416 Mr. Horner on [June, 



This table contains the results of Chapman's determinations 

 for ships of war from his large and expensive work ; and there- 

 fore is by far the most valuable information contained in it. A 

 copy of the original work is in possession of Mr. J. Knowles, 

 Secretary at the Navy Office, whose scientific exertions for the 

 navy are well known ; and whose liberal and obliging conduct 

 in lending his books on nautical subjects, of which he has pur- 

 chased a large collection, is to be highly appreciated : the more 

 so, as the naval college does not contain within its walls a library 

 on the subject. Previous to my going to Paris, and purchasing 

 such works, I was much indebted to that gentleman for the loan 

 of his books. 



Article III. 



On the Use of continued Fractions with unrestricted Numerators 

 in >Sum>nation of' Series.* By W. G. Horner, Esq. 



(To the Editors of the Annals of Philosophy.) 



GENTLEMEN, Bath, jlpriin, 1826. 



1. While the labours of Lagrange have reduced the theory 

 of continued fractions, with the constant numerator 1, to a state 

 little short of perfection, scarcely any thing of consequence, to 

 the best of my knowledge, has been done to render these frac- 

 tions in their unrestricted form available to any useful purpose. 

 The object, however, is not undeserving of attention, since the 

 fractions of Lagrange are applicable almost solely to numerical 

 solution ; while theothers apply even to series in their literal or 

 algebraic form. 



If these fractions have not all the clearness of convergency 

 which Lagrange's possess, yet where these are not attainable, 

 they are much preferable to expressions in a finite fraction, on 

 account of thefacilities they afford for simplifyingthe reductions. 



The formul'de of Brouncker, who started the discovery of this 

 calculus, have been introduced by Euler, in his Analysis Injini- 

 toruvi, in elucidation of a general method of turning any given 

 series with alternate signs into the form of a continued fraction. 

 The first of Euler's formulaj is contained in the following propo- 

 sition, and when aided by a reduction that will presently be 

 described, it is adequate to all the others. 



* Substituted for the investigation mentioned in my last paper, which more properly 

 belongs to another course of essays, and would be less interesting in a detached fonn. 



