Line.-This is the simplest type of locus. Also x2 + y2 +2Ax+2By+C =0 the simplest type of equation in x and y'is Ar+By+C =0, one of Conversely every equation of this form represents a circle: we have the first degree. Here the coefficients A, B, C are constants. They only to take -A, -B, A++B2-C for h, k, på respectively, to obtain are, like the current coordinates, x, y, numerical. But, in giving its centre and radius. But this statement must appear too un. interpretation to such an equation, we must of course refer to restricte Ought we not to require A*+BP-C to be positive? numbers Ax, By, C of unit magnitudes of the same kind, of units Certainly, if by circle we are only to mean the visible round cirof counting for instance, or unit lengths or unit squares. It will cumference of the geometrical definition. Yet, analytically, we now be seen that every straight contemplate altogether imaginary circles, for which pd is negative, line has an equation of the first and circles, for which pro, with all their reality condensed into degree, and that every equation thcir centres. Even when på is positive, so that a visible round of the first degree represents a circumference exists, we do not regard this as constituting the straight line. whole of the circle. Giving to x any value whatever in (x-h)? + It has been seen (8 7) that lines (y-k) = p?, we obtain two values of ý, real, coincident or imaginary, parallel to the axes have equa- each of which goes with the abscissa x as the ordinate of a point, tions of the first degree, free real or imaginary, on what is represented by the equation of the from one of the variables. Take circle. now a straight line ABC inclined The doctrine of the imaginary on a circle, and in geometry generto both axes. Let it make a | ally, is of purely algebraical inception; but it has been in its entirety given angle a with the positive accepted by modern pure geometers, and signal success has attended direction of the axis of x, i.e. in the efforts of those who, like K. G. C. von Staudt, have striven to fig. 50 let this be the angle base its conclusions on principles not at all algebraical in form, through which Ax must be re- though of course cognate to those adopted in introducing the volved counter-clockwise about imaginary into algebra. A in order to be made coin- A circle with its centre at the origin has an equation x+y? = p. cident with the line. Let C, of In oblique coordinates the general equation of a circle is coordinates (h, k), be a fixed point x+2xy cos w ty* +2Ax+2By+C =0. on the line, and P (x, y) any other point upon it. Draw the ordinates 10. The conic sections are the next simplest loci; and it will be CD, PM of C and P, and let the parallel to the axis of x through C seen later that they are the loci represented by equations of the meet PM, produced if necessary, in R. The right-angled triangle second degree. Circles are particular cases of conic sections; and B В M FIG. 50. measure A FIG. 52. B they have just been seen to have for their equations a particular, sections, we can, by drawing circles which meet each of them in class of equations of the second degree. Another particular class rcal points, construct the radical axis of the first-mentioned two of such equations is that included in the form (Ax+By+C) (A'x+ circles. B'y+C') = 0, which represents two straight lines, because the product 13. The principle employed in showing that the equation of the on the left vanishes is, and only if, one of the two factors does, i.e. common chord of two circles is S-S'=o is one of very extensive if, and only if, (x, y) lies on one or other of two straight lines. The application, and some more illustrations of it may be given. condition that ax' +2hxy+by+2gx+2fy+c=0, which is often Suppose S=0, S' =0 are lines (that is, let S, S now denote linear written (a, b, c,fig, h) (x, y, 1) = 0, takes this form is abc +afgh-af?. functions Ax+By+C, A'x+B'y+C'), then S-kS' =0 (k an arbitrary bge-ch? = 0. Note that the two lines may, in particular cases, be constant) is the equation of any line passing through the point parallel or coincident. of intersection of the two given línes. Such a line may be made to Any equation like F1(x, y) Fr(x, y) . . . F.(x, y)=0, of which pass through any given point, say the point (to yo); i.e. if Sa. S'e are the left-hand side breaks up into factors, represents all the loci what S, S' respectively become on writing for (x, y) the values (X®, ye), separately represented by F1(x, y) =0, F2(x, y) = 0, : .. Fn(x, y) = 0. then the valuc of kis k = So+S'. The cquation in fact is SS.