and the reasoning in the last paragraph shews that whatever dif- 

 ferential disturbances may be produced, must be compensated by a 

 number of readings, for the disturbing forces are manifestly func- 

 tions of the angular distance from the vertical diameter. 



This is perfectly consistent with experiment, for the largest circle 

 in the world appears to have the same error of collimation from the 

 horizon to the zenith, which is highly improbable on any other 

 supposition. A rigorous determination of the shape which an 

 instrument would assume by its own weight is extremely difficult ; 

 the necessity of considering the transverse elasticity of the arms 

 and circumferences (or that which resists flexure) leads to an 

 unmanageable equation of mixed differences ; but by neglecting it 

 as small in comparison of the forces which resist longitudinal 

 extension or compression, perhaps the following rude method may 

 determine with sufficient accuracy, a certain number of points in 

 the deformed circle. 



Let the circle be considered a polygon of as many sides as it 

 has arms : part of the weight of each arm is borne by the centrcj 

 and the remainder =W acts at its extremity, pulling the sides of 

 the polygon that are above, and pressing those that are below it. 

 If the material were inextensible, the first alone would happen, if 

 incompressible, the second ; both take place and beginning at op- 

 posite extremities of the vertical diameter, follow the same law, 



with opposite signs. Their sum therefore gives the actual tension 

 or compression of each side, and compounding two adjacent, we 

 get the force exerted on an arm in the direction of its length. 



