and, therefore, PY: PS :: PS: the length of the

curve.

PROP. LV.

To find the differential of the area of a spiral.

Let SPQ be a spiral, SL the initial line, the angle LSP=0, the angle PSQ=1, SP=p, and SQ another radius vector, from S as a centre at the distances SP and SQ, describe two circular arcs PT, Qp.

Then the area of the sector SPT =

M

P

W

2

ρ

(Prop. 39.

2

SQ2.

2

2

Cor. 3.). Also, the area of the sector SQp

=

&c.)

+ &c.

pdp.+&c. (4).

+

d Ꮎ

Now, the area SMP of the spiral, is a function of the angle LSP or (0), and if we represent it by (0), we have

area PSQ= (0 + 1) − & (0)

=ƒ. ¥ (0) . 1+ƒ3. ¥ (0)

2

+ &c. (B).

1 2

Now, since the area PSQ lies between the sectors SPT, SQр, the series (B) will lie between the series

SECT. VII.

ON THE CURVATURE OF CURVES.

DEF. A CURVE is said to have at any point a greater, or a less degree of curvature, according as it deflects more or less from the tangent to that point.

It is manifest, that the same curve may have different degrees of curvature at different points. In the hyperbola, the curvature is greatest at the vertex; as the distance of any point from the vertex increases, the form of the curve manifestly approximates to that of a straight line; and, consequently, its deviation from its tangent continually decreases, that is, its curvature becomes less and less.

In the following propositions, we shall assume these four axioms.

1. The curvature of the circumference of a circle is the same throughout.

The truth of this is manifest.

For at every

point the deviation from the tangent is the same.

2. Of two circles, the circumference of that which has the greater radius, has a less degree of curvature. The truth of this will immediately appear, if we suppose the circles to touch each other internally, for in that case, the smaller circle falls entirely within the other, and if a tangent be drawn touching them at the point, where they touch each other, the

deviation of the circumference of the inner circle from this tangent, is manifestly greater than the deviation of the circumference of the other circle.

3. If a circle and a curve have at any point a common tangent, and if the circle at the touching point falls entirely within the curve, the curvature of the circle at that point cannot be less than that of the curve.

Thus, if the line TAT" touch the curve CAC', and the circle BAB' at the point A, and the two arcs AB, AB' of the circle measured from A, fall both within the curve CAC', it is manifest,

T

T

that if the deviation of the circle BAB at the point A from the tangent TAT' be not greater*, it certainly cannot be less than that of the curve at the same point; that is, the curvature of the circle at A, is not less than that of the curve.

4. If a circle and a curve have at any point a common tangent, and if at that point the curve falls entirely within the circle, the curvature of the circle cannot be greater than that of the curve at that point. Suppose the straight line T'AT

to touch the circle BAB', and the curve CAC' at the point A, and let that part of the curve CAC', which is immediately adjacent to the point A, fall entirely within the circle; then

T

B

B'

T

* A circle may be drawn touching an ellipse at the extremity of the axis minor, which, at the point of contact, shall fall entirely within the ellipse, and yet have its curvature at that point not greater than, but equal to, the curvature of the ellipse. If, however, the curvature of a circle, at that point, be less than that of the ellipse, the circle must fall externally.

it is manifest, that if at this point, the deviation of the curve from the tangent be not greater, at least it is not less than the deviation of the circle; and, consequently, the curvature of the circle cannot be greater than that of the curve.

PROP. LVI.

Having given the first and second differential coefficients of the ordinate of a circle, considered as a function of the abscissa; to find the radius of the circle.

Let the first and second differential coefficients of the ordinate= (p) and (q) respectively; let r=the radius of the circle, and let (a) and (B) = the coordinates of the centre; then we have, (by Prop. 10.)

Eliminating the quantities ( y − a), (x − ẞ), and sub