SECT. 12.] Theory of the Average. 477 



This then is the number in a complete cycle of the results 

 taken two and two together. 



12. So much for their number ; now for their arrange 

 ment or distribution. What we have to ascertain is, firstly, 

 how many times each possible pair of observations will present 

 itself; and, secondly, where the new results, obtained from 

 the combination of each pair, are to be placed. With regard 

 to the first of these enquiries ; it will be readily seen that 

 on one occasion we shall have F repeated twice ; on 20 occa 

 sions we shall have F combined with E (for F coming first 

 we may have it followed by any one of the 10 at E, or any 

 one of these may be followed by F) ; E can be repeated in 

 10 x 10, or 100 ways, and so on. 



Now for the position of each of these reduced observations, 

 the relative frequency of whose component elements has thus 

 been pointed out. This is easy to determine, for when we 

 take two errors there is (as was seen) scarcely any other 

 mode of treatment than that of selecting the mid-point be 

 tween them; this mid-point of course becoming identical 

 with each of them when the two happen to coincide. It will 

 be seen therefore that F will recur once on the new arrange 

 ment, viz. by its being repeated twice on the old one. G, 

 midway between E and F, will be given 20 times. E, on 

 our new arrangement, can be got at in two ways, viz. by its 

 being repeated twice (which will happen 100 times), and by 

 its being obtained as the mid-point between D and F (which 

 will happen 90 times). Hence E will occur 190 times alto 

 gether. 



The reader who chooses to take the trouble may work out 

 the frequency of all possible occurrences in this way, and if 

 the object were simply to illustrate the principle in accord 

 ance with which they occur, this might be the best way of 

 proceeding. But as he may soon be able to observe, and as 



