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taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. Take, for instance, the equation y=mx, where m is a given constant. It is satisfied by the coordinates of every point P, which is such that, in fig. 48, the distance MP, with its proper sign, ism times the distance OM, with its proper sign, i.e. by the coordinates of every point in the straight line through O which we arrive at by making a line, originally coincident with x'Ox, revolve about O in the direction opposite to that of the hands of a watch through an angle of which m is the tangent, and by those of no other points. That line is the locus which it represents. Take, more generally, the equation y=4(x), where (x) is any given non-ambiguous function of x. Choosing any point M on x'Ox in fig. 1, and giving to x the value of the numerical measure of OM, the equation determines a single corresponding y, and so determines a single point P on the line through M parallel to y'oy. This is one point whose coordinates satisfy the equation. Now let M move from the extreme left to the extreme right of the line x'Ox, regarded as extended both ways as far as we like, i.e. let x take all real values from too. With every value goes a point P, as above, on the parallel to y'Oy through the corresponding M; and we thus find that there is a path from the extreme left to the extreme right of the figure, all points P along which are distinguished from other points by the exceptional property of satisfying the equation by their coordinates. This path is a locus; and the equation y=(x) represents it. More generally still, take an equation f(x, y)=0 which involves both x and y under a functional form. Any particular 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 ing a point or points (x, y), of which the coordinates satisfy the equation f(x, y)=o. Here again, as x takes all values, the point or points describe a path or paths, which constitute a locus represented by the equation. Except when y enters to the first degree only in 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 it is not uncommon for all to be imaginary for some ranges of values of x. The locus may largely consist of continua of imaginary points; but the real parts of it constitute a real curve or real curves. Note that we have to allow x to admit of all imaginary, as well as of all real, values, in order to obtain all imaginary parts of the locus.

A locus or curve may be algebraically specified in another way; viz. we may be given two equations x=f(0), y= F(0), which express the coordinates of any point of it as two functions of the same variable parameter 0 to which all values are open. As takes all values in turn, the point (x, y) traverses the curve.

It is a good exercise to trace a number of curves, taken as defined by the equations which represent them. This, in simple cases, can be done approximately by plotting the values of y given by the equation of a curve as going with a considerable number of values of x, and connecting the various points (x, y) thus obtained. But methods exist for diminishing the labour of this tentative process. Another problem, which will be more attended to here, is that of determining the equations of curves of known interest, taken as defined by geometrical properties. It is not a matter for surprise that the curves which have been most and longest studied geometrically are among those represented by equations of the simplest character.

B

O

8. The Straight Line.-This is the simplest type of locus. Also the simplest type of equation in x and y is Ax+By+C=o, one of the first degree. Here the coefficients A, B, C are constants. They are, like the current coordinates, x, y, numerical. But, in giving interpretation to such an equation, we must of course refer to numbers Ar, By, C of unit magnitudes of the same kind, of units of counting for instance, or unit lengths or unit squares. It will now be seen that every straight line has an equation of the first degree, and that every equation of the first degree represents a straight line.

It has been seen (§ 7) that lines parallel to the axes have equations of the first degree, frce from one of the variables. Take now a straight line ABC inclined to both axes. Let it make a given angle a with the positive direction of the axis of x, i.e. in fig. 50 let this be the angle through which Ax must be revolved counter-clockwise about A in order to be made coincident with the line. Let C, of coordinates (h, k), be a fixed point on the line, and P (x, y) any other point upon it. Draw the ordinates CD, PM of C and P, and let the parallel to the axis of x through C meet PM, produced if necessary, in R. The right-angled triangle

FIG. 50.

D

M

CRP tells us that, with the signs appropriate to their directions attached to CR and RP,

RP=CR tan a, i.e. MP-DC=(OM —OD) tan a, and this gives that

y-k-tan a (x-h),

an equation of the first degree satisfied by x and y. No point not on the line satisfies the same equation; for the line from C to any point off the line would make with CR some angle ẞ different from a, and the point in question would satisfy an equation y-k=tanẞ(x-h), which is inconsistent with the above equation.

The equation of the line may also be written y=mx+b, where m=tan a, and b-k-h tan a. Here b is the value obtained for y from the equation when o is put for x, i.e. it is the numerical measure, with proper sign, of OB, the intercept made by the line on the axis of y, measured from the origin. For different straight lines, m and b may have any constant values we like.

