or finally it is Ax+By+C-0, showing that the point Q lies in a lins 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-0 just obtained is in fact the line joining the point Q with the point R which is the intersection of AB and A'B'.

10. The equation Ax+ By +C=0 of a line contains in appearance 3, but really only 2 constants (for one of the constants can be divided out), and a line depends accordingly upon 2 parameters, or can be made to satisfy 2 conditions. Similarly, the equation (a, b, c, f, g, hxx, y, 1)2=0 of a conic contains really 5 constants, and the equation (*)(x, y, 1)3 = 0 of a cubic contains really 9 constants. 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 cubic curves S=0, S'= 0, these intersect in 9 points, and through these 9 points we have the whole series of cubics 8-kS'-0, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (k-So÷So, if the coordinates are (, ), and So, So 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 (o, 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.

Metrical Theory.

And again, by projecting on a line Q1, inclined at the angle a' to QR, we have p cos (a-a') - Cos a +7 sin a'; and by substituting for ¿, n their foregoing values, cos (a- a) cos a cos a' + sin a sin a'.

It is to be remarked that, assuming only the theory of similar triangles, we have herein a proof of Euclid, Book I., Prop 47; in fact, the same as is given Book VI., Prop. 31; and also a proof of the trigonometrical formula for cos(a-a'). The formulæ for cos(a + a') and sin (a + a') could be obtained in the same manner.

spectively, so that we have now the quadrilateral QRPTQ, or, what Draw PT at right angles to Qr, and suppose QT, TP-1, ni reis the same thing, the two broken lines QRP and QTP, each extending from Q to P. Projecting on the four sides successively, we

have

where the third equation is that previously written
pcos (a-a)- cos a+ sin a.

Equations of Right Line and Circle.-Transformation
of Coordinates..

12. The required formulæ are really contained in the foregoing results. For, in fig. 11, supposing that the axis of x is parallel to QR, and taking a, b for the coordinates of Q, and (x, y) for those of P, then we have

x-up cos a, y-b-p sin
p2 = (x-a)3 + (y − b)2.'
Writing the first two of these in the form
x-α y-b
cos a sin a

a,

11. The foundation of the metrical theory consists in the simple theorem that if a finite line PQ (fig. 10) be projected, n=x-a, y-b respectively; and therefore upon any other line 00' by lines perpendicular to 00', then the length of the projection P'Q' is equal to the length of PQ into the cosine of its inclination to P'Q'; or, what is the same thing, that the perpendicular distance P'Q' of any two parallel lines is equal to the inclined distance PQ into the cosine of the inclination. It at once follows that the algebraical sum of the projections of the sides of a closed polygon upon any line is = 0; or, reversing the signs of certain sides, and considering the

(-p),

we may regard Q as a fixed point, but P as a point moving in the direction to P, so that a remains constant, and then, omitting the equation (p), we have a relation between the coordinates ,y of the point P thus moving in a right line, that is, we have the equation of the line through the given point (a, b) at a given

inclination a to the axis of x. And, moreover, if, using this equa-
tion (p), we write x-a+p cos a, y=b+p sin a, then we have
expressions for the coordinates x, y of a point of this line, in terins
of the variable parameter
Again, take the point f
to be fixed, but consider the point P as
moving in the line TP at right angles to QT. If instead of we
take p for the distance QT, then the equation - cos a' + sind
will be
x-a) cos a' + (y - b) sin a' -p;

that is, this will be the equation of a line such that its perpendi-
cular distance from the point (a, b) is-p, and that the inclination
(of this distance to the axis of x is

a

down we can, by giving to a a proper value (in fact = 0), make the (x − a) and (y-b) equal to 201 and /1 respectively; in the other system we could only mako them equal to x1,y1, or -x, y, respectively. But for the very reason that the second system can be so easily derived from the first, it is proper to at end exclusively to the first system,—that is, always to take the new axes to that the two sets adinit of being brought into coincidence. In the foregoing system of two pairs of equations, the first pair give the original coordinates x, y in terms of the new coordinates 1, 1; the second pair the new co

The formule involve (a, b), the original coordinates of the now origin; it would be easy instead of these to introduco (a, b), the new coordinates of the origin. Writing (a, b) = (0, 0), we have, of course, the formule for trans formation between two sets of rectangular axes haring the same origin, and it is as well to write the formulæ in this more simple form: the subsequent transformation to s new origin, but with axes parallel to the original axes, can then be effected without any difficulty.

