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(4.) As an example, let the function fq be the reciprocal, q-1; then (comp. 339, III.) its mth differential is (for dq= const.),

XIII... dmfq=dm. q ̄1= 2.3...m.q1(-r)", if r=dq.q1;

and it is easy to prove, without differentials, that

XIV...(g+rq)=g-1(1+r)-1=g-1{1 − r + r2 — . . + (− r)m + (− r)m· 1 (1 + r)1 }; we have therefore here

XV... qm = q ̄1(− r)TM, Tm=- qmr (1+r) ̄', _T(TMm : qm) = Tr. T(1+r) ̄1; and this last tensor indefinitely diminishes with Tdq, the quaternion q being supposed to have some given value different from zero.

(5.) In general, if we establish the following equation,

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as a definitional extension of the equation 325, V.; and if we suppose that neither the function fq itself, nor any one of its differentials as far as dm-1fq is infinite; the result contained in the limiting equation XI. may then be expressed by the formula, XVII... f) (q, dq) = dmfq;


which for the particular value m = 1, if we suppress the upper index, coincides with the form 325, VIII. of the definition dfx, but for higher values of m contains a theorem namely (when dmfq is supposed neither to vanish, nor to become infinite), what we have called Taylor's Theorem adapted to Quaternions.

343. That very important theorem may be applied to cases, in which a quaternion (as in 327, (5.)), or a vector (as in 337), is expressed as a function of a scalar; also to transcendental forms (333), whenever the differentiations can be effected; and to those new forms (334), which result from the peculiar operations of the present Calculus itself. A few such applications may here be given.

(1.) Taking first this transcendental and quaternion function of a variable scalar, I. q=ftat, with Ta= 1, da = 0, dt = = const.,

we have, by 333, VIII., the general term,

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dividing then. at by at, we obtain an infinite series, which is found to be correct, and convergent; namely (comp. 308, (4.)),

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(2.) Correct and finite expansions, for S(q+dq), V(g+dq), K(q + dq), and N(+ dq), are obtained when we operate with ed on Sq, Vq, Kq, and Nq; for example (dq being still constant), the third and higher differentials of Nq vanish by 334, XI., and we have

IV... &1Nq = (1 + d +}d2) Nq = Ng + 2S(Kg. dq) + Ndq= N(q + dq);

an expression for the norm of a sum, which agrees with 210, XX., and with 200, VII.

(3.) To develope, on like principles, the tensor and versor of a sum, let us again writer for dq: q, and denote the scalar and vector parts of this quotient by s and v; so that, by 334, XIII. and XV.,

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(4.) Then writing also, for abridgment, as in a known notation of factorials,


VII... [1]=(-1). (-2). (-3).... (m),

we shall have, by 342, XIII., dq being still treated as constant, the equation,

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of which it is easy to separate the scalar and vector parts; for example,

IX... ds = S. (s + v)2 = − (s2 + v2) ;

(5.) We have also, by V. and VI.,

'. (s + v)2 = − 2sv.

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the notation being such that we have, for instance, by IX.,

XII. . . (s + d) 1 = s;

XIII... (v + d) 1=v;

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(s + d)2 1 = (s + d) s = s2 + ds = − v2 ;
(v + d)o 1 = (v + d) v = v2 + dv = v2 − 2sv.

(6.) The exponential formula 342, I., gives, therefore,

XIV. T(q+dq) = ɛdTq = ɛs+d1. Tq ;

XV... U(g+dq)= edUq=d1.Uq;

or, dividing and substituting,

XVI... T(1 + s + v) = xs+d1;

XVII... U(1 + s + v) = εe+d1 ;

s and v being here a scalar and a vector, which are entirely independent of each other; but of which, in the applications, the tensors must not be taken too large, in order that the series may converge.

