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V-1 in the imaginary part of each expression, represents the length of a tangent
E'E" to the sphere, drawn from that external point, or foot, E'.

(5.) In fact, if we write OE'=' + sẞ, we shall have

E'E = -¿' = − B = BS = = projection of OE on OB;

which proves the Ist assertion (4.), whether the points P1, P1 be real or imaginary. And because

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and the IInd assertion (4.) is justified.

(6.) An expression of the form (4.), or of the following,

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in which ẞ and y are two real vectors, while V - 1 is the (scalar) imaginary of algebra, and not a symbol for a geometrically real right versor (149, 153), may be said to be a BIVECTOR.

(7.) In like manner, an expression of the form (3.), or x' = s + t V −1, where s and t are two real scalars, but V — 1 is still the ordinary imaginary of algebra, may be said by analogy to be a BISCALAR. Imaginary roots of algebraic equations are thus, in general, biscalars.

(8.) And if a bivector (6.) be divided by a (real) vector, the quotient, such as

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in which go and qı are two real quaternions, but V-1 is, as before, imaginary, may be said to be a BIQUATERNION.*

215. The same distributive principle (212) may be employed in investigations respecting circumscribed cones, and the tangents (real or ideal), which can be drawn to a given sphere from a given point.

(1.) Instead of conceiving that o, A, B are three given points, and that limits of position of the point E are sought, as in 214, (2.), which shall allow the points of intersection Po, P1 to be real, we may suppose that o, A, E (which may be assumed to be collinear, without loss of generality, since a enters only by its tensor) are now the data of the question; and that limits of direction of the line OB are to be assigned, which shall permit the same reality: EPOP1 being still drawn parallel to OB, as in 214, (1.).

(2.) Dividing the equation Ta= T(ε + xẞ) by Te, and squaring, we have

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* Compare the second Note to page 131.

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and its roots are real and unequal, or real and equal, or imaginary, according as

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(3.) If E be interior to the sphere, then Tɛ <Ta, T(a: e) > 1; but TVUg can never exceed unity (by 204, XIX., or by 210, XV., &c.); we have, therefore, in this case, the first of the three recent alternatives, and the two roots of the quadratic are necessarily real and unequal, whatever the direction of ẞ may be. Accordingly it is evident, geometrically, that every indefinite right line, drawn through an internal point, must cut the spheric surface in two distinct and real points.

(4.) If the point E be superficial, so that Tɛ = Ta, T(a:ɛ)=1, then the first alternative (2.) still exists, except at the limit for which ẞ ε, and therefore TVU(ẞ: ε) = 1, in which case we have the second alternative. One root of the quadratic in x is now = 0, for every direction of ß; and the other root, namely x = 28(ε: ẞ), is likewise always real, but vanishes for the case when the angle EOB is right. In short, we have here the same system of chords and of tangents, from a point upon the surface, as in 213; the only difference being, that we now write E for A, or ɛ for a.

(5.) But finally, if E be an external point, so that Te > Ta, and T(a: ɛ) < 1, then TVU (B: e) may either fall short of this last tensor, or equal, or exceed it; so that any one of the three alternatives (2.) may come to exist, according to the vary. ing direction of B.

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through E, drawn parallel (as before) to OB, either cuts the sphere, or touches it, or does not (really) meet it at all. (Compare the annexed Fig. 52.)




(7.) If E be still an external point, the cone of tangents which can be drawn from it to the sphere is real; and the equation of this enveloping or circumscribed come, with its vertex at E, may be obtained from that of the recent cone (6.), by simply changing p to p-; it is, therefore, or at least one form of it is,

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if P' be the foot of the perpendicular let fall from P on OE; and in fact the first quotient is evidently = sin OEP.

(9.) We may also write,



TV2=T2.T(2-1); or 0=(82) - N2 + N ? (N? - 28 ? + 1 );

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to express that the point P is on the enveloped sphere, as well as on the enveloping cone, we find the following equation of the plane of contact, or of what is called the polar plane of the point E, with respect to the given sphere:

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while the fact that it is a plane of contact" is exhibited by the occurrence of the exponent 2, or by its equation entering through its square.