Now the general equation of the first degree Ax+By+C=o may be written y=-x-B, unless B=0, in which case it represents a A C

line parallel to the axis of y; and -A/B, -C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line. It is important to notice that the general equation, which in appearance contains three constants A, B, C, in effect depends on two only, the ratios of two of them to the third. In virtue of this last remark, we see that two distinct conditions suffice to determine a straight line. For instance, it is easy from the above to see that

+=1

I

that it makes intercepts OA, OB on the two axes, of which a and b is the equation of a straight line determined by the two conditions are the numerical measures with proper signs. note that in fig. 50 a is negative. Again,

y-y1=2(x-x1),

i.e.

(11-12)x-(x1-X2)Y+X1у2-X2Y1=0,

represents the line determined by the data that it passes through two given points (x, y) and (x, y). To prove this find m in the equation y-y=m(x-x1) of a line through (x, y), from the condition that (x2, y2) lies on the line.

In this paragraph the coordinates have been assumed rectangular. Had they been oblique, the doctrine of similar triangles would have given the same results, except that in the forms of equation y-k= m(x-h), y=mx+b, we should not have had m=tan a.

9. The Circle.-It is easy to write down the equation of a given circle. Let (h, k) be its given centre C, and p the numerical measure of its given radius. Take P (x, y) any point on its circumference, and construct the triangle CRP, in fig. 50 as above. The fact that this is right-angled tells us that

CR2+RP2 CP2, and this at once gives the equation (x−k)2+(y−k)2 = p2.

A point not upon the circumference of the particular circle is at some distance from (h, k) different from p, and satisfies an equation inconsistent with this one; which accordingly represents the circumference, or, as we say, the circle. The equation is of the form

x2+y2+2Ax+2By+C=0.

Conversely every equation of this form represents a circle: we have only to take A, B, A2+B-C for h, k, p2 respectively, to obtain its centre and radius. But this statement must appear too unrestricted. Ought we not to require A+B-C to be positive? Certainly, if by circle we are only to mean the visible round circumference of the geometrical definition. Yet, analytically, we contemplate altogether imaginary circles, for which p is negative, and circles, for which p=0, with all their reality condensed into their centres. Even when p is positive, so that a visible round circumference exists, we do not regard this as constituting the whole of the circle. Giving to x any value whatever in (x − h)2+ (y-k)= p2, we obtain two values of y, real, coincident or imaginary, each of which goes with the abscissa x as the ordinate of a point, real or imaginary, on what is represented by the equation of the circle.

The doctrine of the imaginary on a circle, and in geometry gener ally, is of purely algebraical inception; but it has been in its entirety accepted by modern pure geometers, and signal success has attended the efforts of those who, like K. G. C. von Staudt, have striven to base its conclusions on principles not at all algebraical in form, though of course cognate to those adopted in introducing the imaginary into algebra.

A circle with its centre at the origin has an equation + y2 = p2. In oblique coordinates the general equation of a circle is x+2xy cos w+y+2Ax+2By+C=0.

10. The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Circles are particular cases of conic sections; and

they have just been seen to have for their equations a particular class of equations of the second degree. Another particular class of such equations is that included in the form (Ax+By+C) (A'x+ B'y+C')=0, which represents two straight lines, because the product on the left vanishes if, and only if, one of the two factors does, i.e. if, and only if, (x, y) lies on one or other of two straight lines. The condition that ax2+2hxy+by2+2gx+2fy+c=0, which is often written (a, b, c, f, g, h) (x, y, 1) =o, takes this form is abc +2fgh-af2bg-cho. Note that the two lines may, in particular cases, be parallel or coincident.

Any equation like F(x, y) F(x, y) . . . Fn(x, y) =0, of which the left-hand side breaks up into factors, represents all the loci separately represented by F(x, y) =0, F2(x, y) =o, . . . F,(x, y) =o. In particular an equation of degree n which is free from x represents n straight lines parallel to the axis of x, and one of degree n which is homogeneous in x and y, i.e. one which upon division by x" becomes an equation in the ratio y/x, represents n straight lines through the origin.

Curves represented by equations of the third degree are called cubic curves. The general equation of this degree will be written (*) (x, y, 1)3 =0.

11. Descriptive Geometry.-A geometrical proposition is either descriptive or metrical: in the former case the statement of it is independent of the idea of magnitude (length, inclination, &c.), and in the latter it has reference to this idea. The method of coordinates seems to be by its inception essentially metrical. Yet in dealing by this method with descriptive propositions we are eminently free from metrical considerations, because of our power to use general equations, and to avoid all assumption that measurements implied are any particular measure

ments.