(7.) The symbolical expressions, XVI. and XVII., for those two series, may be developed by (4.) and (5.); thus, if we only write down the terms which do not exceed the second dimension, with respect to s and v, we have by XII. and XIII. the development,

XVIII. . . T (1 + s + v) = 1 + s − } v2 + . . .,

XIX... U(1 + s + v) = 1 + v + (3v2 − sv) + ... ... · ;

of which accordingly the product is 1+s+v, to the same order of approximation.
(8.) A function of a sum of two quaternions can sometimes be developed, with-
out differentials, by processes of a more algebraical character; and when this hap-
pens, we may compare the result with the form given by Taylor's Series, as adapted
to quaternions in 342, and so deduce the values of the successive differentials of the
function; for example, we can infer the expression 342, XIII. for d. q ̈1, from the
series 342, XIV., for the reciprocal of a sum.



(9.) And not only may we verify the recent developments, XVIII. and XIX., by comparing them with the more algebraical forms,

XX. . . T(1 + s + v) = (1 + s + v )3 (1 + s − v)3,

XXI. . . U(1 + s + v ) = ( 1 + s + v ) 3 (1 + s − v)3,

but also, if the first of these for example (when expanded by ordinary processes, which are in this case applicable) have given us, without differentials,

XXII. . . T(q + q') = (1 + s − 4v2 . .)Tq, where s = Sq'q-1, and v=

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we can then infer the values of the first and second differentials of the tensor of a quaternion, as follows:

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whereof the first agrees with 334, XII. or XIII., and the second can be deduced from it, under the form,

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(10.) In general, if we can only develope a function ƒ(q+q′) as far as the term or terms which are of the first dimension relatively to q', we shall still obtain thus an expression for the first differential dfq, by merely writing dq in the place of qʻ. But we have not chosen (comp. 100, (14.)) to regard this property of the differential of a function as the fundamental one, or to adopt it as the definition of dfq; because we have not chosen to postulate the general possibility of such developments of functions of quaternion sums, of which in fact it is in many cases difficult to discover the laws, or even to prove the existence, except in some such way as that above explained.

(11.) This opportunity may be taken to observe, that (with recent notations) we have, by VIII., the symbolical expression,

XXV... εs+v+d1=1+s+o;

or XXVI. . . .


344. Successive differentials are also connected with successive differences, by laws which it is easy to investigate, and on which only a few words need here be said.

(1.) We can easily prove, from the definition 324, IV. of dfq, that if dq be constant,

I. . . d2ƒq = lim . n2 { ƒ(q + 2n−1 dq) − 2ƒ (q + n ́1dq) +ƒq};


with analogous expressions for differentials of higher orders.

(2.) Hence we may say (comp. 340, X.) that the successive differentials, II... dfq, d2fq, d3fq,.. for d2q=0,

are limits to which the following multiples of successive differences,

III... nAfq, n2▲2ƒq, n3▲3ƒq,.. for A2q=0,

all simultaneously tend, when the multiple nAq is either constantly equal to dq, or at least tends to become equal thereto, while the number n increases indefinitely.

(3.) And hence we might prove, in a new way, that if the function f(q+dq)

can be developed, in a series proceeding according to ascending and whole dimensions with respect to dq, the parts of this series, which are of those successive dimensions, must follow the law expressed by Taylor's Theorem* adapted to Quaternions (342).

345. It is easy to conceive that the foregoing results may be extended (comp. 338), to the successive differentiations of functions of several quaternions; and that thus there arises, in each such case, a system of successive differentials, total end pariial: as also a system of partial derivatives, of orders higher than the first, when a quaternion, or a vector, is regarded (comp. 337) as a function of several scalars.

(1.) The general expression for the second total differential,

I... d2 Q = d2 F(q, r, . .),

involves d2q, d2r, . . ; but it is often convenient to suppose that all these second differentials vanish, or that the first differentials dq, dr, . . are constant; and then dm Q, or dmF(q, r, . .), becomes a rational, integral, and homogeneous function of the mth dimension, of those first differentials dq, dr, . which may (comp. 329, III.) be thus denoted,

II. . . dm Q = (dq + dr+....)” Q; or briefly, III. . . dm = (dą + dr + . .)TM,

in developing which symbolical power, the multinomial theorem of algebra may be employed because we have generally, for quaternions as in the ordinary calculus, ¡V. . . d-d2 = dqdr.

(2.) For example, if we denote dq and dr by q' and r', and suppose

V... Q = rgr,



VI... dqQ=rq'r; VII. . . d, Q = r'gr + rqr';
VIII... d,dqQ= dqd, Q = r'q'r +rq'r'.

