SECT. 3.] ANY NUMBER OF OBSERVATIONS. 267 



S3&quot; Q + STP, 



so that S(, 51 , S3 , 21&quot;, S3&quot;, &quot; may be determinate quantities. We shall have, 

 therefore (by restricting the number of unknown quantities to four), 



) &quot; 31&quot; 33&quot; S&quot; 



Hence we deduce the following conclusion. The most probable values of the 

 unknown quantities p, q, r, s, etc., to be derived by elimination from the equations 



P= 0, Q = 0, R = 0, 8= 0, etc., 



will, if P, Q) R, S, etc., are regarded for the time as indeterminate, be expressed 

 in a linear form by the same process of elimination by means of P, Q, R, 8, etc., 

 so that we may have 



p = L+ AP + BQ+ CR + DS+ etc. 



q = L + AP + B Q+C R+D S-\- etc. 



r = L&quot;-\-A P-\-B&quot;Q + C&quot;R+& S-}- etc. 



s =L &quot;+A&quot;P+B&quot; Q + C &quot;R+iy&quot;S+ etc. 



etc. 



This being done, the most probable values of p, q, r, s, etc., will evidently be 

 L, L , L&quot;, L &quot;, etc., respectively, and the measure of precision to be assigned to 

 these determinations respectively will be expressed by 



_L J_ J_ 1 



p ^&quot; v/c&quot;&quot; Jiy 7 &quot; 



the precision of the original observations being put equal to unity. That which 

 we have before demonstrated concerning the determination of the unknown 

 quantity s (for which -^ answers to D &quot;) can be applied to all the others by the 

 simple interchange of the unknown quantities. 



184. 



In order to illustrate the preceding investigations by an example, let us sup 

 pose that, by means of observations in which equal accuracy may be assumed, 

 we have found 



