150 ESSAY ON PROBABILITIES. 



the long run. Suppose, for example, that (as in the 

 transit-instrument) the correctness of the observations 

 depends (partly) upon the line of sight of a telescope heing 

 always exactly perpendicular to the axis upon which 

 the telescope turns. Such exact perpendicularity is 

 a mathematical fiction, which was never yet realised : 

 the telescope will incline more or less to the right 

 or to the left. But if the telescope he fixed to its 

 axis, and if the axis itself rest on pivots, from which 

 it can be taken off and the position of the instrument 

 reversed, it is obvious that such a reversal of the ends 

 of the axis will alter the error of the instrument, 

 throwing the line of sight as much to the left as it 

 was before to the right, or vice versa. The average of 

 a large number of observations will now present no 

 signs of fixed error, arising from this cause at least : 

 provided that the numbers of observations made in the 

 two different positions be equal. The chance of the 

 average of s observations in each position lying within 

 a given degree of nearness to the truth is precisely that 

 of twice s observations made with no fixed error, and 

 the same tendency to positive and negative casualties 

 as before. When the result has been obtained by 

 combination of the different sets, the fixed error of 

 the instrument may be ascertained by comparing the 

 combined average with the separate averages of each 

 set. If the observations be numerous, and the reversal 

 of the method of observing introduce no new errors, 

 then the combined average will be an arithmetical 

 mean (or nearly so) between the other averages, and 

 the difference between the former and either of the 

 latter will be the fixed error required. 



Independently of one or other of these two methods, 

 the only result directly furnished by the observations 

 (except the average affected by the fixed error) is 

 their weight, which is obtained precisely as in p. 137. 

 It will be seen that either of the preceding methods 

 introduces entirely new elements ; in the first we have 

 previous truths for comparison with subsequent ob- 



