484 Theory of the Average. [CHAP. xix. 



with the Binomial Law, it will suffice to say that in this case 

 there is no doubt that the arithmetical average is not only 

 the simplest and easiest to deal with, but is also the best in 

 the above sense of the term. And since this Binomial Law, 

 or something approximating to it, is of very wide prevalence. 

 a strong prima facie case is made out for the general employ 

 ment of the familiar average. 



19. The analysis of a few pages back carried the 

 results of the averaging process as far as could be con 

 veniently done by the help of mere arithmetic. To go 

 further we must appeal to higher mathematics, but the 

 following indication of the sort of results obtained will 

 suffice for our present purpose. After all, the successive 

 steps, though demanding intricate reasoning for their proof, 

 are nothing more than generalizations of processes which 

 could be established by simple arithmetic 1 . Briefly, what we 

 do is this : 



(1) We first extend the proof from the binomial form, 

 with its finite number of elements, to the limiting or ex 

 ponential form. Instead of confining ourselves to a small 

 number of discrete errors, we then recognize the possibility 

 of any number of errors of any magnitude whatever. 



(2) In the next place, instead of confining ourselves to 

 the consideration of an average of two or three only, 

 already, as we have seen, a tedious piece of arithmetic, we 

 calculate the result of an average of any number, n. The 

 actual result is extremely simple. If the modulus of the 

 single errors is c, that of the average of n of these will be 

 c-r*fn. 



(3) Finally we draw similar conclusions in reference to 

 the sum or difference of two averages of any numbers. Sup- 



1 The reader will find the proofs in Galloway on Probability, and in 

 of these and other similar formulae Airy on Errors. 



