. The London, Edinburgh and Dublin philosophical magazine and journal of science. already stated, 11-^ The case of w = f is analogous. Fig. 2 is designed to re-present this type. When u= Vf, it should seem that onlyhalf the original curve is reproduced in the transformation;the other half becoming impossible. The transformed curve is \7TC * Ante, p. 436. f Ibid. of Error and Correlated Averages. 521 having two branches; between which the values of V + £should be equally distributed. The curve has two maximum Fig. 0 X points ti = +\/ -q, and is concave from zero up to a distance from the ce
. The London, Edinburgh and Dublin philosophical magazine and journal of science. already stated, 11-^ The case of w = f is analogous. Fig. 2 is designed to re-present this type. When u= Vf, it should seem that onlyhalf the original curve is reproduced in the transformation;the other half becoming impossible. The transformed curve is \7TC * Ante, p. 436. f Ibid. of Error and Correlated Averages. 521 having two branches; between which the values of V + £should be equally distributed. The curve has two maximum Fig. 0 X points ti = +\/ -q, and is concave from zero up to a distance from the centre, on either side, \/2 times as great as that ofthe maximum-point ; after which the curve becomes the maximum-point is near the Median of the correspondinglimb, which is (corresponding to £= ±476.,.c,which is the quartile of the original curve), and the curve isconcave about that central region, its general appearance willnot differ very sensibly from that of the result will not be very different if for the square root be Fig. substituted the tth root. Fig. 3 is designed to illustratethis type. The probable errors, both of the original and the 522 Prof. F. Y. Edgeworth on the Law transformed curve, are in the neighbourhood of the maximum-points Q1? Q2. The question may now be raised: whether one of thesetypes—especially the more distinctive one, where F (X) = 0—is likely to be realized in the (exceptional) case of the actionof numerous independent agencies not resulting in the pro-bability-curve. The following problem may have wide ana-logies. Let a body be moved from rest in a right line by thesimultaneous action of numerous small impulses, each as likelyto be in the positive as in the negative direction. The lawof frequency for the momentum of the body will be the proba-bility-curve ; for the energy of the body is the curve representedin fig. 2. V. So far we have supposed the elements to assumedifferent values in virtue e
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