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IX... (q+n1q′) = &q+n ̄1¥n (9, 9′) ;


hence generally, for all values of the number n, as well as for all values of the two independent quaternions, q, q', and for all forms of the two functions, f, p, we may write,

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an equation of which the limiting form, for n = ∞, is (with the notations used) the equation VII. which was to be proved.

(5.) It is scarcely worth while to verify the general formula X., by any particular example: yet, merely as an exercise, it may be remarked that if we take the forms,

XI... fq=q2, $q = q2, 4q=q^,

of which the two first give, by 325, VI., the common derived form,

XII. . . ƒn (q, q) = ¢n(I, q) = qq′ + qʻq + n-1 q'2,

the formula X. becomes,

XIII... ¥n(q, q′) = $n(q2, ¶¶′ + qʻq+n−1q'2)

= q2 (qg' + qʻq + n ́1 q2) + (qq' + qʻq+n ̄1q′3) q2 +n ̄1 (qq′ + q′g + n ̄1 q'2)2 ; which agrees with the value deduced immediately from the function g or q1, by the definition 325, IV., namely,

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(6.) In general, the theorem, or rule, for differentiating as in (1.) a function of a function, of a quaternion or other variable, may be briefly and symbolically expressed by the formula,

XV. . . d(pƒ)q = d¢(fq) ;

and if we did not otherwise know it, a proof of its correctness would be supplied, by the recent proof of the correctness of the equivalent formula VII.

SECTION 4.-Examples of Quaternion Differentiation.

332. It will now be easy and useful to give a short collection of Examples of Differentiation of Quaternion Functions and Equations, additional to and inclusive of those which have incidentally occurred already, in treating of the principles of the subject.

(1.) If c be any constant quaternion (as in 330), then

I... dc = 0; II. . . d( fq + c) = dfq;
III... d. cfq=cdfq; IV... d(fq. c) = dfq. c.

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and similarly for a product of more functions than two: the rule being simply, to differentiate each factor separately, in its own place, or without disturbing the order

of the factors (comp. 318, 319); and then to add together the partial results (comp.


(4.) In particular, if m be any positive whole number,


VIII... d. q = qm-1 dq + qm-2 dq. q..+qdq. qm-2 + dq . qm−1 ;

and because we have seen (324, (2.)) that

IX... d. q-1=- q1. dq. q ́1,

we have this analogous expression for the differential of a power of a quaternion, with a negative but whole exponent,

X...d.q- =-q


=-q1dq.qm- q-2 dq. q1-m — q1-m dq. q ̃o — q-m dq . q ̈1.

(5.) To differentiate a square root, we are to resolve the linear equation,*
XI... q. d. q + d. q1. q1 = dq; or XI'. . . rr' + r'r = q',

if we write, for abridgment,

XII. . . r = q1,

q'=dq, r'=d. qa = dr.

XIII... 8 = = Kr= K. q1,

(6.) Writing also, for this purpose,

whence (by 190, 196) it will follow that

XIV. . . rs = Nr =Tr2 = Tq, and XV. . . r + s = 2Sr = 2S . qt,

the product and sum of these two conjugate quaternions, r and s, being thus scalars (140, 145), we have, by XI.,

whence, by addition,

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XVII. . . q' + r1q's = (r + s ) r ' + r' (r + 8) = 2r' (r + s) ;

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an expression for the differential of the square-root of a quaternion, which will be found to admit of many transformations, not needful to be considered here.

(7.) In the three last sub-articles, as in the three preceding them, it has been supposed, for the sake of generality, that q and dq are two diplanar quaternions; but if in any application they happen, on the contrary, to be complanar, the expressions are then simplified, and take usual, or algebraic forms, as follows:


XX... d.qm = mqm-1 dq;
XXII... d. q = qdq,

XXI... d.qm :
m=- - mq ̃m-1dq;
if XXIII. . . dg ||| 2 (123);

* Although such solution of a linear equation, or equation of the first degree, in quaternions, is easily enough accomplished in the present instance, yet in general the problem presents difficulties, without the consideration of which the theory of differentiation of implicit functions of quaternions would be entirely incomplete. But a general method, for the solution of all such equations, will be sketched in a subsequent Section.

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because, when q' is complanar with q, and therefore with q, or with r, in the expression XVIII., the numerator of that expression may be written as r¬1q′ (r + s). (8.) More generally, if x be any scalar exponent, we may write, as in the ordinary calculus, but still under the condition of complanarity XXIII.,

XXIV... d. q = xqx-1 dq;

or XXV... qd. q* = xq* dx.

333. The functions of quaternions, which have been lately differentiated, may be said to be of algebraic form; the following are a few examples of differentials of what may be called, by contrast, transcendental functions of quaternions: the condition of complanarity (dq || q) being however here supposed to be satisfied, in order that the expressions may not become too complex. In fact, with this simplification, they will be found to assume, for the most part, the known and usual forms, of the ordinary differential calculus.

