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

or,

P'Q PQ :: PM: MT

dy: dx :: y : MT

dx

MT=Y 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

T

P

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 Curces.

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 polar curve.

Let LZ be a curve referred to Polar Co-ordi

nates.

Let S be the pole and SZ the straight line from which the angles are measured.

Take any point P, and draw a tangent PT.
Let SP r, angle ZSP = 0.

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.

P

Q R

de

The following expressions are much employed in the investigations of phys cal astronomy.

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

SY = p, SP=r, OP = g.

Now while the arc of curve receives the increment PQ, and SP varies from SP to SQ; the point O remains fixed, and .. OP and SO remain constant. But

SO2 = SP2 + PO2 — 2PO . PN
= r2+ ç2 — 2pg

.. differentiating o = rdr

dp

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.

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/

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

INTEGRAL CALCULUS.

CHAPTER I.

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

This primitive function is called the integral of the prognosed diferential caefficient, and is obtained by the application of the different principies established in finding differential co-efficients and by various transformacions. In vie to avoid the embarrassment which would arise from the perpetual changes af the independent variable, which it would be necessary to effect if ve marett ourselves to the use of differential co-efficients alone, we shail genera y empuny differentials according to the infinitesimal method explained in the preceding chapter.

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

y = azt

We know that dy = 4ar' dr

and

If then, the quantity 4a23 dz be given in the course of any tarzda we are desirous to indicate that the prim tive funcing fra vara i La V derived is ar', we express this by writing

Star' dz = er'

When constant quantities are combined with varihin ost.. so far HEM + or – we know that they duappear in taking tâm e foronina mate.com and therefore they must be restored in taking the margra

Hence in taking the integral of any fezrine it is poupay them in wh constant quantity, which is usually repermented by the of stint i

be required to find the integral of a quanczy mors) 200