The distance CS = CH = c, is called the eccentricity of the hyperbola, and the ratio c to a is usually designated by the symbol e; hence we have

If we wish to obtain the equation to the hyperbola referred to the vertex A, as the origin of co-ordinates, since this new origin is situated on the axis of x's, at a distance + a from the former origin; if we substitute (x + a) for x in equation (D), we obtain

which is the equation required.

y2 =

b2 a2

(2ax + x)....

..... (F)

On the Transformation of Co-ordinates.

When we reflect upon the equations to the straight line and circle, and consider the different forms which these equations assume according to the different positions of these lines with regard to the axes of co-ordinates; we perceive that the same line may be represented by different equations which will be more or less simple, according as the position of the line is more or less simple relatively to the axes, and according as the axes themselves are rectangular or oblique. Thus, the most general equation to a straight line being

y = ax + b.

The equation to a straight line passing through the origin, is

y = ax;

a having in each of the above equations a different signification when the axes are oblique, from that which is attributed to it when the axes are rectangular. In like manner, the most general equation to a circle when referred to oblique

- — −

(x — xv)2+(y y')2 + 2(x — x) (y—y') cos. = p2 which becomes

x2 + y2 = r2

when the circle is referred to rectangular co-ordinates, and the centre is the origin.

It is easy to conceive, that, when the position of a curve upon a plane is fixed by means of an equation, if we perceive that the position of the curve with regard to two new straight lines, is more simple than with regard to the axes to which it is referred by the equation in question; it would greatly facilitate our investigations respecting the properties of the curve, if we could deduce an equation to the curve, referred to these new straight lines as axes, from that equation to the curve which we actually possess.

Such then is the object of the problem which is proposed in the transforma. tion of co-ordinates, which may be enunciated in its most general terms, as follows:

Given an equation to a curve referred to any two axes whatever, to find the equation to the same curve when referred to two new axes.

Before however proceeding to solve the problem in its most general form, we shall consider one or two particular cases which are of most frequent

Occurrence.

I. Let the new axes be parallel to the former ones.

Let AX, AY, be the original axes;

Let AX', AY', be the new axes parallel to the former.

Let x, y, be the co-ordinates of a point P, referred to the old axes;

Let x', y', be the co-ordinates of a point P, referred to the new axes.

Let a, ß, be the co-ordinates of the new origin A'.
Draw PM and A'N, parallel to AY.

A

N M

Then,

AM = x, MP=y, A'M' = x, PM'y', AN = ∞, A'N = 3.

If therefore, in the equation to the proposed curve, we substitute a + a for 2, and y' for y; we shall obtain a relation between x and y', which will be the equation to the curve referred to the new axes.

COR.

If the new origin be on the axis AX, then ß = 0;
If the new origin be on the axis AY, then a = 0.

II.

To pass from one system of rectangular axes, to another also rectangular.

= a + x cos. (xx') — y' sin. (xx') ·· PM'S = 90° —(xx').

MP or y MT+TS + SP

Substituting therefore these values of x and y in the equation proposed, we shall obtain a relation between ' and y', which will be the equation to the curve referred to the axes AX, AY'.

COR. If the new origin be coincident with the old, then = 0, 60, and the above equations become

x = x' cos. (xx') — y′ sin (xx')
yx sin. (xx') + y' cos. (xx1).

II.

We may now proceed to the solution of the general problem, viz.—To pass from one system of axes inclined at any given angle to another system; also inclined at any given angle.

Let AX, AY, be the original system; A'X', A'Y', the new.

In addition to the former notation,

Let the inclination of A'X' to AX be called (xx')

Let the inclination of A'Y' to

Let the inclination of AY

&c.

AX be called (xy') to AX be called (xy).

Then, AM or x = AN+ NM

&c.

=AN+AT+ M'S

MP or y MR+RS+ SP

AN+M'T + SP

YA

P

X'

sin. (xy.)

Such is the most general formula for the transformation of co-ordinates, from which it is easy to deduce the formulas corresponding to all positions of a new origin, and to the different inclinations of the new axes compared with the old ones, by giving proper values either positive or negative to a and ß, and any value to the angles (xx'), (xy'), from 0 up to 90o.

As to the angle xy, it is always given à priori, since it is the angle contained by the original axes.

We can easily deduce from the general formula, the results already obtained in cases I. II.

REMARKS.

(1). In general we distinguish between two different species of the transformation of co-ordinates; The change in the position of the origin, and the change in the direction of the axes. When the problem proposed requires this double transformation, it is frequently more advantageous to execute them in succession than at first.

(2). Since we have frequently occasion, in the same question, to effect several transformations of co-ordinates, it is convenient to suppress the accents of x', y', in the second member of the formulas which relate to these transformations; that is to say, we may designate both the old and new co-ordinates by x and y, although their values are different, but the circumstance of using the different formulæ in succession will be sufficient to point out, that the curve after having

been referred to one system of axes, is afterwards referred to a second, to z third, and so on.

