The geometrical and kinematical interpreta tions of geometric addition and subtraction involve no special difficulties. They are fully explained in a variety of treatises in which this subject has its natural place, particularly in those parts of mathematical physics where the composition of velocities and of forces is explained. The first full exposition of the subject was given in the celebrated trische of Mains, published in 1827, and was, therefore, well known about 16 years before the invention of the calculus of qua ternions by Hamilton.
The non-commutative character of multi plication greatly enlarges its powers of ex pression and discrimination, but it also intro duces troublesome transformation processes and calls for extended analysis. Fully ex pressed, the quaternion product has the double form q( )q, and involves eight scalar quan tities. A first step in its analysis resolves it, in either of two important ways, into simpler factors 1. Since by the law of association q( l[q( q[1()q,]1, it consists of the two special commutative operators q( )1, 1( )q,, applied in succession, or simultaneously, at pleasure.
2. It is always possible to determine four versor quaternions r, s, r,, Si, that satisfy the following conditions: Uq-=. rs sr, and Sr = Sri, Ss— Ssi.t The given product then assumes the form ( )o= mr[s( )ri where m is a number. It consists of a tensor factor and two rotational transformations r( )ri s( )s, which are commutative and may be applied simultaneously, or in succession.
The interpretation of the special products q( )1, 1( )0, r( )r, — s( as motions, *turns,* and °rotations,* generally in a space of four dimensions, has been successfully achieved.* An adequate account of them, however, would expand this notice beyond permissible limits. No accurate description of the motion represented by the general product q ( ) q, has yet been given.
A simpler case than any of the preceding is the product q( )q Its complete inter pretation as a rotation in three-dimensional space was an early achievement by Hamilton himself. The motion consists of a rotation through twice the angle being given in the form m[cos +p sin 0]) about the vector of q as an axis; that is, any vector,placed at the origin, moves through the angle 20, on the surface of a cone having its vertex at the origin.t
Quaternion multiplication may be expressed in terms of matrices. This fact was first es tablished by Peirce,f af tem% ard verified by Spot tiswoode,§ but more explicitly by Cayley.11 It has also been shown that every similarity transformation in four variables is expressible as a quaternion product.ff Such facts as these bring out clearly the multiple character of the quaternion algebra.
The further development of the theory of quaternions, which follows the exposition of fundamental principles outlined in the pre ceding paragraphs, includes the deduction of the formula of scalar and vector products and quotients, the interpretation and transforma tion of quaternion expressions in general, the differentiation of quaternions, the solution of equations, especially of equations of the first degree, and the analytical theory of quater nions.
But although• the works of Hamilton and Tait contain a large mass of material concern ing particular quaternion transformations, the general Analytical Theory of Quaternions has yet to be written. A brief outline sketch of a method is given by Cayley in the sixth chapter of Tait's Treatise.
Applications.— It is impossible, in the space permitted to this notice, to do more than name some of the subjects in which the method of quaternions finds its applications. The most important of these to which writers have thus far given attention are the following: The theory of matrices; orthogonal trans formation; geometry of the straight line, of the plane, of the sphere, of the cyclic cone, of surfaces of the second order, of curves and sur faces in general, and of planes in four dimen sional space; kinematics of a point, of a rigid system and of deformable systems; axes and moments of inertia; statics of a rigid system,• kinetics of a rigid system; precession and notation; the problems of the pendulum; geometrical and physical optics; electro dynamics; the solenoid; applications of the d d koPerator, V .= y j — , and of La.
dx ds place's operator v. to physical analogies, to line, surface and volume integrals, to the stress function; application of the v integrals to magnetic problems; the hydrokinetic equa tions; applications of v in connection with Taylor's theorem and the calculus of varia tions.