SECT. 10.] Arrangement and Formation of the Series. 37 



Now if all deflections from a mean were brought about in 

 the way just indicated (an indication which must suffice for 

 the present) we should always have one and the same law of 

 arrangement of frequency for these deflections or errors, viz. 

 the exponential 1 law mentioned in 5. 



10. It may be readily admitted from what we know 

 about the production of events that something resembling 



1 A definite numerical example of 

 this kind of concentration of fre 

 quency about the mean was given in 

 the note to 4. It was of a binomial 

 form, consisting of the successive 

 terms of the expansion of (1 + l) m . 

 Now it may be shown (Quetelet, Let 

 ters, p. 263 ; Liagre, Calcul des Proba- 

 bilites, 34) that the expansion of such 

 a binomial, as m becomes indefinitely 

 great, approaches as its limit the 

 exponential form ; that is, if we take 

 a number of equidistant ordinates 

 proportional respectively to 1, m, 

 m(m- 1) 



1-2 



&c. , and connect their ver 



tices, the figure we obtain approxi 

 mately represents some form of the 

 curve y = Ae- hx&amp;lt;2 , and tends to become 

 identical with it, as m is increased 

 without limit. In other words, if 

 we suppose the errors to be produced 

 by a limited number of finite, equal 

 and independent causes, we have an 

 approximation to the exponential 

 Law of Error, which merges into 

 identity as the causes are increased 

 in number and diminished in magni 

 tude without limit. Jevons has given 

 (Principles of Science, p. 381) a dia 

 gram drawn to scale, to show how 

 rapid this approximation is. One 



point must be carefully remembered 

 here, as it is frequently overlooked 

 (by Quetelet, for instance). The co 

 efficients of a binomial of two equal 

 terms as (l + l) m , in the preceding 

 paragraph are symmetrical in their 

 arrangement from the first, and very 

 speedily become indistinguishable in 

 (graphical) outline from the final ex 

 ponential form. But if, on the other 

 hand, we were to consider the suc 

 cessive terms of such a binomial as 

 (l + 4) m (which are proportional to 

 the relative chances of 0, 1, 2, 3,... 

 failures in m ventures, of an event 

 which has one chance in its favour 

 to four against it) we should have an 

 unsymmetrical succession. If how 

 ever we suppose m to increase with 

 out limit, as in the former supposi 

 tion, the unsymmetry gradually dis 

 appears and we tend towards pre 

 cisely the same exponential form as 

 if we had begun with two equal terms. 

 The only difference is that the posi 

 tion of the vertex of the curve is no 

 longer in the centre : in other words, 

 the likeliest term or event is not an 

 equal number of successes and fail 

 ures but successes and failures in 

 the ratio of 1 to 4. 



