of Astronomical Instntments. 11 



part compressed ; the greatest degree of extension taking place 

 at A, the greatest degree of compression at B. 



These derangements may be estimated according to the rules 

 estimated by writers on the resistance of solid bodies. 

 Put r = the radius E D of the circle ; 



rg = the distance of the centre of gravity G from the 

 centre of the circle ; 

 b = the thickness of the metal ; 

 s = the specific weight of a cubic inch in pounds ; 

 y= the strain in lbs. upon a square inch at the points 



of greatest strain ; 

 m = the height in feet of the modulus of elasticity of 



the metal; and 

 p = the circumference of a circle of which the diame- 

 ter is unity. 

 The weight of one of the semi-circles will be, in this no- 

 tation, - — — ; and, by rules of the resistance of solids, there 

 will be an equilibrium between the stress and strain when, 



—J— - —^ ; or, / - - . 



But the force F that woidd produce an extension e may be 

 found by direct experiment ; consequenUy 



F:^^^,(=/)::.:/=^^^ (E) 



where t is the extension produced by the stress on a division 

 situate at A, or the compression on a division situate at B. 



Hence it appears that were the natural length of a division 

 at A or B, unity when the plane of the circle was horizontal ; 

 by changing it to a vertical position the length of the division 

 would become 



I ^ Jll£i • the negative sign being used when 



the division is compressed, as at B; and the positive when it 

 is extended, as at A. 



Therefore, the difference between the divisions at A and B 



will be —^5^, on the assumption that they were equal when 



the plane of the circle was horizontal. 



. F F 



Now as the height of the modulus m is -7:77;-; "'" '2/m — • * 



the difference between a division at A and one at B is 



Jl^ (F) 



Since the derangement is directly as the radms of the cncle, 

 it is indiflerent, as to this source of error, whether circles be 



B 2 large 



