CHAP. II.] CASE OF DEPRESSED EQUATION. 449 (3.) If then we have not only m = 0, as in I., but also m′ = 0, as in III., the condition XIII. is satisfied, by XV.; and the two planes, above referred to, are generally rectangular. (4.) We might indeed propose to satisfy that condition XIII., by supposing that we had always, XVII... =0, that is, XVII'... p = 0, for every direction of p; but in this case, the quaternion constant q2 would vanish (by 349, II.); and therefore the constant m', as being its scalar part (by 350, III.), would still be equal to zero. (5.) The particular supposition XVII. would however alter completely the geometrical character of the question; for it would imply (comp. 351, (2.)) that the directions of the lines op and pʻp (when not evanescent) are fixed, instead of those lines having only certain planes for their loci, as before. (6.) On the side of calculation, we should thus have, for the two conjugate functions, pp and pʻp, monomial expressions of the forms, XVIII. . . op = ẞSap, p'p=aSßp; whence, by 347, XVIII., and 350, VII., we should recover the equations, p=0 and m' = 0. (7.) We should have also, in this particular case, so that op now vanishes, if p be any line in the fixed plane perpendicular to a; and in like manner op is a null line, if p be in that other fixed plane, which is at right angles to the other given line, B. (8.) These two planes, or their normals a and ẞ, or the fixed directions of the two lines o'p and pp, will be rectangular (comp. (1.)), if we have this new equation, XXI. . . $2 = 0, or XXI'... p2p = 0, for every direction of p; and accordingly the expression XVIII. gives p2p=Saß.pp=0, if Ba, and reciprocally. (9.) Without expressly introducing a and ẞ, the equation 350, XXIII. shows that when 0, and therefore also m′ = 0, as in (4.), the symbol o satisfies (comp. (2.)) the new quadratic or depressed equation, XXII... 0 = p2 — m ̋p ; which is accordingly a factor of the cubic IV., but to which that cubic is not reducible, unless we have thus = 0, as well as m' = 0. (10.) The condition, then, of the existence and rectangularity of the two planes (7.), for which we have respectively op=0 and 'p=0, without op generally vanishing (a case which it would be useless to consider), is that the four following equations should subsist: XXIII. . . m = =0, m'=0, m"= 0, and XVII... = 0; or that the cubic IV., and its quadratic factor XXII., should reduce themselves to the very simple forms, XXIV... p3 = 0, and XXV. . . $2= 0; the cubic in having thus its three roots equal, and null, and up vanishing. (11.) We may also observe that as, when even one root of the general cubic 350, I. is zero, that is when m= 0, the vector equation XXVI... op=0 was seen (in 351) to be satisfied by one real direction of p, so when we have also m' = 0, or when the cubic in has two null roots, or takes the form IV., then the two vector equations, XXVII. . . op=0, 4p=0, are satisfied by one common direction of the real and actual line p; because we have, by 350, XVII. and XX., the general relation, ψρ=ηρ - χφρ. (12.) And because, by 350, XV., we have also the relation xp=m"p - øp, it follows that when the three roots of the cubic all vanish, or when the three scalar equations XXIII. are satisfied, then the three vector equations, XXVIII. . . pp = 0, p=0, xp=0, have a common (real and actual) vector root; or are all satisfied by one common direction of p. (13.) Since m"-=x, the cubic IV. may be written under any one of the following forms, XXIX... 0=42x = 0x¢ = x2 = $.&x=&c., in which accented may be substituted for unaccented symbols: and its geometrical signification may be illustrated by a reference to certain fixed lines, and fixed planes, as follows. (14.) Suppose first that m and m' both vanish, but that m" is different from zero, so that the cubic in ø is reducible to the form IV., but not to the form XXIV.; and that the operation, which is here equivalent to - 4x, or to − x, does not annihilate every vector p, so that (comp. (4.) (5.) (6.)) øp and o'p have not the directions of two fixed lines, but have only (comp. (1.) and (3.)) two fixed and rectangular planes, II and II', as their loci; and let the normals to these two planes be denoted by A and X', so that these two rectangular lines, A