416 
so that we deal, in it, with a new square root, cube root, and 
Jifth root, of positive unity ; the latter root being the product 
of the two former, when taken in an order assigned, but not 
in the opposite order. From these simple assumptions (A), a 
long train of consistent calculations opens itself out, for every 
result of which there is found a corresponding geometrical 
interpretation, in the theory of two of the celebrated solids of 
antiquity, alluded to with interest by Plato in the Timeus ; 
namely, the Icosaedron, and the Dodecaedron: whereof the 
angles may now be unequal. By making \‘=1, the author 
obtains other symbolical results, which are interpreted by the 
Octaedron and the Hexaedron. The Pyramid is, in this 
theory, almost too simple to be interesting: but it is dealt 
with by the assumption, A* = 1, the other equations (A) being 
untouched. As one fundamental result of those equations 
(A), which may serve as a slight specimen of the rest, it 
is found that if we make cx* = x, we shall have 
w= 1, w= AA, A= pp 
so that this new fifth root « has relations of perfect reciprocity 
with the former fifth rootA. But there exist more general 
results, including this, and others, on which Sir W. R. H. 
hopes to be allowed to make a future communication to the 
Academy: as also on some applications of the principles 
already stated, or alluded to, which appear to be in some 
degree interesting. 
The following donations were presented :— 
1. By Corry Connellan, Esq.:—A copy of Sir Martin A. 
Shee’s portrait of the late Thomas Moore, Esq. 
2. By Edward Bewley, M. D. :—An autograph letter of 
Dr. Charles Lucas, of which the following is a copy :— 
‘* By this time, I may congratulate my worthy, honest 
friend, first, on his safe arrival with his fair convoy and then, 
on their kind reception and assured success, in Dublin. I am 