12. It is worth while to illustrate this by the instance of the well-known

theorem of the radical centre

of three circles. The theorem
is that, given any three circles
A, B, C (fig. 51), the common
chords aa, BB, YY' of the
three pairs of circles meet in
a point.

The geometrical proof is
metrical throughout:-

FIG. 51. Take O the point of intersection of aa', BB', and joining this with y', suppose that 'O does not pass through y, but that it meets the circles A, B in two distinct points 72, 71 respectively. We have then the known metrical property of intersecting chords of a circle; viz. in circle C, where aa', 88, are chords meeting at a point O, Oa.Oa' ÓB.Oẞ', where, as well as in what immediately follows, Oa, &c., denote, of course, lengths or distances. Similarly in circle A,

B

sections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the first-mentioned two circles.

13. The principle employed in showing that the equation of the common chord of two circles is S-S'-o is one of very extensive application, and some more illustrations of it may be given.

Suppose S=0, S'=0 are lines (that is, let S, S' now denote linear functions Ax+By+C, A'x+B'y+C'), then S-kS'=0 (k an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be made to pass through any given point, say the point (x, y); i.e. if So. S', are what S, S' respectively become on writing for (x, y) the values (xe, ye). then the value of k is k=So+S'o. The equation in fact is SS'-SS' = 0; and starting from this equation we at once verify it a posteriori; the equation is a linear equation satisfied by the values of (x, y) which make S=0, S'=0; and satisfied also by the values (xo, yo); and it is thus the equation of the line in question.

If, as before, S=0, S'o represent circles, then (k being arbitrary) S-kS'=o is the equation of any circle passing through the two points of intersection of the two circles; and to make this pass through a given point (xo, yo) we have again k=S,÷S'. In the particular case k=1, the circle becomes the common chord (more accurately it becomes the common chord together with the line infinity; see § 23 below).

If S denote the general quadric function,
S=ax2+2hxy+by2+2fy+2gx+c.

B

then the equation So represents a conic; assuming this, then, if
S'o represents another conic, the equation S-kS-o represents
any conic through the four points of intersection of the two conics.
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 O (fig. 52), we draw
through O any two lines meeting
the circle in the points A, A' and
B, B' respectively, and then
taking. Q as the intersection of
the lines AB' and A'B, the
theorem is that the locus of the
point Q is a right line de-
pending only upon O and the circle, but independent of the
particular lines OAA' and OBB'.

FIG. 52.

Taking O as the origin, and for the axes any two lines through O
at right angles to each other, the equation of the circle will be
x+y2+2Ax+2By+C=0;

and if the equation of the line OAA' is taken to be y=mx, then the
points A, A' are found as the intersections of the straight line with
the circle; or to determine x we have
x2(1+m2)+2x(A+Bm)+C=0.

If (x, y1) are the coordinates of A, and (x, y) of A', then the roots
of this equation are x1, x2, whence easily
A+Bm

I
X1 X2

= + 1/1/1

=-2

equation of the line OBB' is taken to be y=m'r, B, B' to be (xs, y1) and (x, y1) respectively,

A+Bm'

OB.OB'=Oy2.Or',

and in circle B,

Oa.Oa'=Oyi.Oy'.

And similarly, if the
Consequently Ori.Oy'-Oy Oy', that is, Oy Oy, or the points and the coordinates of
Y and coincide; that is, they each coincide with 7.
We contrast this with the analytical method:-

then

Here it only requires to be known that an equation Ax+By+C=o represents a line, and an equation x2+y2+Ax+By+C=o represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x2+y+Ax+By+C, the equation of a circle is S=o. Let the equation of any other circle be S', x2+y2+A'x+ B'y+C'=0; the equation S-S'o is a linear equation (S-S' is in fact=(A-A')x+(B−B') y+C−C'), and it thus represents a line; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points $=0 and S'o, therefore also S-S'=0); hence the equation S-S'o is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S",= x2+y2+A′′x+By+C"=o, the equations of the common chords are S-S'=o, S-S" -o, S'-S" =0 (each of these a linear equation); at the intersection of the first and second of these lines SS' and S=S", therefore also S'S", or the equation of the third line is satisfied by the coordinates of the point 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 So, So do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S-S'=0; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary inter

