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when n is a whole number greater than three; nor in interpreting, in connexion therewith, the equations,

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Denoting, for the moment, what we shall call the principal nth root of a quaternion Q by the symbol/Q, we have, on this plan (comp. 231, VIII.),

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VII... V(-1):i>0;

this last condition, namely that there shall be a positive (scalar) coefficient y of i, in the binomial (or couple) form x + iy (228), for the quaternion / 1, thus serving to complete the determination of that principal nth root of negative unity; or of any other negative scalar, since - 1 may be changed to -a, if a > 0, in each of the two last formulæ. And as to the general nth root of a quaternion, we may write, on the same principles,

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where the factor 15, representing the general nth root of positive unity, has n different values, depending on the division of the circumference of a circle into n equal parts, in the way lately illustrated, for the case n=3, by Figure 54; and only differing from ordinary algebra by the reality here attributed to i. In fact, each of these nth roots of unity is with us a real versor; namely the quotient of two radii of a circle, which make with each other an angle, equal to the nth part of some whole number of circumferences.


(1.) We propose, however, to interpret the particular symbol in, as always denoting the principal value of the nth root of i; thus writing,

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whence it will follow that when this root is expressed under the form of a couple (228), the two constituents x and y shall both be positive, and the quotient y: shall have a smaller value than for any other couple x + iy (with constituents thus positive), of which the nth power equals i.

(2) For example, although the equation


is satisfied by the two values, ±(1 + i): V2, we shall write definitely,

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is satisfied by the three distinct and real couples, (i + √3): 2, and -i, we shall adopt only the one value,



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and this root, in, thus interpreted, denotes a versor, which turns any line on which it operates, through an angle equal to the nth part of a right angle, in the positive direction of rotation, round the given axis of i.

234. If m and n be any two positive whole numbers, and q any quaternion, the definition contained in the formula 233, II., of the whole power, q", enables us to write, as in algebra, the two equations:

I... qmq" = qmn; II... (q)m=qmn;

and we propose to extend the former to the case of null and negative whole exponents, writing therefore,

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We shall also extend the formula II., by writing

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whether m be positive or negative; so that this last symbol, if m and n be still whole numbers, whereof n may be supposed to be positive, has as many distinct values as there are units in the denominator of its fractional exponent, when reduced to its

Compare the Note to page 121.


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least terms; among which values of q", we shall naturally consider as the principal one, that which is the mth power of the principal nth root (233) of q.

(1.) For example, the symbol q* denotes, on this plan, the square of any cuberoot of q; it has therefore three distinct values, namely, the three values of the cuberoot of the square of the same quaternion q; but among these we regard as principal, the square of the principal cube-root (231) of that proposed quaternion.

(2.) Again, the symbol q is interpreted, on the same plan, as denoting the square of any fourth root of q; but because (1)2 = 1 = +1, this square has only two distinct values, namely those of the square root q, the fractional exponent being thus reduced to its least terms; and among these the principal value is the square of the principal fourth root, which square is, at the same time, the principal square root (199, (1.), or 227) of the quaternion q.

(3.) The symbol q denotes, as in algebra, the reciprocal of a square-root of q; while q2 denotes the reciprocal of the square, &c.

(4.) If the exponent t, in a symbol of the form q', be still a scalar, but a surd (or incommensurable), we may consider this surd exponent, t, as a limit, towards which a variable fraction tends: and the symbol itself may then be interpreted as the corresponding limit of a fractional power of a quaternion, which has however (in this case) indefinitely many values, and can therefore be of little or no use, until a selection shall have been made, of one value of this surd power as principal, according to a law which will be best understood by the introduction of the conception of the amplitude of a quaternion, to which in the next Section we shall proceed.

(5.) Meanwhile (comp. 233), (4.) ), we may already definitely interpret the symbol i' as denoting a versor, which turns any line in the given plane, through t right angles, round Ax. i, in the positive or negative direction, according as this scalar exponent, t, whether rational or irrational, is itself positive or negative; and thus may establish the formula,

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SECTION 3. On the Amplitudes of Quaternions in a given Plane; and on Trigonometric Expressions for such Quaternions, and for their Powers.

