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In this second interpretation, which is found to agree in all its results with the first, but is better adapted to an extension of the theory, as in the following Sections, to ternary products of vectors, a product of two vectors is treated as the product of the two right quaternions, of which those vectors are the indices (II. i. 5). It is shown that, on the same plan, the Sum of a Scalar and a Vector is a Quaternion. SECTION 6.-On the Interpretation of a Product of Three or more Vectors as a Quaternion,

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. 316-330

This interpretation is effected by the substitution, as in recent Sections, of Right Quaternions for Vectors, without change of order of the factors. Multiplication of Vectors, like that of Quaternions, is thus proved to be an Associative Operation. A vector, generally, is reduced to the Standard Trinomial Form,

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in which i, j, k are the peculiar symbols already considered (II. i. 10), but are regarded now as denoting Three Rectangular Vector-Units, while the three scalars x, y, z are simply rectangular co-ordinates; from the known theory of which last, illustrations of results are derived. The Scalar of the Product of Three coinitial Vectors, OA, OB, OC, is found to represent, with a sign depending on the direction of a rotation, the Volume of the Parallelepiped under those three lines; so that it vanishes when they are complanar. Constructions are given also for products of successive sides of triangles, and other closed polygons, inscribed in circles, or in spheres; for example, a characteristic property of the circle is contained in the theorem, that the product of the four successive sides of an inscribed quadrilateral is a scalar: and an equally characteristic (but less obvious) property of the sphere is included in this other theorem, that the product of the fire successive sides of an inscribed gauche pentagon is equal to a tangential vector, drawn from the point at which the pentagon begins (or ends). Some general Formulæ of Transformation of Vector Expressions are given, with which a student ought to render himself very familiar, as they are of continual occurrence in the practice of this Calculus; especially the four formulæ (pp. 316, 317):

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pSaßy = VẞySap + VyaS3p+ VaßSyp;

in which a, ẞ, y, p are any four vectors, while S and V are signs of the operations of taking separately the scalar and vector parts of a quaternion. On the whole, this Section (III. i. 6) must be considered to be (as regards the present exposition) an important one; and if it have been read with care, after a perusal of the portions previously indicated, no difficulty will be experienced in passing to any subsequent applications of Quaternions, in the present or any other work.


SECTION 7.-On the Fourth Proportional to Three Diplanar
SECTION 8.-On an Equivalent Interpretation of the Fourth
Proportional to Three Diplanar Vectors, deduced from
the Principles of the Second Book, .

SECTION 9.-On a Third Method of interpreting a Product
or Function of Vectors as a Quaternion; and on the
Consistency of the Results of the Interpretation so ob-
tained, with those which have been deduced from the
two preceding Methods of the present Book,





. . 361-364

These three Sections may be passed over, in a first reading. They contain, however, theorems respecting composition of successive rotations (pp. 334, 335, see also p. 340); expressions for the semi-area of a spherical polygon, or for half the opening of an arbitrary pyramid, as the angle of a quaternion product, with an extension, by limits, to the semiarea of a spherical figure bounded by a closed curve, or to half the opening of an arbitrary cone (pp. 340, 341); a construction (pp. 358360), for a series of spherical parallelograms, so called from a partial analogy to parallelograms in a plane; a theorem (p. 361), connecting a certain system of such (spherical) parallelograms with the foci of a spherical conic, inscribed in a certain quadrilateral; and the conception (pp. 353, 361) of a Fourth Unit in Space (u, or + 1), which is of a scalar rather than a vector character, as admitting merely of change of sign, through reversal of an order of rotation, although it presents itself in this theory as the Fourth Proportional (ijk) to Three Rectangular Vector Units.

SECTION 10.-On the Interpretation of a Power of a Vector as a Quaternion,

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It may be well to read this Section (III. i. 10), especially for the Exponential Connexions which it establishes, between Quaternions and Spherical Trigonometry, or rather Polygonometry, by a species of extension of Moivre's theorem, from the plane to space, or to the sphere. For example, there is given (in p. 381) an equation of six terms, which holds good for every spherical pentagon, and is deduced in this way from an extended exponential formula. The calculations in the sub-articles to Art. 312 (pp. 375-379) may however be passed over; and perhaps Art. 315, with its sub-articles (pp. 383, 384). But Art. 314, and its sub-articles, pp. 381-383, should be read, on account of the exponential forms which they contain, of equations of the circle, ellipse, logarithmic spirals (circular and elliptic), helix, and screw surface.

