260 DETERMINATION OF AN ORBIT FROM [BoOK II. 



equal when we have Ad = h S . Now, therefore, if for example h = 2 h, a double 

 error can be committed in the former system with the same facility as a single 

 &quot;error in the latter, in which case, according to the common way of speaking, a 

 double degree of precision is attributed to the latter observations. 



179. 



We will now develop the conclusions which follow from this law. It is evi 

 dent, in order that the product 



may become a maximum, that the sum 



vv + v v + v&quot;v&quot; + etc., 



must become a minimum. Therefore, that will be the most probable system of values of 

 the unknown quantities p, q, r, s, etc., in which the sum of the squares of the differences 

 between the observed and computed values of the functions V, V, V&quot;, etc. is a minimum, if 

 the same degree of accuracy is to be presumed in all the observations. This prin 

 ciple, which promises to be of most frequent use in all applications of the mathe 

 matics to natural philosophy, must, everywhere, be considered an axiom with 

 the same propriety as the arithmetical mean of several observed values of the 

 same quantity is adopted as the most probable value. 



This principle can be extended without difficulty to observations of unequal 

 accuracy. If, for example, the measures o f precision of the observations by 

 means of which V=M, V = 3/ , V&quot; = M&quot;, etc. have been found, are expressed, 

 respectively, by h, h , h&quot;, etc., that is, if it is assumed that errors reciprocally pro 

 portional to these quantities might have been made with equal facility in those 

 observations, this, evidently, will be the same as if, by means of observations of 

 equal precision (the measure of which is equal to unity), the values of the func 

 tions hV, h V, h&quot;V&quot;, etc., had been directly found to be hM, h M ,h&quot;M&quot;, etc.: 

 wherefore, the most probable system of values of the quantities p, q, r, s, etc., 

 will be that in which the sum of hhvv -f- h h v v -\- h&quot;h&quot;v&quot;v&quot; -)- etc , that is, in which 

 /lie sum of the squares of tlie differences between the actually observed and computed values 

 multiplied by numbers tJiat measure the degree of precision, is a minimum. In this way it 



