130 Proceedings of the Royal Irish Academy. 



thus the following final equations for determining the unknown 

 quantities : — 



+ 33-0000 rf. A8o + 0-3172 k - 8-2603 d.^' - l-27037r = + 0"-2400 

 + 0-3172 „ +3-1428,,+ 4-6399 „ - 2-5263 „= + -7179 



- 8-2603 „ +4-6399 ,, +20-5923 „ + 0-1074 „= + 2 -4580 



- 1-2703 „ -2-5263,,+ 0-1074 „ + 12-4156 „= + -3016. 



The solution of these equations gives the following values : — 



<?.ASo = + 0"-036 

 d^! = +Q -0978 

 K = + -112 

 7r = + -047. 



The errors which remain in the equations of condition after the 

 substitution of these values are given in the last column of the pre- 

 ceeding Table. 



The squares of the errors are thereby reduced from 0"-899 to 0-532, 

 which gives for the probable error of one observation the value ± 0"-09, 

 and for the probable errors of the quantities above : — 



(^.ASo) . . 



. . ± 0"-018 



d^' . . 



. . ± -0759 



K . 



. . ± -029 



77 . 



. . ± -030. 



