J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 339 



If we now deduct every number from the following one, and 

 multiply the remainder by the number of the observations = 470, 

 we find 



Number of errors. 



0-20186 



0-18916 



0-16603 



0-13667 



0-10532 



0-07607 



0-05150 



0-03265 



0-01940 



0-01117 



0-01017 



In other examples also it is found, for the most part, that the 

 larger errors occur somewhat more frequently in experience 

 than according to theory, a proof that the assumption of the 

 limits — CO and + co has not misled us ; for, if it had, the con- 

 trary would have been the case. This deviation is easily ex- 

 plained from the circumstance, that the larger errors suppose 

 in the rule a very unusual combination of unfavourable influ- 

 ences, and, indeed, are frequently occasioned by occurrences so 

 insulated that no theory could subject them to calculation. 



The determinate value of one of the constants h or r in a set 

 of observations can, however, only be deduced from actual ex- 

 perience, or from a series of errors which have been found to 

 occm- in this set of observations. We must first learn how to 

 proceed, in order to obtain in the; given observations those errors 

 which approximate most nearly to the true erroi's ; and we must 

 then see how the function 4) A is to be determined numerically 

 in all its parts from those errors. We may begin with the most 

 simple case. It wiU afterwards be more easy to take a view of 

 the rules for the more general and complicated cases, as the ge- 

 neral fundamental propositions remain unaltered. 



For the value of an unknown magnitude x, let direct observa- 

 tion, repeated m number of times, in the same manner and 

 under completely equal circumstances, have given m values 



n, n', n", n'", &c. 

 Each insulated observation will have given an approximate value 

 by virtue of the equations of condition, 



