7. To find the co-ordinates of the point cutting in a given ratio mn, the line joining two given points x'y', x"y".

Let x, y be the co-ordinates of the point R which we seek to determine, then

If the line were to be cut externally in the given ratio, we

It will be observed that the formulæ for external section are obtained from those for internal section by changing the sign of the ratio: that is, by changing m:+n into m: -n. In fact, in the case of internal section, PR and RQ are measured in the same direction, and their ratio (Art. 6) is to be counted as positive. But in the case of external section PR and RQ are measured in opposite directions, and their ratio is negative.

Ex. 1. To find the co-ordinates of the middle point of the line joining the points x'y', x"y". x' + x" y'+y" Ans. x =

2

Ex. 2. To find the co-ordinates of the middle points of the sides of the triangle, the co-ordinates of whose vertices are (2, 3), (4, — 5), (— 3, -6).

Ex. 3. The line joining the points (2, 3), (4, 5), is trisected; to find the coordinates of the point of trisection nearest the former point. Ans. x, y = },

Ex. 4. The co-ordinates of the vertices of a triangle being x'y', x"y", x""y"", to find the co-ordinates of the point of trisection (remote from the vertex) of the line joining any vertex to the middle point of the opposite side.

Ans. x = }(x' + x' + x'''); y = } (y' + y′′ + y′′'').

Ex. 5. To find the co-ordinates of the intersection of the bisectors of sides of the triangle, the co-ordinates of whose vertices are given in Ex. 2. Ans. x=1, y = — §.

Ex. 6. Any side of a triangle is cut in the ratio m: n, and the line joining this to the opposite vertex is cut in the ratio m +n: 7; to find the co-ordinates of the point of section. lx' + mx" + nx"

Ans. x =

1 + m + n

, y=

ly' + my" + ny"" l + m + n

TRANSFORMATION OF CO-ORDINATES.*

8. When we know the co-ordinates of a point referred to one pair of axes, it is frequently necessary to find its coordinates referred to another pair of axes. This operation is called the transformation of co-ordinates.

We shall consider three cases separately: first, we shall suppose the origin changed, but the new axes parallel to the old; secondly, we shall suppose the directions of the axes changed, but the origin to remain unaltered; and thirdly, we shall suppose both origin and directions of axes to be altered. First. Let the new axes be parallel to the old.

Let Ox, Oy be

the old axes, O'X, O'Y the new axes. Let the co-ordinates of the new origin referred to the old be x', y', or O'S = x', O'R=y'. Let the old co-ordinates be x, y, the new X, Y, then we have

that is,

OM=OR+RM, and PM=PN+ NM,

x=x+X, and y=y'+Y.

These formulæ are, evidently, equally true, whether the axes be oblique or rectangular.

9. Secondly, let the directions of the axes be changed, while the origin is unaltered.

*The beginner may postpone the rest of this chapter till he has read to the end of Art. 41.

y

Let the original axes be Ox, Oy, so that we have OQ=x, PQ=y. Let the new axes be OX, OY, so that we have ON=X, PN=Y. Let OX, OY make angles respectively a, B, with the old axis of x, and angles a', B' with the old axis of y; and if the angle x Oy between the old axes be w, we have obviously a+a'=w,

S

R Q

M

since XOx+XOy=xOy; and in like manner ẞ + B' = w.

The formulæ of transformation are most easily obtained by expressing the perpendiculars from P on the original axes, in terms of the new co-ordinates and the old. Since

sinw.

PM=PQ sin PQM we have PM= y sin w.

But also PM=NR+PS=ON sin NOR+ PN sin PNS.

or

Hence y sin∞ = X sina + Y sin ß.

In like manner

x sin∞ = X sina' + Y sinß';

x sin∞ = X sin(-a) + Y sin (w-B).

