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like manner, as the general Division of Quaternions was seen (in 191) to admit of being reduced to an arithmetical division of tensors, and a geometrical division of versors, so we may now (by 197, III., and 206, IV.) reduce, generally, the Subtraction of Quaternions to (Ist) an algebraical subtraction of scalars, and (IInd) a geometrical subtraction of vectors: this last operation being again constructed by a parallelogram, or even by a plane triangle (comp. Art. 4, and Fig. 2). And because the sum of any given set of vectors was early seen to have a value (9), which is independent of their order, and of the mode of grouping them, we may now infer that the Sum of any number of given Quaternions has, in like manner, a Value (comp. 197, (1.)), which is independent of the Order, and of the Grouping of the Summands: or in other words, that the general Addition of Quaternions is a Commutative* and an Associative Operation.

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is now seen to hold good, for any number of quaternions, independently of the arrangement of the terms in each of the two sums, and of the manner in which they may be associated.

(2.) We can infer anew that

K (g' + q)=Kq' + Kq, as in 195, II.,

under the form of the equation or identity,

S(g' + q)-V (q + q) = (Sq' - Vg') + (Sq - Vg).

(3.) More generally, it may be proved, in the same way, that

K&q= EKq, or briefly, KΣ = 2K,

whatever the number of the summands may be.

208. As regards the quotient or product of the right parts, Vq and Vq', of any two quaternions, let t and t' denote the tensors of those two parts, and let x denote the angle of their indices, or of their axes, or the mutual inclination of the axes, or of the planes,†.of the two quaternions q and q' themselves, so that (by 204, XVIII.),

* Compare the Note to page 175.

Two planes, of course, make with each other, in general, two unequal and supplementary angles; but we here suppose that these are mutually distinguished, by taking account of the aspect of each plane, as distinguished from the opposite aspect : which is most easily done (111.), by considering the axes as above.

CHAP. 1.] QUOTIENT OR PRODUCT of right parts.


t=TVq=Tq.sin q, t' = TVq' = Tq' . sin ≤ q',

x= (IVq': IVq) = L (Ax. q': Ax. q). Then, by 193, 194, and by 204, XXXV., XXXV'.,

I... Vq': Vq=IVq': IVq = + (TVq': TVq) . (Ax. q': Ax. q);


II. . . Vg'.Vq=IVq :I√q

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and therefore (comp. 198), with the temporary abridgments pro

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XIII... V (Vq': Vq)=0, and XIV... V (Vq'. Vq)=0,

if q'q (123);

because (comp. 191, (6.), and 204, VI.) the quotient or product of the right parts of two complanar quaternions (supposed here to be both non-scalar (108), so that t and t' are each > 0) degenerates (131) into a scalar, which may be thus expressed :

XV... Vq': Vq=+t't', and XVI... Vq'.Vq=- t't, if x=0; but

XVII. . . Vq': Vq=-t't', and XVIII. . . Vq'. Vq = +t't, if x=π; the first case being that of coincident, and the second case that of opposite axes. In the more general case of diplanarity (119), if we denote by & the unit-line which is perpendicular to both their axes, and therefore common to their two planes, or in which those planes intersect, and which is so directed that the rotation round it from Ax. q to Ax. q' is positive (comp. 127, 128), the recent formulæ I., II. give easily,

XIX... Ax. (Vq': Vq) =+d;

XX... Ax. (Vq'.Vq) =~ô;

and therefore (by IX., XI., and by 204, XXXV.), the indices of the right parts, of the quotient and product of the right parts of any two diplanar quaternions, may be expressed as follows:

XXI... IV (Vq': Vq) = +8.t't1 sin x;

XXII. . . IV (Vq'. Vq) = - ô.t't sin x.

(1.) Let ABC be any triangle upon the unit-sphere (128), of which the spherical angles and the corners may be denoted by the same letters A, B, C, while the sides shall as usual be denoted by a, b, c; and let it be supposed that the rotation (comp. 177) round a from o to B, and therefore that round в from A to c, &c., is positive, as in Fig. 43. Then writing, as we have often done,

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we easily obtain the the following expressions for the three scalars t, t', x, and for the vector :

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whence t and t' are as just stated. Also if A', B', c' be (as in 175) the positive poles of the three successive sides BC, CA, AB, of the given triangle, and therefore the points A, B, C the negative poles (comp. 180, (2.)) of the new arcs B'c', c'A', A'B', then

Ax. q = oc', Ax. q' = os';

but a and are the angle and the axis of the quotient of these two axes, or of the quaternion which is represented (162) by the arc c'A'; therefore x is, as above stated, the supplement of the angle B, and d is directed to the point upon the sphere, which is diametrically opposite to the point B.

