19. Find the normal of a face which is equally inclined to three given faces.

20. Shew that the rhombic dodecahedron may be derived from the cube, or from the octahedron, by truncation of the edges.

21. Find the form whose faces replace, symmetrically, the edges of the rhombic dodecahedron.

22. Shew how the two kinds of hemihedral forms are indicated by the quaternion expressions.

23. Shew that the cube may be produced by truncating the edges of the regular tetrahedron.

24. Point out the modifications in the auxiliary vector function required in passing to the pyramidal and prismatic systems respectively.

25. In the rhombohedral system the auxiliary quaternion operator assumes a singularly simple form. Give this form, and point out the results indicated by it.

26. Shew that if the hodograph be a circle, and the acceleration be directed to a fixed point; the orbit must be a conic section, which is limited to being a circle if the acceleration follow any other law than that of gravity.

27. In the hodograph corresponding to acceleration ƒ(D) directed towards a fixed centre, the curvature is inversely as D2f(D).

28. If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut by any two common orthogonals, the sum of the two times of hodographically describing the two intercepted arcs (small or large) will be the same for the two hodographs. (Hamilton, Elements, p. 725.)

29. Employ the last theorem to prove, after Lambert, that the time of describing any are of an elliptic orbit may be expressed in terms of the chord of the arc and the extreme radii vectores.

30. If q(q1 be the operator which turns one set of rectangular unit-vectors a, ẞ, y into another set a1, B1, 71, shew that there are three equations of the form

4 Sq Syg

βαβιβα
Saß-Spa-

Tq2

CHAPTER XI.

PHYSICAL APPLICATIONS.

373.] WE propose to conclude the work by giving a few instances of the ready applicability of quaternions to questions of mathematical physics, upon which, even more than on the Geometrical or Kinematical applications, the real usefulness of the Calculus must mainly depend-except, of course, in the eyes of that section of mathematicians for whom Transversals and Anharmonic Pencils, &c. have a to us incomprehensible charm. Of course we cannot attempt to give examples in all branches of physics, nor even to carry very far our investigations in any one branch: this Chapter is not intended to teach Physics, but merely to shew by a few examples how expressly and naturally quaternions seem to be fitted for attacking the problems it presents.

We commence with a few general theorems in Dynamics-the formation of the equations of equilibrium and motion of a rigid system, some properties of the central axis, and the motion of a solid about its centre of inertia.

374.] When any forces act on a rigid body, the force ẞ at the point whose vector is a, &c., then, if the body be slightly displaced, so that a becomes a + da, the whole work done is

Σάβδα.

This must vanish if the forces are such as to maintain equilibrium. Hence the condition of equilibrium of a rigid body is

Σάβδα = 0.

For a displacement of translation da is any constant vector, hence

For a rotation-displacement, we have by § 350, being the axis, and Te being indefinitely small,

da = Vea,

and ES.ẞVea 28.eVaß S.eΣ (Vaß) = 0,
Σδ.βΓεα =

whatever be E, hence

=

These equations, (1) and (2), are equivalent to the ordinary six equations of equilibrium.

375.] In general, for any set of forces, let

Σβ = βυ

Σ.Γαβ = 1,

it is required to find the points for which the couple a, has its axis coincident with the resultant force B1. Let y be the vector of such a point.

Then for it the axis of the couple is

a straight line (the Central Axis) parallel to the resultant force. 376.] To find the points about which the couple is least.

=

where is any vector whatever. It is useless to try yẞ1, but we may put it in succession equal to a, and Va11. Thus

or the force must be in the plane of the couple. If this be the case,

378.] To assign the values of forces έ, §1, to act at €, 1, and be equivalent to the given system.

Similarly for .

The indefinite terms may be omitted, as they must evidently be equal and opposite. In fact they are any equal and opposite forces whatever acting in the line joining the given points.

379.] For the motion of a rigid system, we have of course

ΣS (mä-ẞ) da = 0,
(mä—ß)

by the general equation of Lagrange.

Suppose the displacements da to correspond to a mere translation, then da is any constant vector, hence

Σ (mä-B) = 0,
β)

or, if a, be the vector of the centre of inertia, and therefore

and the centre of inertia moves as if the whole mass were concentrated in it, and acted upon by all the applied forces.

380.] Again, let the displacements da correspond to a rotation about an axis e, passing through the origin, then

da = Vea,

it being assumed that Te is indefinitely small.

Hence

ES.Va (ma-3)= 0,

for all values of e, and therefore

Σ.Va (mä—ß) = 0,

which contains the three remaining ordinary equations of motion. Transfer the origin to the centre of inertia, i. e. put a = a1+☎, then our equation becomes

ΣV (a1 + ∞) (mä1+mä— ß) :

Or, since Σm = 0,

= 0.

Σπ(mä-β)+ Vαι (ä, Ση-Σβ) = 0.

But ä1Σm-Σß = 0, hence our equation is simply

Again, as the motion considered is relative to the centre of inertia, it must be of the nature of rotation about some axis, in general variable. Let e denote at once the direction of, and the angular velocity about, this axis. Then, evidently,

But, by operating directly by 2/Sedt upon the equation (1), we get

Σm (Vew)2=-h2 + 2/Seødt..
(Γεπ) =-h2+2/Seødt........

(2) and (4) contain the usual four integrals of the first order.

382.] When no forces act on the body, we have therefore

Στο Γεπ

Σm2 = 2m (Ve∞)2 = — h2,
Seyh2.

and, from (5) and (6),

(4)

0,

and

(5)

(6)

(7)

One interpretation of (6) is, that the kinetic energy of rotation remains unchanged: another is, that the vector e terminates in an ellipsoid whose centre is the origin, and which therefore assigns the angular velocity when the direction of the axis is given; (7) shews that the extremity of the instantaneous axis is always in a plane fixed in space.

Also, by (5), (7) is the equation of the tangent plane to (6) at the extremity of the vector e. Hence the ellipsoid (6) rolls on the plane (7).

From (5) and (6), we have at once, as an equation which e must satisfy, y2 Σ.m (Vew)2 = — h2 (Σ.mw Ve∞)2.

This belongs to a cone of the second degree fixed in the body. Thus all the ordinary results regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively and the only difficulties to be met with in more complex properties of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. R. I. A. 1848.)

383.] Let a be the initial position of , q the quaternion by which the body can be at one step transferred from its initial position to its position at time t. Then