-S.S' = 0; In particular an equation of degree n which is free from x represents and starting from this equation we at once verify it a posteriori; n straight lines parallel to the axis of x, and one of degree n which the equation is a lincar equation satisfied by the values of (x, y) is homogeneous in x and y, i.e. one which upon division by be- which make S=0, S' = 0; and satished also by the values (Is yu); comes an equation in the ratio y/x, represents a straight lines through and it is thus the equation of the line in question. the origin. II, as before, S=0, S'= 0 represent circles, then (k being arbitrary) Curves represented by equations of the third degree are called S-kS' =0 is the equation of any circle passing through the two cubic curves. The general equation of this degree will be written points of intersection of the two circles; and to make this pass (*) (x, y, 1)' = 0. through a given point (xo. Yo) we have again k = So+S'. In the 11. Descriptive Geometry:-A geometrical proposition is either particular case k=1, the circle becomes the common chord (more descriptive or metrical: in the former case the statement of it is accurately it becomes the common chord together with the line independent of the idea of magnitude (length, inclination, &c.), infinity; see $ 23 below), and in the latter it has relerence to this idea. The method of co- If S denote the general quadric function, ordinates seems to be by its inception essentially metrical. Yet S=ex' +2hxy+byl +afy+2gx+, in dealing by this method with descriptive propositions we are eminently free from metrical considerations, because of our power to then the equation S=0 represents a conic; assuming this, then, if use general equations, and S'-o represents another conic, the equation S-kS'=0 represents to avoid all assumption that any conic through the four points of intersection of the two conics. measurements implied are 14. The object still being to illustrate the mode of working with coordinates for descriptive pur- poses, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point (hg. 52), we draw the circle in the points A, A' and B, B' respectively, and then A, B, C (fig. 51), the common taking. Q as the intersection of the lines AB' and A'B, the dea point. point Q is a right line The geometrical proof is pending only upon, and the circle, but' independent of the metrical throughout particular lines OAA' and OBB'. Take the point of inter Taking O as the origin, and for the axes any two lines through O section of aa'. BB', and joining at right angles to each other, the equation of the circle will be this with y', suppose that y'd does not pass through y, but that it x + y2 +2Ax+2By+C=0; meets the circles A, B in two distinct points 7.Vi respectively. Wc and is the equation of the line taken to be y=mx, then the have then the known metrical property of intersecting chords of a points A, A' are found as the intersections of the straight line with circle; viz. in circle C, where aa', BB', are chords meeting at a point 0, the circle; or to determine x we have Oa.Oa' =OB.OB', x*(1 +m?) +2x(A+Bm) +C =0. where, as well as in what immediately follows, Oa, &c., denote, of If (x, yı) are the coordinates of A, and (:, y) of A', then the roots course, lengths or distances. Similarly in circle A, of this equation are X1, X3, whence casily OB.OB' = Oyz.Or. A+Bm and in circle B, Oa.Oa' = Or. Oy'. Consequently Or.. Oy' =Oy..Oy', that is, Ori=Oys. or the points and the coordinates ol B, B' to be (x3, ys) and (x..yu) respectively, And similarly, if the equation of the line OBB' is taken to be y=m'r, ni andy coincide; that is, they each coincide with y. then We contrast this with the analytical method: A+B m' Here it only requires to be known that an equation Ax+By+C =0 + represents a line, and an equation x + y2 +Ax+By+C =0 represents a circle. A, B, C have, in the two cases respectively, metrical We have then by g 8 significations, but these we are not concerned with. Using S to *(y1 - y.) - y(x,-*) +*194-*0=0, denote the function x2 + y +Ax+By+C, the equation of a circle is x(y: - ys) – y(xx-*3)+xzY3-IY:=0, S=0. Let the equation of any other circle be S', = x? +32 +A'x+ B'y+C =0; the equation S-S'=o is a lincar equation (S-S' is in by means of the relations yi-mx, =0, y:-mxz=0, 9:-*'x;=0, as the equations of the lines AB' and A'B respectively. Reducing this equation is satisfied by the coordinates of each of the points of Yo-m'xo=0, the two equations become intersection of the two circles (for at each of these points S=0 and x(mx; -m'xa) – y(x2-x^)+(m'-m)x1x1=0, S' = 0, therefore also S-S=0); hence the equation S-S'=0 is x(mxz - m'xz) – y(x2-x)+(m' -m)x;X;=0, that of the line joining the two points of intersection of the two circles, and if we divide the first of these equations by x1x4, and the second or say it is the equation of the common chord of the two circles. by Xzx), and then add, we obtain Considering then a third circle S',= x + y2 +A'x+Boy+C"=0, the equations of the common chords are S-Ś=, S-S"=0, S'-S'=0 (each of these a linear equation); at the intersection of the first and +20'-2m=0, second of these lines S=S' and S=S', therefore also S'=S", or the equation of the third line is satisfied by the coordinates of the point or, what is the same thing, in question; that is, the three chords intersect in a point O, the coordinates of which are determined by the equations S=S' =S'. It further appears that if the two circles S=0.S' =o do not intersect which by what precedes is the equation of a line through the point Q. in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line Substituting herein for + to their foregoing values, the 31 represented by the equation S-S'=0; or that two circles, whether their intersections be real or imaginary, have always a real common equation becomes chord (or radical axis), and that for any three circles the common - (A+Bm) (y-m'x)+(A+Bm') (y-mx)+C( mm)=0; chords intersect in a point (of course real) which is the radical centre. that is, And by this very theorem, given two circles with imaginary inter (m-m') (Ar+By+C)=0; FIG. 51. (+) 6-m's) – (4 +on) 6-mx)+zm’ –2m =0, I T I X M' А M Fig. 53. or finally it is Ax+By+C =0, showing that the point Q lies in a line Suppose, for instance, that we want to take for new origin the the position of which is independent of the particular lines OAA', point o' of old coordinates OA=h, AOʻ=k, and for new axes of OBB' used in the construction. It is proper to notice that there is X and Y lines through O' obtained by rotating parallels to the old no correspondence to each other of the points A, A' and B. B'; the axes of x and y, through an angle 0 counter-clockwise. Construct grouping might as well have been A, A' and B. B; and it thence (hig: 53) the old and new coappears that the line Ax+By+C=o just obtained is in fact the line ordinates of any, point P. Ex- y joining the point with the point R which is the intersection of pressing that the projections, AB and A'B'. first on the old axis of x and 15. In $ 8 it has been seen that two conditions determine the secondly on the old axis of y, of equation of a straight line, because in Ax+By+C=o one of the OP are equal to the sums of the coefficients may be divided out, leaving only two parameters to be projections, on those axes respecdetermined. Similarly five conditions instead of six determine an tively, of the parts of the broken equation of the second degree (a, b, c, f, g, h) (x, y, 1) =0, and nine line OO'M'P, we obtain: instead of ten determine a cubic (R) (x, y, 1))=0. It thus appears x=k+X cos 0+Y cos (0+1=) = that a cubic can be made to pass through 9 given points, and that k+X cos 0-Y sin e, the cubic so passing through 9 given points is completely determined. and There is, however, a remarkable exception. Considering two given y=k+x cos(4:-0)+Y cos 0 = cubic curves S=0, S'=0, these intersect in 9 points, and through ktX sin 0 +Y cose. these 9 points we have the whole series of cubics S-ks' =0, where k is an arbitrary constant: k may be determined so that the cubic Be careful to observe that these shall pass through a given tenth point (k = So+So, if the coordinates formulae do not apply to every are (rm yo), and So. S. denote the corresponding values of S, S'). conceivable change of reference from one set of rectangular axes to The resulting curve SS'o-S'S. =0 may be regarded as the cubic another. It might have been required to take OʻX, O'Y' for the determined by the conditions of passing through 8 of the 9 points positive directions of the new axes, so that the change of directions and through the given point (xo yo); and from the equation it of the axes could not be effected by rotation. We must then write thence appears that the curve passes through the remaining one of -Y for Y in the above. the 9 points. In other words, we thus have the theorem, any cubic Were the new axes oblique, making angles a, 8 respectively with curve which passes through 8 of the 9 intersections of two given the old axis of x, and so inclined at the angle B-a, the same method cubic curves passes through the 9th intersection. would give the formulae The applications of this theorem are very numerous; for instance, *=h+X cos a+Y cos B, y=k+X sin a +Y sin B. we derive from it Pascal's theorem of the inscribed hexagon. Con 18. The Conic Sections. The conics, as they are now called, were sider a hexagon inscribed in a conic. The three alternate sides at first defined as curves of intersection of planes and a cone; but constitute a cubic, and the other three alternate sides another cubic. Apollonius substituted a definition free from reference to space of The cubics intersect in 9 points, being the 6 vertices of the hexagon, three dimensions. This, in effect, is that a conic is the locus of a and the 3 Pascalian points, or intersections of the pairs of opposite point the distance of which from a given point, called the focus, has sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 (see CONIC Section). Ife: 1 is the ratio, e is called the eccentricitv. a given ratio to its distance from a given line, called the directrix of the 9 points of intersection, and it therefore passes through the The distances are considered signless. remaining point of intersection--that is, the third Pascalian point; and since obviously this does not lie on the conic, it must lie on the directrix." The absolute values of v{(x-h)+(-k)} and p-x cos a. Take (h, k) for the focus, and x cos a ty sin a-p=0 for the line-that is, we have the theorem that the three Pascalian points y sin a are to have the ratio e : 1; and this gives (or points of intersection of the pairs of opposite sides) lie on a line. (x-h)2+(-k)* =*[p-x cos a- y sin a)* 16. Metrical Theory resumed. Projections and Perpendiculars.-It as the general equation, in rectangular coordinates, of a conic. is a metrical fact of fundamental importance, already used in § 8, It is of the second degree, and is the general equation of that that, if a finite line PQ be projected on any other line 00' by per- degree. If, in fact, we multiply it by an unknown , we can, by pendiculars PP', 99 to OO, the length of the projection P'd' is solving six simultaneous equations in the six unknowns a, k, k, e, pa, equal to that of PO multiplied by the cosine of the acute angle so choose values for these as to make the coefficients in the equation between the two lines. Also the algebraical sum of the projections equal to those in any equation of the second degree which may be of the sides of any closed polygon upon any line is zero, because as a given. There is no failure of this statement in the special case point goes round the polygon, from any vertex A to A again, the when the given equation represents two straight lines, as in § 10, point which is its projection on the line passes from A' the projection but there is speciality: if the two lines intersect, the intersection of A to A' again, i.e. traverses equal distances along the line in and either bisector of the angle between them are a focus and positive and negative senses. If we consider the polygon as con directrix; if they are united in one line, any point on the line and a sisting of two broken lines, each extending from the same initial perpendicular to it through the point are: if they are parallel, to the same terminal point, the sum of the projections of the lines the case is a limiting one in which e and h2+k* have become infinite which compose the one is equal, in sign and magnitude, to the sum while ?(+k) remains finite. In the case ($ 9) of an equation of the projections of the lines composing the other. Observe that such as represents a circle there is another instance of proceeding the projection on a line of a length perpendicular to the line is to a limit: e has to become o, while ep remains finite: moreover a is indeterminate. The centre of a circle is its focus, and its directrix Let us hence find the equation of a straight line such that the has gone to infinity, having no special direction. This last fact perpendicular OD on it from the origin is of length p taken as illustrates the necessity, which is also forced on plane geometry by positive, and is inclined to the axis of x at an angle 20D=, three-dimensional considerations, of treating all points at infinity measured counter-clockwise from Ox. Take any point P (x, y) on in a plane as lying on a single straight line. the line, and construct OM and MP as in fig. 48.' The sum of the Sometimes, in reducing an equation to the above focus and directrix projections of OM and MP on OD is OD itself; and this gives the form, we find for h, k, e, p. tan e, or some of them, only imaginary equation of the line values, as quadratic equations have to be solved; and we have in x cos a ty sin a= p. fact to contemplate the existence of entirely imaginary conics. Observe that cos a and sin a here are the sin a and -cos a, or the For instance, no real values of x and y satisfy **++3=0. Even - sin a and cos a of $ 8 according to circumstances. when the locus represented is real, we obtain, as a rule, four sets of We can write down an expression for the perpendicular distance values of h, k, e, p, of which two sets are imaginary: a real conic from this line of any point (x, y) which does not lie upon it. If the has, besides two real foci and corresponding directrices, two others parallel through (y) to the line meet OD in E, we have x' cos at that are imaginary. sin a=0E, and the perpendicular distance required is OD-OE, In oblique as well as rectangular coordinates equations of the 1.e. Par cos ay sin a; it is the perpendicular distance taken second degree represent conics. positively or negatively according as (x, y) lies on the same side 19. The three Species of Conics.