XXII. On quaternions; or on a new system of imaginaries in algebra
1846; Taylor & Francis; Volume: 29; Issue: 192 Linguagem: Inglês
10.1080/14786444608645590
ISSN1941-5974
Autores Tópico(s)History and Theory of Mathematics
ResumoTHE geometrical considerations of the foregoing article may often suggest algebraical transformations of functions of the new imaginaries which enter into the present theory.Thus, if we meet the function S. ~r dr _ d S. d t ~, ...... (1.) we may see, in the first place, that in the recent notation this function is algebraically a pure imaginary, or vector form, which may be constructed geometrically in this theory by a straight line having length and direction in space; because the three symbols ~, d, d ~ are supposed to be themselves such vector forms, or to admit of being constructed by three such lines; while S. d a" and S. ~,r~ are~ in the same notation, two scalar forms, and denote some two real numbers, positive, negative, or zero.We may therefore equate the proposed function (1.) to a new snmll Greek letter, accented or unaccented, for example to d", writing d" = o~ S. i d t --d S. d I ~ ...... (~.) Multiplying this equation by ¢d, and taking the scalar parts of the two members of the product, that is, operating on it by the characteristic S. d l; and observing that, hy the properties of scalars, S. dt ~S.d d t = S. dr ~. S. d d r =S.dr d. S. dl~__ -S.dl d S. dt ~, in which the notation S.d t~ S. dd
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