258 DETERMINATION OF AN ORBIT FROM [BOOK II. 



this cannot be defined a priori, we will, approaching the subject from another 

 point of view, inquire upon what function, tacitly, as it were, assumed as a 

 base, the common principle, the excellence of which is generally acknowledged, 

 depends. It has been customary certainly to regard as an axiom the hypothesis 

 that if any quantity has been determined by several direct observations, made 

 under the same circumstances and with equal care, the arithmetical mean of the 

 observed values affords the most probable value, if not rigorously, yet very 

 nearly at least, so that it is always most safe to adhere to it. By putting, 

 therefore, 



V=V =V&quot; Qte.=p, 

 we ought to have in general, 



9 (Mp) + &amp;lt;? (M f p) + 9 (M&quot; p) + etc. = 0, 

 if instead of p is substituted the value 



wnatever positive integer /a expresses. By supposing, therefore, 



M&quot;= etc. =M 



we shall have in general, that is, for any positive integral value of 



whence it is readily inferred that ^ must be a constant quantity, which we will 

 denote by Jc. Hence we have 



-\- Constant, 



denoting the base of the hyperbolic logarithms by e and assuming 



Constant = log K. 



Moreover, it is readily perceived that Tt must be negative, in order that /2 may 

 really become a maximum, for which reason we shall put 



i# hh; 

 and since, by the elegant theorem first discovered by LAPLACE, the integral 



