148 ESSAY ON PROBABILITIES. 



the result which the instrument would give, in the long 

 run. In p. 137. is shewn the method of determining 

 the chance that in s observations, the average should lie 

 within e of the final average, that is, within e of A -f P. 

 This ascertains the chance that the final average just 

 mentioned lies between A + P -f e and A +- P e. Let 

 R be the result shown by the instrument, the truth A 

 being otherwise known. Then if R lies between 

 A -f- P e and A -f- P -f- e, it follows that P lies between 

 R A 4- e and R A e, and the chance of the first 

 is that of the second. 



EXAMPLE. The truth being known to be 30, and 

 the average of 20 observations giving 31, what is the 

 chance that there is in the instrument a fixed error 

 lying between 1 + 1 and 1 J, or between -J and J. 



The weight of the observations must first be found, 

 which is done by summing the squares of the errors, 

 taking the average given by the instrument as true, 

 precisely in the manner used in p. 137. Suppose this 

 weight to be 10; then e or -, multiplied by the square 

 root of 10, or 3'l62, is '79, to which in table I., the 

 value of H is "74. It is therefore about 3 to 1 that 

 the instrument, in the long run, would give a result 

 between 31 -f- and 31 -J- ; that is, that there is a 

 fixed error in the instrument lying between 1 + -J- and 



!-i- 



This first method, then, of ascertaining the fixed 



error of a set of observations, supposes that there are 

 cases in which the result is known beforehand, so that 

 the instrument may first be read by the aid of phe- 

 nomena, instead of phenomena by the instrument. The 

 first observation is that of the error of the latter, found 

 by comparing its indications with the known truth ; 

 the second, the observation of unknown phenomena, 

 follows : accurate results being obtained, not by altering 

 the instrument, but by applying the correction to the 

 observations which the preceding class of observations 

 has rendered necessary. 



A reader unused to astronomical works, on opening 



