and the Elimination of Chance. 323 



mean error is small. But if the total range is unity/ the mean 

 square of error is less than the mean error. Whence the pro- 

 position. Also the Lagrangian * method may be applied to the 

 investigation of explicit expressions for the extreme terms 

 of the compound facility-curve. Comparing these extreme 

 terms with the corresponding ordinates of the representative 

 probability-curve, we see that the former are apt to be of a 

 higher order than the latter. Hence the representative curve 

 does not in general disengage itself from the real locus before 

 the extremity of the latter has been reached. The received 

 formula, therefore, does not afford a superior limit to the pro- 

 bability of the observed divergence occurring by chance. 

 Again, by parity of reasoning, by way of a partial differential 

 equation, it may be seen that when the modulus is small, 

 cceteris paribus, the range and degree of accuracy is dimi- 

 nished. But a remedy for this defect is not attainable, owing 

 to the unmanageability of the explicit expression for the 

 general term. 



a. There remains the class of infinite facility-curves. This 

 is of the less importance in that the attribute infinity must be 

 construed strictly in order that the class should be exceptional. 

 If it is merely meant that the range of the elementary facility- 

 curve is very large, then the peculiarities which I have else- 

 where f ascribed to this class will not occur. Consider, for 



example, the curve y= —r\ ^r. If it is merely meant that 



r ' * 7T(1 + ^ 2 ) j 



this form holds for a long way, then it constitutes no excep- 

 tion to the law of convergence that the arithmetical mean has 

 a less dispersed facility-curve than the primary or elementary 

 curve. Convergence will set in at once and continue until 

 any assigned extent of the original form has been converted 

 into that of the probability-curve. It is not easy to under- 

 stand how such strictly infinite errors as here postulated can 

 exist in observations or statistics. The only case in which it 

 occurs to me that such forms might arise is where the regis- 

 tered " errors " or variations are not the raw material of obser- 

 vations, but functions of a datum given by observation, and 

 where the function becomes infinite for certain finite values of 

 its variable. Suppose, for example, an angle to be the direct 

 object of observation; and that it is as likely to have one 



value as another between limits ± «-« («Zl). Then it may 



be shown that the facility-curve which represents the possible 



* Above. 



t PM1. Mag. Oct. 1883 ; Carnb. Phil. Trans. 1885. 

 2 A2 



