Errors in the Use of Reflecting Circles. 163 



Then lBkV = a, Z BAQ = 60°-o, and Z BAO = 60^-f «; also 

 . BQ,BP, BO are sines of BAQ, B AP and B AO, to the radius BA. 

 But sine 60°— a -f- sine a= sine 60 + fl ; wherefore, generally, 

 BQ + BP= BO. Now BQ, BP, and BO are sines of LF, D'I, 

 and HN, to the radius of the circle. Hence sine DI+ sine LF = 

 sine HN ; and consequently chord 2.Di + chord 2.LF = chord 

 2.HN. But the sum of the chords of two arcs is greater than 

 the chord of their sum; therefore HN is greater than Di + LF, 

 or the correction is theoretically imperfect. 



Again: When the sum of the sines of two arcs is equal to the 

 sine of a third arc, the sum of the two arcs is a minimum, and 

 consequently the difference between that sum and the third arc 

 is a maximum, when the two sines or when the two arcs are 

 equal. 



Hence the error of the correction is a maximum when BQ = 

 BP, or when the A CAI = 30°, or when one of the indexes re- 

 volving round the excentric point is perpendicular to the line 

 drawn through that point and the centre. 



Having now, I trust, satisfactorily proved the existence of the 

 error in theory, it is proper that I should add, that as the instru- 

 ment is actuailv made, the maxinnmi error is too minute to be 

 computed by common trigonometrical tables. If we suppose an 

 excentricity of -'^, the maximum error will be about 26', but 

 if we suppose an excentricity of -j--tro5 ^'^^ maximum error will be 

 a very small fraction of a second. 



And wlien we consider that tlie actual excentricity is probably, 

 in all cases, a very small part of what we have last supposed, and 

 reflect on the other important purposes to which the contrivance 

 in question is made subservient ; we shall still have sufficient rea- 

 son to admire the ingenuity of its author, thougii he was wrong 

 in supposing that one of the advantages which he proposed irom 

 its adoption would in strict theory be always produced. 



But that a mistaken idea should have been generally enter- 

 tained respecting the theory of an instrument of such importance, 

 is a circumstance which I'hope will apologize for my troui)ling 

 you with this communication, and requesting you to give it a 

 place in the Philosophical Magaicine. 



Your obedient servant, 



Edward Riddle, 



Aug.27, 18iy. Master of the Tniiity-Uouso School, 



Ncwcustle-ou-Tyiie. 



L2 XXX. Expe- 



