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

«->(£) + (|)= <">

Now x, y, z, being the coordinates to the point of contact, are constant for a given point, and so are (^)> (^)' (^) which

are functions of x,y,z; and £, T/, ( 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

(J) = v> (£) = v, (£) = Wi

us + va + wa = Q2;

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(15)

(16) (17)

(18)

333.] On comparing (14) with (6), and with equations (7), if a, j3, 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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Also if the surface is expressed by an algebraical equation of the form,

F (x, y, z) = u„ + «„_! + +«! + «o = 0, (23)

where u„, ... «i, % 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 equation to the tangent plane is

= - {m»-i + 2«„_2 + + (n-l)Mj + nM0}, (24)

and is therefore an equation of only (« — 1) dimensions in terms of x, y, and z.

In the equation to the tangent plane, considering £, T}, (to be constant, and the coordinates of a given point through which a series of tangent planes is drawn, x, y, z refer to the points of contact on the surface; hence we have the following theorems:

If through a given point planes are drawn, touching a given surface of the nth order, the points of contact lie on a surface of the (» —l)th order; and therefore

If through a given point planes are drawn touching a surface of the second order, all the points of contact lie in one plane.

In harmony with the nomenclature of Art. 226, the point (f. 1i 0 whence the tangent planes are drawn is called the pole: the surface whose equation is (24) is called the first polar surface, and the surface y (x, y,z) = c is called the base-surface. Also in the same way that the first polar surface is derived from the base-surface and is of the (n l)th order, so may other and successive polar surfaces with reference to the same pole be derived, and these will be of the (» —2)th, («—3)th, order.

Want of space however precludes me from entering on these subjects, although they are replete with interest.

336.] To find the equations to a normal of a curved surface.

A normal is a straight line drawn through any point of a curved surface, and at right angles to the tangent plane at that point.

Let £, rj, ( be the current coordinates of the normal, and x, y, z the coordinates to the point where it meets the surface; then, by Art. 333, the direction-cosines of the normal being

proportional to (t-)> {~r)> (;r~)> ^s equations are

Also the form of the equations to the normal shews that it is the longest or the shortest line which can be drawn from a point on it to the surface.

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(25)

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If the equation to the surface is given in the explicit form, these equations, by means of Art. 334, become

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In these equations, if f, 77, ( are constant, ,r, y, z refer to the points on a surface where normals drawn through a given point meet it, and the equations (25) or (26) are those to a curve in space which is the locus of such points of contact.

337.] From (25) it follows, that the equations to a line passing through the origin, and at right angles to the tangent plane, are e .

= (27)

W \dy> \dz>

13y means of which equations, combined with those to the tangent plane and to the surface, we may determine the equation to the surface, which is the locus of the point of intersection of a tangent plane, with the perpendicular drawn to it from the origin.

338.] Examples illustrative of the preceding Articles. Ex. 1. The ellipsoid whose equation is

r(»,y,«) = + £ = 1. (28)

/dr\ _ 2x /dr\ _ 2y /dv\ _ 2z
\dx> ~ 7?' \dy' ~ V W ~ c*'

therefore by equation (14) the equation to the tangent plane is

a* + A? + 7» ~ l>

which is plainly the equation, since the equation to the surface is a homogeneous function of two dimensions.

Also if (f, 7j, () is the pole, the equation to the first polar is

+ + ii <30>

a2 oa c' which is the equation to a plane.

Hence also we have by (22),

P - \v* + T* + Ta (31)

And the equations to the normal are

a2 A* c1

—it-*) = —(i-y) = ((-*)■ (32)

w y z

The equations therefore to a line through the origin, and perpendicular to the tangent plane, are

x y z

Whence may be found the equation to the surface, which is the locus of the point of intersection of these lines with the tangent planes.

For f, ri, C being the same in (29) and (33), we have

^l = ^L=^i=il = tL = cS= (a* (* + bW+ x y z x y z_

a b c

P n2 <**
{x rjy (z
a* b* c*

... (f2+|JI + fl}I _ + (34)

which is the equation to the surface required.
Ex. 2. The elliptic paraboloid whose equation is

y°- «■ n

4a 4b

W ' Kdyl~ 2a W~ 2b' therefore the equation to the tangent plane is

'+-E-B-0' <*>

and the equations to the line through the origin, and perpendicular to the tangent plane, are

(= _^!? = (36)

y z

PRICE, VOL. I. 3 T

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