And in general, each of the two equated symbols IV. gives, by its operation on F(q, r), the limit of this other function, or product (comp. 344, I.),

IX. . . nn' { F(q+n-1dq, r+n'-1 dr) − F(q, r + n' ́1dr) − F(q + n−1 dq, r) + F(q, r)}; in which the numbers n and n'are supposed to tend to infinity. (3.) We may also write, for functions of several quaternions,

or briefly,

X... Q+AQ= F(q + dq, r + dr, • .) = €'1q+d,+• • F(q, r);

XI. 1+ A = €+d+ed;

with interpretations and transformations analogous to those which have occurred already, for functions of a single quaternion.

(4.) Finally, as an example of successive and partial derivation, if we resume the vector expression 308, XVIII. (comp. 315, XII. and XIII.), namely,

XII... p=rkt jskj ́sk-*,

* Some remarks on the adaptation and proof of this important theorem will be found in the Lectures, pages 589, &c.




which has been seen to be capable of representing the vector of any point of space, we may observe that it gives, without trigonometry, by the principle mentioned in 308, (11.), and by the sub-articles to 315, not only the form,

XIII. . . p = rktj2sk1-t, as in 308, XIX.,

but also, if a be any vector unit,


XIV... p = rkt+1j-28k-t = rkt (kS. a2 + iS. a2s-1). kt;

XV... p = rV. k2s+1 +rk2tV. ¿2s, as in 315, XII.

(5.) We have therefore the following new expressions (compare the sub-articles to 337), for the two partial derivatives of the first order, of this variable vector p, taken with respect to s and t:

XVI... D. πrktjeij ̄sk¬t ——πpkijk‍t,

with the verification, that


XVII... PD1 = πr2. kt jskj ̄sk ̄t‚ k1jsij ̄sk ̄t = πr2k1jk-t;

XVIII. . . Dip = πrk2tV.j22 = πrk2jS. a2s-1 = r ̄lpDsp. S. a25-1,





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pDip=-rDsp. S. a2s-1, and XX. Dsp. Dip=rps.a2s-1; XXI... Drp=r ̄1p=ktjskj-sk-t, as in 337, XXV.;

so that we have the following ternary product of these derived vectors of the first order,


XXII... Drp. Dsp. Dip π2p2S. a2s-1 = πr2D ̧S. a28;

the scalar character of which product depends (comp. 299, (9.)) on the circumstance, that the vectors thus multiplied compose (337, (10.)) a rectangular system.

(6.) It is easy then to infer, for the six partial derivatives of p, of the second order, taken with respect to the same three scalar variables, r, s, t, the expressions: XXIII. . . D‚2p =0; DrDsp = D ̧D‚p=r ̄1Ð ̧p; D‚Dtp = D¿Drp =r-1Dtp ; XXIV. . . D ̧3p =-π2p; D ̧D1p = D¿D1 = π2rk21V.j2s+1; Di2p=— π2rk2tV. i2. (7.) The three partial differentials of the first order, of the same variable vector P, are the following:

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XXVII. drp. dsp. dip = πr2dr. dS. a2. dt.

(8.) These differential vectors, drp, dẹp, dɩp, are (in the present theory) generally finite; drp, like D,p, being a line in the direction of p, or of the radius of this sphere round the origin, at least if dr, like r, be positive; while dp, like Dap, is (comp. 100, (9.)) a tangent to the meridian of that spheric surface, for which r and t are constant; but dip, like Dep, is on the contrary a tangent to the small circle (or parallel), on the same sphere, for which r and s are constant.

(9.) Treating only the radius r as constant, and writing p=OP, if we pass from the point P, or (s, t), to another point q, or (s + As, t), on the same meridian, the chord PQ is represented by the finite difference, Asp; and in like manner, if we pass from P to a point R, or (s, t + At), on the same parallel, the new chord PR is represented by the other partial and finite difference, Atp; while the point (s + As, t + At) may be denoted by s.

(10.) If now the two points q and R be conceived to approach to P, and to come to be very near it, the chords PQ and PR will very nearly coincide with the two cor

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