stituted for x and y in the equation to a curve, answer its conditions, then two straight lines, one = X, and measured along the horizontal axis to the left of the first point, and the other Y, and measured downward from the first point along the perpendicular axis, will be co-ordinates to a point in the curve.

=

COR. 1. Hence if the equation to any curve be such, that (y) can be generally found in terms of (x), the curve may be traced, or its form ascertained. For assuming (x) of all positive values, we may find all the corresponding values of (y), and measuring the positive values of (y) above the horizontal axis, and the negative below, the form of that part of the curve which lies to the right of the perpendicular axis is traced. In the same manner assuming (x) of all possible negative values, that form of the remaining part of the curve may be known.

PROP. X.

Having given the nature and properties of a

curve, to find its equation.

Ex. 1. Let the curve be a circle, whose radius is (a), and the co-ordinates to whose centre

A

D

M

m

이

E 172

RN B

circle, take also OR, Or, ON, On co-ordinates to

the centre C, and the point P respectively;

then OR=a, Or=ß, ON=x, OM=y,

and Pn manifestly=y-B, and Cn=x-a,

but Pn2 + Cn2 = CP2,

or (y −ß)2+(x — a)2 = a2,

which is the general equation to the circle.

COR. 1. If we suppose ß=0, and a=0, or the circle be referred to two diameters of its own as axes, we have

y2+x2=a2, or y = ± √ a2 − x2.

COR. 2. If we suppose ẞ=0, but a=a, or refer the circle to two axes, one of which is a diameter FC, and the other a straight line drawn from the point F perpendicular to it, we have

y2 + (x − a)2 = a2, or y=±√√2ax - x2.

Ex. 2. Let the curve be a parabola. Let PO be a parabola, whose focus is S, referred to two axes, one of which is its diameter, and the other a perpendicular drawn to it from its vertex.

M

A

n 0 SN

B

Produce BO to m; make mO = OS, and through (m) draw mM parallel to OA; join PS, and from P draw PM perpendicular to mM, then by the nature of the parabola,

SP=PM,

now let PN=y, NO=x, OS= a;

then SP2 = SN2 + PN2 = (x − a)2 + y2,

PM2 = (x+a)2;

whence (x-a)2+y2 = (x+a)', or y2=4ax.

Ex. 3. Let the curve be an hyperbola. Let the hyperbola be referred to its major and minor axes produced, as axes. Let the major axis = 2a, the minor = 26, and distance between the foci = 2c, then calling and the distances of any point from the outer and inner force respectively, we have

z"2 = y2 + (x + c) ̊,

x2 = y2 + (x − c)2,

z-z=2a,

b2 = c2 - a2.

Whence eliminating c, x and x', we have

SECT. I.

Ir two variable quantities so depend upon each other, that when one of them is given, the other must have some determinate value, the two quantities are said to be functions of each other.

Thus if (x) and (y) be two variable quantities, so dependent upon each other, that when (x) is assumed of any particular value, there are some corresponding values, which (y) must assume, in consequence of the relation subsisting between it and (x), then (y) and (x) are said to be functions of each other.

If, in any proposition or problem, one quantity is considered as a function of another, the latter quantity is sometimes by distinction termed the root of the former.

The relation between a function and its root, may sometimes be expressed by an equation, which gives the value of the function in terms of the root. Thus, let y±√2ax-x®,

In these equations, if we suppose the quantities (a), (b), (c), (m), (n), &c. to be constant, it is manifest, that if we substitute any quantity for (x), (y) must assume some particular values, in order that the

equation may be satisfied. Hence, (y) is a function of (x) according to the definition; and since (y) can be expressed in terms of (x), it is called an explicit function of (x).

Sometimes the relation between a function and its root is expressed by such equations as these y3 ±axy + x3 = 0,

y3 + (a + x) y* + (b+cx+kx2) y3 +gx1y +x3 =0;

where, as before, to every value of (x), it is manifest, that there must be some corresponding values of (y), and (y) consequently is a function of (x); but, as (y) is not known in terms of (x), it is termed an implicit function of (x).

Sometimes the relation between a function and its root cannot be expressed by any equation, the number of whose terms are finite. Thus, if

it is still manifest, that every value of (a) has some value or values of (y) corresponding to it, and that consequently, (y) is still a function of (x); but, since, if the quantities sin x, sin-'x, &c. are developed in terms of (x), the number of those terms will be infinite, there is no algebraical equation, which