Reduction of Observations. 137 



one on a large scale practicable. We started with the relative 

 method* lately ; we shall now pursue the absolute. 



As before, let us follow in the footsteps of Laplacef, merely 

 diverging from him in this respect — that whereas he employs 

 a definite function, a " representative particular/' as Berkeley 

 might call it, to express the detriment of error, we shall em- 

 ploy the general undetermined function F. Let us suppose 

 with Laplace the facility-curves for all the observations iden- 

 tical and known, say (£. Let 



y' = H</>(V — «%) x <£(V — os 2 ) . . . , 



where a ly oc 2 . . . are the observations. Let I be the sought, 

 most advantageous, value. Let us put for the detriment in- 

 curred by taking I for the real point, when it really is of, not 

 with Laplace the particular function I — «/, but the general 

 function F(7— a/). Integrating the detriment incurred in 

 the long run, we find by steps indicated by Laplace (remem- 

 bering that F(0) = 0) that the most advantageous value is 

 given by the equation 



Y(rf -T)tf drf = f F(Z~0#W ; 



j "F(^-%W=f* 



where + co are used to denote the extreme limits of the facility- 

 curves. The solution of this equation is the central pointy of 

 the curve ?/•; if, as Laplace argues, we may regard that curve 

 as capable of being represented by 



Laplace's argument is in effect that, in integration, a factor of 

 the expression to be integrated may be neglected, when for 

 small values of the variable the factor is nearly unity, and for 

 large values factor and multiplicand each nearly zero. Now 

 this is the very argument which is employed to prove § the 

 exponential law of error as valid in general ; the argument 

 which is questioned by Mr. Glaisher||, and which, if our 

 remarkslf on the law of error are correct, is proved, not only 

 by theory but example, not to hold universally. I submit 

 therefore, with great deference, that Laplace's argument holds 

 (at least is known to hold) only in two cases, precisely those 

 for which the exponential law of error holds: first, where 

 the facility- curves are of the family named u Probability " ; 

 secondly, when the facility-curves, however irregular, have 

 limits indefinitely small relatively to the number of observa- 



* Phil. Mag. November 1833, p. 361. t Theorie Anal. p. 365. 



X Supposing the function of detriment to be symmetrical. 

 § JE. g. Todhunter's i History of Probabilities/ art. 1002, p. 556. 

 || Phil. Mag. 1872, p. 199. ' If Ibid. October 1883. 



