1842.] On Equations of Condition for a Quadrilateral. 33 



each of the 12 angles concerned. I do not see how any of them can 

 fairly be omitted : for although any one of them may involve all the others 

 when the angles are free from error, such is not necessarily the case 

 when the angles, as happens in fact, are mixed up with errors we know 

 not how. I know not of any way by which a fair judgment can be 

 *ormed as to the goodness or badness of observations, besides that re- 

 sulting from the amount of minimum alteration required to make the 

 whole consistent among themselves. It is quite possible, and will gene- 

 rally happen, that in every one of the above equations taken singly, 

 the errors will be so mixed up in two or more of the quantities concern- 

 ed as in a greater or less degree to destroy the effect of each, which 

 errors will become sufficiently apparent when the quantities are other- 

 wise combined. Using the whole of the equations, if the correction for 

 any one quantity retains the same signs throughout, while on another 

 quantity the correction is in a great measure destroyed, being some- 

 times -|- and sometimes — , we may fairly infer that in the former case 

 the observed quantity is erroneous, and in the latter that it approaches 

 to its true value ; the errors being in proportion to the algebraic sum 

 of all the corrections. 



The sinal and angular equations of figure being quite independent of 

 each other, I am not aware of any reason for preferring the one set of 

 them to the other ; it appears to me that both ought to be taken into 

 account simultaneously, giving equal weight to the mean error as 

 found from each set. By any other method, the ultimate corrections 

 will depend on the arbitrary order in which the equations may have 

 been applied. It may, however, be expedient to apply the totopartial 

 equations, which are independent of figure, after having taken the mean 

 of the others. 



The practical use of these equations in the method above sketched, 

 when we retain only what is necessary, though still somewhat long, is by 

 no means very difficult. The most convenient way would be to take the 

 sums of the effective probabilities, and the sums of the errors, and get 

 the correction by common Rule of Three, by the help of a sliding rule. 



Writers on the doctrine of probabilities direct that when several in- 

 dependent quantities occur, they should be combined according to their 

 weights, or inversely as their probabilities of error, as found by the com- 

 mon rule. This applied to each of the above equations would give rise 



