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THE following detached notes are designed to supplement papers contributed to former numbers of this Journal.

I.* My first task is to test the conclusions reached in my paper on the Statistics of Unprogressive Communities, which appeared in the last number of the Journal. It may be remembered that the most recherché investigation there attempted was to determine the average duration of a wasp's absence from its nest, by observing the time and number of those going in and out, en masse, without distinguishing individuals. One method is to observe the number issuing and entering each minute for an hour or so before closing time; and, if the number of exits during that period is n, to subtract the sum of the times of these n exits from the sum of the times of the last n entrances, and put this difference, divided by n, for the required mean duration of a voyage. Applying this method to a nest on the banks of the Cherwell (Oxford), on 20th July, 1896, I found that between 7.57 P.M., when I began to observe, and closing time, which occurred at 8.46 P.M., there were 262 exits. The sum of the times of these exits, measured from 7.57 as zero, was 3,368 minutes. The corresponding sum of the last 262 entrances was 7,303. Whence for the mean duration of a voyage we have (7,303 - 3,368) ÷ 262 = 15 minutes. For the first half of the period the mean duration proved to be 16 minutes, for the first quarter of the period, 17 minutes.

The corresponding operation proper to the morning having been performed on a nest at Edgeworthstown, 14th September, 1896, between 5.10 A.M. and 6.50 A.M., during which period there occurred 142 entrances, the mean duration of a voyage proved to be 12 minutes for the whole period; for the latter half of the period 17 minutes, and for the last quarter of the period 19 minutes.

These results are in accordance with those obtained by similar methods before.1

* Read before Section D of the British Association, 22nd September, 1896. 1 Journal of the Statistical Society, June, 1896, Tables VIII, X, and XII.

But, when I employed the method proper to observations made in the middle of the day, I obtained this year, as sometimes last year, results differing from the above. Thus, in a nest on the banks of the Cherwell, on 22nd July, 1896, about 5 P.M., the flow of wasps issuing and entering was at the rate of about 425 per minute, 38 having been observed to enter, 30 to issue during a period of 8 minutes. Whence it might have been expected that, if the time of absence was about a quarter of an hour, there would have been 64 wasps out. But, having stopped the hole, I intercepted 92 returning, and there still remained over, when I had to leave the scene, a good number, say 20. These statistics point to

a mean duration of voyage longer than 20 minutes.

I obtained similar statistics at Belton, near Great Yarmouth. There, at a nest in the garden of an eminent statistician, I observed in the course of half an hour, on 31st July, after 10 A.M., 55 entrances and 38 exits, giving a flow of not quite 15 per minute. Also the number excluded by suddenly stopping the hole proved to be 68. From which data it would seem that the mean duration of a voyage was about 45 minutes.

It may be suspected that the determination of the flow in the two preceding instances, being based on comparatively few observations, is inaccurate. But this suspicion is less appropriate in the following instance:-At Edgeworthstown, on 30th August, 1896, a wasps' nest, observed for 36 minutes after 12.4 P.M., presented the following data:

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The total number of exits in the 36 minutes was 281; the total number of entrances was 285.

Whence with some confidence we may put, for the flow per minute, approximately 8.

The hole having been plugged at 12.47 P.M., the excluded were slaughtered as they appeared on the scene, in numbers and periods which are stated in the annexed table. The horizontal lines which demarcate the entries in the table indicate breaches of the continuity of observation, when I had to leave the scene for some time.

2 Loc. cit., Table IV.

Not the one before mentioned.

♦ } (55 + 38) ÷ 30 = 1·5 nearly.

The minutes were not quite successive, owing to my having occasionally missed count.

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The total number, 289, divided by 8 gives about 36 minutes for the mean voyage, a result which is paralleled by some of last year's observations, which I had regarded as inexact.

The next experiment is even more decisive against the hypothesis that the normal duration of a voyage in the daytime is only a quarter of an hour. At Edgeworthstown, 8th September, in 34 (not quite consecutive) minutes, after 10.33 A.M., the following observations were inade:

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Whence the flow per minute = † (449 + 453) ÷ 34 = 13·3. The steadiness of the flow is shown by its approximate equality for all the periods. Also the nest having been closed at 11.12 A.M., the arrivals were intercepted as follows:

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Whence for the mean duration of a voyage we have 740 ÷ 13·3 = 56 minutes nearly!

