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TRANSACTIONS 
OF 
THE SECTIONS. 
1. MATHEMATICS. 
Professor Hamilton, Astronomer Royal for Ireland, gave an 
account of a Memoir by James Mac Cullagh, F.T.C.D., On 
the Attractions of Spheroids, which had lately been presented 
to the Royal Irish Academy, and in which is given a very 
simple demonstration of a rigorous theorem corresponding to a 
celebrated and contested approximate theorem of Laplace; to¬ 
gether with a geometrical construction of the quantity neglected 
in that approximate theorem, which is shown to be the function 
of a certain small solid assigned by the author. He likewise 
gave an outline of a manuscript Memoir on Numeral Evolution , 
by Dr. Allman, Professor of Botany in the University of 
Dublin, which related principally to a new method for the 
arithmetical calculation of logarithms. 
Professor Hamilton also stated a general Theorem of his 
own respecting differences and differentials of Functions of 
Zero, which he had presented to the Royal Irish Academy, and 
to which he had been led by meditating on a method given by 
Sir John Herschel for the development of exponential functions. 
He gave a verbal account of his view of Mathematical Optics, of 
which the following is an abstract. 
On a View of Mathematical Optics. By William R. Hamilton, 
Royal Astronomer of Ireland, fyc, 
“ The Memoirs on Systems of Rays, which have been pre¬ 
sented by me to the Royal Irish Academy, and of which some 
have been published in the XVth and XVIth volumes of the 
Transactions of that Academy, contain a view of mathematical 
optics, which appears to me to be analogous to the view taken by 
Descartes of algebraical geometry, and likely to lead those who 
shall adopt it to analogous changes of method. It has been 
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