-A real conic, which does not of the line as the origin or not. degenerate into straight lines, is called an ellipse, parabola or hyper: The general equation Ax+By+C=0 may be given the form bola according as e<, -, or >1. To trace the three forms it is cos a ty sin a-p=0 by dividing it by v(A? +B:). Thus (Ax' + best so to choose the axes of reference as to simplify their equations. By+C) (A+B) is in absolute value the perpendicular distance In the case of a parabola, let 2c be the distance between the given of (x, y) from the line Ax+By+C=0. Remember, however, that focus and directrix, and take axes referred to which these are the there is an essential ambiguity of sign attached to a square root. point (,0) and the line x=-c. The equation becomes (x-c)' +y = The expression found gives the distance taken positively, when(x+c), i.e. 3 = 4cx. (2.39 is on the origin side of the line, is the sign of C is given to In the other cases, take a such that ale~-1) is the distance of focus V(A+B*). from directrix, and so choose axes that these are (ae, o) and x=ar', 17. Transformation of Coordinates. We often need to adopt new thus getting the equation(x-ge) + 3* = e*(x-ce-1), i.e. (1-¢*)x* + y2 = axes of reference in place of old oncs; and the above principle of a'(1 –c).. When e<1, i.e. in the case of an ellipse, this may be projections readily expresses the old coordinates of any point in written x*/a? + y2lb2 = 1, where b = a*(1 -e*); and when <>!, i.e. terms of the new. in the case of an hyperbola, x7/q?-y2/b2 = 1, where be=0*(6-1). tero. Fig. 54. FIG. 55 The axes thus chosen for the ellipse and hyperbola are called the to a fixed line Ox, determine the point: 4, the numerical measure principal axes. of OP, the radius vector, and 0, the circular measure of x0P, the In figs. 54, 55, 56 in order, conics of the three species, thus referred, inclination, are called polar coordinates of P. The formulae x = are depicted. 7 cos 0, y=, sin 6 connect Cartesian and polar coordinates, and make The oblique straight lines in fig. 56 are the asymptoles x/a = *y/b transition from either system to the other easy. In polar coordinates of the hyperbola, lines to which the curve tends with unlimited the equations of a circle through O, and of a conic with O as focus, take the simple forms r = 20 cos (-a), rf1-e cos (2-a)} =. The el use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon, 19 such curves for instance as r=cos mo, with m integral, and also the curves called spirals, which have equations giving , as functions of O itself, and not merely of sin 0 and cos e. In the geometry of motion under central forces the advantage of working with polar coordinates is great. 0 23. Trilinear and Areal Coordinates. Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by A the area, of the triangle. We might, as in 86, take CA, CB as axes of x and y, inclined at an angle c. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin c from CA and CB. Call these B and a respectively. The signs of B and a are those of y and x, i.e. B is positive or negative according as P lies 19 on the same side of CA as B does or the opposite, and similarly for a An equation in (x, y) of any degree may, upon replacing in it í andy by a cosec C and B cosec C, be written as one of the same degree in (a, b). Now let y be the perpendicular distance of P from the third side AB, taken as positive or negative as P is on the C side of AB or not. The geometry of the figure tells us that ca+68+cy=24. By means of this relation in a B, y we can give an equation considered countless other forms, involving two or all of , B, 4. In particular we may make it homogeneous in a, B; y: to do this we have only to multiply the terms of every degree less than the highest present in the equation by a power of (aa+bB+c9)/24 just sufficient to raise them, in each case, to the highest degree. Fig. 56. We call (a, B, Y) Erilinear coordinates, and an equation in them the trilinear equation of the locus represented. Trilinear equations closeness as it goes to infinity. The hyperbola would have an equa- are, as a rule, dealt with in their homogeneous forms. An advantage tion of the form xy=c is referred to its asymptotes as axes, the co thus gained is that we need not mean by (a, B. y) the actual measures ordinates being then oblique, unless a =b, in which case the hyperbola of the perpendicular distances, but any properly signed numbers is called rectangular. An ellipse has two imaginary asymptotes. which have the same ratio two and two as these distances. In particular a circle 3*+ y2 = a*, a particular ellipse, has for asymp- In place of a, b, u it is lawful to use, as coordinates specifying totes the imaginary lines x= Eyv-1. These run from the centre the position of a point in the plane of a triangle of reference ABC, to the so-called circular points at infinity. any given multiples of these. For instance, we may use x = 