(3.) Hence, by III. V. VII. VIII. IX. XI., for any triangle ABC on the unitsphere, with a=OA, &c., we have the formulæ :

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(4.) Also, by XIX. XX. XXI. XXII., if the rotation round в from a to c be still positive,

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(5.) If, on the other hand, the rotation round в from A to c were negative, then writing for a moment a1 =-a, ẞ1=-ß, y1=−y, we should have a new and opposite triangle, A1B1C1, in which the rotation round B1 from A to C1 would be positive, but the angle at B1 equal in magnitude to that at B; so that by treating (as usual) all the angles of a spherical triangle as positive, we should have B1 = B, as well as c1 = c, and a1 = a; and therefore, for example, by XXXI.

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the four formulæ of (4.) would therefore still subsist, provided that, for this new direction of rotation in the given triangle, we were to change the sign of ß, in the second member of each.

(6.) Abridging, generally IVq: Sq to (IV: S)q, as TVq: Sq was abridged, in 204, XXXIV'., to (TV: S) q, we have by (5.), and by XXIV., XXXII., this other general formula, for any three unit-vectors a, ẞ, y, considered still as terminating at the corners of a spherical triangle ABC:

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the upper or the lower sign being taken, according as the rotation round B from A to c, or that round ẞ from a to y, which might perhaps be denoted by the symbol aẞy, and which in quantity is equal to the spherical angle B, is positive or negative.

209. When the planes of any three quaternions q, q', q", considered as all passing through the origin o (119), contain any common line, those three may then be said to be Collinear* Quaternions; and because the axis of each is then perpendicular to that line, it follows that the Axes of Collinear Quaternions are Complanar: while conversely, the complanarity of the axes insures the collinearity of the quaternions, because the perpendicular to the plane of the axes is a line common to the planes of the quaternions.

(1.) Complanar quaternions are always collinear; but the converse proposition does not hold good, collinear quaternions being not necessarily complanar.

(2.) Collinear quaternions, considered as fractions (101), can always be reduced to a common denominator (120); and conversely, if three or more quaternions can be so reduced, as to appear under the form of fractions with a common denominator ε, those quaternions must be collinear: because the line is then common to all their planes.


(3.) Any two quaternions are collinear with any scalar; the plane of a scalar being indeterminate† (131).

(4.) Hence the scalar and right parts, Sq, Sq', Vq, Vq', of any two quaternions, are always collinear with each other.

(5.) The conjugates of collinear quaternions are themselves collinear.

Quaternions of which the planes are parallel to any common line may also be said to be collinear. Compare the first Note to page 113.

+ Compare the Note to page 114.

210. Let q, q', q' be any three collinear quaternions; and let a denote a line common to their planes. Then we may determine (comp. 120) three other lines ẞ, 7, 8, such that

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and thus may conclude that (as in algebra),

I... (q' + q) q' = q'q" + qq",

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In like manner, at least under the same condition of collinearity,* it may be proved that

II. . . (q' − q) q'' = q'q′′ — XX" ·

Operating by the characteristic K upon these two equations, and attending to 192, II., and 195, II., we find that

III... Kq". (Kq' + Kq) = Kq". Kq' + Kq". Kq;

IV... Kq".(Kq' - Kq)=Kq". Kq' - Kq".Kq;

where (by 209, (5.)) the three conjugates of arbitrary collinears, Kq, Kq', Kq", may represent any three collinear quaternions. We have, therefore, with the same degree of generality as before,

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If, then, q, q', q'', q''' be any four collinear quaternions, we may establish the formula (again agreeing with algebra):

VII... (q'"+q'') (q' + q) = q'"'q' + q'q'+q""q + q′′q;

and similarly for any greater number, so that we may write briefly, VIII... £q'. £q = Eq'q,

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m and n being any positive whole numbers. In words (comp. 13), the Multiplication of Collineart Quaternions is a Doubly Distributive Operation.

It will soon be seen, however, that this condition is unnecessary.

+ This distributive property of multiplication will soon be found (compare the last Note) to extend to the more general case, in which the quaternions are not colli


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