238 ABSTRACTS - ABSTRACTS - KURZE MITTEILUNGEN KURZE MITTEILUNGEN OLOF TEDIN: Sma II sam pie s f a Poi s son s e r i e s In his »Statlstical methods for research workers FISHER presents a method by which it is possible to ascertain whether the variation within small samples agrees with the theoretical distribution of a Poisson series For each S(x -X)! x - sample an index of dispersion is calculated by the formula and, if the samples belong to a true Poisson series, the indices are distributed as l' for n-I degrees of freedem, if n = the number of variates in the sample For a single sample the test is of small value, but if a large number of samples have been obtained under comparable conditions the distribution of the indices may be used to test the reliability of the methods used If the nature of the material studied is such that the variation should follow a Poisson series but the distribution of the indices does not agree with l2, it may be concluded that the method of sampling or of analysis is at fault FISHER (I c., sect 16) recommends the method for, e g., counts of bacterial colonies in dilution plates where 'we may have only parallel plates, bearing perhaps 200 colonies apiece', The present author has tried to apply the method to material where by routine a large number of different but comparable biological populations are sampled, the n of the samples varying from upwards but usually being less than 10 In this material the m of the presumed Poisson series is usually low, rarely above and often less than On such material the method of FISHER cannot be used indiscriminately The relative frequencies of the variate values 0, 1, 2, etc for a few low values of m are given below Relative frequencies of variate values 0123456789 m 0,1 0,9048 0,0905 0,0045 0,0002 0,5 0,6065 0,3033 0,0758 0,0126 0,0016 0,0002 1,0 0,3679 0,3679 0,1839 0,0613 0,0153 0,0031 0,0005 0,0001 2,0 0,1353 0,2707 0,2707 0,1805 0,0902 0,0361 0,0120 0,0034 = 0,0009 0,0002 When m is as low as 0,1 and n 2, no less than (0,9048)2 or 81,9 % of all samples will show the values 0,0 Even if n 10, (0,9048) 10 or 37 % of the samples will show only the value O For this low value of m there is no agreement with the distribution of X2 unless, perhaps, when n is large enough to allow a direct comparison with a Poisson series For any value of m and any value of n the relative frequencies of samples with the index of dispersion may be obtained by summing the nth powers of the relative frequencies of the variates 0, 1, 2, etc For three values of m tabulated above these frequencies are as follows: = = Relative frequencies of dispersion indices = when n is equal to m 10 0,5 0,4658 0,2514 0,1438 0,0847 0,0506 0,0304 0,0184 0,0111 0,0067 1,0 0,3084 0,1060 0,0377 0,0136 0,0048 0,0018 0,0006 0,0002 2,0 0,2068 0,0486 0,0123 0,0031 0,0008 0,0002 ABSTRACTS - 239 KUnZE MITTEILUNGEN This simple test shows that the frequency of low values of the index is altogether too large, when m is small, if n is not sufficiently large, but it also indicates that with increasing values of m the frequencies of indices = Q drop off rapidly In a few cases the total distribution of the index has been calculated, with the result that when the value is in excess the rest of the distribution also differs widely from that of X2 • It would be of considerable value to know at which combinations of m and n the distribution of the indices reaches so fair an agreement with that of x2 that the test of FISHER may be used with reasonable safety The author has made one attempt in this direction by calculating the distribution when m and n The 4th power of the decanomial showing variate frequencies of at least 0,0001 when m includes 715 terms Of these 244 have frequency values of at least 0,0001, the sum of these frequencies is 0,9915 and the remaining 0,0085 is obviously the sum of frequencies below 0,00005 The index value has been determined for each of the 244 terms mentioned, and these indices have been grouped in accordance with the X2 table of FISHER Multiplying the frequencies of each group so as to make their sum 1, the values given below have been obtained The corresponding frequencies of the X2 distribution are given for comparison ° = = = = Frequencies of dispersion indices between the values index 0,352 0,584 1,005 1,424 2,366 3,665 4,642 6,251 7,815 0,0377 0,0367 0,1405 0,0476 0,2423 0,1820 0,1l81 0,1202 0,0384 0,0361 0,05 0,05 0,10 0,10 0,20 0,20 0,10 0,10 0,05 0,05 The agreement is fair but not good The fraction 0,0123 of the indices is 0, whereas only 0,01 of X2 is less than 0,1l5 In spite of this slight excess in the very lowest class there are too few index values below the limit 0,352 that indicate the 0,05 point of the X2 distribution The distribution of the indices is markedly discontinuous, and if the classes are narrow