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CHAP. I.] QUATERNION EQUATION of the ellipsoid.


represents the locus of all such ellipses (13.), and will be found to be an adequate representation, through quaternions, of the general ELLIPSOID (with three unequal axes): that celebrated surface being here referred to its centre, as the origin o of vectors to its points; and the six scalar (or algebraic) constants, which enter into the usual algebraic equation (by co-ordinates) of such a central ellipsoid, being here virtually included in the two independent vectors, a and 3, which may be called its two Vector-Constants.*

(15.) The equation (comp. (12.)),


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represents a cylinder of revolution, circumscribed to the ellipsoid, and touching it along the ellipse which answers to the value x = 0, in (13.); so that the plane of this ellipse of contact is represented by the equation,



the normal to this plane being thus (comp. 196, (17.)) the vector a, or oa; while the axis of the lately mentioned enveloping cylinder is ß, or OB.

(16.) Postponing any further discussion of the recent quaternion equation of the ellipsoid (14.), it may be noted here that we have generally, by XXII., the two following useful transformations for the squares, of the scalar Sq, and of the right part Vq, of any quaternion q:

XXXI... Sq2 = Tq2 + Vq2;

XXXII... Vq2 = Sq2 — Tq2.

(17.) In referring briefly to these, and to the connected formula XXII., upon occasion, it may be somewhat safer to write,"

(S)2 = (T)2 + (V)2, (V) (S)-(T)2,

(T)2 = (S)2 - (V)3,

than S2 = T2 + V3, &c.; because these last forms of notation, S2, &c., have been otherwise interpreted already, in analogy to the known Functional Notation, or Notation of the Calculus of Functions, or of Operations (comp. 187, (9.); 196, VI.; and 204, IX.).

(18.) In pursuance of the same analogy, any scalar may be denoted by the general symbol,


because scalars are the only quaternions of which the right parts vanish.


(19.) In like manner, a right quaternion, generally, may be denoted by the sym


and since this includes (comp. 204, I.) the right part of any quaternion, we may establish this general symbolic transformation of a Quaternion:


(20.) With this form of notation, we should have generally, at least for real† quaternions, the inequalities,

It will be found, however, that other pairs of vector-constants, for the central ellipsoid, may occasionally be used with advantage.

Compare Art. 149; and the Notes to pages 90, 134.

(V-10)2>0; (S-10) <0;

so that a (geometrically real) Quaternion is generally of the form:

Square-root of a Positive, plus Square-root of a Negative.

(21.) The equations 196, XVI. and 204, XIX. give, as a new link between quaternions and trigonometry, the formula:

XXXIII... tanq=TVUq: SUq= TVq: Sq.

(22.) It may not be entirely in accordance with the theory of that Functional (or Operational) Notation, to which allusion has lately been made, but it will be found to be convenient in practice, to write this last result under one or other of the abridged forms: *

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which have the advantage of saving the repetition of the symbol of the quaternion, when that symbol happens to be a complex expression, and not, as here, a single let

ter, q.

(23.) The transformation 194, for the index of a right quotient, gives generally, by II., for any quaternion q, the formula:

XXXV... IVq = TVq. Ax. q ;

XXXVI... IUVq = Ax. 9 ;

XXXVI'. . . IUV = Ax.

so that we may establish generally the symbolicalt equation,

(24.) And because Ax. (1: Vq) = — Ax. Vq, by 135, and therefore II., we may write also, by XXXV.,

XXXV'. . . I (1: Vq) = Ax. q: TVq.

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205. If any parallelogram OBDC (comp. 197) be projected on the plane through o, which is perpendicular to oa, the projected figure OB"D"C" (comp. 11) is still a parallelogram; so that

OD" = OC" + OB" (6), or d′′=y"+ ß" ;

and therefore, by 106,

S" : a = (y" : a) + (B′′ : a).

Hence, by 120, 202, for any two quaternions, q and q', we have the general formula,

I. . . V (q' + q) = Vq' + Vq ;

Compare the Note to Art. 199.

At a later stage it will be found possible (comp. the Note to page 174, &c.), to write, generally,

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and then (comp. the Note in page 118 to Art. 129) the recent equations, XXXVI., XXXVI., will take these shorter forms:

Ax. q = UVq; Ax. = UV.

CHAP. I.] RIGHT part of a sum OR DIFFERENCE.

with which it is easy to connect this other,

II. . . V (q' − q) = Vq' – Vq.

Hence also, for any three quaternions, q, q', q”,


V{q" + (g' + q)} = Vq" + V (q' + q) = Vq" + (Vq' + Vq) ; and similarly for any greater number of summands: so that we may write generally (comp. 197, II.),

III... VΣg = ΣVq, or briefly III'... V2 = 2V; while the formula II. (comp. 197, IV.) may, in like manner, be thus written,

IV... VA AV;


IV... VAq=AVq, or the order of the terms added, and the mode of grouping them, in III., being as yet supposed to remain unaltered, although both those restrictions will soon be removed. We conclude then, that the characteristic V, of the operation of taking the right part (202, 204) of a quaternion, like the characteristic S of taking the scalar (196, 197), and the characteristic K of taking the conjugate (137, 195*), is a Distributive Symbol, or represents a distributive operation: whereas the characteristics, Ax., 4, N, U, T, of the operations of taking respectively the axis (128, 129), the angle (130), the norm (145, (11.)), the versor (156), and the tensor (187), are not thus distributive symbols (comp. 186, (10.), and 200, VII.); or do not operate upon a whole (or sum), by operating on its parts (or summands).

