OF TESTIMONIES OR JUDGMENTS. 623 



26. Such is the final general expression for the probable altitude of the star. 

 The following observations may throw light upon its real nature : — 



1st, In the analysis by which this expression was obtained, p x and p 2 are the 

 observed altitudes of the star, a quadrant of the celestial arc being taken as 

 unity. Considered, however, as the expression, not of a probability, but of the 

 most probable measure of a physical magnitude, the truth of the formula will of 

 course be independent of the unit of magnitude. 



Idly, The formula is independent of mechanical analogy. We may place it in 

 the well-known form 



r=W lPl + W 2 p 2 (1) 



in which, as the subject is usually treated W\ and W 2 are called the weights of the 

 observations. Here, however, these quantities are determined as functions of the 

 initial data — these data being probabilities. We have 



1 — Ob x 1 — a 2 



~\ 1 "l ^2 



W,= Cl W= 2 . (2) 



l—a x l — tf 2 2 1 — «j 1 — a 2 



r=^" Cl + T^7 2 ° 2 r 11 ^ 1 + 1^2 ° 2 



?>dly, The initial probabilities, of which W, and W 2 are functions, are neither 

 foreign nor imaginary elements. They may be difficult to determine, but theo- 

 retically their determination rests upon considerations which are entirely proper 

 to the subject. When an observation has been made, the question whether 

 it is correct or not is a question of probability. We can never predicate absolute 

 correctness. We can seldom affirm absolutely that an observation is incorrect. 

 Our knowledge of the circumstances of the observation, Art. 22, leads us to regard 

 the probability in question as sometimes greater, sometimes less. To suppose it 

 capable of a numerical value, as we have done, by the introduction of the con- 

 stants c x c 2 , is then perfectly legitimate. It has been said that an estimate of the 

 correctness of the observation rests upon the circumstances by which it was ac- 

 companied. These circumstances, taken in the aggregate, are themselves a sub- 

 ject of probability. This we express by the introduction of the constants a x a r 

 The probability after an observation is made that it is correct, and the probability 

 before it is made that the state of things shall be such as to give to the result that 

 particular probability of correctness, are quite different things. 



4:thly, In the same course of observations made by the same individual with 

 consciously uniform regard to personal and instrumental accuracy the values of 

 a t and a 2 would be sensibly equal. The formula (10) would thus reduce to the 

 following, viz. : — 



r= J _£] l —^ (3) 



1 — C 1 1 — Cj 



