270 DETERMINATION OF AN ORBIT FROM [BoUK II. 



less than another system, is without doubt to be preferred to the latter, still the 

 choice between two systems, one of which presents a better agreement in some 

 observations, the other in others, is left in a measure to our judgment, and innu 

 merable different principles can be proposed by which the former condition is 

 satisfied. Denoting the differences between observation and calculation by A, 

 ,/, ,/ , etc., the first condition will be satisfied not only Mi. A A -f A A -)- A&quot; A&quot; + 

 etc., is a minimum (which is our principle), but also if // 4 -f- ./* -(- //&quot; 4 -J- etc., or 

 j _|_ j 6 _|_ //&quot; 6 -|- etc., or in general, if the sum of any of the powers with an 

 even exponent becomes a minimum. But of all these principles ours is the most sim 

 ple ; by the others we should be led into the most complicated calculations. 



Our principle, which we have made use of since the year 1795, has lately 

 been published by LEGENDRE in the work Nouvclles mcthodes pour la determination des 

 orbites des cometes, Paris, 1806, where several other properties of this principle have 

 been explained, which, for the sake of brevity, we here omit. 



If we were to adopt a power with an infinite even exponent, we should be 

 led to that system in which the greatest differences become less than in any other 

 system. 



LAPLACE made use of another principle for the solution of linear equations the 

 number of which is greater than the number of the unknown quantities, which 

 had been previously proposed by BOSCOVICH, namely, that the sum of the errors 

 themselves taken positively, be made a minimum. It can be easily shown, that a 

 system of values of unknown quantities, derived from this principle alone, must 

 necessarily* exactly satisfy as many equations out of the number proposed, as 

 there are unknown quantities, so that the remaining equations come under consid 

 eration only so far as they help to determine the choice : if, therefore, the equation 

 V = M, for example, is of the number of those which are not satisfied, the sys 

 tem of values found according to this principle would in no respect be changed, 

 even if any other value N had been observed instead of M, provided that, denot 

 ing the computed value by n, the differences M n, N n, were affected by the 

 same signs. Besides, LAPLACE qualifies in some measure this principle by adding 



* Except the special cases in which the problem remains, to some extent, indeterminate. 