Thus, in order to pass from a rectangular or oblique system to another system parallel to it, we may, in the equation to the curve, substitute x + a for æ, and y+ 3 for y, and the x and y of the second equation will represent the co-ordinates referred to the new axes, the co-ordinates of whose origin, referred to the former origin, are a, ß. In like manner we may proceed in all other cases, and thus simplify our calculations by avoiding the use of numerous accents.

(3). The quantities a, ß, (xx), (xy'), &c. which enter into the above formulas. are constants whose value fixes the position of the new origin and the direction of the new axes with reference to the original axes, whose inclination to each other is expressed by (xy). The quantities a, ß, (xx'), (xy'), &c. must be regarded as known and given a priori, whenever we wish to refer the curve to new axes whose position with regard to the proposed curve has been discovered to be more simple than that of the old axes.

It frequently happens, however, that we perform a transformation of co-ordinates when our object in so doing is to make some specific change in the form of the equation to the curve, for example, to make certain terms disappear. In this case a, B, (xx'), (xy'), &c. are constants which are, for the time being, inde terminate; and whose values we afterwards endeavour to calculate in such a manner as to simplify the equation in the manner required. With regard to the angle (xy) we cannot employ it in this manner, since it is the angle contained by the old axes, and is in every case supposed to be known a priori.

The number of terms which it is our wish to remove from the equation, will indicate the number of indeterminate quantities which we must introduce into our calculation, and therefore the system of formulas which we must employ. These remarks will be better understood when applied to particular examples.

ON POLAR CO-ORDINATES.

We have hitherto supposed the position of a curve upon a plane to be determined by means of an equation between variables, expressing the distances of each of the points in the curve from two fixed straight lines, the distances being reckoned parallel to these lines. There is, however, another method for determining the position of a point or of a series of points which in certain cases is more convenient.

To explain this mode of representing curves analytically, let us consider any curve Pp.

Let SO be a given straight line in the plane of the

curve, and S a given point in that line.

From S draw a straight line SP to any point P in the curve.

Let SP be called r, and the angle between SP and SO be é,

It is evident that, if we can obtain a relation between r and which holds good for every point in the curve, the curve will be entirely determined, for

S

f we give to a succession of values 01, 02, 0s, &c. we shall obtain from the equation between r and a series r1, 72, 71, &c. of corresponding values of r. Making therefore at the point S the angle Q2SO, Q,SO, Q,SO, &c. respectively equal to 61, 62, 03, &c. and taking SP1, SP, SP, &c. equal to the corresponding values of r, we shall obtain the points P1, P2, P., &c. which belong to the curve.

S

P3

P2

The variable quantities r and are called Polar Co-ordinates, the point S is called the Pole, r the Radius Vector, and the relation between r and is termed the Polar Equation to the curve.

A curve being traced upon a plane, we may, from some known property of the curve, determine the polar equation at once, more usually, however, we have the position of the curve determined by an equation between rectilinear co-ordinates, and it is required to deduce the equation between polar co-ordinates. This can be easily effected by a transformation of co-ordinates, which we shall now proceed to explain.

Let us begin with two of the most simple and

useful cases:

1. Let Pp be the curve whose equation is given in terms of rectangular co-ordinates, AX and AY being the axes. Let it be required to determine the polar equation, S being the pole and SO parallel to AX. Let the co-ordinates of the point Sreferred to the axes AX, AY, be a, Во

Take any point P in the curve, draw PM perpendicular to AX, join SP, draw SN perpendicular to AX.

Then,

AM = x, MP = y, SP = r, PSO = 8, AN = α, SN = ß

AM or ≈ = AN + NM

=arcos.

.........(a)

MP or y MR + RP

=ẞr sin. 8.

Substituting.. these values of x and y in the equation to the curve, we shall obtain a relation between r and which will be the polar equation required. 2. Let SO coincide with AX, the point S with A, in this case, a, ß = 0, and the above formula becomes

xr cos.

y = r sin. 0} (b).

The general problem is, given the equation to a curve referred to any system of axes, to find the polar equation; the position of the pole being any whatever. Let Pp be curve referred to the axes AX, AY. Take any point S as the pole, and let SO be the fixed straight line, and let the co-ordinates of S referred to AX, AY, be a, ß.

Through S draw Sr, Sy parallel to AX, AY. Draw PM parallel to AY, join S, P; draw SN parallel to AY.

Then, AM=x, MP=y, AN=a, SN= ß, SP =r, PSO = 0.

Let the angle between the axes AX, AY, be de

S

R

M

P

noted by (x, y,) the angle between the fixed line SO and the axis AX being 0.