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IX. Researches respecting Quaternions. First Series. By Sw WitrtAm Rowan 
Hamitton, LL.D., V.P.R.1.A., Fellow of the American Society of Arts and 
Sciences ; of the Society of Arts for Scotland; of the Royal Astronomical So- 
ciety of London ; and of the Royal Northern Society of Antiquaries at Copen- 
hagen ; Corresponding Member of the Institute of France; Honorary or 
Corresponding Member of the Royal or Imperial Academies of St. Petersburgh, 
Berlin, and Turin ; of the Royal Societies of Edinburgh and Dublin ; of the 
Cambridge Philosophical Society ; the New York Historical Society ; the So- 
ciety of Sciences at Lausanne ; and of other scientific Societies in British and 
foreign Countries ; Andrews’ Professor of Astronomy in the University of 
Dublin ; and Royal Astronomer of Ireland. 
Read November 13, 1843. 
THE researches respecting Quaternions, of the first series of which an account 
is submitted in the following pages, are to be considered as being, at least in their 
first aspect and conception, a continuation of those speculations concerning 
algebraic Couples, and respecting Algebra itself, regarded as the science of Pure 
Time, which were first communicated to the Royal Irish Academy in November, 
1833, and were published in the year 1835 in the seventeenth Volume of its 
Transactions. The author has thus endeavoured to fulfil, at least in part, the 
intention which he expressed in the concluding sentence of his former Essay, 
in the volume just referred to, of publishing, at a time then future, some appli- 
cations of the same view of algebra to a theory of sets of moments, steps, and 
numbers, which should include that former theory of couples. Some general 
remarks on this whole train of speculation, and on its application to geometrical 
and physical questions, will be offered at the end of this paper. And the author 
indulges a hope that the papers containing an account of those subsequent inves- 
tigations respecting Quaternions, which he has made, and (in part) communicated 
WON.) SRT 2k 
