the Operation of taking the Scalar of a Quaternion is a Distributive Operation (comp. 13). As to the general Subtraction of any one quaternion from any other, there is no difficulty in reducing it, by the method of Art. 120, to the second general formula of 106; nor in proving that the Scalar of the Difference is always equal to the Difference of the Scalars. In symbols, or briefly, III... S (q-q) = Sq' - Sq; when A is used as the characteristic of the operation of taking a difference, by subtracting one quaternion, or one scalar, from another. (1.) It has not yet been proved (comp. 195), that the Addition of any number of Quaternions, q, q', I′′, .. is an associative and a commutative operation (comp. 9). But we see, already, that the scalar of the sum of any such set of quaternions has a value, which is independent of their order, and of the mode of grouping them. (2.) If the summands be all right quaternions (132), the scalar of each separately vanishes, by 196, VII.; wherefore the scalar of their sum vanishes also, and that sum is consequently itself, by 196, XIV., a right quaternion: a result which it is easy to verify. In fact, if ẞa and + Y α, then γ +ẞa, because a is then perpendicular to the plane of ß and y; hence, by 106, the sum of any two right quaternions is a right quaternion, and therefore also the sum of any number of such quaternions. (3.) Whatever two quaternions and q' may be, we have always, as in algebra, the two identities (comp. 191, (7.)): 198. Without yet entering on the general theory of scalars of products or quotients of quaternions, we may observe here that because, by 196, XV., the scalar of a quaternion depends only on the tensor and the angle, and is independent of the axis, we are at liberty to write generally (comp. 173, 178, and 191, (1.), (5.)), II... S. q (q': q) = Sq'; I... Sqq' Sq'q; = the two products, qq' and q'q, having thus always equal scalars, although they have been seen to have unequal axes, for the general case of diplanarity (168, 191). It may also be noticed, that in virtue of what was shown in 193, respecting the quotient, and in 194 * Examples have already occurred in 196, (2.), (5.), (16.). CHAP. I.] SCALAR of a product, qUOTIENT, OR SQUARE. 185 respecting the product, of any two right quaternions (132), in connexion with their indices (133), we may now establish, for any such quaternions, the formula: III. . S (q': q) = S (Ig': Iq) = T (q': q). cos ≤ (Ax. q': Ax. q); == =- Tqq. cos (Ax. q': Ax. q); where the new symbol Iq is used, as a temporary abridgment, to denote the Index of the quaternion q, supposed here (as above) to be a right one. With the same supposition, we have therefore also these other and shorter formulæ : V... SU(q': q)=+ cos (Ax. q': Ax. q); VI... SUq'q= cos 2 (Ax. q': Ax. q); which may, by 196, XVI., be interpreted as expressing that, under the same condition of rectangularity of q and q', In words, the Angle of the Quotient of two Right Quaternions is equal to the Angle of their Axes; but the Angle of the Product, of two such quaternions, is equal to the Supplement of the Angle of the Axes. There is no difficulty in proving these results otherwise, by constructions such as that employed in Art. 193; nor in illustrating them by the consideration of isosceles quadrantal triangles, upon the surface of a sphere. 199. Another important case of the scalar of a product, is the case of the scalar of the square of a quaternion. On referring to Art. 149, and to Fig. 42, we see that while we have always T (q2) = (Tq)2, as in 190, and U(q2) = U(q)2, as in 161, we have also, I... 4 (q)2 = 2 ≤ q, and Ax. (q) = Ax. q, if but, by the adopted definitions of q (130), and (127, 128), II... 4 (q2)=2(π-Lq), Ax. (q2) = π 2 of Ax. q - Ax. q, if 497 2 π In each case, however, by 196, XVI., we may write, III... SU (q) = cos 2 (q) = cos 2 Lq; a formula which holds even when 4 q is 0, or which gives, IV... SU (q2) = 2 (SUq)2 – 1. Hence, generally, the scalar of q2 may be put under either of the two following forms: - V... S(q) = Tq'. cos 2 49; VI. . . S (q2) = 2 (Sq)' – Tq3 ; where we see that it would not be safe to omit the parentheses, without some convention previously made, and to write simply Sq, without first deciding whether this last symbol shall be understood to signify the scalar of the square, or the square of the scalar of q: these two things being generally unequal. The latter of them, however, occurring