Mr. Ivory oti the Method of the Least Squares. 167 



preponderaney may be estimated by the distance, from the 

 common fulcrum, of the centre of gravity of the weights liang- 

 ing from tlieir new points of suspension. But whatever the 

 supposed variations are, it will be admitted that the errors 

 may undergo opposite variations, so as to acquire an equal 

 and contrary preponderaney; and it is obvious that the system 

 of the least squares is an exact mean between the two oppo- 

 site systems. We cannot therefore but conclude that the 

 system in which the sum of the stjuares is a minimum, which 

 occupies the mean place among all the possible svstems, is 

 preferable to every other. 



When the rule of the least squares is demonstrated in a 

 satisfactory manner from the nature of the equations of condi- 

 tion, it has been shown that the errors can follow only one 

 law of probability. But it would be in vain to attempt to 

 verity this law in a direct manner, or to show that the errors 

 of any set of observations exactly agreed with it, or even ap- 

 pi'oximated to it. A very cautious inquirer might therefore 

 wish to compare the results obtained by the doctrine of pro- 

 babilities with the like results deduced immediately from the 

 equations of condition by the ordinary processes of investiga- 

 tion. There is no doubt that a very exact coincidence would 

 be found between the conclusions deduced from the two me- 

 thods ; but we cannot enter upon this discussion. 



The practice which is universally followed of taking the 

 arithmetical mean of a set of observations is comprehended 

 in the genertil method we have been considering, as we have 

 always supposed ; lor it is the particular case when the weights 

 a, a\ a" &c. are all equal, and the sum of the errors is equal 

 to zero. It may not, however, be improper to prove this more 

 particularly; and for this purpose we must go back to the 

 original meaning of the symbols. We have 

 e = V -o 



-17 -IV d'^' 1r/ 



V = V -f — — J- = V -\- ax : 



ax 



now when a, or -— , remains invaiiable from one observa- 

 tion to another, the value of V will likewise be constantly tlie 

 same. WMiereibre the errors are respectively 



e =Y -o 



e' =Y - o' 



e" =.W - o" 



&c. 

 and wlien the sum of the errors is zero, their number being 

 n, we obtain -y _ " -f »' + »" -f & »-•• _ 



In 