0/14, 20. Tangents and Curvature. -Let (x", y) and (x'+h, y+k) be y=bB/24, 2=57/24; the properly signed ratios of the triangular two neighbouring points P, P' on a curve. The equation of the line areas PBC, PCA, PAB to the triangular area ABC. These are called on which both lie is h(y-y')=k(x-x'). Now keep P fixed, and let the areal coordinates of P. In areal coordinates the relation which P' move towards coincidence with it along the curve. enables us to make any equation homogeneous takes the simple necting line will tend towards a limiting position, to which it can form x+y+s=1; and, as before, we need mean by x, y, s, in a never attain as long as P and P are distinct. The line which homogeneous equation, only signed numbers in the right ratios occupies this limiting position is the langent at P. Now if we sub- Straight lines and conics are represented in trilinear and in areal, tract the equation of the curve, with (x", y') for the coordinates in it, because in Cartesian, coordinates by equations of the first and from the like equation in (x'th, y'+k), we obtain a relation in h second degrees respectively, and these degrees are preserved when and k, which will, as a rule, be of the form o=Ah+Bk + terms of the equations are made homogeneous. What must be said about higher degrees in h and k, where A, B and the other coefficients points infinitely, far off in order to make universal the statement, involve x' and y'. This gives k/h=-A/B + terms which tend to to which there is no exception as long as finite distances alone are vanish as h and k do, so that -A : B is the limiting value tended to considered, that every homogeneous equation of the first degree by k : h. Hence the cquation of the tangent is B(y-7) +A(x—X') = 0. represents a straight line? Let the point of areal coordinates The normal at (x", y') is the line through it at right angles to the (x", y', z') move infinitely far off, and mean by x, y, 2 finite quantities tangent, and its equation is Aly-y'), B(x– x')=0. in the ratios which x, y, s tend to assume as they become infinite. In the case of the conic (a, b, c, f, g, h) (x, y, 1)2 =0 we find that The relation x+y+z=1 gives that the limiting state of things A/B = (ax' thy' +ş)/(hx'+by' +). tended to is expressed by x+y+z=o. This particular equation of We can obtain the coordinates of Q, the intersection of the normals the first degree is satisfied by no point at a finite distance; but we QP, QP' at (x, y) and (x' +h, y'+k), and then, using the limiting see the propriety of saying that it has to be taken as satisfied by Value of k : h, deduce those of its limiting position as P' moves up all the points conceived of as actually at infinity. Accordingly the to P. This is the centre of curvature of the curve at P (x", y'), and special property of these points is expressed by saying that they lie is so called because it is the centre of the circle of closest contact on a special straight line, of which the areal equation is x+y+s=0. with the curve at that point. That it is so follows from the facts in trilinear coordinates this line ai infinity has for equation ca+13+ that the closest circle is the limit tended to by the circle which touches cy=0. the curve at P and passes through P', and that the arc from P to P' 'On the one special line at infinity parallel lines are treated as of this circle lies between the circles of centre Q and radii QP, QP', meeting. There are on it two special (imaginary) points, the circular which circles tend, not to different limits as P moves up to P, but points at infinity of $ 19, through which all circles pass in the same to one. The distance from P to the centre of curvature is the radius In fact if S=O be one circle, in areal coordinates, of curvature. S+(x+y+z) (lx+my+nz)=0 may, by proper choice of l, m, n, be 21. Differential Plane Geometry.-The language and notation of the made any other; since the added terms are once lx+my+ns, and differential calculus are very useful in the study of tangents and have the generality of any expression like a'r+by+c' in Cartesian Denoting by (É, ) the current coordinates, we find, coordinates. Now these two circles intersect in the two points where as above, that the tangent at a point (x, y) of a curve is my=. either meets x+y+z=o as well as in two points on the radical axis (4x)dy/dx, where dy/dx is found from the equation of the curve. If Ix+my+ns=0. this be f(x, y) =o the tangent is (-x) (Of/0x)+(-y) (aflay) =0. If 24. Let us consider the perpendicular distance of a point (a'.8°, 77 and (a, B) are the radius and centre of curvature at (x, y), we find that from a line la + mB+ny. We can take rectangular axes of Cartesian q(a-x) =-P(1 +p2).9(B-y) = 1 +p?, q*p* = (1 +p2)3, where p, a denote coordinates (for clearness as to equalities of angle it is best to dy!dx, dzy/dx? respectively. (See INFINITESIMAL CALCULUS.) choose an origin inside ABC), and refer to them, by putting expres In any given case we can, at all events in theory, eliminate x, y sions p-x cos 0-y sin 0, &c., for a &c.: we can then apply $ 16 to between the above equations for a--* and B-y, and the equation get the perpendicular distance; and finally revert to the trilinear of the curve. The resulting equation in (a, b) represents the locus notation. The result is to find that the required distance is of the centre of curvature. This is the evolute of the curve. (la'+ms' +ny')/{l, m, n}, 22. Polar Coordinates.