138 Mr. F. Y. Edgeworth on the 



tions*. For (1) in the general case let the facility-functions first 

 be symmetrical. And, according to the usual proof of the law 

 of error, multiply each by cos ax, and integrate with regard to x 

 between extreme limits, and multiply together the reciprocal 

 functions so obtained. The resulting function in a, say 0( ), 

 will, in general, differ from the type £-i s * 2 * 2 (where k 2 is the 

 mean square of error) by a remainder R which is of the form 

 r x a^ + r 2 a 6 + &c, where r x , r 2 , &c. are not, in general, known 

 to vanish f . Therefore the law of error will differ from the 



exponential type by a remainder 1 R cos axda, which is not 



in general known to be insensible. For example, if the facility- 

 functions are of the general reproductive type symbolized 

 below on page 140, when t is sensibly different from 2, the law 

 of error employed in the method of least squares is sensibly 

 erroneous. And at the same time it appears that the absolute 

 method falls along with the relative. For if the y of Laplace 

 could in general be regarded as a probability-curve, then, con- 

 sidering the reciprocals of the facility-curves, we see that their 

 0(a) is of the form e~ Ca2 ; their law of error must be expo- 

 nential, which has been shown not to be in general true. 

 A fortiori in the case of unsymmetrical functions: — (2) In 

 the particular case where the limits of the facility- functions 

 are indefinitely small, let them first be symmetrical of the 

 form A — Cx 2 , powers of the variable above the second being 

 neglected. Then, according to the absolute method, the quse- 

 situm is rightly given by the equation 



C C 



j~(x—x 1 )+ j£(x—x 2 ) + &c. = Q, 



C 

 where it is easily seen that . is proportional to the inverse 



mean square. And at the same time it appears that the expo- 

 nential law of error employed in the relative method (the 

 method of least squares) is in this case correct. For in this 

 case 6(a) must be of the required type; Rmust be zero, since 

 the inverse mean powers are negligible. When the functions 

 are not symmetrical, say A — Bx— Cx 2 , where the origin is 



taken so that S -r- = 0, the preceding statements still hold good ; 



C 

 except that -r- no longer is proportional to the inverse mean 



square. Thus, if either of the two conditions is fulfilled, Laplace's 



* Postscript to "Law of Error;' Phil. Mag. November 1883. 

 i Traversing Ellis, Camb. Phil, Trans, p. 210 ; cf. ibid. p. 215; Camb. 

 Math. Journ. iv. p. 132. 