and X', are situated respectively in the planes II' and II. (15.) Then it is easily shown (comp. 351) that the operation & destroys the line X' itself, while it reduces* every other line (that is, every line which is not of the form xX', with Vx = 0) to the plane IIA; and that it reduces every line in that plane to a fixed direction, μ, in the same plane, which is thus the common direction of all the lines p2p, whatever the direction of p may be. And the symbolical equation, x. 20, expresses that this fixed direction μ of p2p may also be denoted by x 10; or that we have the equation, XXX... 0 = xμ=m"μ-oμ, if μ=p2p, which can accordingly be otherwise proved: with similar results for the conjugate symbols, p' and x'. * We propose to include the case where an operation of this sort destroys a line, or reduces it to zero, under the case when the same operation reduces a line to a fixed direction, or to a fixed plane. CHAP. II.] CASE OF MONOMIAL FORM. 451 (16.) For example, we may represent the conditions of the present case by the following system of equations (comp. 351, V. VII. IX. X., and 350, VI. VII. X. XI): (op = ẞSap + B'Sa'p, 'p=aSßp + a'Sß'p, XXXI... 0=m' = S(Vaa'.VB'B) = Saß Sa'ß' – Saß'Sa'ß, m" =Saß + Sa'ß'; XXXII... α [ xp = V(aVẞp + a'Vẞ′p) = m′′p - pp, - &p=&xp=X&p= Vaa'Sßß'p, and may then write (not here supposing X' Vμv, &c.), XXXIII. ... Sλ=Vßß', X'=Vaa', SXX'= 0, \μ = øß || $ß', μ'=q'a' || p'a,_S\\μ = SX'μ' = 0 ; after which we easily find that XXXIV... φλ=0, φορ || μ, όμ=m"μ, χμ=0; (17.) Since we have thus x'' = 0, where μ' is a line in the fixed direction of '2p, we have also the equation, the locus of xp is therefore a plane perpendicular to the line '; and in like manner, μ is the normal to a plane, which is the locus of the line xp. And the symbolical equations, .$x = 0, 42. x = 0, may be interpreted as expressing, that the operation reduces every line in this new plane of xp to the fixed direction of 4-10, or of X'; and that the operation 2 destroys every line in this planeμ'; with analogous results, when accented are interchanged with unaccented symbols. Accordingly we see, by XXXII., that xp has the fixed direction of Vaa', or of X'; and that 4.4XP = 0, because X' = 0. (18.) We see also, that the operation x, or xo, destroys every line in the plane II, to which the operation reduces every line; and that thus the symbolical equations, x. 0, xp. 40, may be interpreted. = (19.) As a verification, it may be remarked that the fixed direction X', of $XP or xp, ought to be that of the line of intersection of the two fixed planes of op and Xp; and accordingly it is perpendicular by XXXIII. to their two normals, A and : with similar remarks respecting the fixed direction λ, of 'x'p or x'o'p, which is perpendicular to X' and to μ. (20.) Let us next suppose, that besides m = 0, and m' = 0, we have = 0, but that m" is still different from zero. In this case, it has been seen (6.) that the expression for pp reduces itself to the monomial form, ßSap; and therefore that the operation & destroys every line in a fixed plane (+ a), while it reduces every other line to a fixed direction (|| ẞ), which is not contained in that plane, because we have not now Saẞ = 0. (21.) In this case we have by (16.), equating a′ or ẞ′ to 0, the expressions, XXXVI... {#p=ßSap, p'p=aSßp, m"=Saß≥0, | xp=V.aVßp = (m” −4)p, x'p=V. ßVap = (m” — 4′)p, so that the equations XVIII. are reproduced; and the depressed cubic, or the quadratic XXII. in p, may be written under the very simple form, XXXVII... 0=&x=xP. (22.) Accordingly (comp. (5.) and (7.)), the operation & here reduces an arbitrary line to the fixed direction of ß, while x destroys every line in that direction; and conversely, the operation x reduces an arbitrary line to the fixed plane perpendicular to a, and destroys every line in that fixed plane. But because we do not here suppose that m" = 0, the fixed direction of op is not contained in the fixed plane of xp; and (comp. (8.) and (10.)) the directions of øp and o'p are not rectangular to each other. (23.) On the other hand, if we suppose that the three roots of the cubic in ø ranish, or that we have m=0, m' = 0, and m” = 0, as in XXIII., but that the equation p 0 is not satisfied for all directions of p, then the binomial forms XXXI. of op and p'p reappear, but with these two equations of condition between their vector constants, whereof only one had occurred before: XXXVIII. . . 