- In plane geometry the distance of any where Il, m, n =P +m2+n2 - 2mn cos A-2nl cos B-2lm cos C. point P from a fixed origin (or pole) O, and the inclination xOP of OP In areal coordinates the perpendicular distance from (x, y, ') The con sense. curvature. M N FIG. 57 FIG. 58. to la+my+na=o is 24(lx' +my'+n2')}(el, bm, cn}. ' In both cases that follows, three which are at right angles two and two. They the coordinates are of course actual values. intersect, two and two, in lines x'Ox, y'oy, z'Oz, called the axes Now let trs be the perpendiculars on the line from the vertices of x, y, z respectively, and divide all space into eight parts called A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in octants. If from any point p in space we draw PN parallel to accord with a convention that oppositeness of sign implies dis- Oz' to meet the plane xOy in N, and then from N draw NM parallel tinction between one side of the line and the other. Three applications of the result above give El=24/{al, bm, cr} = n/m=3/n; and we thus have the important fact that Ex' +ny' +53' is the perpendicular distance between a point of areal coordinates (x'y's') and a line on which the perpendiculars from A, B, C are g, n, Š respectively. We have also that &x+ny+$z=o is the areal equation of the line on which the perpendiculars are 5, n. 5; and, by equating the two expressions for the perpendiculars from (x', 'y', 's') on the line, that in all cases fas, bn, C3)* = 44%. 25. Line-coordinates. Duality:-A quite different order of ideas may be followed in applying analysis to geometry. The notion of a straight line specified may precede that of a point, and points may be dealt with as the intersections of lines. The specification of a line may be by means of coordinates, and that of a point by an equation, satisfied by the coordinates of lines which pass through it. N Systems of line-coordinates will here be only briefly considered. Every such system is allied to some system of point.coordinates: to yoy to meet x'Ox in M, the coordinates (x, y, z); of P are the and space will be saved by giving prominence to this fact, and not recommencing ab initio. numerical measures of OM, MN, NP; in the case of rectangular Suppose that any particular system of point-coordinates, in which coordinates these are the perpendicular distances of P from the three li+my+n3 =0 may represent any straight line, is before us: notice planes of reference. The sign of each coordinate is positive or that not only are trilinear and areal coordinates such systems, but negative as P lies on one side or the other of the corresponding Cartesian coordinates also, since we may write xlz, y/2 for the plane. In the octant delineated the signs are taken all positive. Cartesian x, y, and multiply through by z.. The line is exactly to the plane zOx, the points of a solid figure being projected on that In fig. 57 the delineation is on a plane of the paper taken parallel assigned if l, m, n, or their mutual ratios, are known, Call (b, m, n) the coordinates of the line. Now keep x, y, s constant, and let the plane by parallels to some chosen line through in the positive coordinates of the line vary, but always so as to satisfy the equation octant. Sometimes it is clearer to delineate, as in. fig. 58. by proThis equation, which we now write x1 + ym +on=0, is satisfied by jection parallel to that line in the octant which is equally inclined to the coordinates of every line through a certain fixed point, and by Ox, Qy: Oz upon a plane of the paper perpendicular to it. It is those of no other line; 'it is the equation of that point in the line possible by parallel projection to delineate equal scales along Ox, coordinates l, m, nh. Oy, Os by scales having any ratios we like along lines in a plane Line-coordinates are also called tangential coordinates. A curve having any mutual inclinations we like. is the envelope of lines which touch it, as well as the locus of points suffices to delineate the sections by the coordinate planes; and, in For the delineation of a surface of simple form it frequently which lie on it. A homogeneous equation of degree above the first in l, m, n is a relation connecting the coordinates of every line which particular, when the surface has symmetry about each coordinate touches some curve, and represents that curve, regarded as an plane, to delineate the beenvelope. For instance, the condition that the line of coordinates quarter-sections ll , m, n), i.e. the line of which the allied point-coordinate equation longing to a single octant. Thus fig: 59 is lo+my+n3=0, may touch a conic (a, b, c, f. 8. h)(x, y, z)=0, is readily found to be of the form (A, B, C, F, G, H), m, n)?