CHAP. II.] GEOMETRICAL ILLUSTRATIONS, QUADRATICS.

269

supposed construction of the oval: there is therefore at least one real position P, upon that curve, for which pp or fq=1; so that, for this position of that point, the equation 249, III., and therefore also the equation 248, I., is satisfied. The theorem of Art. 248, and consequently also, by 247, the theorem of 244, with its transformations. 245 and 246, is therefore in this manner proved.

253. This conclusion is so important, that it may be useful to illustrate the general reasoning, by applying it to the case of a quadratic equation, of the form,

I... fq=2(2-1) = 1; or II...

.

φρ=

응(0-1)

1).

ОР AP

=

OS

= 1. ОА

σ a

We have now to prove (comp. 250, VIII.) that a (real) point P

exists, which renders the fourth proportional (226) to the three

lines OA, OP, AP equal to a given line os, or AB, if this lat

ter be drawn = os; or which

P

A

Fig. 55.

as part of the locus) of P, by means of this equality IV., with the additional condition

*This curve of the fourth degree is the well-known Cassinian; but when it breaks up, as in Fig. 56, into two separate ovals, we here retain, as the oval of the proof, only the one round o, rejecting for the present that round a.

M', N', we reject all but the point м which is nearest to o, as not belonging (comp. 251, XII.) to the oval here considered. Then while P moves upon that oval, in the positive direction relatively to o, from м to N, and from N to м again, so that the ray op performs one positive revolution, and the amplitude of the factor OP: os increases continuously by 2, the ray AP performs in like manner one positive revolution, or (on the whole) does not revolve at all, and the amplitude of the factor AP: OA increases by 27 or by 0, according as the point a is interior or exterior to the oval. In the one case, therefore, the amplitude am op of the product increases by 47 (as in Fig. 55, bis); and in the other case, it increases by 27 (as in Fig. 56); so that in each case, it passes at least once through a value of the form 2pm, whatever its initial value may have been. Hence, for at least one real position, P, upon the oval, we have V... am op = 1, and therefore VI... Upp=1; VII... Top 1,

but

=

throughout, by the construction, or by the equation of the locus IV.; the geometrical condition op = 1 (II.) is therefore satisfied by at least one real vector p; and consequently the quadratic equation fq = 1 (I.) is satisfied by at least one real quaternion root, q=p:λ (250, VII.). But the recent form I. has the same generality as the earlier form,

VIII... F2q = q2 + 919 + 92 = 0 (comp. 245),

where qi and q2 are any two given, real, actual, and complanar quaternions; thus there is always a real quaternion in the given plane, which satisfies the equation,

VIII'. . . F2q' = q'2 + 919' + 92 = 0 (comp. 247);

subtracting, therefore, and dividing by q-q', as in algebra (comp. 224), we obtain the following depressed or linear equa

"

=

tion q, IX. . . q+q'+q=0, or IX... q-q-q-q. (comp. 246). The quadratic VIII. has therefore a second real quaternion root, q", related in this manner to the first; and because the quadratic function Faq (comp. again 245) is thus decomposable into two linear factors, or can be put under the form,

CHAP. II.] RELATIONS BETWEEN THE ROOTS.

271

it cannot vanish for any third real quaternion, q; so that (comp. 244) the quadratic equation has no more than two such real roots.

(1.) The cubic equation may therefore be put under the form (comp. 248),

X... F39 = q3 + 919a + 929 + 93 = 9 (q − q′) (a − q′′) + 93 = 0 ;

it has therefore one real root, say q', by the general proof (252), which has been above illustrated by the case of the quadratic equation; subtracting therefore (compare 247) the equation F3q' = 0, and dividing by q-q', we can depress the cubic to a quadratic, which will have two new real roots, q" and q"; and thus the cubic function may be put under the form,

XI. . . F3q = (q − q') (a − q'') (a — q``),

which cannot vanish for any fourth real value of q; the cubic equation X. has therefore no more than three real quaternion roots (comp. 244): and similarly for equations of higher degrees.

(2.) The existence of two real roots q of the quadratic I., or of two real vectors, p and p', which satisfy the equation II., might have been geometrically anticipated, from the recently proved increase = 47 of amplitude op, in the course of one circuit, for the case of Fig. 55, bis, in consequence of which there must be two real positions, P and P', on the one oval of that Figure, of which each satisfies the condition of similarity III.; and for the case of Fig. 56, from the consideration that the second (or lighter) oval, which in this case exists, although not employed above, is related to a exactly as the first (or dark) oval of the Figure is related to o; so that, to the real position P on the first, there must correspond another real position P', upon the second.

(3.) As regards the law of this correspondence, if the equation II. be put under the form,

for comparison with the form VIII.; and then the recent relation IX'. (or 246) between the two roots will take the form of the following relation between vectors,

XV... p+p' = a; or XV'... OP' = p' = a − p = PA;

so that the point p' completes (as in the cited Figures) the parallelogram OPAP', and the line PP' is bisected by the middle point c of OA. Accordingly, with this position of P', we have (comp. III.) the similarity, and (comp. II. and 226) the equation, XVI... A AOP' & P'AB; XVII. . . pp' = $(a− p) = øp = 1.

