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(q′′ + q) q = (Sq′′ + Sq'). Sq + (Vq" + Vq′). Sq + Vq. (Sq" + Sq′) + (Vq′′ + Vq). Vq = (Sq′′. Sq + Vq”. Sq +Vq. Sq" + Vq'. Vq) + (Sq'. Sq + Vq'. Sq + Vq. Sq' +Vq'.Vq) =q"q + q'q; so that the formula 210, I., and therefore also (by conjugates) the formula 210, V., is valid generally.

212. The General Multiplication of Quaternions is therefore (comp. 13, 210) a Doubly Distributive Operation; so that we may extend, to quaternions generally, the formula (comp. 210, VIII.),

I. . . Σ' . Σq = Σ44:

however many the summands of each set may be, and whether they be, or be not, collinear (209), or right (211).

(1.) Hence, as an extension of 210, XX., we have now,

II... NΣq=2Nq + 2ΣSqKq' ;

where the second sign of summation refers to all possible binary combinations of the quaternions q, q', . .

(2.) And, as an extension of 210, XXIX., we have the inequality,

III... Tq> TΣq,

unless all the quaternions q, q', bear scalar and positive ratios to each other, in which case the two members of this inequality become equal: so that the sum of the tensors, of any set of quaternions, is greater than the tensor of the sum, in every other case.

(3.) In general, as an extension of 210, XXVII.,

IV... (ETq) (TΣq)=2(T-S)qKq'.

(4.) The formulæ, 210, XVIII., XIX., admit easily of analogous extensions. (5.) We have also (comp. 168) the general equation,

in which, by 210, IX.,

V... (Eq)-(g2) = E (qq′ + q'q);

VI... qq' + q'q=2(Sq. Sq' + Vq. Sq' + Vq'. Sq + S (Vg'. Vg) );

because, by 208, we have generally

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213. Besides the advantage which the Calculus of Quaternions gains, from the general establishment (212) of the Distributive Principle, or Distributive Property of Multiplication, by being, so far,

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CHAP. 1.] INTERsections of right lines and sPHERES. 215

assimilated to Algebra, in processes which are of continual occurrence, this principle or property will be found to be of great importance, in applications of that calculus to Geometry; and especially in questions respecting the (real or ideal*) intersections of right lines with spheres, or other surfaces of the second order, including contacts (real or ideal), as limits of such intersections. The following Examples may serve to give some notion, how the general distributive principle admits of being applied to such questions: in some of which however the less general principle (210), respecting the multiplication of collinear quaternions (209), would be sufficient. And first we shall take the case of chords of a sphere, drawn from a given point upon its surface.

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(1.) From a point A, of a sphere with o for centre, let it be required to draw a chord AP, which shall be parallel to a given line OB; or more fully, to assign the vector, P=OP, of the extremity of the chord so drawn, as a function of the two given vectors, a = OA, and ẞ=0B; or rather of a and Uß, since it is evident that the length of the line ẞ cannot affect the result of the construction, which Fig. 51 may serve to illustrate.

(2.) Since AP || OB, or p-a || B, we may begin by writing the expression,

p = a + xẞ (15),

P, E


Fig. 51.

which may be considered (comp. 23, 99) as a form of the equation of the right line AP; and in which it remains to determine the scalar coefficient r, so as to satisfy the equation of the sphere,

In short, we are to seek to satisfy the equation,

Tp Ta (186, (2.)).

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by some scalar x which shall be (in general) different from zero; and then to substitute this scalar in the expression p=a+xß, in order to determine the required vector p.


(3.) For this purpose, an obvious process is, after dividing both sides by Tẞ, to square, and to employ the formula 210, XXI., which had indeed occurred before, as 200, VIII., but not then as a consequence of the distributive property of multiplication. In this manner we are conducted to a quadratic equation, which admits of division by x, and gives then,

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2x Sa+

*Compare the Notes to page 90, &c.

the problem (1.) being thus resolved, with the verification that ẞ may be replaced by Uẞ, in the resulting expression for p.

(4.) As a mere exercise of calculation, we may vary the last process (3.), by dividing the last equation (2.) by Ta, instead of Tẞ, and then going on as before. This last procedure gives,

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a new expression for p, in which it is not permitted generally, as it was in (3.), to

treat the vector ẞ as the multiplier,* instead of the multiplicand.

(6.) It is now easy to see that the second equation of (2.) is satisfied; for the expression (5.) for p gives (by 186, 187, &c.),

Tp = T 2. TB = Ta,

as was required.

