37 



The tension at any point is — ' gin^'a '' ^"^ therefore its effect in 



increasing the divisions is as in the last section — *' ""' "^ : it may 



also be remarked that in this case the force required to change the 

 curvature of the arc s does not interfere with the above equation 

 only changing the value of «'. j^' ; ♦: 



Even should one of the arms be stronger than the rest, yet the 

 change of figure which this produces must be propagated through 

 the others till it reaches the opposite point of the circle, and its 

 effect at each arm depending on the curvature there, can evidently 

 be expressed in terms of ^, and therefore this as well as the other 

 errors produced by flexure, vanish on the supposition so often re- 

 ferred to of a =r -— . ' 



n 



In the course of this investigation we have seen that generally all 

 errors which have reference to the position of the circle are corrected 

 in this way, except the effect of expansion on the microscopes. It 

 may also be remarked that two microscopes must be sufficient in 

 most instances, for the co-efficients e, t and s' are so small that their 

 second powers can have no sensible effect ; yet we can conceive 

 cases where they might fail. If the instrument were an ellipse con- 

 centric with the original circle, as in it the difference between s and 

 p is as sin. 2 <{>, three would be required, and such an occurrence is 

 by no means impossible. Circles are generally turned on their 

 pivots, and of course any imperfection in the latter is transferred to 

 the former. In this process, any jarring or shake has a tendency to 

 propagate itself round the circumference in equidistant waves, 

 and the number of these gives that multiple of <p on whose 

 sine or cosine the error to be corrected depends. When this is 

 divisible by the number of microscopes, the error is not corrected, 



VOL. XV. a 



