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Proceedings of the Royal Society 
given by the great but prematurely removed mathematician 
Eisenstein. These results of Eisenstein’s were left by him almost 
entirely without demonstration, and their connection among them- 
selves or with other known theorems was very obscurely indicated. 
So much was this the case, that several mathematicians had endea- 
voured unsuccessfully to prove the truth or falsity of the results. 
Mr Muir, guided mainly by the general ideas gained in the 
research just mentioned, takes up these theorems of Eisenstein’s, 
and succeeds with apparent ease in not only proving them, but in 
showing their relation to each other and to other known results. 
There is yet another of Mr Muir’s papers on the subject of con- 
tinued fractions which deserves especial mention at this moment. 
I allude to his paper on Continuants, in the eighth volume of the 
Society’s Proceedings, Passing over the remarkable general 
theorems regarding the special form of determinants called continu- 
ants, I would especially direct your attention to the remarkable 
theorems there arrived at regarding the expression of quadratic surds 
by means of continued fractions. Mr Muir finds a general expres- 
sion for any integer whose square root is expressible by means of a 
continued fraction having unit numerators and a given recurring 
cycle of denominators. And he finds the general condition, that 
any given periodic fraction may represent a quadratic surd. 
Another branch of analysis in which Mr Muir has equally distin- 
guished himself is the theory of Determinants, which I may call 
the chief handmaiden of the tlieory of algebraic forms. In this 
subject Mr Muir may be classed as a worthy follower of our great 
countrymen, Sylvester and Cayley. I need only allude at present 
to three of the papers on this subject. The papers in volumes 
xxix. and xxx. of the Transactions contain two most valuable 
generalisations in this theory, viz., Mr Muir’s Extension of Lap- 
lace’s Law, and his Theorem of Extensible Minors. These results 
are characteristic of Mr Muir’s work, the constant tendency of 
which is the attainment of that higher kind of simplicity which 
results from greater generalisation. 
Concerning the last paper of the series (not, I ani glad to see by 
the evidence of the billet for to-night, the last from Mr Muir to the 
Society), On a Class of Permanent Symmetric Functions, I have 
simply to say, that in it, by the evolution of a few simple general 
