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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 consideringonly 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)=-x(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.
Compound series. 1 |10| 46120 210 251 208 120 46 10 10 10
10 45 120 210 252210120 10 10
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*. IN N 1842 Professor Joseph Henryt 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 g 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 revolving 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 conditious, 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 Nylands. 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. + Proc. Amer. Phil. Soc.
I“ Ueber die electrische Flaschenentladung," Pogg. Ann. vol. cxvi. p
132. § Journal de Physique, vol. ii. p. 252. ! Onderzoekingen gedaan in het Physiologisch Laboratorium der Utrechtsche Hoogeschool, 1868-69.
1 Archives Néerlandaises des Sciences exactes et naturelles, vol. v. p. 292.
volved about twenty times per second, it is rendered very fat by centrifugal action. It can then be brought between points or balls, even when the latter are separated by no more than
millim. When in this position, the discharge between the points or balls perforates the disk and leaves a permanent record of its character, of the duration of the whole discharge, and of the intervals separating its constituent flashes and sparks. To obtain the time of rotation of the disk, I use the method invented by Young in 1807 (see his “Natural Philosophy,' vol. i. p. 191); that is, I present momentarily to the rotating disk a delicate point which is attached to a vibrating tuningfork. The number of vibrations per second of this fork has been determined to the last degree of precision by means of a breakcircuit clock, which sends at each second a spark from an inductorium through the fork's sinuous trace on blackened paper covering a revolving cylinder. The axis of the sinuous line on the disk is traced with a needle point; and then, on drawing radii through symmetrical intersections of this axis on the sinuous line, we divide the disk off into known fractions of time. The disk is now removed from the rotating apparatus, and the carbon is fixed by floating the disk for a moment on thin spiritvarnish. When the disk is dry and flat it is centred on a divided circle provided with a low-power reading-microscope ; and the duration of the whole discharge, and the intervals separating its components, can be determined to the color of a second.
Many results have been obtained with this apparatus. I defer their publication until I have carefully examined them and bave extended this research with the study not only of the discharge of the inductorium, but also of the frictional machine, of the Leyden jar, and of the Holtz machine, under every condidion of charged surface and of striking-distance, and when the current is flowing freely over a conductor and when it is doing work. I here present, merely as examples of the value of the method, the results I have obtained in three conditions of experiment.
1. Discharge of large inductorium* between platinum points one millim. apart. No jar in the circuit.
The platinum electrodes were neatly rounded and formed on wire millim. in diameter. After the discharge through the rotating disk, nothing was visible on it except a short curve formed of minute, thickly set white dots ; but on holding the disk between the eye and the light, it was found to be perforated with thirty-three clean round holes with the carbon undis
• The striking-distance of this coil between brass points was 45 centims.
turbed around their edges. The portion of the discharge which makes these holes lasts 3 second ; and the holes are separated by intervals which gradually decrease in size toward the end of the discharge, so that the last spark-holes are separated about one half of the distance which separates the holes made at the beginning of the discharge. The average interval between the spark-boles is 7o second. After this portion of the discharge has passed there is a period of quiescence lasting about too second; then follows a shower of minute sparks, which forms the short dotted line above spoken of. This spark-shower lasts sto of a second, and is formed of 30 sparks; bence the average interval separating these sparks is gobo second. The intervals separating these sparks, however, are not uniform, but are smaller in the middle of the spark-shower than at the beginning and at the end of this phenomenon. The spark-shower, indeed, is a miniature of the phenomenon obtained when a Leyden jar is placed in the circuit of the coil, and which is described below. The above numbers were determined as the average measures on six disks. It is bere to be remarked that all the discharges studied in this paper were made by suddenly depressing the platinum-faced break” of the primary circuit, and the break was held in this position until the disk had been removed from between the points or balls.
2. Discharge of large inductorium between platinum points one millim. apart, with a Leyden jar of 242 sq. centims. connected with the terminals of the secondary coil.
After this discharge through the disk a very remarkable appearance is presented, the full description of which I reserve for a more extended paper. The discharge in its path around the disk dissipates little circles of carbon. There are 91 of these circles, each perforated by 4, 3, 2, or 1 hole. I shall here have to adopt a new nomenclature for the description of this complex phenomenon. I call the whole act of discharge of the coil the discharge. Those separate actions which form the little circles by the dissipation of the carbon I denominate flashes;
and the perforations in these circles I call sparks. The discharge in the above experiment lasts of a second. The flashes at the beginning of the discharge are separated by intervals averaging 55 second up to about the tenth flash; after this the intervals of the flashes rapidly close up, so that during the fourth fifth of the discharge they follow at each gosa second. During the last fifth of the discharge the intervals between the flashes gradually increase, and the last flash is separated from its predecessor by Tour of a second.
3. Discharge of large inductorium between brass balls, one centim. in diameter, separated one millim., with a Leyden jar of
Phil. Mag. S. 4. Vol. 49. No. 322. Jan. 1875. E