We have then by § 8

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=-2

Q

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as the equations of the lines AB' and A'B respectively. Reducing by means of the relations y-mx=0, y-mx2=0, y1—m'x1 =0, Y4-m'x=0, the two equations become

x(mx-m'x1)-y(xɩ −x1)+(m′—m)x1x1=0, x(mx2-m'x1)- y(x2 −x3)+(m′ —m)x2x3=0, and if we divide the first of these equations by xx, and the second by x2x3, and then add, we obtain

or, what is the same thing,

[ocr errors]

(¦1 + ;-;) (v — m'x) − ( +1, + +-) (y—mx)+2m'—2m =0,

which by what precedes is the equation of a line through the point Q. + their foregoing values, the

Substituting herein for +

I I I
X1 X2 X3 X4

equation becomes
-(A+Bm) (y-m'x)+(A+Bm') (y−mx)+C(m'—m)=0;
(m-m') (Ax+By+C)=0;

that is,

or finally it is Ax+By+C=o, showing that the point Q lies in a line the position of which is independent of the particular lines OAA', OBB' used in the construction. It is proper to notice that there is no correspondence to each other of the points A, A' and B, B'; the grouping might as well have been A, A' and B', B; and it thence appears that the line Ax+By+C=o just obtained is in fact the line joining the point Q with the point R which is the intersection of

AB and A'B'.

and

15. In 8 it has been seen that two conditions determine the equation of a straight line, because in Ax+By+C-o one of the coefficients may be divided out, leaving only two parameters to be determined. Similarly five conditions instead of six determine an equation of the second degree (a, b, c, f, g, h) (x, y, 1)2=o, and nine instead of ten determine a cubic (*) (x, y, 1)=0. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely determined. There is, however, a remarkable exception. Considering two given y=k+X cubic curves S=0, S'o, these intersect in 9 points, and through these 9 points we have the whole series of cubics S-kS'-o, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (k=So+S'o, if the coordinates are (x, y), and So. S'o denote the corresponding values of S, S'). The resulting curve SS'.-S'S=0 may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (xo, yo); and from the equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection.

The applications of this theorem are very numerous; for instance, we derive from it Pascal's theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 of the 9 points of intersection, and it therefore passes through the 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 line-that is, we have the theorem that the three Pascalian points (or points of intersection of the pairs of opposite sides) lie on a line.

16. Metrical Theory resumed. Projections and Perpendiculars.-It is a metrical fact of fundamental importance, already used in § 8, that, if a finite line PQ be projected on any other line OO' by perpendiculars PP', QQ' to OO', the length of the projection P'Q' is equal to that of PQ multiplied by the cosine of the acute angle between the two lines. Also the algebraical sum of the projections of the sides of any closed polygon upon any line is zero, because as a point goes round the polygon, from any vertex A to A again, the point which is its projection on the line passes from A' the projection of A to A' again, i.e. traverses equal distances along the line in positive and negative senses. If we consider the polygon as consisting of two broken lines, each extending from the same initial to the same terminal point, the sum of the projections of the lines which compose the one is equal, in sign and magnitude, to the sum of the projections of the lines composing the other. Observe that the projection on a line of a length perpendicular to the line is

zero.

Suppose, for instance, that we want to take for new origin the point O' of old coordinates OA=h, AO'=k, and for new axes of X and Y lines through O' obtained by rotating parallels to the old axes of x and y through an angle counter-clockwise. Construct (fig. 53) the old and new coordinates of any point P. Expressing that the projections, first on the old axis of x and secondly on the old axis of y, of OP are equal to the sums of the projections, on those axes respectively, of the parts of the broken line OO'M'P, we obtain: x=h+X cos 0+Y cos (0+1)= h+X cos 0-Y sin 0, cos(0)+Y cos 0= k+X sin @+Y cos 0. Be careful to observe that these formulae do not apply to every conceivable change of reference from one set of rectangular axes to another. It might have been required to take O'X, O'Y' for the positive directions of the new axes, so that the change of directions of the axes could not be effected by rotation. We must then write -Y for Y in the above.

FIG. 53.

Let us hence find the equation of a straight line such that the perpendicular OD on it from the origin is of length p taken as positive, and is inclined to the axis of x at an angle xOD=a, measured counter-clockwise from Ox. Take any point P (x, y) on the line, and construct OM and MP as in fig. 48. The sum of the projections of OM and MP on OD is OD itself; and this gives the equation of the line x cos a+y sin a=p. Observe that cos a and sin a here are the sin a and -cos a, or the -sin a and cos a of § 8 according to circumstances.

