new axes, x, y the co-ordinates of a point referred to the former, x, y those of the same point when referred to the latter.

Since the nature of a curve is independent of any particular system of axes, the object of the problem is to determine how x and y must be expressed in terms of x' and y', in order that, when the latter is substituted for the former in a given equation between x and y, the degree of the resulting equation may remain unaltered. This, it is evident, can only take place, when the co-ordinates of the one system are linear functions of those of the other.

For example, let the proposed equation be that to the straight line or circle, then, in order that the transformed equations may continue to be of the first or second degree, respectively, x' and y' must necessarily be linear functions of x and y.

We shall assume, therefore, generally, that the relation between the primitive and the new co-ordinates may be thus expressed, x=mx+ny' and y=m'x'n'y'.. (1),

m, n; m', n' being constant quantities, which are now to be determined.

(1) Let y'=0; therefore the point will be situated on AX' as at P; draw PM parallel to AY.

(2) In like manner, by supposing x=0, we obtain

Therefore, m, m'; n, n', being replaced by the values just obtained, equation (1) becomes

Hence, any equation between x and y may, by substitution of these values, be transformed to an equation between ' and y': as was required.

73. COR. 1. Let the primitive axes be rectangular, and the new ones oblique: then, sin x, y = 1.

74. COR. 2. Let both systems be rectangular; therefore, the formulas become

x='x' cos x, x-y' sin x', x,

y=x' sin x', x+y' cos x', x.

75. COR. 3. Let the primitive axes be oblique, and the new ones rectangular.

Then, sin y', y cos x, y, and sin y', x = cos x', x;

76. If, besides changing the direction of the axes, it be required at the same time to displace their origin, we have only in the foregoing formulas to add the new abscissa to the value of x, and the new ordinate to the value of y.

Thus, if a, b, be the co-ordinates of the new origin, then the general formulas will become

When the origin alone is displaced, and the new axes supposed to remain parallel to the primitive ones, then the formulas are in this case,

77. PROB. 2.

polar co-ordinates.

x = a + x'

y = b + y

To pass from one system to another, of

Let a, b, be the rectilinear co-ordinates of the new pole S, and let the axis to which the radius vector is referred make with the primitive axis an angle a: then, the formulas of Art. 46. become, by substitution,

78. COR. Let the axis to which r is referred be parallel to AX, then a=0, and the formulas become

By adopting the usual notation, these formulas may be thus

expressed,

M

79. Let the axes be rectangular, then the formulas in Art. 77 and 78, become

80. By supposing the angle which the radius vector makes with the axis, to pass through all degrees of magnitude between O and 360°, the signs of the trigonometrical functions of that angle will determine the position of the point P. Hence, in the application of the foregoing formulas, the radius vector

must always be considered as positive; and those values of (r), which are negative, must be rejected. See Scholium, p. 35.

SECTION III.

ON THE DISCUSSION AND PROPERTIES OF LINES OF
THE SECOND ORDER.

СНАР. І.

ON LINES OF THE SECOND ORDER IN GENERAL.

81. A CURVE, whose equation is of the nth degree, is called a line of the nth order.

Agreeably to this definition, the straight line, since it is characterized by an equation of the first degree, is a line of the first order, and the circle, for a similar reason, is a line of the second order.

82. PROP. 1. To find the locus of the equation of the second degree between two variables.

The general form of this equation is

ay2+ bxy+cx2+dy+ex+f=0,

in which a, b, c... are constant quantities that may be positive or negative, fractional or integral.

The equation being resolved in terms of x and y successively, we have

1