of

179

CHAP. I.]

GEOMETRICAL EXAMPLES.

same plane locus for P, as that which is represented by any one of the four other equations of 186, (6.); or by the equation,

expresses that BPOA; or that the points B and P have the same projection on oa; or that the locus of P is the plane through B, perpendicular to the line OA.

expresses (comp. 132, (2.)) that P is on one sheet of a cone of revolution, with o for vertex, and oa for axis, and passing through the point B.

(4.) The other sheet of the same cone is represented by this other equation,

expresses that the locus of P is the plane through A, perpendicular to the line os; because it expresses (comp. XIX.) that the projection of or on oA is the line on it

expresses that the projection of OB on OP is OP itself; or that the angle OPB is right; or that the locus of P is that spheric surface, which has the line OB for a diameter. (7.) Hence the system of the two equations,

represents the circle, in which the sphere (6.), with OB for a diameter, is cut by the plane (5.), with oa for the perpendicular let fall on it from o.

obtained by multiplying the two last, represents the Cyclic Cone (or cone of the

Historically speaking, the oblique cone with circular base may deserve to be named the Apollonian Cone, from Apollonius of Perga, in whose great work on Co

second order, but not generally of revolution), which rests on this last circle (7.) as its base, and has the point o for its vertex. In fact, the equation (8.) is evidently satisfied, when the two equations (7.) are so; and therefore every point of the circular circumference, denoted by those two equations, must be a point of the locus, represented by the equation (8.). But the latter equation remains unchanged, at least essentially, when p is changed to xp, x being any scalar; the locus (8.) is, therefore, some conical surface, with its vertex at the origin, o; and consequently it can be none other than that particular cone (both ways prolonged), which rests (as above) on the given circular base (7.).

(in writing the first of which the point may be omitted,) represents a conic section; namely that section, in which the cone (8.) is cut by the new plane, which has oc for the perpendicular let fall upon it, from the origin of vectors o.

(10.) Conversely, every plane ellipse (or other conic section) in space, of which the plane does not pass through the origin, may be represented by a system of two equations, of this last form (9.); because the cone which rests on any such conic as its base, and has its vertex at any given point o, is known to be a cyclic cone.

(11.) The curve (or rather the pair of curves), in which an oblique but cyclic cone (8.) is cut by a concentric sphere (that is to say, a cone resting on a circular base by a sphere which has its centre at the vertex of that cone), has come, in mo. dern times, to be called a Spherical Conic. And any such conic may, on the foregoing plan, be represented by the system of the two equations,

the length of the radius of the sphere being here, for simplicity, supposed to be the unit of length. But, by writing Tp = a, where a may denote any constant and positive scalar, we can at once remove this last restriction, if it be thought useful or convenient to do so.

(12.) The equation (8.) may be written, by XII. or XII'., under the form (comp. 191, VII.):

:( (株)

2

a

=

1,

nics (kwvikõv), already referred to in a Note to page 128, the properties of such a cone appear to have been first treated systematically; although the cone of revolution had been studied by Euclid. But the designation "cyclic cone" is shorter; and it seems more natural, in geometry, to speak of the above-mentioned oblique cone thus, for the purpose of marking its connexion with the circle, than to call it, as is now usually done, a cone of the second order, or of the second degree: although these phrases also have their advantages.

so that a' and B are here the lines oa' and OB', of Art. 188, and Fig. 48.

(13.) Hence the cone (8.) is cut, not only by the plane (5.) in the circle (7.), which is on the sphere (6.), but also by the (generally) new plane, S = 1, in the

α

(generally) new circle, in which this new plane cuts the (generally) new sphere, B S=1; or in the circle which is represented by the system of the two equations,

(14.) In the particular case when ẞ || a (15), so that the quotient ß: a is a scalar, which must be positive and greater than unity, in order that the plane (5.) may (really) cut the sphere (6.), and therefore that the circle (7.) and the cone (8.) may be real, we may write

and the circle (13.) coincides with the circle (7.).

(15.) In the same case, the cone is one of revolution; every point P of its circular base (that is, of the circumference thereof) being at one constant distance from the vertex o, namely at a distance = aTa. For, in the case supposed, the equations (7.) give, by XIL,

(16.) Conversely, if the cone be one of revolution, the equations (7.) must conduct to a result of the form,

Ρ P

which can only be by the line ẞ– a2a vanishing, or by our having ß= a2a, as in (14.); since otherwise we should have, by XIV., p + ẞ − a2a, and all the points of the base would be situated in one plane passing through the vertex o, which (for any actual cone) would be absurd.