We can write down an expression for the perpendicular distance from this line of any point (x, y) which does not lie upon it. If the parallel through (x, y) to the line meet OD in E, we have x' cos a+ y' sin a=OE, and the perpendicular distance required is OD-OE, i.e. p-x cos a-y sin a; it is the perpendicular distance taken positively or negatively according as (x', y') lies on the same side of the line as the origin or not.

The general equation Ax+By+Co may be given the form x cos aty sin a-p-o by dividing it by √(A+B). Thus (Ax'+ By+C)+(A+B) is in absolute value the perpendicular distance of (x, y) from the line Ax+By+Co. Remember, however, that there is an essential ambiguity of sign attached to a square root. The expression found gives the distance taken positively when | (x, y) is on the origin side of the line, if the sign of C is given to √(A+B).

17. Transformation of Coordinates.-We often need to adopt new axes of reference in place of old ones; and the above principle of projections readily expresses the old coordinates of any point in

terms of the new.

X

144

0

A

M

Were the new axes oblique, making angles a, ß respectively with the old axis of x, and so inclined at the angle ẞ-a, the same method would give the formulae

x=h+X cos a+Y cos ẞ, y=k+X şin a+Y sin ẞ.

18. The Conic Sections.-The conics, as they are now called, were at first defined as curves of intersection of planes and a cone; but Apollonius substituted a definition free from reference to space of three dimensions. This, in effect, is that a conic is the locus of a point the distance of which from a given point, called the focus, has a given ratio to its distance from a given line, called the directrix (see CONIC SECTION). Ife: 1 is the ratio, e is called the eccentricity. The distances are considered signless.

directrix. The absolute values of v{(x-h)+(y-k)) and p-x cosa.-
Take (h, k) for the focus, and x cos a+y sin a-p-o for the
y sin a are to have the ratio e: 1; and this gives
(x−h)2+(y−k)2=e2(p-x cos a-y sin a)

as the general equation, in rectangular coordinates, of a conic.
It is of the second degree, and is the general equation of that
degree. If, in fact, we multiply it by an unknown λ, we can, by
solving six simultaneous equations in the six unknowns A, h, k, e, p, a,
so choose values for these as to make the coefficients in the equation
equal to those in any equation of the second degree which may be
given. There is no failure of this statement in the special case
when the given equation represents two straight lines, as in § 10,
but there is speciality; if the two lines intersect, the intersection
and either bisector of the angle between them are a focus and
directrix; if they are united in one line, any point on the line and a
perpendicular to it through the point are: if they are parallel,
the case is a limiting one in which e and h2+k2 have become infinite
while (+) remains finite. In the case (89) of an equation
such as represents a circle there is another instance of proceeding
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
has gone to infinity, having no special direction. This last fact
illustrates the necessity, which is also forced on plane geometry by
three-dimensional considerations, of treating all points at infinity
in a plane as lying on a single straight line.

Sometimes, in reducing an equation to the above focus and directrix form, we find for h, k, e, p, tan a, or some of them, only imaginary values, as quadratic equations have to be solved; and we have in fact to contemplate the existence of entirely imaginary conics. For instance, no real values of x and y satisfy x+2y+3=0. Even when the locus represented is real, we obtain, as a rule, four sets of values of h, k, e, p, of which two sets are imaginary; a real conic has, besides two real foci and corresponding directrices, two others that are imaginary.

In oblique as well as rectangular coordinates equations of the second degree represent conics.

19. The three Species of Conics.-A real conic, which does not degenerate into straight lines, is called an ellipse, parabola or hyperbola according as e<,=, or >1. To trace the three forms it is best so to choose the axes of reference as to simplify their equations.

In the case of a parabola, let 2c be the distance between the given focus and directrix, and take axes referred to which these are the point (c, o) and the line x=-c. The equation becomes (x−c)2+y= (x+c)2, ie. y2 = 4cx.

In the other cases, take a such that a(ee) is the distance of focus from directrix, and so choose axes that these are (ae, o) and xae1. thus getting the equation(x-ae)2+y=e2(x — ae ̄1)a‚i.e. (1 − e2)x2+y2= a2(1-e). When e<1, ie. in the case of an ellipse, this may be written x/a2+y2/1, where ba2(1-e); and when e>1, i.e. in the case of an hyperbola, x2/a2-y2/b2=1, where ¿2=a2 (e2 — 1).

In figs. 54, 55, 56 in order, conics of the three species, thus referred, are depicted.

