of Edinburgh^ Session 1883 - 84 , 
563 
be explored under bis leadership, or I might take your fancy a 
promenade on that most fascinating new pleasure-ground of 
naturalists, the bottom of the sea. 
But, in the case of a pure mathematician’s work there is, to use 
the words of one of the greatest living cultivators of the analytic 
art, no such appeal to the immediate Utilitarianism so dear to the 
Philistine soul. If, however, the public to whom I can appeal is 
small, I feel that their judgment is sure ; and, when I have 
reminded those who represent that public here, of the succession of 
papers which Mr Muir has contributed to our Transactions and 
Proceedings^ I am contident that they will sanction with an 
emphatic approval the decision of the Council of the Koyal Society 
to confer upon him the Keith Prize. 
One of the most interesting branches of analysis is the theory 
of Continued Fractions. The subject has a double interest, because 
it is connected on the one hand with that most pure of all the 
branches of pure mathematics, I mean the theory of numbers, a 
sanctuary into which the profane foot of even the applied mathe- 
matician scarcely enters; and, on the other, with the theory of 
forms, for I need scarcely remind my mathematical hearers that the 
algorithms of the greatest common measure and of the calculation 
of the convergent to a continued fraction are formally identical ; 
and that this algorithm is also that by which Sturm deduced the 
functions which bear his name, and which play so important a part 
in the theory of equation. 
Mr Muir has pursued the theory of continued fractions, and has 
obtained some very important results in both its branches. 
In the first paper of the series for wdiich the Keith Prize is now 
awarded him, published in the twenty-seventh volume of our 
Transactions, he gives a perfectly general solution of the problem 
of transforming an infinite series into a continued fraction. Special 
cases of this transformation were known before, but no one had 
been able to assign the general form of the expression until Mr 
Muir successfully attacked the problem. . 
His second paper, in volume xxviii. of the Transactions, is to my 
mind the most noteworthy of the various pieces of work now under 
review. He there takes up a number of remarkable transformations 
of various series into the form of continued fractions, which were 
2 0 
VOL. XII. 
