x{a1(b2c3 - b3c2) + a2(b3c1 - b1c3) + a3(b1c2 - b2c1)}
= d1(b2c3 - b3c2) + d2(b3c1 - b1c3) + d3(b1c2 - b2c1)
The coefficient of x, and the right-hand member, are here determinants of the third order, and the result is written
| x | a1, b1, c1 a2, b2, c2 a3, b3, c3 | = | d1, b1, c1 d2, b2, c2 d3, b3, c3 | . |
When the coefficient of x is expanded fully, it consists of six terms, half of them positive and half negative, each term being the product of three letters, one from each row and one from each column. Thus in every term, such as the leading term a1b2c3, the three letters a, b, c, and the three suffixes 1, 2, 3, all occur once and once only. With regard to the signs, we note that if the letters a, b, c are always kept in this order, an interchange of two suffixes changes the sign. Thus, e.g. we may start from + a1b2c3, interchange the suffixes 2 and 3, and thus find - a1b3c2. Again, if in the latter term - a1b3c2 we interchange the suffixes 1 and 3, we find + a3b1c2. It is easy to verify that after any number of interchanges of this kind a particular term will always come up with the same sign.
Thus the interchange of two rows (and similarly of two columns) changes the sign of the determinant. It follows that if two rows (or two columns) are identical, the determinant is zero. It is now easy to define functions of n2 letters, arranged in a square array of n rows
and n columns, with similar properties to those observed above in the case of determinants of the third order. Determinants are very extensively used in higher algebra, co-ordinate geometry, and other branches of mathematics. See also Elimination.—Bibliography: C. Smith, Algebra; T. Muir, Theory of Determinants.
Determinants, in biology, the name applied by Weismann to hypothetical particles contained in the nuclei of germ-cells, which determine the existence and nature of the various parts of the body of the embryo. The theory of determinants is part of an elaborate attempt to explain the facts of heredity on a mechanical basis. Although it may serve as a working hypothesis, it is regarded with disfavour by many experts, and may be regarded as an elaboration of the 'provisional theory of pangenesis' advanced by Darwin.
Deter´minism, a term employed by recent writers, especially since J. Stuart Mill, to denote a philosophical theory which holds that the will is not free, but is invincibly determined either—according to the older form of the theory—by a motive furnished by Providence, or—according to the modern form—by the aggregation of inherited qualities and tendencies. Biological determinism maintains that each of our voluntary acts finds its sufficient and complete cause in the physiological conditions of the organism. Psychological determinism ascribes efficiency to the psychical antecedents. Opposed to determinism is the doctrine of indeterminism or indifferentism. See Free-will.
Det´inue, in law, the form of action whereby a plaintiff seeks to recover a chattel personal unlawfully detained.