SECTION 11.-On Powers and Logarithms of Diplanar Quaternions; with some Additional Formulæ,

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It may suffice to read Art. 316, and its first eleven sub-articles, pp. 384-386. In this Section, the adopted Logarithm, lq, of a Quaternion q, is the simplest root, q', of the transcendental equation,

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and its expression is found to be,

lg = ITq + q .UVq,


in which T and U are the signs of tensor and versor, while ▲ q is the
angle of q, supposed usually to be between 0 and π.
Such logarithms
are found to be often useful in this Calculus, although they do not gene-
rally possess the elementary property, that the sum of the logarithms
of two quaternions is equal to the logarithm of their product: this ap-
parent paradox, or at least deviation from ordinary algebraic rules,
arising necessarily from the corresponding property of quaternion
multiplication, which has been already seen to be not generally a com-
mutative operation (qʻq" not q'q', unless q' and q′′ be complanar).
And here, perhaps, a student might consider his first perusal of this
work as closed.*




It has been already said, that this Chapter may be omitted in a first perusal of the work.

SECTION 1.-On the Definition of Simultaneous Differentials,



* If he should choose to proceed to the Differential Calculus of Quaternions in the next Chapter (III. ii.), and to the Geometrical and other Applications in the third Chapter (III. iii.) of the present Book, it might be useful to read at this stage the last Section (I. iii. 7) of the First Book, which treats of Differentials of Vectors (pp. 98-102); and perhaps the omitted parts of the Section II. i. 13, namely Articles 213-220, with their subarticles (pp. 214-233), which relate, among other things, to a Construction of the Ellipsoid, suggested by the present Calculus. But the writer will now abstain from making any further suggestions of this kind, after having indicated as above what appeared to him a minimum course of study, amounting to rather less than 200 pages (or parts of pages) of this Volume, which will be recapitulated for the convenience of the student at the end of the present Table.


SECTION 2.-Elementary Illustrations of the Definition, from Algebra and Geometry,

In the view here adopted (comp. I. iii. 7), differentials are not necessarily, nor even generally, small. But it is shown at a later stage (Art. 401, pp. 626-630), that the principles of this Calculus allow us, whenever any advantage may be thereby gained, to treat differentials as infinitesimals; and so to abridge calculation, at least in many applications.

SECTION 3.-On some general Consequences of the Defini

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Partial differentials and derivatives are introduced; and differentials of functions of functions.





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One of the most important rules is, to differentiate the factors of a quaternion product, in sitû; thus (by p. 405),

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for the differential of the reciprocal of a quaternion (or vector), is also very often useful; and so are the equations (p. 413),

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9 being any quaternion, and a any constant vector-unit, while t is a variable scalar. It is important to remember (comp. III. i. 11), that we have not in quaternions the usual equation,

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unless q and dq be complanar; and therefore that we have not generally,

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if p be a variable vector; although we have, in this Calculus, the scarcely less simple equation, which is useful in questions respecting orbital motion,

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if a be any constant vector, and if the plane of a and p be given (or constant).

SECTION 5.-On Successive Differentials and Developments, of Functions of Quaternions,.


In this Section principles are established (pp. 423-426), respecting quaternion functions which vanish together; and a form of development (pp. 427, 428) is assigned, analogous to Taylor's Series, and like it capable of being concisely expressed by the symbolical equation, 1+▲ = ed (p. 432). As an example of partial and successive differentiation, the expression (pp. 432, 433),

p=rktjskj-ak ̄t,

which may represent any vector, is operated on; and an application is made, by means of definite integration (pp. 434, 435), to deduce the known area and volume of a sphere, or of portions thereof; together with the theorem, that the vector sum of the directed elements of a spheric segment is zero: each element of surface being represented by an inward normal, proportional to the elementary area, and corresponding in hydrostatics to the pressure of a fluid on that element.

SECTION 6.-On the Differentiation of Implicit Functions of Quaternions; and on the General Inversion of a Linear Function, of a Vector or a Quaternion: with some connected Investigations,

In this Section it is shown, among other things, that a Linear and Vector Symbol, p, of Operation on a Vector, p, satisfies (p. 443) a Symbolic and Cubic Equation, of the form,




0=m- m'p+m"p2 − 03;

mp-1 = m' — m”p + p2 = 4,



= another symbol of linear operation, which it is shown how to deduce otherwise from ø, as well as the three scalar constants, m, m', m”. The connected algebraical cubic (pp. 460, 461),

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is found to have important applications; and it is proved† (pp. 460, 462) that if SAøp = Spøλ, independently of X and p, in which case the function is said to be self-conjugate, then this last cubic has three real roots, c1, c2, c3; while, in the same case, the vector equation,

Τρφο = 0,

(P) is satisfied by a system of Three Real and Rectangular Directions: namely (compare pp. 468, 469, and the Section III. iii. 7), those of the axes of a (biconcyclic) system of surfaces of the second order, represented by the scalar equation,

* At a later stage (Art. 375, pp. 509, 510), a new Enunciation of Taylor's Theorem is given, with a new proof, but still in a form adapted to quaternions. † A simplified proof, of some of the chief results for this important case of self-conjugation, is given at a later stage, in the few first subarticles to Art. 415 (pp. 698, 699).

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