III... at S. at + aS. at-1; IV... at S. at - aS. at-1; V... (S. a)2+(S. at-1)2 = ata ̄t=1. (2.) Let a and be any two unit-vectors, and let t be still any scalar; then V. at aS. at-1; VI... S. at S.; = VII... VIII... aV. at = a2S. at-1 = S.at+1. (3.) Hence, by the laws of i, j, k, IX... iV.it=jV.jt = kV.kt = S. at+1. (4.) We have also, by the same principles and laws, X... iV.jt=V.k'; jV.kt V.it; kV.i=V.jt; XI. . . jV. ¿t = — V.kt; kV.jt =-V.i'; iV.k' =— V. jt. (5.) The expression 308, (10.), for an arbitrary vector p, may be put under the following form: XII. p=r√. k2s+1+rkaV.¿a3. (6.) And it may be expanded as follows: XIII... p = r{(i cos tπ+j sin tπ) sin sπ+k cos Sπ}. (7.) We shall return, briefly, in the Second Chapter of this Book, on some of these last expressions, in connexion with differentials and derivatives of powers of vectors; but, for the purposes of the present Section, they may suffice. SECTION 11.-On Powers and Logarithms of Diplanar Quaternions; with some Additional Formule. 316. We shall conclude the present Chapter with a short Supplementary Section, in which the recent definition (308) of a power of a vector, with a scalar exponent, shall be extended so as to include the general case, of a Power of a Quaternion, with a Quaternion Exponent, even when the two quaternions so combined are diplanar: and a connected definition shall be given (consistent with the less general one of the same kind, which was assigned in the Second Chapter of the Second Book), for the Logarithm of a Quaternion in an arbitrary Plane:* together with a few additional Formulæ, which could not be so conveniently introduced before. q being any quaternion, and ɛ being the real and known base of the natural (or Napierian) system of logarithms, of real and positive scalars: so that (as usual), *The quaternions considered, in the Chapter referred to, were all supposed to be in the plane of the right versor i. But see the Second Note to page 265. (2.) We shall also write, for any quaternion q, the following expression for what we shall call its principal logarithm, or simply its Logarithm: III... 1q = ITq + ≤ q.UVq; and thus shall have (comp. 243) the equation, IV... ε14 = q. (3.) When q is any actual quaternion (144), which does not degenerate (131) into a negative scalar, the formula III. assigns a definite value for the logarithm, lq; which is such (comp. again 243) that V... Slq = ITq; VI... Vlg = q. UVq; VII... UVlq=UVq; VIII. . . TVlq = Lq; the scalar part of the logarithm being thus the (natural) logarithm of the tensor; and the vector part of the same logarithm lq being constructed by a line in the direction of the axis Ax. q, of which the length bears, to the assumed unit of length, the same ratio as that which the angle L q bears, to the usual unit of angle (comp. 241, (2.), (4.)). (4.) If it were merely required to satisfy the equation, IX... ¿9' = 9, in which q is supposed to be a given and actual quaternion, which is not equal to any negative scalar (3.), we might do this by writing (compare again 243), X... q' = (log q)n = lq + 2nπUVq, where n is any whole number, positive or negative or null; and in this view, what we have called the logarithm, lq, of the quaternion q, is only what may be considered as the simplest solution of the exponential equation IX., and may, as such, be thus denoted: XI... lq = (log q)0. (5.) The excepted case (3.), where q is a negative scalar, becomes on this plan a case of indetermination, but not of impossibility: since we have, for example, by the definition III., the following expression for the logarithm of negative unity, which in its form agrees with old and well-known results, but is here interpreted as signifying any unit-vector, of which the length bears to the unit of length the ratio of to 1 (comp. 243, VII.). (6.) We propose also to write, generally, for any two quaternions, q and q', even if diplanar, the following expression (comp. 243, (4.)) for what may be called the principal value of the power, or simply the Power, in which the former quaternion q is the base, while the latter quaternion q' is the exponent : XIII... q?' = ε9'19 ; and thus this quaternion power receives, in general, with the help of the definitions I. and III., a perfectly definite signification. (7.) When the base, q, becomes a rector, p, its angle becomes a right angle; the definition III. gives therefore, for this case, and this is the quaternion which is to be multipled by q', in the expression, (8.) When, for the same vector-base, the exponent q' becomes a scalar, t, the last formula becomes: XVI... pt = t1=Tpt. ExUp, if 2x=tπ; and because, by I., the relation (Up)2 = − 1 gives, or briefly, cosx+ Up sin x, XVII'. . . ¿¤Uo = cpsx, XVII... εUp: we see that the former definition, 308, I., of the power at, is in this way reproduced, as one which is included in the more general definition XIII., of the power q'; for we may write, by the last mentioned definition, with the recent values XVI. and XVII., of x and Up. (9.) In the present theory of diplanar quaternions, we cannot expect to find that the sum of the logarithms of any two proposed factors, shall be generally equal to the logarithm of the product; but for the simpler and earlier case of complanar quaternions, that algebraic property may be considered to exist, with due modifications for multiplicity of value.