The axes thus chosen for the ellipse and hyperbola are called the | to a fixed line Ox, determine the point: r, the numerical measure principal axes. of OP, the radius vector, and 0, the circular measure of xOP, the inclination, are called polar coordinates of P. The formulae x= 7 cos 0, y=r sin @ connect Cartesian and polar coordinates, and make transition from either system to the other easy. In polar coordinates the equations of a circle through O, and of a conic with O as focus, take the simple forms r=2a cos (-a), r(1-e cos (-a)}=1. The use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon, such curves for instance as a cos me, with m integral, and also the curves called spirals, which have equations giving r as functions of itself, and not merely of sin @ and cos . In the geometry of motion under central forces the advantage of working with polar coordinates is great.

The oblique straight lines in fig. 56 are the asymptotes x/ay/b of the hyperbola, lines to which the curve tends with unlimited

y

0

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I

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 § 6, 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 ẞ and a are those of y and x, i.e. B is positive or negative according as P lies 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 x and y 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 aa+b+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 a, 8, 7. 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+b+cy)/2A just sufficient to raise them, in each case, to the highest degree.

We call (a, B, y) trilinear coordinates, and an equation in them the trilinear equation of the locus represented. Trilinear equations are, as a rule, dealt with in their homogeneous forms. An advantage thus gained is that we need not mean by (a, ẞ, y) the actual measures of the perpendicular distances, but any properly signed numbers which have the same ratio two and two as these distances.

FIG. 56.

closeness as it goes to infinity. The hyperbola would have an equation of the form xy=c if referred to its asymptotes as axes, the coordinates being then oblique, unless a=b, in which case the hyperbola is called rectangular. An ellipse has two imaginary asymptotes. In particular a circle x2+ya, a particular ellipse, has for asymptotes the imaginary lines xy-1. These run from the centre to the so-called circular points at infinity.

The con

20. Tangents and Curvature.-Let (x, y) and (x'+h, y′+k) be two neighbouring points P, P' on a curve. The equation of the line on which both lie is hy-y')=k(x-x'). Now keep P fixed, and let P' move towards coincidence with it along the curve. necting line will tend towards a limiting position, to which it can never attain as long as P and P' are distinct. The line which occupies this limiting position is the tangent at P. Now if we subtract the equation of the curve, with (x', y') for the coordinates in it, from the like equation in (x+h, y+k), we obtain a relation in and k, which will, as a rule, be of the form o= Ah+Bk+ terms of higher degrees in h and k, where A, B and the other coefficients involve x and y. This gives k/h-A/B+ terms which tend to vanish as h and k do, so that A: B is the limiting value tended to by kh. Hence the equation of the tangent is B(y-y)+A(x−x')=0. The normal at (x, y) is the line through it at right angles to the tangent, and its equation is A(y-y)-B(x-x)=0.

In the case of the conic (a, b, c, f, g, h) (x, y, 1)2=0 we find that A/B (ax+hy+g)/(hx′+by+f).

=

We can obtain the coordinates of Q, the intersection of the normals QP, QP' at (x', y') and (x'+h, y+k), and then, using the limiting value of kh, deduce those of its limiting position as P' moves up to P. This is the centre of curvature of the curve at P (x, y), and is so called because it is the centre of the circle of closest contact with the curve at that point. That it is so follows from the facts that the closest circle is the limit tended to by the circle which touches the curve at P and passes through P', and that the arc from P to P' of this circle lies between the circles of centre Q and radii QP, QP', which circles tend, not to different limits as P moves up to P, but to one. The distance from P to the centre of curvature is the radius of curvature.

In place of a, B, y it is lawful to use, as coordinates specifying the position of a point in the plane of a triangle of reference ABC, any given multiples of these. For instance, we may use x=Ga/2A, y=bB/2A, z=cy/2A, the properly signed ratios of the triangular areas PBC, PCA, PAB to the triangular area ABC. These are called the areal coordinates of P. In areal coordinates the relation which enables us to make any equation homogeneous takes the simple form x+y+1; and, as before, we need mean by x, y, z, in a homogeneous equation, only signed numbers in the right ratios.

Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? Let the point of areal coordinates (x, y, z) move infinitely far off, and mean by x, y, z finite quantities in the ratios which x, y, tend to assume as they become infinite. The relation x+y+2=1 gives that the limiting state of things tended to is expressed by x+y+2=0. This particular equation of the first degree is satisfied by no point at a finite distance; but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity. Accordingly the special property of these points is expressed by saying that they lie on a special straight line, of which the areal equation is x+y+3=0. In trilinear coordinates this line at infinity has for equation da+b+

cy=0.