From either form it appears that the equation of a line is, in fact, a linear equation of the form Ax+By+C=0.ordinates x, y, in terms of the original coordinates r, y. It is important to notice that, starting from this equation, we can determine conversely tho a but not the (a, b) of the form of equation which contains these quantities; and in like manner the a' but not the (a, b) or p of the other form of equation. The reason is obvious. In each case (a, b) denote the coordinates of a point, fixed indeed, but which is in the first form any point of the line, and in the second form any point whatever. Thus, in the second form the point from which the perpendicular is let fall may be the origin. Here (a, b) = (0, 0), and the equation is x cos a' + y sin a' - p = 0. Comparing this with Ax+By+C=0, we have the values of cos a', sin a', and p.

13. The equation

each set being obviously at once deducible from the other one. In these formule (a, b) are the xy-coordinates of the new origin Q,, and a is the inclination of Qx, to Ox. It is to be noticed that Q, Q are so placed that, by moving O to Q, and then turning the axes Ox1, Oy, round Q (through an angle a measured in the sense Ox to Oy), the original axes Or, Oy will come to coincide with QQ respectively. This could not have been done if Qy had been drawn (at right angles always to Qx1) in the reverse direction, we should then have had in the formulay instead of 1. The new formule which would be thus obtained are of an essentially distinct form the analytical test is that in the formulae as written

15. All questions in regard to the line may be solved by means of one or other of the foregoing formsAx+By+C-0, y-Ax+B,

y-b 08 a sin a

A

B

representing the line through the points (a, 0) and (0, b), or, what is the same thing, the line moeting the axes of x and y at the distances from the origin a and b respectively. lt may be noticed that, in the form Az+By+C=0, denotes the tangent of the inclination to the axis of z, or we may say that B÷ √Ã2+12 and − A÷ √A2+B2 denote respectively the cosine and the sine of the inclination to the axis of x. A better form is this: A+ A+D and B+ A2+ B2 denote respectively the cosine and the sine of inclination to the axis of x of the perpendicular upon the line. So of course, in regard to the form y = Ar+B, A is here the tangent of the inclination to the axis of r; 1÷√A2+1 and A÷A+1 are the cosine and sine of this inclination, &c. It thus appears that the condition in order that the lines Ax + By +CO and A'z + B'y-C' = 0 may meet at right angles is AA'+ BB'=0; so when the equations are y=Ax+B, y=A'r + B', the condition is AA'+1=0, or say the value of A' is - 1 A..

=

The perpendicular distance of the point (a, b) from the line Ax+By+C=0 is (Aa + Bb + C) ÷ √A+B. In all the formulæ involving A+B or A2+1, the mdical should be written with the sign, which is essentially indeterminate: the like indeterminateness of sign presents itself in the expression for the distance of two points p= ± √ √(x − a)2 + ( y − h); if, as before, the points are Q, P, and the indefinite line through these is 'Q'z, then it is the same thing whether we measure off from Q along this line, considered as drawn from z' towards a, a positive

Supposing positive and <1, then, writing m
equation becomes
that is,

"(1-6), the

(1 − c2)(x − a )2 + y2 — a2(1 − c2),

บริ a1 a2(1 −

distance k, or along the line considered as drawn reversely
from a towards z, the equal negative distance – k, and the
expression for the distance p is thus properly of the form
It is interesting to compare expressions which do
not involve a radical: thus, in seeking for the expression
for the perpendicular distance of the point (a, b) from a
given line, let the equation of the given line be taken or, changing the origin and writing ¿a—a2(1 − e2), this is
in the form, x cos a+ y sin a-p=0 (p being the perpen-
dicular distance from the origin, a its inclination to the
axis of x): the equation of the line may also be written
(-a) cos a+(y-b) sin a-p1-0, and we have thence
PP-u cos a-b sin a, the required expression for the
distance : it is here assumed that p, is drawn from
(b) in the sante sense as p is drawn from the origin,
and the indeterminateness of sign is thus removed.