this becomes very apparent Small changes in the value of m, however, may change the frequencies in neighbouring classes considerably This indicates the possibility that, whereas the distribution of indices disagrees with that of 1.2 when m and n are small, the sum of indices, obtained from a large number of samples from biological populations with varying true values of m, may agree with the X' value appropriate to the accumulated number of degrees of freedom As a matter of fact, in a material of 44 samples with altogether 191 degrees of freedom the sum of the indices was 192,50, in close agreement with the theoretical value of X2 • In 10 samples with n 2, the sum of indices was 4,48, in samples with n 3, the sum of indices was 16,91, etc., the sum of indices in all cases falling within reasonable limits The difficulty offered by samples all the variates of which are should be pointed out It is impossible to know whether such samples represent a population with the true value of m = 0, or whether they are random samples of populations with higher values of m If they are all included in a comprehensive test, they may wrongly increase the frequency of low values of the index; if they are excluded, the result will most probably be a deficiency in such low values = = = 240 ABSTRACTS - KURZE MITTEILUNGEN Not being a mathematician, the author has made no further attempts to solve the problem here barely outlined It would certainly be of great value if a modification adequate to small values of m and n could be given of the test of FISHER The Poisson series seems to be comparatively little used in genetical research, but if its true possibilities became more generally known, there is no doubt that it might be used in many instances where less adequate or no statistical methods at all are now employed LABS EHRENBERG: The s hap e f the s pin die a t met a p has e is d i t ion e d b y the s hap e fit s mol e cui e s In a series of experiments not yet completed the shape of the spindle at metaphase has been studied on the basis of statistical methods It was found that the Curvature of the outline of the spindle as it appears when the spindle is studied at right angles to the direction of the nuclear division was a measurable property The material consisted of fixed and stained rootmeristems of Salix [raqili« X alba The investigations have so far demonstrated: (1) The curvature of the outline of the spindle increases with a fall of temperature within the range 30° to 10° C (2) The curvature of the outline of the spindle is greater in the cells of the plerome than in those of the periblem Simultaneously it was observed that the hydrogen ion activity in the plerome cells was about 0,1 of that of the periblern cells (3) Treatment of the roots with small concentrations of substances which in higher concentrations have a colchicine effect induces a greater curvature of the outline of the spindle So far it has been demonstrated that colchicine and ethyl alcohol have this property Pure hydrocarbons likewise show a tendency in this direction It can be demonstrated that the greater curvature of the outline of the spindle is most likely due to a contraction and thickening of its constituents, which consist of oblong polypeptide chains In connection with this observation the chemical powers which might induce the contraction of the protein molecules may be discussed The results of the investigation suggest that the formation and breaking up of hydrogen bonds whithin and between the polypeptide chains have an important function in this process These bonds most likely arise between the> CO and> NH groups of the main polypeptide chains as well as between the oxygen and nitrogen carrying sidechains Other powers, however, may also playa part in the contraction of the molecules Among these the attraction between the lipophil groups within the molecules, a power which according to OSTERGREN'S hypothesis might explain the colchicine-mitosis, seems to play the most important part Although conclusive evidence has so far not been furnished, several factors suggest that the speed of the anaphase movements of the chromosomes is lowest in the spindle that shows the greater degree of curvature in its outline At the present stage of the investigations it may, therefore, be assumed that the breaking up and reformation of hydrogen bonds play some part in the process of mitosis, a hypothesis which needs further verification Institute of Forest Tree Breeding, Ekebo, Kallstorp Sweden, October, 1914 ... are excluded, the result will most probably be a deficiency in such low values = = = 240 ABSTRACTS - KURZE MITTEILUNGEN Not being a mathematician, the author has made no further attempts to solve.. .ABSTRACTS - 239 KUnZE MITTEILUNGEN This simple test shows that the frequency of low values of the index is