235. Using the binomial or couple form (228) for a quaternion in the plane of i (225), if we introduce two new and real scalars, r and z, whereof the former shall be supposed to be positive, and which are connected with the two former scalars x and y by the equations,

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we shall then evidently have the formulæ (comp. 228, (5.)): II. . . Tq = T (x + iy) = r ;

III... Uq= U (x + y) = cos z + i sin z;

which last expression may be conveniently abridged (comp. 233, XII'., and 234, VIII.) to the following:

IV... Uq = cis z ; so that V... q = r cis z.

And the arcual or angular quantity, z, may be called the Amplitude of the quaternion q; this name being here preferred by us to "Angle," because we have already appropriated the latter name, and the corresponding symbol q, to denote (130) an angle of the Euclidean kind, or at least one not ex. ceeding, in either direction, the limits 0 and ; whereas the amplitude, z, considered as obliged only to satisfy the equations I., may have any real and scalar value. We shall denote this amplitude, at least for the present, by the symbol,† am.q, or simply, am q; and thus shall have the following formula, of connexion between amplitude and angle,

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The symbol V was spoken of, in 202, as completing the system of notations peculiar to the present Calculus; and in fact, besides the three letters, i, j, k, of which the laws are expressed by the fundamental formula (A) of Art. 183, and which were originally (namely in the year 1843, and in the two following years) the only peculiar symbols of quaternions (see Note to page 160), that Calculus does not habitually employ, with peculiar significations, any more than the five characteristics of operation, K, S, T, U, V, for conjugate, scalar, tensor, versor, and vector (or right part): although perhaps the mark N for norm, which in the present work has been adopted from the Theory of Numbers, will gradually come more into use than it has yet done, in connexion with quaternions also. As to the marks, 4, Ax., I, R, and now am. (or am,), for angle, axis, index, reciprocal, and amplitude, they are to be considered as chiefly available for the present exposition of the system, and as not often wanted, nor employed, in the subsequent practice thereof; and the same remark applies to the recent abridgment cis, for cos + i sin; to some notations in the present Section for powers and roots, serving to express the conception of one ath root, &c., as distinguished from another; and to the characteristic P, of what we shall call in the next section the ponential of a quaternion, though not requiring that notation afterwards. No apology need be made for employing the purely geometrical signs, —, I, I, for perpendicularity, parallelism, and complanarity: although the last of them was perhaps first introduced by the present writer, who has found it frequently useful.

CHAP. II.] ADDition and subtraction of AMPLITUDES. 253

the upper or the lower sign being taken, according as Ax. q = + Ax. i; and n being any whole number, positive or negative or null. We may then write also (for any quaternion q || ¿) the general transformations following:


VII... Uq cis am q;

VIII... q = Tq.cis am q.

(1.) Writing q = ß: a, the amplitude am. q, or am (3 : a), is thus a scalar quantity, expressing (with its proper sign) the amount of rotation, round Ax. i, from the line a to the line ß; and admitting, in general, of being increased or diminished by any whole number of circumferences, or of entire revolutions, when only the directions of the two lines, a and ß, in the given plane of i, are given.

(2.) But the particular quaternion, or right versor, i itself, shall be considered as having definitely, for its amplitude, one right angle; so that we shall establish the particular formula,

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(3.) When, for any other given quaternion q, the generally arbitrary integer a in VI. receives any one determined value, the corresponding value of the amplitude may be denoted by either of the two following temporary symbols,* which we here treat as equivalent to each other,

amn., or Ln9;

so that (with the same rule of signs as before) we may write, as a more definite formula than VI., the equation:

X... amn.q= Ln q = 2nπ + Lq;

and may say that this last quantity is the nth value of the amplitude of q; while the zero-value, am。 q, may be called the principal amplitude (or the principal value of the amplitude).

(4.) With these notations, and with the convention, am。 (− 1) = +π, we may



XI... amo q = Log=±49;

XII. . . am, a = am, 1 = 4, 1=2nπ, if a>0;

XIII... am,, (-a) = am, (− 1) = ≤n (− 1) = (2n + 1) π,

if a be still a positive scalar.

236. From the foregoing definition of amplitude, and from the formerly established connexion of multiplication of versors with composition of rotations (207), it is obvious that (within the given plane, and with abstraction made of tensors) multiplication and division of quaternions answer respectively to

* Compare the recent Note, respecting the notations employed.

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