16. As an instance of the mode of using the formulæ, take the problem of finding the locus of a point such that its distance from a given point is in a given ratio to its distance from a given line.

We take (a, b) as the coordinates of the given point, and it is convenient to take (,) as the coordinates of the variable point, the locus of which is required: it thus becomes necessary to use other letters, sy (X, Y), for current coordinates in the equation of the given line. Suppose this is a line such that its perpendicular distance from the origin is = p, and that the inclination of p to the axis of x is a; the equation is X cos a+ Y sin a-p=0. In the result obtained in § 15, writing (x, y) in place of (4, 6), it appears that the perpendicular distance of this line from the point (r, y) is

The Conics (Parabola, Ellipse, Hyperbola).

17. The conics or, as they were called, conic sections were originally defined as the sections of a right circular cone; but Apollonius substituted a definition, which is in fact that of the last example: the curve is the locus of a point such that its distance from a given point (called the focus) is in a given ratio to its distanco from a given line (called the directrix) taking the ratio as 1, then e is called the eccentricity.

Take FD for the perpendicular from the focus F upon the diTetrix, and the given ratio being that of e: 1 (e>, - or < 1,

at positive), and let the distance FD be divided at O in the given

ratio, say

wo have OD—m,

OFcm, where is positive;

y

then the origin may be taken at O, the axis Ox being in the direc
tion OF (that is from 0 to F), and the axis Oy at right angles to it.
The distance of the point (x, y) from F is =√(x-emp+y, its
distance from the directrix is -+m; the equation therefore is
(x − em)2 + y2 = e2 (x + m )2 ; '

or, what is the same thing, it is

which is the ellipse.

which is the hyperbola.

=

_a(e-1),

18. The general equation ax2 + 2hxy+by2 + 2ƒy + Qys + c = 0, or as it is written (a, b, c, f, I, k)(x, y, 1)2 = 0, may be such that the quadric function breaks up into factors, — (ax+ By + y)(a'x +By+y); and in this case the equation represents a pair of lines, or (it may be) two coincident lines. When it does not so break up, the function can be put in the form A{(x − u' )2 + (y − b')2 — e2(x cosa + y sina - p)}, or, equating the two expressions, there will be six equa tions for tho determination of A, a, b, e, p, a; and by what precoles, if ', b', e, p, a are real, the curve is either a parabola, ellipse, or hyperbola. The original coefficients (a, b, c, f, g, h) may be such as not to give any system of real values for a' b', e, p, a ; but when this is so the equation (a, b, c, f, g, h)(x, y, 1)2 = 0 does not represent a real curve; the imaginary curve which it represents is, however, regarded as a conic. Disregarding the special cases of the pair of lines and the twice repeated line, it thus appears that the only real curves represented by tho general equation (a, b, c, f, g, h)(x, y, 1) 0 are the parabola, the ellipse, and the hyperbola. The circle is cou sidered as a particular case of the ellipse.

The same result is obtained by transforming the equation (a, b, c, f, g, h)(x, y, 1)2=0 to new axes. If in the first place the origin be unaltered, then the directions of the new (rectangular) axes Ox, Oy, can be found so that h (the coefficient of the term x,y) shall be 0; when this is done, then either one of the coefficients of x2, y2 is and the curve is then a parabola, or neither of these coofficients is 0, and the curve is then an ellipse or hyperbola, according as the two coefficients are of the same sign or of opposite signs.

and it will be noticed how the form of the last equation
puts in evidence the two asymptotes
hyperbola. Referred to the asymptotes (as a set of

It is proper to remark that, when (a, b, c, f, g, h) (x, y, 1)3 −0 does represent a real curve, there are in fact fouf systems of values of a', b', e, p, a, two real, the other two imaginary; we have thus two real equations and two imaginary equations, each of them of the form (x -- a')3 + (y − b′)3 (cos a + y cos 8-p), representing each of them one and the same real curve. This is, consistent with the assertion of the text that the real curve is in every case represented by a real quation of this form.

dy(dx2 + dy2)
- dr(dx3 + dy3)
dxd3y - dyd2x, Y - B~ dxď1y-dyď2x

(dx2 + dy2)3
, -(-a)2+(y-B)", - (dx d'y - dy d'x)*

In the case where y is given directly as a function of 1,
writing for shortness p
(1 + p2) 3

then,

dy

dzy

=

=

dx

, this is dx2'

