SECT. 3.] ANY NUMBER OF OBSERVATIONS. 257 



etc., and, therefore, that all systems of values of these unknown quantities were 

 equally probable previous to the observations, the probability, evidently, of any 

 determinate system subsequent to the observations will be proportional to 2. 

 This is to be understood to mean that the probability that the values of the un 

 known quantities lie between the infinitely near limits p and p-\-dp, q and g-\-dq, 

 r and r-\-dr, s and s-(-ds, etc. respectively, is expressed by 



A.&djod^drds , etc., 



where the quantity A will be a constant quantity independent of p, q, r, s, etc. : 



. * 



and, indeed, ^ will, evidently, be the value of the integral of the order v, 



f v 2dpdgdrds , etc., 



for each of the variables p, q, r, s, etc., extended from the value - - oo to the 

 value -|- oo . 



177. 



Now it readily follows from this, that the most probable system of values of 

 the quantities p, q, r, s, etc. is that in which 12 acquires the maximum value, and, 

 therefore, is to be derived from the v equations 



- = 0, ~ = 0, i== 0, =?= 0, etc. 



dp dy dr ds 



These equations, by putting 



V M= v, V M = v , V&quot; M&quot; = v&quot;, etc., and ^~ = 9 4, 

 assume the following form : 



dv , . dv , f I dv&quot; i n \ , r&amp;gt; 



dv , . dv f , , , dt/ / /, . A 



TqVv + djVv+^Vv +eto.= Q, 



dv , | dv , , , d/ , i A 



dT 9 v + j; 9 v + -^ y v 4- etc. = 0, 



dv , . dv , , , dv i n \ rv 



r.V v + dTVv+d^Vv +eta=a 



Hence, accordingly, a completely determinate solution of the problem can be 

 obtained by elimination, as soon as the nature of the function y is known. Since 



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