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suppose a crown were our "small" unit, and we had a medley of 10 crowns, 33 shillings, and 100 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 fourpenny 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 expressed 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 approximation 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 which 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 à 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 a vast number of cases, and (2) that we may learn what is the greatest possible 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; g-n 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)
Now, if we apply these results to observed facts, we shall rarely find that the series has been due to any large number of equal elements. Thus, in the stature of man the probable error,
is about 30, which makes it impossible that it can be looked upon as due to the effect of more than 200 equally small elements. On consideration, however, it will appear that in certain cases the number may be less, even considerably less, than the tabular value, though it can never exceed it. As an illustration of the principle upon which this conclusion depends, consider what the value of would be in the case
we may of a wall built of 17 courses of stone, each stone being 3 inches thick, and subject to a mean error in excess or deficiency of one fifth of an inch. Obviously the mean height m of the wall would be 3× 17 inches; and its probable error q-m would be very small, being derived from a binomial ogive of 17 elements, each of the value of only one fifth of an inch. Now we saw from our previous calculation that this would be eight fifths, or 1.6
inch, which would give the value to of or about 321; y-m 1.6'
consequently we should be greatly misled if, after finding by observation the value of that fraction, and turning to the Table and seeing there that it corresponded to more than 200 equal elements, we should conclude that that was the number of courses of stones. The Table can only be trusted to say that the number of courses certainly does not exceed that number; but it may be less than that.
The difficulty we have next to consider is that which I first mentioned, but have intentionally postponed. It is due to the presence of influences of extraordinary magnitude, as Aspect in the size of fruit. These influences must be divided into more than two phases, each differing by the same constant amount from the next one, and that difference must not be greater than exists between the opposite phases of the "small" alternatives. If we had to divide an influence into three phases, we should call them "large," "moderate," and "small;" if into four, they would be "very large," " moderately large," " moderately small,” and very small," and so on. Any objects (say, fruit) which are liable to an influence so large as to make it necessary to divide it into three phases, really consist of three series generically different which are entangled together, and ought theoretically to be separated. If there had been two influences of three phases, there would be nine such series, and so on. In short, the fruit, of which we may be considering some hundred or a few thousand
specimens, ought to be looked upon as a multitude of different sorts mixed together. The proportions inter se of the different sorts may be accepted as constant; there is no difficulty arising from that cause. The question is, why a mixture of series radically different, should in numerous cases give results apparently identical with those of a simple series.
For simplicity's sake, let us begin with considering only one large influence, such as aspect on the size of fruit. Its extreme effect on their growth is shown by the difference in what is grown on the north and south sides of a garden-wall, which in such kinds of fruit as are produced by orchard-trees, is hardly deserving of being divided into more than three phases, "large," "moderate," and "small." Now if it so happens that the "moderate" phase occurs approximately twice as often as either of the extreme phases (which is an exceedingly reasonable supposition, taking into account the combined effects of azimuth, altitude, and the minor influences relating to shade from leaves &c.), then the effect of aspect will work in with the rest, just like a binomial of two elements. Generally the coefficients of (a+b) are the same as those of (a+b)n-rx (a+b)". Now the latter factor may be replaced by any variable function the frequency and number of whose successive phases, into which it is necessary to divide it, happen to correspond with the value of the coefficients of that factor.
It will be understood from what went before, that we are in a position to bring these phases to a common measure with the rest, by the process of fictitious grouping with appropriate doses of minute influences, as already described.
On considering the influences on which such vital phenomena depend as are liable to be treated together statistically, we shall find that their mean values very commonly occur with greater frequency than their extreme ones; and it is to this cause that I ascribe the fact of large influences frequently working in together with a number of small ones without betraying their presence by any sensible disturbance of the series.
The last difficulty I shall consider, arises from the fact that the individuals which compose a statistical group are rarely affected by exactly the same number of variable influences. For this cause they ought to have been sorted into separate series. But when, as is usually the case, the various intruding series are weak in numbers, and when the number of variable influences on which they depend does not differ much from that of the main series, their effect is almost insensible. I have tried how the figures would run in many supposititious cases; here is one taken at haphazard, in which I compare an ordinary series due to 10 alternatives, giving 210=1024 events, with a compound series.
The latter also comprises 1024 events; but it is made up of three parts: viz. nine tenths of it are due to a 10-element series; and of the remaining tenth, half are due to a 9 and half to an 11 series. I have reduced all these to the proper ratios, ignoring fractions. It will be observed how close is the correspondence between the compound and the simple series.
It appears to me, from the consideration of many series, that the want of symmetry commonly observed in the statistics of vital phenomena is mainly due to the inclusion of small series of the above character, formed by alien elements; also that the disproportionate number of extreme cases, as of giants, is due to this cause.
The general conclusion we are justified in drawing appears to be, that, while each statistical series must be judged according to its peculiarities, a law of frequency of error founded on a binomial ogive is much more likely to be approximately true of it than any other that can be specified à priori; also that the exponential law is so closely alike in its results to those derived from the binomial ogive, under the circumstances and within the limits between which statisticians are concerned, that it may safely be used as hitherto, its many well-known properties being very convenient in all cases where it is approximately true. Therefore, if we adopt any uniform system (such as already suggested) of denoting the magnitudes of qualities for the measurement of which no scale of equal parts exists, such system may reasonably be based on an inverse application of the law of frequency of error, in the way I have described, to statistical series obtained by the process of intercomparison.
V. On a new Method of investigating the Composite Nature of the Electric Discharge. By ALFRED M. MAYER*.
N 1842 Professor Joseph Henry† observed that when a needle was placed in a helix and magnetized by the discharge of a Leyden jar, the direction of the polarity of the needle varied with the "striking-distance" of the jar; and these observations led Henry to the discovery that the discharge was multiple and oscillatory in its nature. In 1862 Feddersent confirmed Henry's discovery, on examining the nature of the discharge by means of a revolving mirror. Subsequently Rood (in a series of classical researches, published in Silliman's American Journal, in 1869, 1871, 1872) studied the multiple character of the discharge of the inductorium by means of rotating disks perforated with narrow radial slits. In 1873 Cazin § also investigated the discharge with the rotating disk. The method I have devised leads us directly, by the simplest means, to phenomena which cannot be revealed by either revol ving mirror or rotating disk. The first method that occurred to me was to attach a delicate metallic point to a vibrating tuningfork, and to send the discharge from this point through lampblackened paper to a revolving metallic cylinder on which the paper was stretched. We can to some extent analyze the electric discharge, in these conditions, from the series of perforations left in the paper in the trail of the vibrating fork. This method, though beautiful as an illustration, is useless as a means of investigation; for the metal cylinder, the paper, and the fork form a species of Leyden jar, which is always in the circuit of the particular discharge whose nature you would investigate. The above method, though original with me, cannot be claimed as my own, having recently found that it was devised by Donders, and has been used in an investigation by Nyland T. To get rid of inductive action in the registering apparatus, I devised the following method:-A cylinder is covered with thin printing-paper; and the latter is well blackened by rotating the cylinder over burning camphor. The paper is then removed from the cylinder, and cut into disks about 15 centims. in diameter. When one of these disks is re
* From Silliman's American Journal for December 1874.
"Ueber die electrische Flaschenentladung,” Pogg. Ann. vol. cxvi.
§ Journal de Physique, vol. ii. p. 252.
Onderzoekingen gedaan in het Physiologisch Laboratorium der Utrechtsche Hoogeschool, 1868-69.
¶ Archives Néerlandaises des Sciences exactes et naturelles, vol. v. p. 292.