72 , or, as the equation is usually written,

Tangent, Normal, Circle and Radius of Curvature, &c. 20. There is great convenience in using the language and and we then have notation of the infinitesimal analysis; thus we consider on a curve a point with coordinates (x, y), and a consecutive point the coordinates of which are (x + dx, y+dy), or again a second consecutive point with coordinates (x + dx + {d2x, y+dy+}2y), &c.; and in the final results the ratios of the infinitesimals must be replaced by differential coefficients in the proper manner; thus, if x, y are considered as given functions of a parameter 0, then dx, dy dx dy have in fact the values de, de, and (only the ratio do do being really material) they may in the result be replaced dx dy by This includes the case where the equation de' de' of the curve is given in the form y = p(x); 0 is here =X, and the increments de, dy are in the result to be replaced by 1, dx' So also with the infinitesimals of the higher orders d2x, &c.

(1+p2)?

the radius of curvature, considered to be positive or negative according as the curve is concave or

convex to the axis of r.

It may be added that the centre of curvature is the intersection of the normal by the consecutive normal.

from the expressions of a, ẞ regarded as functions of w eliminate x, we have thus an equation between (a, B), which is the equation of the evolute.

The locus of the centre of curvature is the evolute. If

Polar Coordinates.

23. The position of a point may be determined by means of its distance from a fixed point and the inclination of this distance to a fixed line through the fixed point.

an equation which is satisfied on writing therein έ, n= (x, y) | Say we have r the distance from the origin, and the or = (x+dx, y+dy). The equation may be written

inclination of to the axis of x; r and are then the polar coordinates of the point, r the radius vector, and the inclination. These are immediately connected with the Cartesian coordinates x, y by the formulæ x=r cose,

y=r sin ; and the transition from either set of coordinates to the other can thus-be made without difficulty. But the use of polar coordinates is very convenient, as well in reference to certain classes of questions relating to curves of any kind-for instance, in the dynamics of central forces-as in relation to curves having in regard

dy being now the differential coefficient of y in regard to x; and this form is applicable whether y is given directly as a function of x, or in whatever way y is in effect given as a function of x: if as before x, y are given each of them as dy dy dx a function of 0, then the value of is - ÷ which " do do is the result obtained from the original form on writing to the origin the symmetry of the regular polygon (curves

dx

therein

dx dy
de' de'

for dx, dy respectively.

such as that represented by the equation r = cos me), an also in regard to the class of curves called spirals, where

the radius vector r is given as an algebraical or exponential | the three pianes respectively, each distance being considered as positive or negative according as it lies on the one function of the inclination 0.

General Point-Coordinates.-Line-Coordinates. 25. All the coordinates considered thus far are pointcoordinates. More generally, any two quantities (or the ratios of three quantities) serving to determine the position of a point in the plane may be regarded as the coordinates of the point; or, if instead of a single point they determine a system of two or more points, then as the coordinates of the system of points. But, as noticed under CURVE, there are also line-coordinates serving to determine the position the ordinary case is when the line is determined by means of the ratios of three quantities έ, n, (correlative to the trilinear coordinates x, y, z). A linear equation a+by+c=0 represents then the system of lines such that the coordinates of each of them satisfy this relation, in fact, all the lines which pass through a given point; and it is thus regarded as the line-equation of this point; and generally a homogeneous equation (*,)=0 represents the curve which is the envelope of all the lines the coordinates of which satisfy this equation, and it is thus regarded as the line-equation of this curve.

IL SOLID ANALYTICAL GEOMETRY (§§26–40).

or the other side

of the plane. Thus
the coordinates in
the eight octants
have respectively
the signs

position of a point being determined by its three coordi. the point to the plane of xy; this is in effect a representasit. We are here concerned with points in space, the accompanied by the ordinates PN, which serves to refer We consider three coordinate planes, at tion of each point P of the curve, by means of two points into eight portions called octants, the coordinates of a right angles to each other, dividing the whole of space P, N such that the line PN has a fixed direction. point being the perpendicular distances of the point from graphic representations is