we may construct derived units of the third, fourth . . . nth species. Every unit of the r th 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 ei, in which case they are equal or opposite according to the usual rule employed in determinants. Thus, for instance—
e1.e2e3＝e1e2.e3＝e1e2e3＝−e2e1e3＝e2e3e1;
and, in general, the number of distinct units of the r th species in the nth category (r ⪕ n) is Cn,r. Finally, it is assumed that (in the nth category) e1e2e3 . . . en＝1, the suffixes being in their natural order.
Let Ar＝ΣαE(r) and Bs＝ΣβE(s) be two extensive quantities of species r and s; then if r+s ⪕ n, they may be multiplied by the rule
ArBs＝Σ(αβ)E(r)E(s)
where the products E(r)E(s) may be expressed as derived units of species (r+s). The product BsAr is equal or opposite to A(r)B(s), according as rs is even or odd. This process may be extended to the product of three or more factors such as A(r)B(s)C(t) . . . provided that r+s+t+ . . . 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, ArBsCt＝(−1)rs+st+trCtBsAr, with analogous rules for other cases.
If r+s > n, a product such as ErEs, 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 n＝4,
|e1＝e2e3e4, |e1e2＝e3e4 |e2e3e4＝−e1,
and so on. Now when r+s > n, the product Er Es is defined to be that unit of which the supplement is the progressive product |Er|Es. For instance, if n＝4, Er＝e1e2, Es＝e2e3＝e4, we have
|Er|Es＝(−e2e4)(−e1)＝e1e2e4＝|e3
consequently, by the rule of regressive multiplication,
e1e3.e2e3e4＝e3.
Applying the distributive law, we obtain, when r+s>n,
ArBs＝ΣαE(r)ΣβE(s)＝Σ(αβ)ErEs,
where the regressive products ErEs are to be reduced to units of species (r+s−n) by the foregoing rule.
If A＝ΣαE, then, by definition, |A＝Σα|E, and hence
A|(B+C)＝A|B+A|C.
Now this is formally analogous to the distributive law of multiplication; and in fact we may look upon A|B as a particular way of multiplying A and B (not A and B). The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product |AB. An inner product may be either progressive or regressive. In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other. In the first two cases the product is said to be pure, in the third case mixed: A pure product is associative; a mixed product, speaking generally, is not.
The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively. As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units. Only a few illustrations can be given here, Let a, b, c, d, e, f be quantities of the first species in the fourth category; A, B, C . . . quantities of the third species in the same category. Then
(de) (abc)＝(abde)c + (cade)b + (bcde)a
＝ (abce)d − (abcd)e,
(ab)(AB)＝(aA)(bB) − (aB)(bA)
ab|c＝(a|c)b − (b|c)a, (ab|cd)＝(a|c)(b|d) − (a|d)(b|c).
These may be compared and contrasted with such quaternion formulae as
S(VabVcd)＝SadSbc − SacSbd
dSabc＝aSbcd − bScda + cSadb
where a, b, c, d denote arbitrary vectors.
8. An n-tuple linear algebra (also called a complex number system) deals with quantities of the type A＝Σαiei derived from n special units e1, e2 . . . en. The sum and product of two quantities are defined in the first instance by the formulaeLinear Algebras.
Σαe+Σβe＝Σ(α+β)e, Σαiei × Σβjej＝Σ(αi+βj)eiej,
so that the laws a, c, d of § 3 are satisfied. The binary products eiej, however, are expressible as linear functions of the units ei by means of a “multiplication table” which defines the special characteristics of the algebra in question. Multiplication may or may not be commutative, and in the same way it may or may not be associative. The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B. Peirce. Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra. If, in the extensive calculus of the nth category, all the units (including 1 and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2n−1)-tuple linear algebra, which, however, is not wholly associative. It should be observed that while the use of special units, or extraordinaries, in a linear algebra is convenient, especially in applications, it is not indispensable. Any linear quantity may be denoted by a symbol (a1, a2, . . . an) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrae or regulators of certain operations on scalars (see Number). This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative. Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.
9. In ordinary algebra we have the disjunctive law that if ab＝0, then either a＝0 or b＝0. This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general. One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.
10. The algebras discussed up to this point may be considered as independent in the sense that each of them deals with a class of symbols of quantity more or less homogeneous, and a set of operations applying to them all. But when an algebra is used with a particular interpretation, or even in the course of its formal development, it frequently Subsidiary algebras.happens that new symbols of operation are, so to speak, superposed upon the algebra, and are found to obey certain formal laws of combination of their own. For instance, there are the symbols Δ, D, E used in the calculus of finite differences; Aronhold’s symbolical method in the calculus of invariants; and the like. In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of algebra (at present a hopeless task), they would have to be taken into account. Even in ordinary algebra the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by writing √(a + b)＝√a + √b, he is unconsciously emphasizing this want of complete harmony.
Authorities.—A. de Morgan, “On the Foundation of Algebra,” Trans. Camb. P.S. (vii., viii., 1839–1844); G. Peacock, Symbolical Algebra (Cambridge, 1845); G. Boole, Laws of Thought (London, 1854); E. Schröder, Lehrbuch der Arithmetik u. Algebra (Leipzig, 1873), Vorlesungen über die Algebra der Logik (ibid., 1890–1895); A. F. Möbius, Der barycentrische Calcul (Leipzig, 1827) (reprinted in his collected works, vol. i., Leipzig, 1885); W. R. Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866);
