162 



Mr. Troughtoji's Expedient for correcting 



and it is equally true, though not so obvious, that three indices 

 will do the san)e." Meaning, I presume, three placed at equal 

 distances as they are in his circles. 



After this and a further extract from the same letter, the au- 

 thor of the article proceeds to observe, that the truth of Mr. 

 Troughton's statement respecting three indexes admits of geo- 

 metrical demonstration ; and he accordingly adds what he ap- 

 pears to consider a satisfactory investigation of the truth of the 

 property. 



In Dr. Brewster's Evcyclopcedia also, article " Circle," it is 

 stated that " three readings perfectly correct the error arising 

 from any excentric motion of the index." 



■ But notwithstanding all these authorities, I conceive it will be 

 easy to show that, unless one of tlie indexes coincide with the 

 line joining the centre and the excentric point round which the 

 indexes revolve, the required compensation will, as 1 have al- 

 ready said, not be tlieoretically perfect. 



With respect to two opposite verniers, it is indeed abundantly 

 obvious that the correction will in all cases be in theory com- 

 plete ; as it is a well known property of the circle, that if two 

 straight lines intersect in any point within it, the angle which they 

 make with each other will be measured by half the sum of the 

 intercepted arcs. Hence, while the angle is a constant quantity, 

 the sum of the intercepted arcs is also a constant quantity. 



But for three verniers, as they are placed by Mr. Troughton, 

 let A be the centre of the circle ; BC, BE, BG, the indexes 

 which revolve about the excentric 

 point B ; AC, AK, AM, indexes 

 which revolve in a similar manner 

 round the centre A. Then when 

 one of the indexes revolving round 

 B coincides with one of those re- 

 volving round A, as AC and BC 

 do in the figure ; it is obvious 

 that KE, GM, which measure 

 the deviations of BE and BG, are 

 equal and of contrary affections, 

 and the correction is therefore 

 complete. 



Letnow AljAL, AN,be another position of the indexes revolv- 

 ing round A ; and BD, BF, BH, the corresponding positions of 

 those revolving round B. Then if CD + EF + GH is not =3 IC, 

 or if ID + LF is not =HN, the correction is imperfect. 



On 1 A produced let fall from B the perpendicular BP, and on 

 AN, AL the perpendiculars BO, BQ, and put a = the angle CAl. 



Then 



<U^ 