In the figure the angles a, B, w are all measured on the same side of Ox; and a', B', w, all on the same side of Oy. If any of these angles lie on the opposite side it must be given a negative sign. Thus, if OY lie to the left of Oy, the angle B is greater than w, and B' (=w-B) is negative, and therefore ß) the coefficient of Y in the expression for x sino is negative. This occurs in the following special case, to which, as the one which most frequently occurs in practice, we give a separate figure.

To transform from a system of rectangular axes to a new rectangular system making an angle with the old.

the truth of which may also be seen directly, since y = PS+ NR, x=OR- SN, while

PS=PN cos 0, NR=ON sinė; OR=ON cose, SN= PN sin 0.

There is only one other case of transformation which often occurs in practice.

To transform from oblique co-ordinates to rectangular, retaining the old axis of x.

We may use the general formulæ making

a=0, B=90, a' = w, B'=w-90.

But it is more simple to investigate the formulæ directly. We have OQ and PQ for the old x and

R

N

P

M

y, OM and PM for the new; and since PQM=w, we have

Y=y sino, X=x+y cosw:

while from these equations we get the expressions for the old co-ordinates in terms of the new

10. Thirdly, by combining the transformations of the two preceding articles, we can find the co-ordinates of a point referred to two new axes in any position whatever. We first find the co-ordinates (by Art. 8) referred to a pair of axes through the new origin parallel to the old axes, and then (by Art. 9) we can find the co-ordinates referred to the required axes.

The general expressions are obviously obtained by adding x' and y' to the values for x and y given in the last article.

х

what will this become if the origin be transformed to the point (2, 3) ?

Ans. X2+ Y2 = 31. Ex. 2. The co-ordinates of a point to one set of rectangular axes satisfy the relation y2 — x2 6: what will this become if transformed to axes bisecting the angles between the given axes? Ans. XY 3. Ex. 3. Transform the equation 2x2 - 5xy + 2y2 = 4 from axes inclined to each other at an angle of 60°, to the right lines which bisect the angles between the given axes. Ans. X2-27 Y2 + 12 = 0.

Ex. 4. Transform the same equation to rectangular axes, retaining the old axis of x. Ans. 3X2+10Y27XY J3 = 6.

Ex. 5. It is evident that when we change from one set of rectangular axes to another, x2 + y2 must = X2 + Y2, since both express the square of the distance of a point from the origin. Verify this by squaring and adding the expressions for X and Y in Art. 9.

Ex. 6. Verify in like manner in general that

x2 + y2+ 2xy cosxOy = X2 + Y2 + 2XY cos XOY.

If we write X sin a + Y sinẞ=L, X cos a + Y cosẞ= M, the expressions in Art. 9 may be written y sin w = L, x sin∞ = M sin w - L cosa; whence

But

sin2 w (x2 + y2+ 2xy cos w) = (L2 + M2) sin2 w.

L2+ M2 = X2 + y2+ 2XY cos (a - ẞ); and a β = ΧΟΥ.

11. The degree of any equation between the co-ordinates is not altered by transformation of co-ordinates.

m

Transformation cannot increase the degree of the equation: for if the highest terms in the given equation be x, y, &c., those in the transformed equation will be

{x' sin w+x sin(-a)+y sin(w−B)}", (y' sinw+x sina+y sin ß)”, &c., which evidently cannot contain powers of x or y above the mth degree. Neither can transformation diminish the degree of an equation, since by transforming the transformed equation back again to the old axes, we must fall back on the original equation, and if the first transformation had diminished the degree of the equation, the second should increase it, contrary to what has just been proved.

POLAR CO-ORDINATES.

12. Another method of expressing the position of a point is often employed.

If we were given a fixed point O, and a fixed line through it

P

OB, it is evident that we should know the position of any point P, if we knew the length OP, and also the angle POB. The line OP is called the radius vector; the fixed point is called the pole; and this method is called the method of polar coordinates.

θ

B

It is very easy, being given the x and y co-ordinates of a point, to find its polar ones, or vice versâ.

C