The infinitely small triangle P' Q P being similar to the triangle P M T, we have,

P'Q PQ :: PM

MT : MT

MT=Y

dx dy

The subtangent MT being thus known, we can immediately determine the normal and tangent, and the equation to these lines.

To find the differential of an arc of a curve, we may consider the infinitely small arc PP' included between the ordinates

PM, P'M', as a straight line, and calling the whole arc of the curve s, the infinitely small portion PP' will be represented by ds.

The right-angled triangle PP'Q gives

P

T

Q

To find the differential of the area comprised between two ordinates PM, P' M' of a curve which are infinitely near to each other, neglecting the area PP'Q, if we call the whole area A, the area of the rectangle PM' may be taken for dA

dA PM x PQ
=ydx

Polar Curves.

In applying the differential calculus to the theory of curves, we have hitherto considered only such as are referred to rectangular co-ordinates. The various propositions which we have demonstrated may, however, be applied to polar curves also, either directly by Taylor's theorem, or, by adapting the expressions already deduced, by aid of the chapters on the transformation of co-ordinates and the change of the independent variable.

The principles of the infinitesimal calculus may also be employed with much elegance in these investigations. Thus, for example,

To find the angle under the radius vector and a tangent at any point of a

Take a point Q infinitely near to P, then the arc PQ may be considered ultimately as coinciding with tangent and angle PSQ = de, draw QR perpendicular to SP, and SY perpendicular on tangent.

The following expressions are much employed in the investigations of physi

To find the radius and chord of curvature in polar curves.

To find the chord of curvature through S, produce PS and PO to meet the circle of curvature in V and L.

Then since the angle at V is a right angle being in a semicircle, the triangles PVL, PSN are similar.

.: PV: PL:: PN: PS

We shall conclude by showing how the first of the above propositions may be established by the transformation of co-ordinates.

To find the angle under the radius vector and tangent, in a spiral curve.

Let RZ be a spiral curve whose pole is

R

S and equation

If we wish to transform this expression into another in which is the independent variable, we shall have

INTEGRAL CALCULUS.

CHAPTER I.

THE object of the Integral Calculus is to discover the primitive function from which a given differential co-efficient has been derived.

This primitive function is called the integral of the proposed differential coefficient, and is obtained by the application of the different principles established in finding differential co-efficients and by various transformations. In order to avoid the embarrassment which would arise from the perpetual changes of the independent variable, which it would be necessary to effect if we restricted ourselves to the use of differential co-efficients alone, we shall generally employ differentials according to the infinitesimal method explained in the preceding chapter.

When we wish to indicate that we are to take the integral of a function we prefix the symbol. Thus, if

y = ax1

We know that dy = 4ax dx

If then, the quantity 4ax3 da be given in the course of any calculation, and we are desirous to indicate that the primitive function from which it has been derived is ax1, we express this by writing

+

S4ax3 dx = axa

When constant quantities are combined with variable quantities by the signs we know that they disappear in taking the differential co-efficients,

or

and therefore they must be restored in taking the integral.

Hence in taking the integral of any function it is proper always to add a constant quantity, which is usually represented by the symbol C. Thus, if it be required to find the integral of a quantity such as