MEASUREMENT OF VARIABLE PROPERTIES 147 



If we suppose that the number of observations has been 10,240 

 (or in general x), we would expect to find approximately — 



In interval I. : 10 figures ( = xx yw^t) 



II.: 100 „ ( = ^x-3-^) 



III.: 450 „ ( = ^xyA|^) 



etc., the approximation being more and more close the greater 

 the number of observations (x).^ 



REMARKS : (i) The above polynomial represents all the 

 possible events and their frequencies when 10 coins (dice with 

 two faces) are tossed {a = hea,d, 6 = tail). (See § 96.) 



(2) The II terms of the above polynomial represent the 

 relative frequencies of the errors (deviations) in each of the 

 II groups, but they are independent of the absolute values 

 of the measurements. 



(3) In this example two groups of causes (forces, factors) 

 have an influence upon each determination of the density 

 of a. 



In the first group I put one cause or force, which is the density 

 we want to discover. This density is a definite, invariable 

 property of a. It is an invariable force, a constant. The mean 

 value is the measure of this constant. 



The second group includes a number of accidental causes 

 (chance), the combination (resultant) of which varies continu- 

 ally and produces in each determination an error (or deviation). 

 Each observed figure is the constant ± an error. 



(4) Here the importance of the mean value is preponderant : 

 it represents a definite something the existence of which is 

 independent of chance. The extreme errors are of very little 

 importance because they are indefinite. When the operator is 

 skilled and disposes of good instruments, the extreme errors 

 come nearer the mean and nearer each other than when the 

 operator is inexperienced and the instruments mediocre. And, 

 moreover, the possibility always exists that, in a new deter- 

 mination, an error may be committed greater than the greatest 

 one previously committed. In other words, in the given 

 example, the range of deviation is unlimited.^ We will see below 

 that this is not always the case. 



^ In reality, the arithmetical method followed here is not quite accurate : 

 a curve of errors is governed by a more complicated mathematical expression 

 deduced by GAUSS. The series of values (terms) obtained by expanding 

 (a + fe)" approximates more and more closely to the probability curve oj Gauss 

 the greater the value of n. If n =20 the error is not very important, and for 

 a greater value of n the error (difference between the arithmetical curve and 

 the curve of Gauss) becomes rapidly smaller and smaller. I take n =10 (which 

 is too small) in order to avoid an exaggerated number of terms. The arith- 

 metical method is sufficient for the object of the present work. 



" This finds its expression in the curve of Gauss. 