* (10.) The definition III. enables us, however, to establish generally the very simple formula (comp. 243, II. III.): XIX... lq=1(Tq. Uq) = ITq + 1Uq; in which (comp. (3.)), XX... lUq= q. UVq = Vlq ; XXI... TIUq=29; XXII... UlUq = UVq. (11.) We have also generally, by XIII., for any scalar exponent, t, and any quaternion base, q, the power, or briefly, XXIII... q = εtlq = (Tq). (cos t Lq + UVq. sint L q) ; XXIII'... qt = Tq. cvs t Lq, if v = UVq; in which the parentheses about Tq may be omitted, because XXIV...T(q)=(Tq)=Tq* (comp. 237, II.). (12) When the base and exponent of a power are two rectangular vectors, p and p', then, whatever their lengths may be, the product p'lp is, by XIV., a rector; but a is always a versor, XXV... Eα = cos Ta + Ua sin Ta, if a be any vector; we have therefore, *In 243, (3.), it might have been observed, that every value of each member of the formula IX., there given, is one of the values of the other member; and a similar remark applies to the formula I. and II. of 236. CHAP. I. POWERS AND FUNCTIONS OF QUATERNIONS. XXVI... T. pe'= 1, if S. pp' = 0; 387 or in words, the power pp' is a versor, under this condition of rectangularity. (13.) For example (comp. 242, (7.),* and the shortly following formula XXVIII.), XXVII. . . ¿ = ¿11i = -k; jí =¿1W = + k ; and generally, if the base be an unit-line, and the exponent a line of any length, but perpendicular to the base, the axis of the power is a line perpendicular to both; unless the direction of that aris becomes indeterminate, by the power reducing itself to a scalar, which in certain cases may happen. (14.) Thus, whatever scalar e may be, we may write, this power, then, is a versor (12.), and its axis is generally the line k; but in the case when c is any whole and even number, this versor degenerates into positive or negative unity (153), and the axis becomes indeterminate (131). (15.) If, for any real quaternion q, we write again, XXIX... UVq=v, and therefore XXX... vq= qv, and XXXI. . . v2 = − 1, the process of 239 will hold good, when we change i to v; the series, denoted in I. by 9, is therefore always at last convergent,† however great (but finite) the tensor Tq may be; and in like manner the two following other series, derived from it, which represent (comp. 242, (3.)) what we shall call, generally, by analogy to known expressions, the cosine and sine of the quaternion q, are always ultimately convergent: (16.) We shall also define that the secant, cosecant, tangent, and cotangent of a quaternion, supposed still to be real, are the functions: and thus shall have the usual relations, sec q=1: cos &c. (17.) We shall also have, XXXVI. . . ¿q = cos q + v sin q, = cos q - v sin q ; * In the theory of complanar quaternions, it was found convenient to admit a certain multiplicity of value for a power, when the exponent was not a whole number; and therefore a notation for the principal value of a power was employed, with which the conventions of the present Section enable us now to dispense. In fact, it can be proved that this final convergence exists, even when the quaternion is imaginary, or when it is replaced by a biquaternion (214, (8.)); but we have no occasion here to consider any but real quaternions. and therefore, as in trigonometry (comp. 315, (1.)), XXXVII... (cos q)2 + (sin q)2= ε "I ‚ ε - "I = ε ° = 1, whatever quaternion q may be. (18.) And all the formulæ of trigonometry, for cosines and sines of sums of two or more arcs, &c., will thus hold good for quaternions also, provided that the quaternions to be combined are in any common plane; for example, XXXVIII... cos (q' + q) = cos q'cos q- sin q'sin q, if q' || | 9. (19.) This condition of complanarity is here a necessary one; because (comp. (9.)) it is necessary for the establishment of the exponential relation between sums and powers. (20.) Thus, we may indeed write, XXXIX. . . ¿9'+q = εI′. EI,_iƒ_q||| 2; but, in general, the developments of these two expressions give the difference, XL. . . ¿9'+ 9 — EL′ ɛI = 2 and + terms of third and higher dimensions; XLI... § (99' — q'q) =V (Vq.Vq'), an expression which does not vanish, when the quaternions q and q' are diplanar. (21.) A few supplementary formulæ, connected with the present Chapter, may be appended here, as was mentioned at the commencement of this Article (316). And first it may be remarked, as connected with the theory of powers of vectors, that if a, ẞ, y be any three unit-lines, OA, OB, OC, and if o denote the area of the spherical triangle ABC, then the formula 298, XX. may be thus written: (22.) The immediately preceding formula, 298, XIX., gives for any three vectors, the relation: XLIII. . . (Uaßy)2 + (Ußy)2 + (Uay)2 + (Uaß)2 +4Uay. SÚaß. SUßy=-2; for example, if a, ß, y be made equal to i, j, k, the first member of this equation becomes, 1-1-1 1+0=-2. (23.) The following is a much more complex identity, involving as it does not only three arbitrary vectors a, ß, y, but also four arbitrary scalars, a, b, c, and r; but it has some geometrical applications, and a student would find it a good exercise in transformations, to investigate a proof of it for himself. To abridge notation, the three vectors a, ẞ, y, and the three scalars a, b, c, are considered as each composing a cycle, with respect to which are formed sums 2, and products II, on a plan which may be thus exemplified: XLIV... EaVẞy = aVẞy + bVya+cVaß; Пa3 = a2b2c3. + Σ (r2 + a2+a2) {(Vẞy)2 + 2bc(r2 + Sẞy) − r2 (B − y)2}; the sign of summation in the last line governing all that follows it. |