(1.) We may now recover the symbolical equation K2 = 1 (145), under the form (comp. 196, VI.; 202, IV.; and 204, IV. VIII. IX. XI.):

V... K2 = (SV)2 = S2 - SV -- VS + V2 = S + V = 1.

(2.) In like manner we can recover each of the expressions for S2, V2 from the other, under the forms (comp. again 202, IV.):

VI... S2 = (1 − V)2 = 1 − 2V + V2=1-VS, as in 196, VI.;

VII. . . Va = (1 − S)2 = 1 − 28 + S2 = 1 − S = V, as in 204, IX.;

or thus (comp. 196, II'., and 204, XIV.), from the expressions for S and V in terms of K:

Indeed, it has only been proved as yet (comp. 195, (1.)), that KΣq = ΣKq, for the case of two summands; but this result will soon be extended.


VIII... S2 = (1 + K)2 = (1 + 2K + K2) = { (1 + K) = S ;
IX. . . V2 = 4 (1 − K)2 = † (1 − 2K + K2) = } (1 − K) = V.

(3.) Similarly,

X.. . SV =
XI. . . VS =

(1 + K) (1 − K) = 4 (1 − K2) = 0, as in 204, IV.;
(1 − K) (1 + K) = 4(1 − K2) = 0, as in 204, VIII.

206. As regards the addition (or subtraction) of such right parts, Vq, Vq, or generally of any two right quaternions (132), we may connect it with the addition (or subtraction) of their indices (133), as follows. Let OBDC be again any parallelogram (197, 205), but let oa be now an unit-vector (129) perpendicular to its plane; so that


Ta= 1, ≤ (B: a) = L (y: a) = L (8: a) = 2,



Let OB'D'C' be another parallelogram in the same plane, obtained by a positive rotation of the former, through a right angle, round oa as an axis; so that

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Ax. (B': ẞ) = Ax. (y' : y) = Ax. (8' : 8) = a.

Then the three right quotients, ẞ: a, ya, and 8: a, may represent any two right quaternions, q, q', and their sum, q'+q, which is always (by 197, (2.)) itself a right quaternion; and the indices of these three right quotients are (comp. 133, 193) the three lines B', y', d', so that we may write, under the foregoing conditions of construction,

B'=I(ẞ:a), y' = I(y: a), d=I (8: a).

But this third index is (by the second parallelogram) the sum of the two former indices, or in symbols, &'y' +ẞ'; we may therefore write,

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or in words the Index of the Sum* of any two Right Quaternions is equal to the Sum of their Indices. Hence, generally, for any two quaternions, q and q', we have the formula,

II. . . IV (q' + q) = IVg' + IVq,

* Compare the Note to page 174.



because Vq, Vq' are always right quotients (202, 204), and V (q+q) is always their sum (205, I.); so that the index of the right part of the sum of any two quaternions is the sum of the indices of the right parts. In like manner, there is no difficulty in proving that

III. . . I (q' − q) = Iq - Iq, if q = Lq=

and generally, that

IV... IV (q-q) = IVq' - IVq;


the Index of the Difference of any two right quotients, or of the right parts of any two quaternions, being thus equal to the Difference of the Indices.* We may then reduce the addition or subtraction of any two such quotients, or parts, to the addition or subtraction of their indices; a right quaternion being always (by 133) determined, when its index is given, or known.

207. We see, then, that as the MULTIPLICATION of any two Quaternions was (in 191) reduced to (Ist) the arithmetical operation of multiplying their tensors, and (IInd) the geometrical operation of multiplying their versors, which latter was constructed by a certain composition of rotations, and was represented (in either of two distinct but connected ways, 167, 175) by sides or angles of a spherical triangle: so the ADDITION of any two Quaternions may be reduced (by 197, I., and 206, II.) to, Ist, the algebraical addition of their scalar parts, considered as two positive or negative numbers (16); and, IInd, the geometrical addition of the indices of their right parts, considered as certain vectors (1): this latter Addition of Lines being performed according to the Rule of the Parallelogram (6.).† In

Compare again the Note to page 174.

It does not fall within the plan of these Notes to allude often to the history of the subject; but it ought to be distinctly stated that this celebrated Rule, for what may be called Geometrical Addition of right lines, considered as analogous to composition of motions (or of forces), had occurred to several writers, before the invention of the quaternions: although the method adopted, in the present and in a former work, of deducing that rule, by algebraical analogies, from the symbol B − A (1) for the line AB, may possibly not have been anticipated. The reader may compare the Notes to the Preface to the author's Volume of Lectures on Quaternions (Dublin, 1853).

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