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CHAPTER XIV.

APPLICATION OF THE DIFFERENTIAL CALCULUS TO PROPERTIES OF CURVED SURFACES.

331.] AN explanation of the mode of generation and of the equations of such curved surfaces and curves in space as are needed for illustration in this and the following Chapters, requires more room than we can afford to give; but it is the less necessary to introduce it, as the ordinary text-books contain sufficient information. It is however desirable to explain the equations to the straight line and the plane, in the forms which we shall employ, as a familiar knowledge of them is requisite to a due understanding of our processes.

(1) To find the equations to a straight line in space.

Let §, ŋ, Ŝ be the current coordinates to the straight line; x, y, z the coordinates to a point through which the line passes; A, μ, v the direction-angles of the line; that is, the angles between a parallel line through the origin and the coordinates axes. And let be the distance between (x, y z) and (§, 7, 8); then the equations to the line are

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the last of the equalities following by reason of Preliminary Theorem I.

If therefore the equations to a straight line are given under the form

ૐ x n-y

8-2

=

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(2)

each of these equalities is by reason of the same Preliminary Theorem equal to

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(L2 + M2 + N2)

and therefore comparing (1) with (2) and (3),

(3)

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and therefore L, M, N in (2) are proportional to the directioncosines of the line, that is, to the cosines of the angles between the line and the coordinate axes.

(2) To find the equation to a plane.

A plane is a surface generated by a straight line revolving round another straight line which is at right angles to it.

Let the origin be at the point o in the straight line oq, fig. 123, round which the perpendicular and generating line QP turns; and let A, p, v be the direction-angles of oq; let έ, n, be the current coordinates to any point P in the line qP which is in any position; and let opp, oq = 8; then the

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and as this relation is true for every point in QP, and in every position of QP, it is according to our definition, the equation to a plane; λ, μ, v being the direction-angles of the normal to the plane, and d the length of the perpendicular from the origin on the plane.

Equation (5) is evident by the theory of projections, the lefthand side of the equation being the sum of the projections of the broken line OMNPQ on the line oq.

If therefore the equation to a plane is given in the form

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- whence it appears that A, B, C, D are proportional respectively

to the direction-cosines of the normal to the plane, and to the length of the perpendicular on the plane from the origin.

332] To find the equation to a tangent plane to a curved surface at a given point.

Let the equation to the surface be

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Our present object is to shew that if a straight line is drawn through a point on the surface (x, y, z), and through a second point (x+dx, y+dy, z+dz) infinitesimally near to it and also on the surface, the locus of such tangent lines is in general a plane; and is what is called the tangent plane. Of course it is manifest that the number of points (x+dx, y+dy, z+dz) contiguous to the first point is infinite, and so therefore is the number of tangent lines.

Let, n, be the current coordinates to one of the tangent lines, and x, y, z the coordinates to the point of contact on the surface; then the equations to the line are

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as the line passes through the point (x+dx, y+dy, z+dz), we have

dx dy dz

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(10)

(11)

r being the distance of (x, y, z) from any point (, 7, () on the line, and

ds = {dx2 + dy2 + dz2};

(12)

that is, ds is the distance between the two points on the surface through which the line passes; (11) therefore are the equations to any straight line touching a surface at a given point.

But as the second point through which the line passes is on the surface, though it may have any position infinitesimally near to (x, y, z); so dx, dy, dz, must be consistent with the equation to the surface. If therefore at the point under consideration

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multiplying the several terms of which by the terms of the equalities (11), we have

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Now x, y, z, being the coordinates to the point of contact, are

constant for a given point, and so are

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(dr), (dy), (dz)

which are functions of x, y, z ; and §, î, being the current coordinates of the locus, it follows that (14) is of the same form as (6), and therefore represents a plane; and being the locus of the tangent lines to the surface represents the tangent plane.

Let us once for all make certain substitutions which for the purpose of abridging the notation will be convenient both in the sequel of the present volume, and in future parts of our Treatise.

Let
dr

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(18)

udx+vdy+wdz = 0.

so that

and thus (14) becomes

u (§−x) + v(n−y) +w($—≈) = 0.

333.] On comparing (14) with (6), and with equations (7), if a, ß, y are the direction-angles of the normal to the plane, and if p is the perpendicular on the plane from the origin, we have

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of the normal to the tangent plane.

334.] If the equation to the surface is given in the explicit form z = f(x, y); then F(x, y, z) = f (x, y) − z = 0;

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in which case the equation to the tangent plane becomes

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and (19) and 20) must be modified accordingly.

335.] If the equation to the surface is a homogeneous function of n dimensions of the form,

F(x, y, z) = c;

then, since by the property of such functions proved in Art. 82,

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Also if the surface is expressed by an algebraical equation of the form,

F(X, Y, Z) = Un + Un−1 +

......

+ U1 + Uo = 0,

(23) where un, un-1, ... u, u are homogeneous functions of n, n−1, ... 1, 0 dimensions; then by a process exactly similar to that of Art. 222, except that in this case there are three variables, it may be shewn that the cquation to the tangent plane is

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