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, and p, 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. 
2dly, The formula is independent of mechanical analogy. We may place it in 
the well-known form 
r=W, p, t+ We Po : , ; 5 : (1) 
in which, as the subject is usually treated W, and W, 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—a, 1l—a, 
= Le Ie,” 
i I=a l-a Wada I= (2) 
——¢, + -—*e let 2¢ 
l—c, 1 1-e, ? l—c, 1" l-e, ? 
3dly, The initial probabilities, of which W, and W, 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. Wecan 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,, 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 ageregate, are themselves a sub- 
ject of probability. This we express by the introduction of the constants a, a,. 
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. 
Athly, In the same course of observations made by the same individual with 
consciously uniform regard to personal and instrumental accuracy the values of 
a, and a, would be sensibly equal. The formula (10) would thus reduce to the 
following, viz. :— 
oP, + Ps 
r= pe aaa acc : : : : (3) 
4 F255 
Ie 1=¢ 
