33 



change of curvature in two adjoining arcs tend to bend the arm, 

 thus producing a differential ciiange in <p which is of no conse- 

 quence in this enquiry. If the Hmb on the other hand should be 

 so strong in the direction of the radius as to be nearly inflexible, it 

 will command the arm, and in this case 



y = 5.(1? — I T. sin. f ). 



The length of the arc s is measured by the divisions of the instru- 

 ment, which are no longer a scale of equal parts; let R be the 

 number of them corresponding to the arc s, the length of that 

 number at the temperature T would become R + sr. sin. (p as 

 R = ^, nearly. Now it is evident that with the equidistant micros- 

 copes in either of these cases the terms affected with sines must dis- 

 appear, and we shall have r?. ^^n; ?.fnr ?;>;*. ^i '. .>'. .:. 



In practice I suppose the first most likely to occur. 



These expressions coincide with the equation of a focal ellipse 

 having the greater segment downwards, which accords with the 

 determination of Mr. Poisson in the memoil* already referred to. . 



If the temperature be merely uniform at a given height, it is evi- 

 dent that the distorted circle must remain symmetrical to its vertical 

 diameter, which will cut its divisions in the same points as if it 

 were undisturbed. 



Let ACD=<p, we have, supposing arc DA=z(f>-\-f<p 

 where / denotes the function of the angle, which 

 expresses the difference between the angle and its 

 reading 



and as the arc EA + arc EBDA=2 tt, 



VOL. XV. F 