=0, conveniently reprei.e. to be of the second degree in the line-coordinates. "It is not hard sents an octant of the to show that the general equation of the second degree in l, m, n wave surface, which thus represents a conic; but the degenerate conics of line-coordinates plane in a circle and are not line-pairs, as in point-coordinates, but point-pairs. The degree of the point-coordinate equation of a curve is the an ellipse. Or we may, order of the curve, the number of points in which it cuts a straight delineate a series of line. That of the line-coordinate equation is its class, the number contour lines, i.e. secof tangents to it from a point. The order and class of a curve are tions by planes parallel generally different when either exceeds two. to xOy, or some other 26. The system of line-coordinates allied to the areal system of chosen plane; of course point-coordinates has special interest. other sections may be The l, m, n of this system are the perpendiculars g, n, $ of $ 24; indicated too for and x' +'5+:'s=o is the equation of the point of areal coordinates greater clearness. For (4,2), s.c. is a relation which the perpendiculars from the vertices the delineation of a of the triangle of reference on every line through the point, but no curve a good method is other line, satisfy. Notice that a non-homogeneous equation of the to represent, first degree in , 9, does not, as a homogeneous one does, represent P thereof, each accompanied by its ordinate PN, which serves to above, a series of points stancy of the perpendicular distance of the fixed point x'E+y'nt refer it to the plane of my. The employment of stereographic :'1-o from the variable line (5, 5;3), i.e. the fact that (5, n. 5) touches projection is also interesting. a circle with the fixed point for centre. The relation in any,1,5 order, we have only the point and the curve. In solid geometry, 28. In plane geometry, reckoning the line as a curve of the first which enables us to make an equation homogeneous is not linear, reckoning a line as a curve of the first order, and the plane as a surface as in point-coordinates, but quadratic, viz. it is the relation fason of the first order, we have the point, the curve and the surface; = 44 of $24. Accordingly the homogeneous equation of the above circle is but the increase of complexity is far greater than would hence at 44°(x'E+y'n+z's)2 = R'{ag, bm, c5}?. first sight appear. In plane geometry a curve is considered in Every circle has an equation of this form in the present system of is considered in connexion with lines and planes (its tangents and connexion with lines (its tangents); but in solid geometry the curve line-coordinates. Notice that the equation of any circle is satisfied osculating planes), and the surface also in connexion with lines and by those coordinates of lines which satisfy both '+y'n+z's=0, planes (its tangent lines and tangent planes); there are surfaces the equation of its centre, and fag, bn, C5)2 = 0. This last equation, arising out of the line-cones, skew surfaces, developables, doubly of which the left-hand side satisfies the condition for breaking up and triply infinite systems of lines, and whole classes of theories into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass. which have nothing analogous to them in plane geometry: it is thus There is strict duality in descriptive geometry between point-line to in the present article. a very small part indeed of the subject which can be even referred locus and line-point-envelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special a single, or say a onefold relation, which can be represented by a In the case of a surface we have between the coordinates (x, y, z) line only, associated with distance, while of special points, associated single relation f(x, y, 3) = 0; with direction, there are two: moreover the line is real, and the expressed each of them as a given function of two variable para or we may consider the coordinates points both imaginary. meters p, q; the form z=f(x, y) is a particular case of each of these II. Solid Analytical Geometry. modes of representation; in other words, we have in the first mode 27. Any point in space may be specified by three coordinates. f(, y, z) = -f(x, y), and in the second mode x = 0, y=q for the We consider three fixed planes of reference, and generally, as in all | expression of two of the coordinates in terms of the parameters, as FIG. 59. |