ALGEBRA duct of two quaternions are defined by the formulae ?ases + ?/3ses = 2(as + (3s)es where the products ereg are further reduced according to the following multiplication table, in which, for example,
e, Cq, the second line is to be read e1e0 = e1, e1 e,6o = - e». The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative. Thus exe2 = - e2ev and if q, q are any two quaternions, qq1 is generally different from q'q. The symbol e0 behaves exactly like 1 in ordinary algebra; Hamilton writes 1, i, j, Tc instead of e0, ev e2, e3, and in this notation all the special rules of operation may be summed up by the equalities i2 =j2 — k2 — ijk = — 1. Putting q = a + ($i + yj 8k, Hamilton calls a the scalar part of q, and denotes it by
- he also writes Nq for
fti + yj + 8k, which is called the vector part of q. Thus every quaternion may be written in the form y = Sy + Yq, where either 8>q or Yq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions. The equations q +x = q and y + q = q are satisfied by the same quaternion, which is denoted by ^ — q. On the other hand, the equations q'x — q and yq = q have, in general, different solutions. It is the value of y which is generally denoted by q + q'; a special symbol for x is desirable, but has not been established. If we put % ^ Sy - Vy', then q'0 is called the covjugate of q, and the scalar qq0 = q0 q is called the norm of q and written N'?'. With this notation the values of x and y may be expressed in the forms— which are free from ambiguity, since scalars are commutative with quaternions. The values of x and y are different, unless ^(qq'o) = 0. In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by .Hamilton as a biquaternion. Clifford's biquaternions are quantities £q + gr, where q, r are quaternions, and £, r; are symbols (commutative with quaternions) obeying the laws £2 = £, = = 0. 7. In the extensive calculus of the nt category, we have, first of all, n independent “ units,” Grassev e2, .. . en. From these are derived symbols tnann’s extensive of the type calculus. Aj = "b a2^2 "b • • * "b an^n = which we shall call extensive quantities of the first species (and, when necessary, of the wth category). The coordinates oq, .. . an are scalars, and in particular applications may be restricted to real or complex numerical values. If B1 = 2/Ie, there is a law of addition expressed by + B1 = 2(cq + f$i)ei = Bx + Aj^; this law of addition is associative as well as commutative.
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The inverse operation is free from ambiguity, and, in fact, Ax — B1 = ? (otj fiifi. To multiply A1 by a scalar, we apply the rule £AX = Ax^ = 2(^aj)(?t-, and similarly for division by a scalar. All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities. The binary products of the units Ci are taken to satisfy the equalities e i~ — 0) €{6] = — ejCi; this reduces them to ^n(n- 1) distinct values, exclusive of zero. These values are assumed to be independent, so we have hn(n— 1) derived units of the second species or order. Associated with these new units there is a system of extensive quantities of the second species, represented by symbols of the type A2 = SaiEi® [f = 1, 2, . . . {n - 1)], where E1(2), E2,2), etc., are the derived units of the second species. If Ax = SajCj, Bx = 2/IjCi, the distributive law of multiplication is preserved by assuming A1B1 =
it follows that A1B1 = - BjAp and that Ax2 = 0. By assuming the truth of the associative law of multiplication, and taking account of the reducing formula? for binary products, we may construct derived units of the third, fourth. . . nth species. Every unit of the rth species which does not vanish is the product of r different units of the first species; two such units are independent unless they are permutations of the same set of primary units Ci, in which case they are equal or opposite according to the usual rule employed in determinants. Thus, for instance— ^l,e2e3 = ele2‘e3 r = eie2e3 = — e2eie3 = e2e3el ’ and, in general, the number of distinct units of the ?'th species in the nth category (r<n) is C„ir. Finally, it is assumed that (in the nth category) e1e2e3. . . en—l, the suffixes being in their natural order. Let Ar = 2aE'r) and Bs = 2/IE's) be two extensive quantities of species r and s • then if r + s^n, they may be multiplied by the rule ArBg-2(a/3)E(j->Ete> where the products E(r,E,s) may be expressed as derived units of species (r + s). The product BgAr is equal or opposite to ArBg, according as rs is even or odd. This process may be extended to the product of three or more factors such as ArBgCt. . . provided that r + « + «+ ... does not exceed n. The law is associative; thus, for instance, (AB)C = A(BC). But the commutative law does not always hold; thus, indicating species, as before, by suffixes, ArBsCi = (-l)rs+6‘t+<r C(BsAr, with analogous rules for other cases. If r + s>n, a product such as ErEg, worked out by the previous rules, comes out to be zero. A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive. The new rule requires some preliminary explanation. If E is any extensive unit, there is one other unit E', and only one, such that the (progressive) product EE' = 1. This unit is called the supplement of E, and denoted by |E. For example, when w = 4, el ~ e2eZeP leie2 ~ e3e4> e2e3e = ~ eV and so on. Now when r + s>n, the product ErEs is