0 = SaẞSa'ß' - Saß'Sa'ß, (24.) We have also now the expressions, and the cubic in becomes simply 430, as in XXIV.; but it is important to observe that we have not here (comp. (9.)) the depressed or quadratic equation p2= 0, since we have now on the contrary the two conjugate expressions, XL... p = &p=Vaa'Sß'ßp, p = ¥'p = Vß3'Sa′ap, which do not generally vanish. And the equation p3= 0 is now interpreted, by observing that 42 here reduces every line to the fixed direction of p ́10; while reduces an arbitrary vector to that fixed plane, all lines in which are destroyed by $2. (25.) In this last case (23.), in which all the roots of the cubic in & are equal, and are null, the theorem (12.), of the existence of a common vector root of the three equations XXVIII., may be verified by observing that we have now, XLI... Vaa' = 0, Vaa'=0, xVaa'=0; the third of which would not have here held good, unless we had supposed m"= 0. (26.) This last condition allows us to write, by (16.), XLII. . . φμ = 0, φ'μ' = 0, Ψμλ' = 0, μόλ=0, 5μμ = 0, the lines μ' and μ thus coinciding in direction with the normals A and A', to the planes II and II'; if then we write, XLIII, . . v = · Vλλ' || Vμμ', so that Suv = 0, Sμ'v=0, this new vector v will be a line in the intersection of those two rectangular planes, which were lately seen (14.) to be the loci of the lines op and 'p, and are now (comp. (17.)) the loci of xp and x'p; and the three lines μ, μ', v (or λ', λ, v) will compose a rectangular system. (27.) In general, it is easy to prove that the expressions, SB=aß1+bẞ'1, B′ = a'ß1 + b′ß′1, XLIV... a1 = aa + a'a', a'1 = ba+b'a', in which a, ẞ, a', ẞ′ may be any four vectors, and a, b, a', b' may be any four scalars, conduct to the following transformations (in which p may be any vector): CHAP. II. VECtor and quateRNION INVARIANTS. XLV... Saẞ1+ Sa'iẞ'1=Saß + Sa′ß′; XLVI... ẞiSaip + ẞ'iSa'ip = ẞSap + ẞ'Sa'p; XLVII... Va1a'1.Vẞ'1ẞ1 = Vaa'.Vß'ß; 453 so that the scalar, Saß + Sa'ß'; the vector, ẞSap + B'Sa'p; and the quaternion,* Vaa'.Vẞ'ß, remain unaltered in value, when we pass from a given system of four vectors aẞa'B', to another system of four vectors aißia'iß'i, by expressions of the forms XLIV. (28.) With the help of this general principle (27.), and of the remarks in (26.), it may be shown, without difficulty, that in the case (23.) the vector constants of the binomial expression ẞSap + B'Sa'p for pp may, without any real loss of generality, be supposed subject to the four following conditions, XLVIII... 0 Saẞ= Sa'ß = Sßß' = Sa'ß'; which evidently conduct to these other expressions, XLIX. . . 2p = ẞSaß'Sa'p, p3p = 0; and thus put in evidence, in a very simple manner, the general non-depression of the cubic 430, to the quadratic, p2 = 0. == (29.) The case, or sub-case, when we have not only m=0, m' =0, m” = 0, but also = 0, and therefore 2 = 0, as a depressed form of p30, by the linear function op reducing itself to the monomial ẞSap, with the relation Sa3 = 0 between its constants, has been already considered (in (10.)); and thus the consequences of the supposition III., that there are (at least) two equal but null roots of the cubic in 4, have been perhaps sufficiently discussed. (30.) As regards the other principal case of equal roots, of the cubic equation in , namely that in which the vector constants are connected by the relation V., or by the equation of condition, L... 0=m"2 — 4m' = (Saß + Sa'ẞ')2 - 4S (Vaa'.Vß'ß) = (Saß - Sa'ẞ')2 + 4Saß'Sa'ß, it may suffice to remark that it conducts, by VI., or by VII. and IX., to the symbolical equation, LI... 0 = p2, if 4-m"; and that thus its interpretation is precisely similar to that of the analogous equation, x 2 = 0, where x =m"-, as given in (14.), and in the following sub-articles. XXIX., "2 = 4m', 353. When we have m=0, but not m' = 0, nor m" the three roots of the cubic in p are all unequal, while one of them is still null, as before; and the two roots of the quadratic and scalar equation, with real coefficients (347), * We have, in these transformations, examples of what may be called Quaternion Invariants. |