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o== '(t) is the equation of the Hodograph.
P = op' + vap" gives the normal and tangential accelerations.
and the orbit is the section of
uTp = Se (926-1-P)
Epitrochoids, &c. $$ 336-348.
turns the system it is applied to through 28 times the angle of q, about
2V09-1. $$ 349-359.
Displacements of systems of points. Consequent condensation and rotation.
Preliminary about the use of y. $$ 368-371,
Condition of equilibrium of a rigid system is £S.B8a 0, where B is a vector
force, a its point of application. Hence the usual six equations in the
form & B = 0, EVaß = 0. Central axis, &c. $8 373-378.
whence the usual forms. The equation
29 = 90-'(9–79),
dw dx dy dz
2 W X Y Z
Here, if no forces act, W, X, Y, Z are homogeneous functions of the third
degree in w, x, y, z. Equation for precession. $$ 379-401.
General equation of motion of simple pendulum. Foucault's pendulum.
8.p($-p?)-1p =-1, Tp-2-0-1)-$p = 0, 1 =- ppF(T+S) Vap Vup,
The conical cusps and circles of contact. Lines of vibration, &c. $$
the integration extending round the circuit. Mutual action of two closed
closed circuit. Magnetic curves. $$ 428–448.
d d d
Farther examination of the use of v as applied to displacements of groups of
points. Proof of the fundamental theorem for comparing an integral
closed surface with one through its content
SSS S.Vods = SS 8. Uvde.
Hence Green's Theorem. Limitations and ambiguities. $$ 458-476.
Similar theorem for double and single integrals
S.odp = SS s.Uvvods.
Applications of these to distributions of magnetism, and to Ampère's
Directrice. Also to the Stress-function. $$ 477–491.
€-Sovf(p) = f(p+o).
Applications and consequences. Separation of symbols of operation, and
their treatment as quantities. $$ 492-495.
Applications of v in connection with the Calculus of Variations. If
A = SQTdp, 8A = 0 gives Qe')-vQ = 0.
Thomson's Theorem that there is one and but one solution of
8.7 (e vu) 4ar. $ 505.
VECTORS, AND THEIR COMPOSITION.
1.] For more than a century and a half the geometrical representation of the negative and imaginary algebraic quantities, - 1 and ✓ -1, or, as some prefer to write them, – and — , has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in employing such expressions to indicate the direction, not the length, of lines.
2.] Thus it was long ago seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities of the same kind by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics.
3.] Wallis, towards the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going out of the line on which, if real, they would have been laid off. His construction is equivalent to the consideration of 11 as a directed unit-line perpendicular to that on which real quantities are measured.
4.] In the usual notation of Analytical Geometry of two dimensions, when rectangular axes are employed, this amounts to reckoning each unit of length along Oy as + ✓– 1, and on Oy as -N-1; while on Ox each unit is +1, and on Os it is
-1. If we look at these four lines in circular order, i.e. in the order of positive rotation (opposite to that of the hands of a watch), they give 1, V-1, -1, V-1. In this series each expression is derived from that which precedes it by multiplication by the factor ✓– 1. Hence we may consider
✓ - 1 as an operator, analogous to a handle perpendicular to the plane of xy, whose effect on any line is to make it rotate (positively) about the origin through an angle of 90°.
5.] In such a system, a point is defined by a single imaginary expression. Thus a +b V-1 may be considered as a single quantity, denoting the point whose coordinates are a and 1. Or, it may be used as an expression for the line joining that point with the origin. In the latter sense, the expression a +b 1 implicitly contains the direction, as well as the length, of this line; since, as
6 we see at once, the direction is inclined at an angle tan-1 to the axis of ix, and the length is Va2 +62.
6.] Operating on this symbol by the factor ✓– 1, it becomes -6+a V–1; and now, of course, denotes the point whose x and y coordinates are — 6 and a; or the line joining this point with the origin. The length is still Va? +b?, but the angle the line makes with the axis of x is tan-1
1(-6); which is evidently 90° greater than before the operation.
7.] De Moivre's Theorem tends to lead us still farther in the same direction. In fact, it is easy to see that if we use, instead of ✓ –1, the more general factor cos a + V1 sin a, its effect on any line is to turn it through the (positive) angle a in the plane
[Of course the former factor, V-1, is merely the particular case of this, when a =* :]
= Thus (cosa + V1 sin a) (a+bV – 1)
= a cos a-sin a + ✓ - 1 (a sin a +b cos a), by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle a has taken place, if he compares it with the common formulæ for turning the cöordinate axes through a given angle. Or, in a less simple manner, thusLength = v(a cos a-b sin a)2 + (a sin a +b cos a)2 =VQ2 4.62
of X, Y.
8.] We see now, as it were, why it happens that
(cosa + V1 sin a)" = cos mat V 1 sin ma. In fact, the first operator produces m successive rotations in the same direction, each through the angle a; the second, a single rotation through the angle ma.
9.] It may be interesting, at this stage, to anticipate so far as to state that a Quaternion can, in general, be put under the form
N (cos 0 + o sin 0), where N is a numerical quantity, 0 a real angle, and
-1. This expression for a quaternion bears a very close analogy to the forms employed in De Moivre’s Theorem ; but there is the essential difference (to which Hamilton's chief invention referred) that a is not the algebraic ✓– 1, but may be any directed unit-line whatever in space.
10.] In the present century Argand, Warren, and others, extended the results of Wallis and De Moivre. They attempted to express as a line the product of two lines each represented by a symbol such as a+b - 1. To a certain extent they succeeded, but simplicity was not gained by their methods, as the terrible radicals in Warren's Treatise sufficiently proves.
11.] A very curious speculation, due to Servois and published in 1813 in Gergonne's Annales, is the only one, so far as has been discovered, in which the slightest trace of an anticipation of Quaternions is contained. Endeavouring to extend to space the form a +b V-1 for the plane, he is guided by analogy to write for a directed unit-line in space the form
p cos a + q cos B+r cos y, where a, ß, y are its inclinations to the three axes.
He perceives easily that p, q, r must be non-reals : but, he asks, “seraient-elles imaginaires réductibles à la forme générale A+ BV –1?” This he could not answer.
In fact they are the i, j, k of the Quaternion Calculus. (See Chap. II.)
12.] Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions;