OF TESTIMONIES OR JUDGMENTS. 615 



and hence determine u as a function of p, q . . and c. The conditions of possible 

 experience and of limitation will be found by supposing s, t, to admit of a single 

 determination in positive values. Or as before, they may be found indepen- 

 dently, and then applied to limit the solution. 



22. We now proceed to the consideration of the problems referred to in Art. 

 (1.) We shall first examine the problem which has for its object the determina- 

 tion of the most probable measure of a physical magnitude, two conflicting mea- 

 sures of which have been assigned by observation. The problem is not, as has 

 been said, Art. 2, in its immediate presentation, one whose elements are logi- 

 cal, but it admits, as we shall see, of being so represented as to give it this cha- 

 racter. 



Problem 1. 



Two simultaneous observations of a physical magnitude, as the elevation of a star, 

 assign to it the respective values p x and p 2 . The probability , when the first observa- 

 tion has been made, that it is correct, is c v the corresponding probability for the 

 second observation is c 2 . Required the most probable value of the physical magni- 

 tude hence resulting. 



First Solution. 



23. The numerical elements which are not, in their immediate presentation, 

 probabilities, are^ and p 2 . But these become such if we contemplate the pro- 

 blem under another aspect. Let a quadrant be taken as the unit of magnitude, 

 then p x and p 2 are proper fractions ; p x actually expressing the probability af- 

 forded by the first observation, p 2 that afforded by the second observation, that a 

 pointer, directed at random to that quadrant of elevation in which the star, re- 

 garded as a physical point, is situated, will point below the star. The problem 

 thus regarded contains the following logical elements, which we shall express by 

 appropriate symbols, viz. 



The event which consists in the first observation, such as it is, being made =cc. 



The event which consists in the second observation, such as it is, being- 

 made =y. 



The event which consists in the first observation being correct, —w. 



The event which consists in the second observation being correct, ~v. 



The event which consists in a pointer, directed at random to the quadrant in 

 which the star is situated, pointing below that star, =z. 



We must now express symbolically the data, including therein whatever logi- 

 cal connections we can establish among the events, x, y, n\ v, and z. 



The probability that the first observation, when made, is correct, is c x . This is 

 a conditional probability ; or, to adopt a well-known form of expression, it is a 

 probability a posteriori. Viewed from a point of time anterior to the observa- 