The morning operation above referred to as yielding an average of 12 minutes was performed on this nest some days after the operation which has just been described.

7

It may also be sought to account for the discrepancy between the evening (or morning) and mid-day observations by the fact that the result of the former method is made too small by misdating the return of those belated individuals who do not come home till morning. The additional observations which I have made seem to show that some, but not very great, inaccuracy may be caused by this circumstance.

There remains only the hypothesis that the voyages in the day time are longer than those in the early morning or late evening.

The hypothesis that the mean duration of a voyage in the day time is longer than a quarter of an hour, squares with an observation which I have made this year upon the total population, compared with the traffic. In a nest within the precincts of Balliol College, Oxford, having administered cyanide of potassium overnight, I counted 1,237 dead; and there survived a remnant who escaped the influence of the drug, numbering apparently from 50 to 150. Also the flow of wasps entering and issuing had been pretty constantly, at different times on several days, about 30 per minute. Now if we suppose that the majority of the working population, say, at least 900 were engaged in driving the traffic, we find for the duration of a complete cycle-including expedition outside and operations within the nest-about 30 minutes. This gives at least 20 minutes for the procuring of a load outside, if we suppose that the disposal of the load inside cannot occupy more than 10 minutes."

I have obtained some confirmation of the length of voyages by watching wasps at work on trees and plants. One method is to fix attention on a particular individual among those who are found at work, and to note the interval between the moment at which the individual comes under observation and the moment at which it takes its departure. The average of such intervals is equateable to half the time occupied in procuring a load, since the observation is equally likely to have commenced at the end, or at the beginning, or at any intermediate point of that time.

This method proving tedious, as the intervals observed were often over twenty minutes, I adopted the following method. Where there is a limited number of wasps working steadily at an isolated shrub or flower-bed, observe the number of those who retire from the scene-not for a momentary excursion, but as bona fide travellers homewards-during a certain short period, e.g., a quarter of an hour; and put for the number constantly at work during that period, the mean of the number present at the beginning, and at the end of the period. Then-by the principle underlying that method of exclusion which is exemplified in the Loc. cit., pp. 383 and 386.

Loc. cit., p. 377 note.

Loc. cit., Section V.

preceding pages-we have, for the required mean time of presence, the number constantly at work by the number of departures per minute, or the number constantly at work x the number of minutes in the period by the number of departures in that period. For example, this September I observed in 42 not quite consecutive minutes 20 departures from a cluster of snakeweed at which there were about 10 wasps constantly at work. Whence we have 10 x 42 ÷ 20 about 21 minutes. It may be suspected that the result is too small, owing to some of the constant workers escaping notice.

10

It remains only to add that some of the observations made this year confirm the hypothesis that the frequency curve for the duration of a voyage is asymmetrical.

II. The reference to the asymmetrical statistical curves leads me on to supplement my remarks on this subject in this Journal for September, 1895, by calling attention to a remarkable formula for the frequency of incomes which has been lately given by the eminent mathematical economist Professor Pareto in his La Courbe de la Répartition de la Richesse (Lausanne, 1896). Designating each amount of income as x, and the number of incomes equal or superior to a as N, he finds the following simple relation between the logarithms of these quantities

log N =

log A - a log x;

where a is a constant which proves to be much the same for different countries, whence

NA÷xa.

=

This law is of considerable economic interest, as M. de Foville has lately pointed out in the Journal des Economistes. The approximate identity of the law as ascertained for different countries, points to the dependence of the distribution of income upon constant causes not to be easily set aside by hasty reformers.

But we are here concerned with the statistical lessons to be derived from Professor Pareto's formula. It appears to me to confirm the opinion which I expressed last September, that a close fit of a curve to given statistics is not, per se and apart from à priori reasons, a proof that the curve in question is the form proper to the matter in hand. The curve may be adapted to the phenomena merely as the empirically justified system of cycles and epicycles to the planetary movements, not like the ellipse, in favour of which there is the Newtonian demonstration, as well as the Keplerian observations.

It may he objected that Professor Pareto's curve does not fit the phenomena at its lower extremity. For according to the formula given above there ought to be an infinite number of null incomes, and an indefinitely large number of incomes in the neighbourhood of zero. This objection however is not applicable to the formula which is given by Professor Pareto as a second approximation." This formula appears at first sight identical with 11 See Appendix, Note I.

10 Loc. cit., p. 366.

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