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(14.) Accordingly, the three operations, P, P1, P2, by which lines in the three lately determined directions (12.) are destroyed, or reduced to zero, and which at first present themselves under the forms,

XLII. . . Φρ = λ8μρ + μπαρ, Φιρ = λομ + ρΤλμ, Φ2= γλρμ - ρΤλμ, are found to admit of the transformations,

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where V, VI, Y2 have the recent forms XXXVI., and the loci of Op, Pip, Pap compose a system of three rectangular planes.


(15.) In general, the relations (13.) give also (comp. 353, (8.)),

whence also,

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the symbols (in any one system of this sort) admitting of being transposed and grouped at pleasure; if then the roots of M=0 be real and unequal, there arises a system of three real and distinct planes, which are connected with the interpretation of the symbolical equation, Þ¡Þ1⁄2Þ3 = 0, exactly as the three planes in 353, (9.) were connected with the analogous equation, 12 = 0.

(16.) And when the cubic has two imaginary roots, it may then be said that there is one real plane (such as the plane ±y in 353, (18.), (19.)), containing the two imaginary directions which then satisfy the equation I.; and two imaginary planes, which respectively contain those two directions, and intersect each other in one real line (such as the line y in the example cited), namely the one real vector root of the same equation I.

355. Some additional light may be thrown upon that vector equation of the second degree, by considering the system of the two scalar equations,

I... Sxpop = 0, and II... Sλp=0,

and investigating the condition of the reality of the two* directions, P and P2, by which they are generally satisfied, and for each of which the plane of p and pp contains generally the given linea in I., or is normal to the plane locus II. of p. We shall find that these two directions are always real and rectangular (except that they may become indeterminate), when the linear function is its own conjugate; and that then, if à be a root p。 of the vector equation,

III... V pop=0,

* Geometrically, the equation I. represents a cone of the second order, with A for one side, and with the three lines p which satisfy III. for three other sides; and II. represents a plane through the vertex, perpendicular to the side λ. The two directions sought are thus the two sides, in which this plane cuts the cone.

CHAP. II.] NEW PROOF OF EXISTENCE OF THE SYSTEM. 465 which has been already otherwise discussed, the lines p1 and p, are also roots of that equation; the general existence (354) of a system of three real and rectangular directions, which satisfy this equation III. when 'pop, being thus proved anew: whence also will follow a new proof of the reality of the scalar roots of the cubic M = 0, for this case of self-conjugation of ; and therefore of the necessary reality of the roots of that other cubic, M。=0, which is formed (354, IV. or XXII.) from the self-conjugate part go of the general linear and vector function Фа as M=0 was formed from 4.

(1.) Let λ, μ, v be a system of three rectangular vector units, following in all respects the laws (182, 183), of the symbols i, j, k. Writing then,

IV... p = yμ + zv, and therefore, λp=yv- zμ, φρ=φμ + zόν,

the equation II. is satisfied, and I. becomes,

V. . . 0 = y2Sv¢μ+yz (Svov – Sμøμ) — z2Sμ¢v ;

the roots of which quadratic will be real and unequal, if

VI. . . (Sνφν – Εμφμ)3 + 45μφνδνφμ > 0;

and the corresponding directions of p will be rectangular, if

that is, if

VII... 0=S(y1μ + Z1v) (Y2μ + 221) = − (Y1Y2 +Z122);
VIII. . . δνφμ = 5μφν,

at least for this particular pair of vectors, μ and v.

(2.) Introducing now the expression, op = pop + Vyp (349, XII.), the conditions VI. and VIII. take the forms,

ΙΧ. . . (νφον – Εμφομ)2 + 45 (μφον)* > 4(Sγμν), and X... Sγμν= 0; which are both satisfied generally when y = 0, or p=4′ = 0; the only exception being, that the quadratic V. may happen to become an identity, by all its coefficients vanishing but the opposite inequality (to VI. and IX.) can never hold good, that is to say, the roots of that quadratic can never be imaginary, when p is thus selfconjugate.

(3.) On the other hand, when y is actual, or o'p not generally =øp, the condition X. of rectangularity can only accidentally be satisfied, namely by the given or fixed line y happening to be in the assumed plane of μ, v; and when the two directions of p are thus not rectangular, or when the scalar Syμv does not vanish, we have only to suppose that the square of this scalar becomes large enough, in order to render (by IX.) those directions coincident, or imaginary.

(4.) When p', or y=0, we may take μ and for the two rectangular directions of p, or may reduce the quadratic to the very simple form yz = 0; but, for this purpose, we must establish the relations,

ΧΙ. . . Εμφν = νόμ = 0.

(5.) And if, at the same time, λ satisfies the equation III., so that pλ || λ, we shall have these other scalar equations,

whence or,

ΧΙΙ. . . 0 = 5μφλ = Sνφλ = Sλφμ = 5λον ;
φμ || Γιλμ, and φν | Ψλμ || ν,
ΧΙΙΙ. . . 0 = Γλφλ = Τμφμ = Ψνφν;

X, μ, thus forming (as above stated) a system, of three real and rectangular roots of that vector equation III.

(6.) But in general, if III. be satisfied by even two real and distinct directions of p, the scalar and cubic equation M=0 can have no imaginary root; for if those two directions give two unequal but real and scalar values, c and c2, for the quotient pp: p, then c1 and c2 are two real roots of the cubic, of which therefore the third root is also real; and if, on the other hand, the two directions pi and p give one common real and scalar value, such as c1, for that quotient, then op = − c1p, or Pip=(+ci)p = 0, for every line in the plane of p1, p2; so that op must be of the form, - cip + ẞSp1020, and the cubic will have at least two equal roots, since it will take the form,

XIV... 0 = (c-c1)2 (c− c1 + Sp102ß),

as is easily shown from principles and formulæ already established.

