So that (19) becomes, after reduction,

(as+bc1+cB1+ A2) x + (αC1+bB+CA1 + B2) Y

+(a B1+bA1+ CC + C2) ≈ + a Ag + b B2 + CC2 + K = 0;

and as this relation is true independently of x, y, z, the coefficients of the several terms must be equal to zero: whence we have

Of these four conditions the first three shew that the surface has a centre, and that the vertex of the cone, viz. the point (a, b, c), is at the centre; and from them the coordinates of the centre may be determined. And the last shews that in the transformation of the equation to the centre as origin the constant term disappears. Thus the equation to the surface becomes after transformation a homogeneous equation of two dimensions in terms of x, y, z.

356.] To find the equation to cylindrical surfaces.

A cylindrical surface is generated by a straight line which moves parallel to itself, and always passes through a given director curve.

2

Let x, y, z be the current coordinates to the surface; x', y', z' the coordinates to a point on the director; L, M, N proportional to the direction cosines of the generating line; then the equations to the generating line are

from which four equations eliminating x', y', z', we have a result

of the form

F(NX-LZ, NY — Mz) = 0,

which is the general equation to cylindrical surfaces.

(31)

Equation (31) may also be put in the form

x―lz = f(y-mz),

(32)

which is the explicit form of the general equation to cylindrical

surfaces.

Also (31) may be written in the following form,

F(Ny-MZ, LZ-NX, Mx-Ly) = 0.

(33)

Two particular cases require notice. Firstly, let the director be a plane curve in the plane xy, and let its equation be

and if these are substituted in (34), we have

x-lz = f(y-mz).

Again, let us suppose the cylinder to circumscribe a surface whose equation is F1 (x, y, z) = 0; then, as the generator is to be perpendicular to the normal at the point of contact,

from which, the equation to the surface, and the equations to the generators, x, y, z' are to be eliminated; and the resulting equation in terms of x, y, z is that to the cylindrical surface.

357.] Now (32) and (33) are the general equations to all cylindrical surfaces. They contain an undetermined functional symbol, under which the coordinates enter as subject-variables in a particular combination, and which is itself determined when the law of the director curve is given: but whatever that law is, the general form of the function in reference to the subjectvariables is the same. These forms also hold good, whether the function is continuous or discontinuous. If the function is continuous, two partial derived functions, say the z and x, and the z and y, may be formed from (32), and the undetermined function may be eliminated; whereby we shall have

dz 1 dx

+ m

dy

dz
= 1.

(37)

Or we may operate on (33) as in Article 53, and we shall have

which is in fact identical with (37); see Art. 50. Now these equations express properties which are true of all cylindrical surfaces whose directors are continuous curves; and what is the geometrical interpretation of them? Let us take (38); L, M, N are proportional to the direction-cosines of the generating line, and the partial derived functions are proportional to the direction-cosines of the normal at (x, y, z): and (38) shews that these lines are perpendicular to each other: or, in other words, the tangent plane at every point contains the generating line of the cylinder which passes through that point.

358.] Examples of cylindrical surfaces.

Ex. 1. To find the equation to the cylinder whose director is the ellipse,

xo2 yo2
+
a2 b2

= 1.

If we substitute for xo and yo from (35), we have

(Na-L2)2 (NY-M2)2

+

= 1;

a2 N2

b2 N2

(39)

(40)

which is the general equation to an oblique elliptical cylinder. If the generator is perpendicular to the plane of xy, the cylinder is called right; and as in that case L=0, M = 0, the equation is

x2 y2

+ = 1.
62

Hence the equation to an oblique circular cylinder is

(NX-LZ)2 + (Ny — M2)2 = N2 a2.

(41)

Ex. 2. To find the equation to a cylinder circumscribing a given ellipsoid.

Let the equations to the generator and to the ellipsoid be

Now as (42) is to touch (43), the point (§, ŋ, §) is in the tangent plane of (43); and therefore we have

whence, operating on the members of the equality (42),

From this equation it appears that if (, n,

the cylindrical surface which is also on the

n2

+ (−1).

b2

ellipsoid, then

)

is a point on

L& Μη NC

+ = 0;

0, and therefore +

a2 b2 c2

which is the equation to a central plane section of the ellipsoid; and is the polar plane of a pole on the line (42) at an infinite distance.

Ex. 3. To determine the conditions that the general equation of the second degree of three variables may represent a cylinder. Let the general equation be

F (X, Y, Z) = Ax2 + By2+cz2 +2 A1 y z +2 B1 zx+2c1 xy

+2A2x+2B2 Y + 2 C2 z + K = 0; (44)

and therefore equation (38) becomes

L (AX+C1Y + B12 + A2) + M (C1 ≈ +BY+A1 ≈ + B2)

+ N (B2+A1Y+CZ + C2) = 0;

and as this condition is to be satisfied for all values of x, y, z, we have

AL+CM+ B1N =

and from the first three combined with the fourth

(BC A13) A2+ (CA-B12) B2+ (AB-C1) C2 = 0;

PRICE, VOL. 1.

3 Y

and taking the first three two and two together, we have

and substituting these values in the last, we have

(A12 — BC) 1⁄2 A2 + (B12 —CA)✯ B2 + (C12 — A B) C2

Now (45) is the well-known condition that the first six terms of (44) are resoluble into two linear homogeneous factors; and (46) is the additional condition requisite that (44) should be of the form

(ax+by+cz+d) (α1 x + b1y + c1 z + d1) − k = 0 ;

-

(47)

and therefore (47) is the equation to a cylindrical surface of the second degree. Now (47) may be resolved into two factors of the form

α 1 x + b1 y + c 1 z + d1 + mk§ = 0,

where m is an undetermined constant: because if m is eliminated, we obtain the equation (47). Each of these equations is that of a plane, and the generating line is the line of intersection of them; and the surface is generated by the line whose varying position is due to the variation of m. From these two equations it is evident that the direction cosines of the generating line are proportional to b1c-c1b, c1a-a1c, a1b-b1a; and these are proportional to L, M, N. Although at first sight it may appear that (47) is not of the forms (31) or (33), yet it will be found that with these values of L, M and N it satisfies (38), and is therefore the equation of a cylindrical surface.

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The surface in its most general form represents an elliptical or a hyperbolic cylinder. If however (46) is satisfied identically, so that 12 = BC, B12= CA, C12 = AB, then (47) is the equation to a parabolic cylinder.