28 



THE CORRECTION OF SEXTANTS FOR ERRORS 



assumed value of that reading when s = 0, the true value being z -^ x, in 

 ■which X is unknown. Then 



^.(l—r) = U—{z -f .r — r),* 

 I % 



or if ^ — ^' = c, and - = ^>,t 



- c ^= ps — z — X -h r, 



% 



and finally hy making r -\' ps — 2 = d,\ we obtain : 



- c + X = d. 



Each comparison furnishes an equation of this form, and if m com- 

 parisons are made, the normal equations are : 



_nl 



[01 »+ [:]--&]-■ 



[:] 



-\- m X 



- ['0 = 



(11) 



The true reading of a vernier is the product of the actual diflTerence 

 between one of its divisions and a division of the limb, multiplied by the 

 number of divisions embraced in the reading; if then 0' be any reading, 

 o its corrected or true value, and q the number of divisions in the vernier : 



the upper and lower signs of i pertaining to the "short" and " long " 

 forms of vernier respectively. The correction is therefore : 



d — d' = f (i (± 5 _ 1) _ A , 



or since i (± q — 1) = I, and I — I' = c, 



— = ; c, 



(12) 



ill which c is positive for either a "short" vernier which is too short, or 

 a " long" vernier which is too long, and vice versa. 



Although the normal equations are ciisily solved for any values of s, 

 time can be saved, even in this case, by adopting a uniform system of 



* In the apparatus here referred to the circle readings diminish as the readings 

 of the usual or "short" form of vernier increase, 

 f For most sextants p = 59. 

 J For a reversed or " long " vernier .5 is essentially negative. 



