Theory of the Average. [CHAP. xix. 



the mathematician would at once be able to prove, the new 

 law of facility of error can be got at more quickly deduc 

 tively, viz. by taking the successive terms of the expansion of 

 (1 + I) 20 . They are given, below the line, in the figure on p. 476. 

 13. There are two apparent obstacles to any direct 

 comparison between the distribution of the old set of simple 

 observations, and the new set of combined or reduced ones. 

 In the first place, the number of the latter is much greater. 

 This, however, is readily met by reducing them both to the 

 same scale, that is by making the same total number of each. 

 In the second place, half of the new positions have no repre 

 sentatives amongst the old, viz. those which occur midway 

 between F and E, E and D, and so on. This can be met by 

 the usual plan of interpolation, viz. by filling in such gaps 

 by estimating what would have been the number at the 

 missing points, on the same scale, had they been occupied. 

 Draw a curve through the vertices of the ordinates at 

 A, B, C, &c., and the lengths of the ordinates at the in 

 termediate points will very fairly represent the corresponding 

 frequency of the errors of those magnitudes respectively. 

 When the gaps are thus filled up, and the numbers thus 

 reduced to the same scale, we have a perfectly fair basis of 

 comparison. (See figure on next page.) 



Similarly we might proceed to group or reduce three 

 observations, or any greater number. The number of possible 

 groupings naturally becomes very much larger, being (1024) 3 

 when they are taken three together. As soon as we get to 

 three or more observations, we have (as already pointed out) 

 a variety of possible modes of treatment or reduction, of 

 which that of taking the arithmetical mean is but one. 



14. The following figure is intended to illustrate the 

 nature of the advantage secured by thus taking the arith 

 metical mean of several observations. 



