POISSON'S PROB.lBIUrV SUMMATION 619 



siniiUir to examples (b6)-(b9), except that the Hmits of the number 

 of calls per sample were 515±2o. Distribution (c7) was obtained 

 for the number of cutoffs, (c8) for the number of double connections, 

 (c9) for the number of calls for the wrong number, and (clO) for the 

 number of connections to the wrong number. The next distribution 

 (ell) was obtained from a count of the number of party-line sub- 

 scribers listed per page of a large telephone directory and the last 

 distribution of the group (cl2) from a count of the number of ad- 

 vertisements in the "lost and found" column of The New York Times 

 on each of the week-days from January 1, 1924 to August 31, 1924. 



The fourth group contains only five examples, three of which are 

 new. The first two of these present the same material used for ex- 

 ample (b4) differently arranged. The 50,000 logarithms used are 

 divided into 100 groups of 500 logarithms each for example (dl), and 

 into 50 groups of 1,000 logarithms each for example (d2). The third 

 (d3) is the distribution of the number of comets observed per year 

 for the years 1789 to 1888 inclusive.^" The other two distributions 

 have been given by Perrin as typical of the data obtained when, in 

 order to determine the density of the particles of an emulsion at a 

 given depth, he restricted his field of vision to a tiny part of that 

 layer, small enough so that the average number of particles visible 

 was only one or two, and then made a large number of observations 

 of the number of particles in that space at regular intervals.'-^ 



As was to be expected, these observed distributions have not only 

 irregularities due to finite sampling but also in some cases what appear 

 to be definite trends away from the corresponding Poisson distri- 

 butions. In some cases there is an explanation ready at hand. For 

 example, in series (bS), which gives the number of articles lost in the 

 Telephone and Telegraph Building, the average number of articles 

 lost per day might be expected to increase as the population of the 

 building increased in this period following the completion of an addi- 

 tion, and the observed slant to the right is what would be expected. 

 Also in series (dS), which gives the number of comets observed per 

 year, the average would naturally increase steadily as a result of the 

 continual improvement of telescopes and other instruments from 

 1789 to 1888. The curve toward the left in examples (c3) and (c5) 

 might also be predicted because of the fact that the number of calls 

 which could possibly be made in five minutes from a group of two 

 telephones is certainly finite and probably rather small, and in ex- 

 amples (d4) and (d5) because it is difficult to judge by eye the number 



20 "Handbook of Astronomy," by G. F. Chambers, 4th ed., Oxford, 1889. 



" "Brownian Movement and Molecular Reality," by Jean Perrin, London, 1910. 



