ON ERRORS OF OBSERVATION. 143 



the nearest variety of the class under consideration be 

 substituted for it. 



Having then asserted, as a result of investigation, the 

 existence of a standard law of facility of error which 

 not only represents or resembles the impressions which 

 unassisted reason would form a priori, but the results of 

 which are more than sufficient mathematical approxi- 

 mations to truth, whatever (with some easily admissible 

 limitations) the law of error may be, I proceed to 

 describe it more particularly, calling it in future the 

 standard law of facility of error. The sole datum 

 necessary for its specific application, is either of the 

 three, the weight of an observation, the mean risk of 

 error, and the probable error, any two of which may be 

 deduced from the third by the rules in p. 137- 



PROBLEM. Given either of the three data, required 

 (A) the chance that the error of any one observation 

 shall lie between e positive and e negative., or that the 

 observation shall give something between e too much, 

 and e too little (B) the chance that the preceding 

 shall not happen (C) the chance that the error shall 

 be positive, but not exceeding e ; or that it shall be 

 negative, not exceeding e. 



RULE. Multiply e by the square root of the weight, 

 and let the product be t ; then (A) is the H corres- 

 ponding to t in table I., and (B) is the remainder of (A) 

 subtracted from unity ; each of the chances called (C) 

 is one half of (A). 



EXAMPLE. The mean risk of error is 10 ; required the 

 chance of the error lying between -f 15 and 15 ; that 

 is, between 15 too much, and 15 too little. Since the 

 square root of the weight is 200 divided by 709 times 

 the mean risk of error 10, or T 2 %, and since 15 times this 

 result is -^-g-g- or *42 ; the probability required IB '45 ; 

 or 11 to 9 against the event. The probability of a 

 positive error less than 15 is "225, and the same for a 

 negative error within the same limits. 



When all the individual observations are made under 

 the same circumstances, so as to have the same weight, 