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is that of the point E' in which the polar plane (10.) of E cuts perpendicularly the right line OE; and we see that

Tε. Te' = Ta2, or T.OE. T.OE' = (T. oa)2,

as was to be expected from elementary theorems, of spherical or even of plane geometry.

* In fact a modern geometer would say, that we have here a case of two coincident planes of intersection, merged into a single plane of contact.

(12.) The equation (10.), of the polar plane of E, may easily be thus transformed:

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it continues therefore to hold good, when ɛ and p are interchanged. If then we take, as the vertex of a new enveloping cone, any point o external to the sphere, and situated on the polar plane Fr.. of the former external point E, the new plane of contact, or the polar plane DD'. . of the new point c, will pass through the former vertex E a geometrical relation of reciprocity, or of conjugation, between the two points c and E, which is indeed well-known, but which it appeared useful for our purpose to prove by quaternions* anew.

(13.) In general, each of the two connected equations,

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may be said to be a form of the Equation of Conjugation between any two points P and P' (not those so marked in Fig. 52), of which the vectors satisfy it: because it expresses that those two points are, in a well-known sense, conjugate to each other, with respect to the given sphere, TpTa.

(14.) If one of the two points, as P', be given by its vector p', while the other point P and vector p are variable, the equation then represents a plane locus; namely, what is still called the polar plane of the given point, whether that point be external or internal, or on the surface of the sphere.

(15.) Let P, P′ be thus two conjugate points; and let it be proposed to find the points s, s', in which the right line PP' intersects the sphere. Assuming (comp. 25)


Os=σ = xp+yp',


To = Ta,

and attending to the equation of conjugation (13.), we have, by 210, XX., or by 200, VII., the following quadratic equation in y: x,

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(16.) Hence it is evident that, if the points of intersection s, s' are to be real, one of the two points P, P' must be interior, and the other must be exterior to the sphere; because, of the two norms here occurring, one must be greater and the other less than unity. And because the two roots of the quadratic, or the two values of y: x, differ

* In fact, it will easily be seen that the investigations in recent sub-articles are put forward, almost entirely, as exercises in the Language and Calculus of Quaternions, and not as offering any geometrical novelty of result.


EQUATION Of ellipsoid, resumed.


only by their signs, it follows (by 26) that the right line PP' is harmonically divided (as indeed it is well known to be), at the two points s, s' at which it meets the sphere : or that in a notation already several times employed (25, 31, &c.), we have the harmonic formula,


(17.) From a real but internal point P, we can still speak of a cone of tangents, as being drawn to the sphere: but if so, we must say that those tangents are ideal, or imaginary ;* and must consider them as terminating on an imaginary circle of contact; of which the real but wholly external plane is, by quaternions, as by modern geometry, recognised as being (comp. (14.)) the polar plane of the supposed internal point.

216. Some readers may find it useful, or at least interesting, to see here a few examples of the application of the General Distributive Principle (212) of multiplication to the Ellipsoid, of which some forms of the Quaternion Equation were lately assigned (in 204, (14.)); especially as those forms have been found to conduct† to a Geometrical Construction, previously unknown, for that celebrated and important Surface: or rather to several such constructions. In what follows, it will be supposed that any such reader has made himself already sufficiently familiar with the chief formulæ of the preceding Articles; and therefore comparatively few references‡ will be given, at least upon the present subject.

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the two constant vectors a, ẞ being supposed to be real, and to be inclined to each other at some acute or obtuse (but not right§) angle, is a surface of the second order,

* Compare again the second Note to page 90, and others formerly referred to. ↑ See the Proceedings of the Royal Irish Academy, for the year 1846. Compare the Note to page 218.

§ If ẞa, the system I. represents (not an ellipse but) a pair of right lines, real or ideal, in which the cylinder of revolution, denoted by the second equation of that system, is cut by a plane parallel to its axis, and represented by the first equa


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