On the one special line at infinity parallel lines are treated as meeting. There are on it two special (imaginary) points, the circular points at infinity of § 19, through which all circles pass in the same sense. In fact if S-O be one circle, in areal coordinates, S+(x+y+z) (lx+my+nz)=0 may, by proper choice of l, m, n, be made any other; since the added terms are once lx+my+ns, and have the generality of any expression like a'x+b'y+c' in Cartesian coordinates. Now these two circles intersect in the two points where either meets x+y+z-o as well as in two points on the radical axis lx+my+ns=0.

24. Let us consider the perpendicular distance of a point (a', B', y) from a line la+ms+ny. We can take rectangular axes of Cartesian coordinates (for clearness as to equalities of angle it is best to choose an origin inside ABC), and refer to them, by putting expressions p-x cos 0-y sin 0, &c., for a &c.; we can then apply § 16 to get the perpendicular distance; and finally revert to the trilinear notation. The result is to find that the required distance is (la'+mẞ'+ny')/{l, m, n}, where l, m, n}2 = P2+m2+n2−2mn cos A-2nl cos B-2lm cos C. In areal coordinates the perpendicular distance from (x', y', ')

If

P

21. Differential Plane Geometry.—The language and notation of the differential calculus are very useful in the study of tangents and curvature. Denoting by (,) the current coordinates, we find, as above, that the tangent at a point (x, y) of a curve is -y= (x)dy/dx, where dy/dx is found from the equation of the curve. If this be f(x, y) =o the tangent is (§−x) (əƒ{dx)+(n−y) (aƒ/ǝy)=0. and (a, 8) are the radius and centre of curvature at (x, y), we find that q(a-x)=-p(1+p2),q(B−y) = 1+p2, q2p=(1+p2)3, where p, q denote dy/dx, dy/dx respectively. (See INFINITESIMAL CALCULUS.)

In any given case we can, at all events in theory, eliminate x, y between the above equations for a-x and B-y, and the equation of the curve. The resulting equation in (a, 8) represents the locus of the centre of curvature. This is the evolute of the curve. 22. Polar Coordinates.-In plane geometry the distance of any point P from a fixed origin (or pole) O, and the inclination xOP of OP

to lx+my+n=0 is 2A(lx′+my′+n2')/{al, bm, cn}. ` In both cases the coordinates are of course actual values.

Now let E,, be the perpendiculars on the line from the vertices A, B, C, .e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in accord with a convention that oppositeness of sign implies distinction between one side of the line and the other. Three applications of the result above give

E/l=2A/{al, bm, cn} =n/m=5/n;

and we thus have the important fact that x+ny'+' 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 E, n, 's respectively. We have also that Ex+y+z=o is the areal equation of the line on which the perpendiculars are ,, ; and, by equating the two expressions for the perpendiculars from (x', y', s') on the line, that in all cases (a, b, c)=442.

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. Systems of line-coordinates will here be only briefly considered. Every such system is allied to some system of point-coordinates; and space will be saved by giving prominence to this fact, and not recommencing ab initio.

Suppose that any particular system of point-coordinates, in which Ix+my+nz=0 may represent any straight line, is before us: notice that not only are trilinear and areal coordinates such systems, but Cartesian coordinates also, since we may write x/2, y/z for the Cartesian x, y, and multiply through by . The line is exactly

assigned if l, m, n, or their mutual ratios, are known. Call (1, m, n) the coordinates of the line. Now keep x, y, s constant, and let the coordinates of the line vary, but always so as to satisfy the equation. This equation, which we now write xl+ym+zn=0, is satisfied by the coordinates of every line through a certain fixed point, and by those of no other line; it is the equation of that point in the linecoordinates l, m, n.

Line-coordinates are also called tangential coordinates. A curve is the envelope of lines which touch it, as well as the locus of points 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 touches some curve, and represents that curve, regarded as an envelope. For instance, the condition that the line of coordinates (m, n), i.e. the line of which the allied point-coordinate equation is lx+my+nzo, may touch a conic (a, b, c, f, g, h) (x, y, z)=0, is readily found to be of the form (A, B, C, F, G, H), m, no, i.e. to be of the second degree in the line-coordinates. It is not hard to show that the general equation of the second degree in 1, m, n thus represents a conic; but the degenerate conics of line-coordinates are not line-pairs, as in point-coordinates, but point-pairs.

The degree of the point-coordinate equation of a curve is the order of the curve, the number of points in which it cuts a straight line. That of the line-coordinate equation is its class, the number of tangents to it from a point. The order and class of a curve are generally different when either exceeds two.