(1.) Admitting the definitions in 316, and supposing throughout that dg ||| 9. we have the usual expressions for the differentials of ε and lq, namely,

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(2.) We have also, by the same system of definitions (316),

III... d sin q = cos qdq; IV... d cos q = -sin qdq; &c. (3.) Also, if r and dr be complanar with q and dg, then, by 316, IV'. . . d. q” = d. ɛrl9 = q'd.rlq = q′′ (lqdr + q1rdq) ;

or in the notation of partial differentials (329),

V... dq.qrqr-1dq, and VI. . . dr. qr = q'lqdr.

(4.) In particular, if the base q be a given or constant vector, a, and if the exponent r be a variable scalar, t, then (by the value 316, XIV. of lp) the recent formula IV. becomes,

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(5.) If then the base a be a given unit line, so that ITa = 0, and Ua = a, we may write simply,

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(6.) This useful formula, for the differential of a power of a constant unit line, with a variable scalar exponent, may be obtained more rapidly from the equation 308, VII., which gives,

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since it is evident that the differential of this expression is equal to the expression itself multiplied by adt, because a2 = − 1.

(7.) The formula VIII. admits also of a simple geometrical interpretation, connected with the rotation through t right angles, in a plane perpendicular to a, of

which rotation, or version, the power at, or the versor Ua', is considered (308) to be the instrument,* or agent, or operator (comp. 293).

334. Besides algebraical and transcendental forms, there are other results of operation on a quaternion, q, or on a function thereof, which may be regarded as forming a new class (or kind) of functions, arising out of the principles and rules of the Quaternion Calculus itself: namely those which we have denoted in former Chapters by the symbols,

I... Kq, Sq, Vq, Nq, Tq, Uq,

or by symbols formed through combinations of the same signs of operation, such as

II... SUq, VUq, UVq, &c.

And it is essential that we should know how to differentiate expressions of these forms, which can be done in the following manner, with the help of the principles of the present and former Chapters, and without now assuming the complanarity, dq ||| 9.

(1.) In general, let ƒ represent, for a moment, any distributive symbol, so that for any two quaternions, q and q', we shall have the equation,

III. . . ƒ (q + q') = ƒq+fq';

and therefore also† (comp. 326, (5.)),

IV... f(xq)=xfq, if x be any scalar.

(2.) Then, with the notation 325, IV., we shall have

V... fn(q, q')=n{ƒ (q+n ̄1q′) − fa} = ƒd' ;

and therefore, by 325, VIII., for any such function fq, we shall have the differential expression,

VI... dfq=fdq.

(3.) But S, V, K have been seen to be distributive symbols (197, 207); we can therefore infer at once that

VII... dKq = Kdq;

VIII... dSq = Sdq;


IX... dVq= Vdq;

or in words, that the differentials of the conjugate, the scalar, and the vector of a quaternion are, respectively, the conjugate, the scalar, and the vector of the differential of that quaternion.

(4.) To find the differential of the norm, Nq, or to deduce an expression for dNg, we have (by VII. and 145) the equation,

* Compare the second Note to page 133.

In quaternions the equation III. is not a necessary consequence of IV., although the latter is so of the former; for example, the equation IV., but not the equation III., will be satisfied, if we assume fq= qcq'c'q, where e and e' are any two constant quaternions, which do not degenerate into scalars.

CHAP. II.] DIFFERENTIALS OF Tensor and versor.

but and


X... dNq=d. qKq=dq. Kq+q. Kdq;

qKq=K.qKq, by 145, and 192, II.;

(1 + K). q'Kq = 2S. q'Kq = 2S(Kq. q' ́), by 196, II., and 198, I.;

XI... dNq = 2S(Kq. dq).


(5.) Or we might have deduced this expression XI. for dNq, more immediately, by the general formula 324, IV., from the earlier expression 200, VII., or 210, XX., for the norm of a sum, under the form,

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(6.) The tensor, Tq, is the square-root (190) of the norm, Nq; and because Tq and Ng are scalars, the formula 332, XXII. may be applied; which gives, for the differential of the tensor of a quaternion, the expression (comp. 158),

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(7.) The versor Uq is equal (by 188) to the quotient, q: Tq, of the quaternion 9 divided by its tensor Tq; hence the differential of the versor is,

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whence follows at once this formula, analogous to XIII., and like it easily remembered,

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(8.) We might also have observed that because (by 188), we have generally q= Tq. Ug, therefore (by 332, (3.)) we have also,

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if then we have in any manner established the equation XIII., we can immediately deduce XV.; and conversely, the former equation would follow at once from the latter.

(9.) It may be considered as remarkable, that we should thus have generally, or for any two quaternions, q and dq, the formula :*

* When the connexion of the theory of normals to surfaces, with the differential calculus of quaternions, shall have been (even briefly) explained in a subsequent Section, the student will perhaps be able to perceive, in this formula XVIII., a recognition, though not a very direct one, of the geometrical principle, that the radii of a sphere are its normals.

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