(4.) The other relation between the two roots of the quadratic VIII., namely (comp. 246),

and accordingly, the line o, or os, is a fourth proportional to the three lines oa, OP, and AP, or a, p, and - p'.

(5.) The actual solution, by calculation, of the quadratic equation VIII. in complanar quaternions, is performed exactly as in algebra; the formula being,

in which, however, the square root is to be interpreted as a real quaternion, on principles already laid down.

(6.) Cubic and biquadratic equations, with quaternion coefficients of the kind considered in 244, are in like manner resolved by the known formula of algebra; but we have now (as has been proved) three real (quaternion) roots for the former, and four such real roots for the latter.

254. The following is another mode of presenting the geometrical reasonings of the foregoing Article, without expressly introducing the notation or conception of amplitude. The equation øp = 1 of 253 being written as follows,

I...σ = xp =2 (p − a), or II... To=TXp, and III.......U。=Uxp,

a

we may thus regard the vector o as a known function of the vector p, or the point s as a function of the point P; in the sense that, while o and A are fixed, P and s vary together: although it may (and does) happen, that s may return to a former position without P having similarly returned. Now the essential property of the oval (253) may be said to be this: that it is the locus of the points P nearest to o, for which the tensor Typ has a given value, say b; namely the given value of Tʊ, or of os, when the point s, like o and A, is given. If then we conceive the point P to move, as before, along the oval, and the point s also to move, according to the law expressed by the recent formula I., this latter point must move (by II.) on the circumference of a given circle (comp. again Fig. 56), with the given origin o for centre; and the theorem is, that in so moving, s will pass, at least once, through every position on that circle, while P performs one circuit of the oval. And this may be proved by observing that (by III.) the angular motion of the radius os is equal to the sum of the angular motions of the two rays, Op and AP; but this latter sum amounts to eight right angles for the case of Fig. 55, bis,, and to four right angles for the case of Fig. 56; the radius os, and the point s, must therefore have revolved twice in the first case, and once in the second case, which proves the theorem in question.

(1.) In the first of these two cases, namely when A is an interior point, each of the three angular velocities is positive throughout, and the mean angular velocity of

CHAP. II.]

CASSINIAN OVALS, LEMNISCATA.

273 the radius os is double of that of each of the two rays OP, AP. But in the second case, when A is exterior, the mean angular velocity of the ray AP is zero; and we might for a moment doubt, whether the sometimes negative velocity of that ray might not, for parts of the circuit, exceed the always positive velocity of the ray OP, and so cause the radius os to move backwards, for a while. This cannot be, however; for if we conceive P to describe, like P', a circuit of the other (or lighter) oval, in Fig. 56, the point s (if still dependent on it by the law I.) would again traverse the whole of the same circumference as before; if then it could ever fluctuate in its motion, it would pass more than twice through some given series of real positions on that circle, during the successive description of the two ovals by P; and thus, within certain limiting values of the coefficients, the quadratic equation would have more than two real roots: a result which has been proved to be impossible.

(2.) While s thus describes a circle round o, we may conceive the connected point в to describe an equal circle round A; and in the case at least of Fig. 56, it is easy to prove geometrically, from the constant equality (253, IV.) of the rectangles OP. AP and OA. AB, that these two circles (with T'U and T'U' as diameters), and the two ovals (with MN and M'N' as axes), have two common tangents, parallel to the line oA, which connects what we may call the two given foci (or focal points), o and a: the new or third circle, which is described on this focal interval on as diameter, passing through the four points of contact on the ovals, as the Figure may serve to exhibit.

(3.) To prove the same things by quaternions, we shall find it convenient to change the origin (18), for the sake of symmetry, to the central point c; and thus to denote now CP by p, and CA by a, writing also CA = Ta = a, and representing still the radius of each of the two equal circles by b. We shall then have, as the joint equation of the system of the two ovals, the following:

But because we have generally (by 199, 204, &c.) the transformations,

VI... S. q2 = 2Sq2 − Tq2 = Tq2 + 2Vq2 = 2NSq − Nq = Nq − 2NVq,

the square of the equation V. may (by 210, (8.)) be written under either of the two following forms:

VII... (Nq-1)2 + 4NVq = 4c2;

VIII... (Ng+1)2 - 4NSq = 4c2;

whereof the first shows that the maximum value of TVq is c, at least if 2c < 1, as happens for this case of Fig. 56; and that this maximum corresponds to the value Tq = 1, or Tpa: results which, when interpreted, reproduce those of the preceding sub-article.

(4.) When 2c>1, it is permitted to suppose Sq=0, NVq = Nq=2c−1; and then we have only one continuous oval, as in the case of Fig. 55, bis; but if c < 1, though, there exists a certain undulation in the form of the curve (not represented in that Figure), TVq being a minimum for Sq= 0, or for ρ + α, but becoming (as before) a maximum when Tq = 1, and vanishing when Sq2 = 2c+1, namely at the two summits M, N, where the oval meets the axis.

(5.) In the intermediate case, when 2c = 1, the Cassinian curve IV. becomes (as is known) a lemniscata; of which the quaternion equation may, by V., be written (comp. 200, (8.)) under any one of the following forms: