260 The Rev. B. Bronwin on the Coefficients of Sines 



by oxidation of some thousand times its weight of" lead ; steam 

 blows our furnace fires, rolls and pipes our metals, and flies 

 with iron wings on roads more solid than the Appian Way. 

 The world of George Stephenson is much different from that 

 of Julius Agricola ; but some features of the past remain to 

 connect the earliest with the latest aspect of our country : and 

 among these the least altered, and the most instructive, appear 

 to be the mineral products and the mining processes. If by 

 these we judge the great Brigantian tribes which surrounded 

 Isurium, they must be placed far higher on the scale of civi- 

 lization than the place usually accorded by the Saxon to the 

 Celt. 



I presume to think, indeed, that without full attention to the 

 mining history of Britain, as indicated by fragments in classic 

 authors, and illustrated by processes not yet extinct, the opinion 

 which may be formed of the ancient British people would be 

 altogether conjectural, derogatory, and erroneous. 



XXXVI. On the Determination of the Coefficients in any series 

 of Sines and Cosines of Multiples of a variable angle from 

 particular valuesof that series. By the Rev. Brice Bronwin *. 



MLE VERRIER'S method of determining the coeffi- 

 • cients of a series of sines and cosines of the multiples 

 of a variable angle requires a very great amount of labour. 

 Moreover his formulae contain large factorials of sines in their 

 denominators, which endangers the accuracy of the results. 

 The method of Mechanical Quadratures, strictly so called, is 

 attended with some uncertainty ; because each term of the cor- 

 rection may vanish, while the sum of all of them to infinity 

 may have a finite value, owing to the initial and terminal series 

 of differences being divergent. Some more suitable method is 

 therefore wanted. Sir J. W. Lubbock, in some tables which he 

 very kindly sent me a short time ago, has inserted one given by 

 Bessel and Hansen in the Astronomische 'Nachrichlen^ which 

 is perhaps as good a method as we have reason to expect. In 

 what way the authors arrived at it I do not know. As it is 

 important, and may be found in a very simple manner, per- 

 haps the investigation may not be unacceptable. Besides, the 

 formulae given by them, as printed in the tables above-men- 

 tioned, are only one out of several distinct sets, and are not 

 well adapted to all cases j and moreover they may be modified 

 and simplified. 



* Communicated by the Author. 



