with Remarks on the Law of Frequency of Error. 43 



suppose a crown were our "small" unit, and we had a medley 

 of 10 crowns, 33 shillings, and 1 00 fourpenny pieces, with which 

 to make successive throws, throwing the whole number of them 

 at once : we might theoretically sort them into fictitious groups 

 each equivalent to a crown. There would be 29 such groups, 

 viz. : — 10 groups, each consisting of 1 crown ; 6 groups, each of 



5 shillings ; 1 group of three shillings and 6 fourpenny pieces ; 



6 groups each of 15 fourpenny pieces ; and a residue of 4 four- 

 penny pieces, which may be disregarded. Hence, on the already 

 expressed understanding that we do not care to trouble ourselves 

 about smaller sums than a crown, the results of the successive 

 throws of the medley of coins would be approximately the same 

 as those of throwing at a time 29 crowns, and would be ex- 

 pressed by the coefficients of a binomial of the 29th power. 

 Hence I conclude that all miscellaneous influences of a few small 

 and many minute kinds, may be treated for a first approxima- 

 tion exactly as if they consisted of a moderate number of small 

 and equal alternatives. 



The second approximation has already been alluded to ; it 

 consists in taking some account of the minute influences which 

 we had previously agreed to ignore entirely, the effect of whieh 

 is to turn the binomial grades into a binomial ogive. I effect 

 it by drawing a curve with a free hand through the grades, 

 which affords a better approximation to the truth than any other 

 that can a priori be suggested. 



I will now show from quite another point of view (1) that the 

 exponential ogive is, on the face of it, fallacious in avast number 

 of cases, and (2) that we may learn what is the greatest pos- 

 sible number of elements in the binomial whose ogive most 

 nearly represents the generic series we may be considering. The 



value of is directly dependent on the number of elements ; 



hence, by knowing its value, we ought to be able to determine 

 the number of its elements. I have calculated it for binomials 

 of various powers, protracting and interpolating, and obtain the 

 following very rough but sufficient results for their ogives (not 

 grades) : — 



Number of (equal) y&l f m 



elemeuts. q—m 



17 5 



32 10 



65 15 



107 20 



145 25 



186 30 



999 48 