(7.) It is then proved anew, that the equation M=0 has all its roots real, if 'pop; and therefore that the equation Mo=0 (as above stated) can never have an imaginary root.

(8.) And we see, at the same time, how the scalar cubic M=0 might have been deduced from the symbolical cubic 350, I., or from the equation 351, I., as the condition for the vector equation III. being satisfied by any actual p; namely by observing that if op=-cp, then ¢2p=c2p, p3p = − c3p, &c., and therefore Mp = 0, in which p, by supposition, is different from zero.

(9.) Finally, as regards the case* of indetermination, above alluded to, when the quadratic V. fails to assign any definite values to y: 2, or any definite directions in the given plane to p, this case is evidently distinguished by the condition,

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356. The existence of the Symbolic and Cubic Equation (350), which is satisfied by the linear and vector symbol o, suggests a Theoremt of Geometrical Deformation, which may be thus enunciated:

* It will be found that this case corresponds to the circular sections of a surface of the second order; while the less particular case in which o'pop, but not SupμSvov, so that the two directions of p are determined, real, and rectangular, corresponds to the axes of a non-circular section of such a surface.


†This theorem was stated, nearly in the same way, in page 568 of the Lectures; and the problem of inversion of a linear and vector function was treated, in the few preceding pages (559, &c.), though with somewhat less of completeness and perhaps of simplicity than in the present Section, and with a slightly different notation. The general form of such a function which was there adopted may now be thus expressed:

pp = 2ẞSap + Vrp, r being a given quaternion; the resulting value of m was found to be (page 561),





"If, by any given Mode, or Law, of Linear Derivation, of the kind above denoted by the symbol, we pass from any assumed Vecto a Series of Successively Derived Vectors, P1, P2, P3, ... or o1p, 3p, 3p,..; and if, by constructing a Parallelepiped, we decompose any Line of this Series, such as p3, into three partial or component lines, mp, — m'p1, m'p1⁄2, in the Directions of the three which precede it, as here of p, P1, P2; then the Three Scalar Coefficients, m, -m', m", or the Three Ratios which these three Components of the Fourth Line p3 bear to the Three Preceding Lines of the Series, will depend only on the given Mode or Law of Derivation, and will be entirely independent of the assumed Length and Direction of the Initial Vector."

(1.) As an Example of such successive Derivation, let us take the law,

I. . . p1=pp = — Vßpy, P2= p2p=- Vßp1y, &c.,

which answers to the construction in 305, (1.), &c., when we suppose that ẞ and y are unit-lines. Treating them at first as any two given vectors, our general method conducts to the equation,

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as may be seen, without any new calculation, by merely changing 9, λ, and μ, in 354, XXXIII., to 0, ß, and – y.

(2.) Supposing next, for comparison with 305, that

IV. . . ẞ2 = y2 =-1, and Sẞy = − 1,

so that ẞ, y are unit lines, and 7 is the cosine of their inclination to each other, the values III. become,

V... m = :l, m'=-1, m" = -1;

and the equation II., connecting four successive lines of the series, takes the form, VI... pз=lp + pi-lp2, or VII. . . p3 - p1 = −1(p2 − p);

m = ΣSaa'a"SB"ß'ß + ES (rVaa'.Vẞ'ß)+SrΣSaßr - SarSßr + SrTr2; and the auxiliary function which we now denote by was,

mp-1o=40= Vaa'Sß'ßo + EV.aV(Vßo.r) + (VorSr – VrSor); where the sum of the two last terms of o might have been written as orSr – rSør. A student might find it an useful exercise, to prove the correctness of these expressions by the principles of the present Section. One way of doing so would be, to treat 23Sap and r as respectively equal to pop + Vyp and c+; which would transform m and 4o, as above written, into the following,

Mo−S (y+ε) (po + c) (y +ε), and Yoo − (y + ε) S (y + ε) o + Vo (po + c) (y + ε) ;

that is, into the new values which the M and Yo of the Section assume, when p takes the new value, p= (po+ c) p + V(y + ε) p.

a result which agrees with 305, (2.), since we there found that if p = OP, &c., the interval P1P3 was =-1× PP2.

(3.) And as regards the inversion of a linear and vector function (347), or the return from any one line p1 of such a series to the line p which precedes it, our general method gives, for the example I., by 354, (12.),

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a result which it is easy to verify and to interpret, on principles already explained.

357. We are now prepared to assign some new and general Forms, to which the Linear and Vector Function (with real constants) of a variable vector can be brought, without assuming its self-conjugation; one of the simplest of which forms is the following,

I. . . pp = Vqop + Vλpu, with I'... q=g+y;



qo being here a real and constant quaternion, and A, μ two real and constant vectors, which can all be definitely assigned, when the particular form of 4 is given: except that λ and u may be interchanged (by 295, VII.), and that either may be multiplied by any scalar, if the other be divided by the same. It will follow that the scalar, quadratic, and homogeneous function of a vector, denoted by Sppp, can always be thus expressed: II... Sppp = gp2 + Sλpμp ;

or thus,

II'... Spop = g'p2 + 2SλpSup, if g=g-Sλμ;

a general and (as above remarked) definite transformation, which is found to be one of great utility in the theory of Surfaces of the Second Order.

(1.) Attending first to the case of self-conjugate functions pop, from which we can pass to the general case by merely adding the term Vyp, and supposing (in virtue of what precedes) that a1a2a3 are three real and rectangular vector-units, and c1c2c3 three real scalars (the roots of the cubic Mo= 0), such that

In the theory of such surfaces, the two constant and real vectors, A and μ, have the directions of what are called the cyclic normals.

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