(17.) Supposing, then, that we have not ẞ || a, and therefore not a' = a, ß′ = ß, as in (14.), nor even a' || a, ß′ || ß, we see that the cone (8.) is not a cone of revolution (or what is often called a right cone); but that it is, on the contrary, an oblique (or scalene) cone, although still a cyclic one. And we see that such a cone is cut in two distinct series* of circular sections, by planes parallel to the two distinct (and mutually non-parallel) planes, (5.) and (13.); or to two new planes, drawn through the vertex o, which have been called the two Cyclic Planes of the cone, namely, the two following:

* These two series of sub-contrary (or antiparallel) but circular sections of a cyclic cone, appear to have been first discovered by Apollonius: see the Fifth Proposition of his First Book, in which he says, καλείσθω δὲ ἡ τοιαύτη τομὴ ὑπεναντία (page 22 of Halley's Edition).

+ By M. Chasles.

a

=0;

while the two lines from the vertex, OA and OB, which are perpendicular to these two planes respectively, may be said to be the two Cyclic Normals.

(18.) of these two lines, a and ẞ, the second has been seen to be a diameter of the sphere (6.), which may be said to be circumscribed to the cone (8.), when that cone is considered as having the circle (7.) for its base; the second cyclic plane (17.) is therefore the tangent plane at the vertex of the cone, to that first circumscribed sphere (6.).

(19.) The sphere (13.) may in like manner be said to be circumscribed to the cone, if the latter be considered as resting on the new circle (13.), or as terminated by that circle as its new base; and the diameter of this new sphere is the line оB', or ', which has by (12.) the direction of the line a, or of the first cyclic normal (17.); so that (comp. (18.)) the first cyclic plane is the tangent plane at the vertex, to the second circumscribed sphere (13.).

(20.) Any other sphere through the vertex, which touches the first cyclic plane, and which therefore has its diameter from the vertex = b'ß′, where b' is some scalar co-efficient, is represented by the equation,

it therefore cuts the cone in a circle, of which (by (12.)) the equation of the plane is

so that the perpendicular from the vertex is b'a' || ẞ (comp. (5.)); and consequently this plane of section of sphere and cone is parallel to the second cyclic plane (17.). (21.) In like manner any sphere, such as

which touches the second cyclic plane at the vertex, intersects the cone (8.) in a circle, of which the plane has for equation,

and is therefore parallel to the first cyclic plane.

(22.) The equation of the cone (by IX., X., XVI.) may also be thus written:

it expresses, therefore, that the product of the cosines of the inclinations, of any variable side (p) of an oblique cyclic cone, to two fixed lines (a and ß), namely to the two cyclic normals (17.), is constant; or that the product of the sines of the inclinations, of the same variable side (or ray, p) of the cone, to two fixed planes, namely to the two cyclic planes, is thus a constant quantity.

(23.) The two great circles, in which the concentric sphere Tp = 1 is cut by the two cyclic planes, have been called the two Cyclic Arcs* of the Spherical Conic (11.), in

By M. Chasles.

CHAP. I.] SCALAR OF A SUM OR DIFFERENCE.

183

which that sphere is cut by the cone. It follows (by (22.)) that the product of the sines of the (arcual) perpendiculars, let fall from any point P of a given spherical conic, on its two cyclic arcs, is constant.

(24.) These properties of cyclic cones, and of spherical conics, are not put forward as new; but they are of importance enough, and have been here deduced with sufficient facility, to show that we are already in possession of a Calculus, with its own Rules of Transformation, whereby one enunciation of a geometrical theorem, or problem, or construction, can be translated into several others, of which some may be clearer, or simpler, or more elegant, than the one first proposed.

197. Let a, ẞ, y be any three co-initial vectors, oa, &c., and let op=8y+ ß, so that OBDC is a parallelogram (6) then, if we write

B: a=q,

y: a=q, and 8: a = q′′ = q' + q (106), and suppose that B', c', D' are the feet of perpendiculars let fall from the points B, C, D on the line oa, we shall have, by 196, XIX., the expressions,

(OB' =) B' = aSq,

Y' = aSq',

& = aSq" = aS (q' + q).

=

But also OB = CD, and therefore OB' C'D', the similar projections of equal lines being equal; hence (comp. 11) the sum of the projections of the lines ẞ, y must be equal to the projection of the sum, or in symbols,

8' = y' + B',

OD' = OC' + OB', d: a=(y': a) + (ẞ': a)· Hence, generally, for any two quaternions, q and q', we have the formula:

I. . . S (g' + g) = Sq + Sq;

or in words, the scalar of the sum is equal to the sum of the scalars. It is easy to extend this result to the case of any three (or more) quaternions, with their respective scalars; thus, if q" be a third arbitrary quaternion, we may write

S{q" + (q' + q)} = Sq′′ + S (q' + q) = Sq' + (Sq''+ Sq) ; where, on account of the scalar character of the summands, the last parentheses may be omitted. We may therefore write, generally,

II... SZq = ΣSq, or briefly, SΣ= ΣS;

where is used as a sign of Summation: and may say that

* Comp. 145, (10.), &c.