and therefore, if arc AP be reckoned a small quantity of the first order, the chord AP and the tangent AR are also of the first order of smallness.

PR2 is a small quantity of the fourth order, and AR+AP is a small quantity of the first order, we see that AR-AP is of the third order of small quantities.

And similarly for other quantities the order of smallness may be geometrically investigated.

35. The base angles of a triangle being given to be small quantities of the first order, to find the order of the difference between the base and the sum of the sides.

By what has gone before (Art. 33), if APB be the triangle and PM the perpendicular on AB, AP-AM and BP-BM are both small quantities of the second order as compared with AB.

Hence AP+PB-AB is of the second order compared

with AB.

If AB itself be of the first order of small quantities, then AP+PB- AB is of the third order.

36. Degree of approximation in taking a small chord for a small arc in any curve.

Let AB be an arc of a curve supposed continuous between A and B, and so small as to be concave at each point throughout its length to the foot of the perpendicular from that point upon the chord. Let AP, BP be the tangents at A and B. Then, when A and B are taken sufficiently near together, the chord AB and the angles at A and B may each be considered small quantities of at least the first order, and therefore, by what has gone before, AP+PB-AB will be at least of the third order. Now we may take as an axiom that the length of the arc AB is intermediate between the length of the chord AB and the sum of the tangents AP, BP. Hence the difference of the arc AB and the chord AB, which is less than that between AP+PB and the chord AB, must be at least of the third order.

EXAMPLES.

1. Show that, in the figure of Art. 33, the area of the segment bounded by the chord AP and the arc AP is of the third order of small quantities.

2. In the same figure, if PM be drawn perpendicular to AQ, show that the triangle PMQ is of the fifth order of smallness.

3. A straight line of constant length slides between two straight lines at right angles, viz., C'Aa, CbB; AB and ab are two positions of the line and P their point of intersection. Show that, in the limit, when the two positions coincide, we have Aa CB PA CB2 and =

=

Bb CA PB CA2

4. From a point T in a radius OA of a circle, produced, a tangent TP is drawn to the circle, touching it in P; PN is drawn perpendicular to the radius OA. Show that, in the limit, when P moves up to A, NA = AT.

5. If, in the equation sin(w - 0) = sin w cos a, O be very small, show that its approximate value is

2 tan w sin

(1

1- tan2w sin2

sin22).

[I. C. S. EXAM.]

6. Tangents are drawn to a circular arc at its middle point and at its extremities. Show that the area of the triangle formed by the chord of the arc and the two tangents at the extremities is ultimately four times that of the triangle formed by the three tangents. [FROST'S NEWTON.]

7. If G be the centre of gravity of the arc PQ of any uniform curve, and if PT be the tangent at P, prove that, when PQ is indefinitely diminished, the angles GPT and QPT vanish in the ratio of 2 to 3.. [I. C. S. EXAM.]

CHAPTER II.

FUNDAMENTAL PROPOSITIONS.

37. Direction of the Tangent of a Curve at a given point. Let AB be an arc of a curve traced in the plane of the the paper, OX a fixed straight line in the same plane.

Let P, Q, be two points on the curve; PM, QN, perpendiculars on OX, and PR the perpendicular from P on QN. Join P, Q, and let QP be produced to cut OX at T.

When Q, travelling along the curve, approaches indefinitely near to P, the chord QP becomes in the limit the tangent at P. QR and PR both ultimately vanish, but the limit of their ratio is in general finite; for

Lt

RQ

'PR

=

= Lt tan RPQ = Lt tan XTP = tangent of the angle

which the tangent at P to the curve makes with OX.

Ex. 1. Consider the straight line whose equation is y=mx+c.

Let OX, OY, be the axes, and let the co-ordinates of P be x, y. Then, taking the general construction of the preceding article, the intercept OA=c, for y=c when x= =0.

Draw AK parallel to OX to meet MP in K; then, from similar RQ KP MP-OA

triangles,

=

Hence tan XTP=tan RPQ=m.

the

Ex. 2. Consider the parabola referred to its usual axes, viz., axis of the parabola and the tangent at the vertex. With the same