(7.) To interpret the solution (3.), let c in Fig. 51 be the middle point of the chord AP, and let D be the foot of the perpendicular let fall from a on OB; then the expression (3.) for p gives, by 196, XIX.,

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we have only to observe (comp. 138) that the angle AOP' is bisected internally, or the supplementary angle AOP externally, by the indefinite right line OB (see again Fig. 51).

(9.) Conversely, the geometrical considerations which have thus served in (7.) and (8.) to interpret or to verify the two forms of solution (3.), (5.), might have been employed to deduce those two forms, if we had not seen how to obtain them, by rules of calculation, from the proposed conditions of the question. (Comp. 145, (10.), &c.)

(10.) It is evident, from the nature of that question, that a ought to be deduci

*Compare the Note to page 159.

CHAP. 1.]



ble from ẞ and p, by exactly the same processes as those which have served us to deduce o from 3 and a. Accordingly, the form (3.) of p gives,


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And since the first form can be recovered from the second, we see that each leads us back to the parallelism, p − a || ẞ (2.).

(11.) The solution (3.) for x shows that

x=0, p=α, PA, if S(a: ) = 0, or if ẞa.

And the geometrical meaning of this result is obvious; namely, that a right line drawn at the extremity of a radius OA of a sphere, so as to be perpendicular to that radius, does not (in strictness) intersect the sphere, but touches it: its second point of meeting the surface coinciding, in this case, as a limit, with the first. (12.) Hence we may infer that the plane represented by the equation,

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is the tangent plane (comp. 196, (5.)) to the sphere here considered, at the point A. (13.) Since ẞ may be replaced by any vector parallel thereto, we may substitute for it y-a, if y=oc be the vector of any given point c upon the chord AP, whether (as in Fig. 51) the middle point, or not; we may therefore write, by (3.) and (5.),

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214. In the Examples of the foregoing Article, there was no room for the occurrence of imaginary roots of an equation, or for ideal intersections of line and surface. To give now a case in which such imaginary intersections may occur, we shall proceed to consider the question of drawing a secant to a sphere, in a given direction, from a given external point; the recent Figure 51 still serving us for illustration.


(1.) Suppose then that is the vector of any given point &, through which it is required to draw a chord or secant EPOP1, parallel to the same given line ẞ as before. We have now, if po= OPO,

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by transformations* which will easily occur to any one who has read recent articles with attention. And the points Po, Pi will be together real, or together imaginary, according as the quantity under the radical sign is positive or negative; that is, according as we have one or other of the two following inequalities,

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represents a cylinder of revolution, with OB for its axis, and with Ta for the radius of its base. If E be a point of this cylindric surface, the quantity under the radical sign in (1.) vanishes; and the two roots o, x1 of the quadratic become equal. In this case, then, the line through E, which is parallel to OB, touches the given sphere; as is otherwise evident geometrically, since the cylinder envelopes the sphere (comp. 204, (12.)), and the line is one of its generatrices. If E be internal to the cylinder, the intersections Po, P1 are real; but if E be external to the same surface, those intersections are ideal, or imaginary.

(3.) In this last case, if we make, for abridgment,

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s and t being thus two given and real scalars, we may write,

20=s-tv −1; 218+tV-1;

where V-1 is the old and ordinary imaginary symbol of Algebra, and is not invested here with any sort of Geometrical Interpretation.† We merely express thus the fact of calculation, that (with these meanings of the symbols a, ß, ɛ, s and t) the formula Ta=T(ɛ+xß), (1.), when treated by the rules of quaternions, conducts to the quadratic equation,

(x − s)2 + t2 = 0,

which has no real root; the reason being that the right line through E is, in the present case, wholly external to the sphere, and therefore does not really intersect it at all; although, for the sake of generalization of language, we may agree to say, as usual, that the line intersects the sphere in two imaginary points.

(4.) We must however agree, then, for consistency of symbolical expression, to consider these two ideal points as having determinate but imaginary vectors, namely, the two following:

Po= ε+8B-tẞ V-1; P1 = ε+sẞ+tẞ V −1;

in which it is easy to prove, Ist, that the real part ɛ+sẞ is the vector of the foot E' of the perpendicular let fall from the centre o on the line through E which is drawn (as above) parallel to OB; and IInd, that the real tensor tTẞ of the coefficient of

* It does not seem to be necessary, at the present stage, to supply so many references to former Articles, or Sub-articles, as it has hitherto been thought useful to give; but such may still, from time to time, be given.

+ Compare again the Notes to page 90, and Art. 149.

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