SECT. 3.] ANY NUMBER OF OBSERVATIONS. 263 



squares and products may be neglected, by which means the equations become 

 linear. If, after the calculation is completed, the values of the unknown quanti 

 ties j/, &amp;lt;/ , /, /, etc., prove, contrary to expectation, to be so great, as to make it 

 appear unsafe to neglect the squares and products, a repetition of the same pro 

 cess (the corrected values of p, q, r, s, etc. being taken instead of n, %, (&amp;gt;, o, etc.), 

 will furnish an easy remedy. 



181. 



When we have only one unknown quantity p, for the determination of which 

 the values of the functions ap -\- n, up -\- n , a&quot;p -|- n&quot;, etc. have been found, re 

 spectively, equal to M, M , M&quot;, etc., and that, also, by means of observations 

 equally exact, the most probable value of p will be 



, __ a m -|- a mf -f- a&quot;m&quot; -{- etc. 



- ~ 



putting m, m, m&quot;, respectively, for M n, M n , M&quot; n&quot;, etc. 



In order to estimate the degree of accuracy to be attributed to this value, let 

 us suppose that the probability of an error A in the observations is expressed by 



Hence the probability that the true value of p is equal to 4 -\-p will be propor 

 tional to the function 



g-hh ((ap mf+(a p m ?+(a&quot;p-m&quot;f+ etc.) 



if A -\-p is substituted for p. The exponent of this function can be reduced to 

 the form, 



hh (aa -\- ctct -f cl ct + etc.) (pp 2pA-{- B), 



in which B is independent of p : therefore the function itself will be propor 

 tional to 



It is evident, accordingly, that the same degree of accuracy is to be assigned to 

 the value A as if it had been found by a direct observation, the accuracy of which 

 would be to the accuracy of the original observations as h^ (aa-\- a a -}-a&quot;a&quot;-\- etc.) 

 to h, or as y/ (a a -(- do! -\- d d -j- etc.) to unity. 



