296 Sir Wrx1am Rowan Hamirton’s Researches respecting Quaternions. 
October, 1846; and in that for the present month, June, 1847, in which these last sheets 
of the present paper are now passing through the printers’ hands. The articles on 
Symbolical Geometry, in the “* Cambridge and Dublin Mathematical Journal,” are also 
designed to have a certain degree of connexion with this subject. 
The “first letter” to Mr. Graves, referred to in the one here printed, was written on 
the 17th of October, 1843,and has been printed in the Supplementary Number of the same 
Philosophical Magazine for December, 1844. It contained a sketch of the process by 
which the writer had succeeded in combining, through Quaternions, his old conception of 
sets of numbers, derived from the conception of sets of moments of time, with the notion 
of tridimensional space. he former conception had been familiar to him since the year 
1834, about the end of which year, and the beginning of the following one, he tried a 
variety of triplet systems, and obtained several geometrical constructions, but was not 
sufficiently satisfied with any of them to give them publicity ; attaching, perhaps, too 
much weight to the objection or difficulty, that in every such system of pure triplets, the 
product was found to be liable to vanish, while the factors were still different from zero. 
It should be here observed that the ¢rplets described in the author’s two letters of October, 
1843, are really imperfect quaternions ; they are, therefore, strictly speaking, not proper 
triplets, such as he had once sought for (and in some degree found); and they cannot be 
regarded as having at all anticipated the independent discoveries since made by Professor 
de Morgan, nor those made subsequently by John T. Graves, Esq. and the Rev. Charles 
Graves, in 1844, respecting certain remarkable systems of such pure and proper Triplets, 
with products of a triplet form, connected with imaginary cube roots of negative or posi~ 
tive unity. 
The writer hopes that a very interesting theory of octaves, including an extension of 
Euler’s theorem respecting products of sums of squares from four to eight, which was 
communicated to him as an extension of his quaternions, about the end of 1843 and begin- 
ning of 1844, in letters from his friend, Mr. John Graves, will yet be published by that 
gentleman, who has also contributed to the ‘* Philosophical Magazine” for April, 1845, a 
remarkable paper on Couples. Some valuable papers on Couples, Quaternions, and 
Octaves, have also been communicated to the same magazine, since the commencement 
of 1845, by Arthur Cayley, Esq., especially an application of quaternions (which appeared 
in the February of that year) to the representation of the rotation ofa solid body. ‘That 
important application of the author’s principles had indeed occurred to himself previously ; 
but he was happy to see it handled by one so well versed as Mr. Cayley is in the theory 
of such rotation, and possessing such entire command of the resources of algebra and of 
geometry. Any furtherremarks which the writer has to offer on the nature and history of 
this whole train of inquiry, must be reserved to accompany the account of his Second 
Series of Researches respecting Quaternions. : 