rather oftener than the former, it appears convenient to fix on it as that which is to be understood by Sq, while the other may occasionally be written with a point thus, S. q'; and then, with these conventions respecting notation,* we may write : VII... Sq2 (Sq)2; VIII... S. q2S (q2)« But the square of the conjugate of any quaternion is easily seen to be the conjugate of the square; so that we have generally (comp. 190, II.) the formula: IX. . . Kq2 = K (q2) = (Kq)2 = Tq2 : Uq2. (1.) A quaternion, like a positive scalar, may be said to have in general two opposite square roots; because the squares of opposite quaternions are always equal (comp. (3.)). But of these two roots the principal (or simpler) one, and that which we shall denote by the symbol Vq, or Vq, and shall call by eminence the Square Root of q, is that which has its angle acute, and not obtuse. We shall therefore write, generally, * As, in the Differential Calculus, it is usual to write dr2 instead of (dx)2; while d (x2) is sometimes written as d. 2. But as d2x denotes a second differential, so it seems safest not to denote the square of Sq by the symbol S2q, which properly signifies SSq, or Sq, as in 196, VI.; the second scalar (like the second tensor, 187, (9.), or the second versor, 160) being equal to the first. Still every calculator will of course use his own discretion; and the employment of the notation S2q for (Sq)3, as cos 2 is often written for (cos x)2, may sometimes cause a saving of space. CHAP. I.] TENSOR AND NORM OF A SUM. 187 with the reservation that, when ≤ q=0, or, this common axis of q and Vq be- while this scalar of the square root of a quaternion may, by VI., be thus trans- XII... SVq=V { } (Tq + Sq)} ; a formula which holds good, even at the limit q=π. (3.) The principle* (1.), that in quaternions, as in algebra, the equation, is an identity, may be illustrated by conceiving that, in Fig. 42, a point B' is deter- 200. Another useful connexion between scalars and tensors (or norms) of quaternions may be derived as follows. In any plane triangle AOB, we havet the relation, (T. AB)=(T. OA)-2(T. oa). (T. OB). COS AOB + (T. OB)2; in which the symbols T. oa, &c., denote (by 185, 186) the lengths of or I. . . T (q − 1 )2 = 1 − 2Tq. SUq + Tq2 = Tq2 − 2Sq + 1 ; II... N (9-1)= Nq − 2Sq + 1. But q is here a perfectly general quaternion; we may therefore change its sign, and write, III. . . T (1 + q)2 = 1 + 2Sq + Tq2 ; IV... N (1+ q)=1+2Sq + Nq. And since it is easy to prove (by 106, 107) that whatever two quaternions q and q' may be, while we easily infer this other general formula, VII... N (q + q) = Ng' + 2S. qKq' + Nq; which gives, if x be any scalar, VIII... N (q + x) = Nq + 2xSq + x2. alter. В Compare the first Note to page 162. + By the Second Book of Euclid, or by plane trigonometry. (1.) We are now prepared to effect, by rules* of transformation, some other passages from one mode of expression to another, of the kind which has been alluded to, and partly exemplified, in former sub-articles. Take, for example, the formula, which has been seen, on geometrical grounds, to represent a certain locus, namely the plane through o, perpendicular to the line oA; and therefor the same locus as that which is represented by the equation, s=0, of 196, (1.). α To pass now from the former equations to the latter, by calculation, we have only to denote the quotient p: a by q, and to observe that the first or second form, as just now cited, becomes then, which gives the third form of equation, as required. (2.) Conversely, from $0, we can return, by the same general formulæ II. a or to T (pa) T (p + a), or to T p+ a - a T( 2 − 1 ) a 1, as above; and gene while the latter equations, in turn, involve, as has been seen, the former. (3.) Again, if we take the Apollonian Locus, 145, (8.), (9.), and employ the first of the two forms 186, (5.) of its equation, namely, T (pa2a) = aT (p − a), where a is a given positive scalar different from unity, we may write it as T (ga2) = aT (q-1), or as N (q − a2) = a2N (q − 1); or by VIII., Nq- 2a2Sq+a1= a2 (Ng − 2Sq + 1); or, after suppressing - 2a2 Sq, transposing, and dividing by a2 – 1, Nq=a2; or, Np=a2Na; or, Tp=aTa; which last is the second form 186, (5.), and is thus deduced from the first, by calculation alone, without any immediate appeal to geometry, or the construction of any diagram. *Compare 145, (10.); and several subsequent sub-articles. |