Empirica I Proof of tli e Law of Error. 331 



and the same definite and constant function of a number of 

 variable elements. Each variable assumes for different obser- 

 vations different values according to some law of facility. 

 And the number of the variables on which an observation 

 depends must be large. 



This is the general statement of the conditions ; but the 

 nearer definition requires care. Quetelet's simple illustrations 

 have countenanced the supposition that the elementary vari- 

 ables must all obey one and the same law of facility, and that 

 of the simplest kind. Bat it is shown by Poisson's generali- 

 zation of Laplace's theory that the facility-curves may be 

 different and of almost any species. If they are unsymme- 

 trical, the figure in which the observations group themselves 

 will still be a Probability-curve — with respect to the central 

 portion at least ; for the extremities will be apt to differ from 

 the typical form and from each other. It is not even neces- 

 sary to assume the facility-curve for each variable to be a 

 constant function. We might suppose these forms to shift 

 and slide between the times at which different observations 

 are taken, provided there remain constant the centre of gra- 

 vity of all the facility-curves and their average mean-square 

 of error. 



Nor, again, need the observation be the simple sum of the 

 variables or elements. It may be any linear function of them. 

 Now, as by Taylor's theorem any function may be regarded 

 as linear for small values of its variables, it might seem that, 

 whatever the relation of the observation to its constituent 

 elements, the law of facility for the observations must be in 

 the neighbourhood of its centre a probability-curve. It will 

 be found, however, that this proposition holds in general for 

 such small distances about the centre that any other symme- 

 trical curve would equally well represent the grouping. The 

 range over which the law prevails is apt to be of a smaller 

 order than when the elementary variables are combined by 

 simple addition*. 



As we can seldom be certain beforehand how far the neces- 

 sary conditions are fulfilled, the theoretical analysis does not 

 enable us to dispense with the empirical verification. How- 

 ever, the hints afforded by theory assist us in examining 

 experience. We shall not expect Social Statistics, e. g. Bank 

 Reserves, to comply with the Law of Error, when they depend 

 on one or two great causes such as a crisis or a warf . When 



* Cf. Glaisher, f Memoirs of the Astronomical Society/ vol. xl. p. 105. 



t The number of elements necessary for some approximation to the 

 Law of Error is less than might be expected. The present writer has 

 shown, by forming- sums oijlve digits taken at random from mathematical 



Z'2 



