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By W. L. Ferrar

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Algebra: A Text-Book of Determinants, Matrices, and Algebraic Forms

Some of the earliest books, quite these relationship again to the 1900s and sooner than, are actually tremendous scarce and more and more dear. we're republishing those vintage works in reasonable, prime quality, glossy versions, utilizing the unique textual content and paintings.

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201) (202) (203) In like manner, each of the seven other coefficients, n010 , &c. , n000 , &c. We should, therefore, by this method, have obtained equations more numerous and less simple than those which were given by the method of the eighteenth article: which method there is, therefore, an advantage in introducing, even for the case of couples, and much more for the case of quaternions, or other ordinal and numeral sets; although the method above exemplified appears to offer itself more immediately from the principles of the seventeenth article.

On the Extension of the Theory of Multiplication of Quaternions to other numeral Sets. 29. This seems to be a proper place for offering a few remarks on the treatment of the general equation (214), which may assist in the future extension of the present theory of multiplication of quaternions to other numeral sets; and may serve, in the meanwhile, to throw some fresh light on the process which has been employed in the twenty-fourth and twenty-fifth articles, for discovering a mode of satisfying that general equation, in the case when the exponent n of the order of the set is 4.

Yet the trouble of investigating these latter expressions will not have been thrown away: for we may see, by (257), that they will serve, hereafter to express the result of a successive multiplication, or the continued product of three numeral quaternions. And by applying the associative principle, already considered in the twenty-first article, to such successive multiplication, we see that, instead of developing the formula (257) by a process which was equivalent to the development of the system of the two equations, m0 + m1 i + m2 j + m3 k = (a + bi + cj + dk)(m0 + m1 i + m2 j + m3 k), (280) and m0 + m1 i + m2 j + m3 k = (m0 + m1 i + m2 j + m3 k)(m0 + m1 i + m2 j + m3 k), (281) we might have developed the same formula (257) by a different, but analogous process, founded on a different mode of grouping or associating the three quaternions which enter as symbolic factors.

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