and the Elimination of Chance. 



317 



provided that the powers of - constitute a descending series. 



Hence it may be concluded that the probability-curve is for a 

 long way a fairly accurate representative of the real locus. 



It continues for a long way accurate and it is ultimately 

 safe ; and though we have not proved that between the accu- 

 rate tract and the safe tract there may not intervene a tract 

 without either of those attributes, still I think the analogy of 

 the binomial-curves may justify us in jumping over this in- 

 terval, and asserting generally that the received formula is 

 either accurate or at least safe. 



In forming this conclusion for symmetrical finite facility- 

 curves, we have proved the greater part of our thesis*. For 

 it will be admitted that the remaining cases are exceptional. 

 The most important remaining class is that of finite unsym- 

 metrical a {3. This may, like the symmetrical finite class, be 

 divided into binomial and non-binomial. 



a /5 7. The binomial species has an interest beyond games 

 of chance in so far as it may be regarded as the type of the 

 laws of error which occur in rerum natura. Here arises the 

 question: given a set of observations unsymmetrically grouped 

 (such as the entries in the column d x of a Life-table between 

 the ages of 50 and 100), how shall we reason back from this 

 composite datum to the elementary binomial from which it is 

 assumed to have sprung ? How shall we perform the analogue 

 of the operation which Mr. Galtonf performs for symmetrical 

 binomials, when he calculates the number and extent of the 

 elements which underlie a given set of statistics. 



Let us first get a clear idea of the problem. What we are 

 given is a set oin observations grouped unsymmetrically as in 

 the accompanying figure, where the ordinates of the curve 



represent the number of measurements corresponding to each 

 degree of the base. Let the observations range from 0, 

 which we will call zero, over a length / to OL. It is assumed 



* P. 



t PHI. Mag. Jan. 1875, p. 43. 



