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liVI. Memorandum rejecting a new System of Roots of Unity, 

 By Sir William Rowan Hamilton, LL.D., M.R,I,A., 

 F.R.A.S., 8fc., Andrews^ Professor of Astronomy in the Uni- 

 versity oj Dublin, and Royal Astronomer of Ireland^, 



I HAVE lately been led to the conception of a new system, 

 or rather /flmt'/y of systems, of non-commutative roots of unity, 

 which are entirely distinct from the ijk of the quaternions, though 

 having some general analogy thereto ; and which admit, even 

 more easily than the quaternion symbols do, of geometrical inter^ 

 pretation. In the system which seems at present to be the most 

 interesting one, among those included in this new family, I 

 assume three symbols, i, k, \, such that 



^8=1, /^=1, xs=l, 

 \=i/c; 



} .... (A) 



where ifc must be distinguished from kl, since otherwise we should 

 have \^'= 1, \= 1. As a very simple specimen of the symbolical 

 conclusions deduced from these fundamental assumptions, I may 

 mention that if we make 



we shall have alsof 



so that /I is a new fifth root of unity, connected 'with the former 

 fifth root X by relations of perfect reciprocity. A long train of 

 such symbolical deductions is found to follow : and every one of 

 the results may be interpreted, as having reference to the passage 

 irom face to face (or from corner to corner) of the icosahedron 

 (or of the dodecahedron) : on which account, I am at present 

 disposed to give the name of the " Icosian Calculus/^ to this new 

 system of symbols, and of rules for their operation. Some ad- 

 ditional remarks on this subject may soon be offered to the Phi, 

 losophical Magazine, under the title, already sanctioned by the 

 Editors, of '' Extensions of the Quaternions." 



Observatory of Trinity College, Dublin, 

 October ^9, 1866. 



* Communicated by the Author, 

 t Jn faci;, by (A), 



lr=t.tic.JC»=:t(K»t)V=(4K«)», 



8l«q 



