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

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To find the chord of curvature through S proture 26 and 20% a net be circle of curvature in V and L.

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

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We shall conclude by showing how the fint về he anove prignastons may established by the transformation of co-ordinates.

To find the angle under the radius vector and tangent, in s grai an

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

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INTEGRAL CALCULUS.

CHAPTER I.

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

This primitive function is called the integral of the proposed differential caefficient, and is obtained by the application of the different principies established in finding differential co-efficients and by various transtornations. In veter to avoid the embarrassment which would arise from the perpetual changes of the independent variable, which it would be necessary to effect if we married ourselves to the use of differential co-efficients ainne, we stail genera y aguny 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 furton we

prefix the symbol f. Thus, if

y = az1

We know that dy = 4er dr

and

If then, the quantity 4ar dz be given in the care of any cruz. A we are desirous to indicate that the primitive fascina ima vaưa i ke ba derived is ar*, we express this by writing

Ssar di = er

When constant quantities are combined with variano que se faHEHE + or — we know that they disappear in taking tam a. formats metr same and therefore they must be restored in taking the morga

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Hence in taking the integral of any fazrzna it in pougong them in ad constant quantity, which is usually supersented by the of estime

ve required to find the integral of a quantay men zu

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but according to the principles which we have already explained, A is the first differential co-efficient of y or ƒ (r), hence it appears that

If y be a function of x, the first differential co-efficient of y may be cassidered as the ratio of the differentials, or infinitely small increments of y and x; and the differential of y is always equal to the first differential co-efficient of y, multiplied by the differential of x.

In order to find the differential of the product of two variables u and 2, each of which is a function of x, we shall suppose that when z becomes z+d, becomes u + du, and z becomes z + dz.

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But du dz being an infinitesimal of the second order, may be neglected.

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which agrees with the result already found by a different process.

To find the differential of sin. x according to this method.

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Let us now show how we may resolve the problem of tangents by the ne of infinitesimals.

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P'Q which is the infinitely small increment of y will be represented

PQ = MM'

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