26. The system of line-coordinates allied to the areal system of point-coordinates has special interest.

The l, m, n of this system are the perpendiculars E, 7, of § 24; and '+y+'s-o is the equation of the point of areal coordinates (x, y, z), ie. is a relation which the perpendiculars from the vertices of the triangle of reference on every line through the point, but no other line, satisfy. Notice that a non-homogeneous equation of the first degree in E, 7, does not, as a homogeneous one does, represent a point, but a circle. In fact x'+y'n+z's R expresses the constancy of the perpendicular distance of the fixed point x'+y'nt to from the variable line (, 7, 5), i.e. the fact that (E, 7, 5) touches a circle with the fixed point for centre. The relation in any which enables us to make an equation homogeneous is not linear, as in point-coordinates, but quadratic, viz. it is the relation (ag, bn, c=44 of 24. Accordingly the homogeneous equation of the above circle is 442(x′ê+y'n+z′s)2=R2{a§, bn, c}}2.

Every circle has an equation of this form in the present system of line-coordinates. Notice that the equation of any circle is satisfied by those coordinates of lines which satisfy both x+y+25=0, the equation of its centre, and (ag, bn, co. This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass. There is strict duality in descriptive geometry between point-linelocus and line-point-envelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary.

II. Solid Analytical Geometry.

27. Any point in space may be specified by three coordinates. We consider three fixed planes of reference, and generally, as in all

that follows, three which are at right angles two and two. They intersect, two and two, in lines x'Ox, y'Öy, z'Oz, called the axes of x, y, z respectively, and divide all space into eight parts called octants. If from any point P in space we draw PN parallel to Oz' to meet the plane xOy in N, and then from N draw NM parallel

M

N

N
FIG. 58.

FIG. 57.

to yoy' to meet x'Ox in M, the coordinates (x, y, z) of P are the
numerical measures of OM, MN, NP; in the case of rectangular
coordinates these are the perpendicular distances of P from the three
planes of reference. The sign of each coordinate is positive or
negative as P lies on one side or the other of the corresponding
plane. In the octant delineated the signs are taken all positive.
to the plane 20x, the points of a solid figure being projected on that
In fig. 57 the delineation is on a plane of the paper taken parallel
plane by parallels to some chosen line through in the positive
octant. Sometimes it is clearer to delineate, as in fig. 58, by pro-
jection parallel to that line in the octant which is equally inclined to
Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is
possible by parallel projection to delineate equal scales along Ox,
Oy, Oz by scales having any ratios we like along lines in a plane
having any mutual inclinations we like.

suffices to delineate the sections by the coordinate planes; and, in
For the delineation of a surface of simple form it frequently
particular, when the surface has symmetry about each coordinate
plane, to delineate the
quarter-sections
longing to a single
octant. Thus fig. 59
conveniently

be

repre-
sents an octant of the
wave surface, which
cuts each coordinate
plane in a circle and
an ellipse. Or we may
delineate a series of

contour lines, i.e. sec-
tions by planes parallel
to xOy, or some other
other sections may be
chosen plane; of course

for

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FIG. 59.

as

indicated too
greater clearness. For
the delineation of a
curve a good method
above, a series of points
is to represent,
P thereof, each accompanied by its ordinate PN, which serves to
refer it to the plane of xy. The employment of stereographic
projection is also interesting.

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
reckoning a line as a curve of the first order, and the plane as a surface
but the increase of complexity is far greater than would hence at
of the first order, we have the point, the curve and the surface;
first sight appear. In plane geometry a curve is considered in
connexion with lines (its tangents); but in solid geometry the curve
is considered in connexion with lines and planes (its tangents and
osculating planes), and the surface also in connexion with lines and
planes (its tangent lines and tangent planes); there are surfaces
arising out of the line-cones, skew surfaces, developables, doubly
and triply infinite systems of lines, and whole classes of theories
which have nothing analogous to them in plane geometry: it is thus
a very small part indeed of the subject which can be even referred

to present article.

In the case of a surface we have between the coordinates (x, y, z) a single, or say a onefold relation, which can be represented by a single relation f(x, y, z) =0; or we may consider the coordinates expressed each of them as a given function of two variable parameters p, q; the form z=f(x, y) is a particular case of each of these modes of representation; in other words, we have in the first mode f(x, y, z) =z-f(x, y), and in the second mode x=p, y=q for the expression of two of the coordinates in terms of the parameters.

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