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29. If Pdx+Qdy be a perfect differential of some function

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30. If Pdx+Qdy + Rdz can be made a perfect differential of some function of x, y, z by multiplying each term by a common factor, show that

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APPLICATIONS TO PLANE CURVES.

CHAPTER VII.

TANGENTS AND NORMALS.

169. Equation of TANGENT.

It was shown in Art. 38 that the equation of the tangent at the point (x, y) on the curve y=f(x) is

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X and Y being the current co-ordinates of any point on

the tangent.

Suppose the equation of the curve to be given in the

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If f(x, y) be an algebraic function of x and y of degree n, suppose it made homogeneous in x, y, and z by the introduction of a proper power of the linear unit z wherever necessary. Call the function thus altered f(x, y, z). Then f(x, y, z) is a homogeneous algebraic function of the nth degree; hence we have by Euler's Theorem

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by virtue of the equation to the curve.

Adding this to equation (2), the equation of the tangent takes the form

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The equation, when made homogeneous in x, y, z by the introduction of a proper power of z, is

f(x, y, z) = x2+a2xyz2+b3yz3+c+24=0,

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Substituting these in Equation 3, and putting z=1, we have for the equation of the tangent to the curve at the point (x, y)

X(4x3+a2y) + Y(a2x+b3)+2a2xy+3b3y+4c1=0.

With very little practice the introduction of the z can

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