56 Proceedings of the Royal Society 
supported in a specified way, which differ only by small quantities 
from the values obtained by Mr Airy, by a method involving cer- 
tain assumptions, which were introduced in order to avoid the con- 
sideration of elastic yielding. 
2. On the Extension of Brouncker’s Method. 
By Edward Sang, Esq. 
The operation in use by the ancient geometers for finding the 
numerical expression for the ratio of two quantities, was to repeat 
each of them until some multiple of the one agreed with a multiple 
of the other; the numbers of the repetitions being inversely pro- 
portional to the magnitudes. 
The modern process, introduced by Lord Brouncker, under the 
name of continued fractions, is to seek for that submultiple of the 
one which may be contained exactly in the other; the numbers 
being then directly proportional to the quantities compared. 
On applying this method to the roots of quadratic equations, the 
integer parts of the denominators were found to recur in periods ; 
and Lagrange showed that, while all irrational roots of quadratics 
give recurring chain-fractions, every recurring chain-fraction ex- 
presses the root of a quadratic ; and hence it was argued that this 
phenomenon of recurrence is exhibited by quadratic equations alone. 
The author of this paper had supplemented Lagrange’s proposi- 
tion, by showing that when the progression of fractions converging 
to one root of a quadratic is continued backwards, the convergence 
is toward the other root. The singularity of this exclusive property 
of quadratic equations led him to consider whether some analogous 
property may not be possessed by equations of higher degrees. 
Putting aside the idea of the chain-fraction as being merely acci- 
dental to the subject, and attending to the series of converging 
fractions, he came upon a kind of recurrence which extends to 
equations of all orders ; and which proceeds by operating on two, 
three, or more contiguous terms according to the rank of the equa- 
tion. In this way a ready means of approximating tp the greatest 
and to the least root of any equation was obtained. 
The following cases were cited : — 
If we begin with the terms ^ , and form a progression by 
