106 BELL SYSTEM TECHNICAL JOURNAL 



t of the (fc + l)st degree. R k may then be determined by (15) and 

 Qk+i by (14). From these results, the next set of coefficients may be 

 found, and so on. The values of the coefficients for k — 2 (L 2 , Mi, Ri, 

 Q 2 ) have been found, and equations (8)-(10) are valid for the par- 

 ticular values of n utilized in the above method. Hence the next set 

 of coefficients (L 3 , Mi, R 2 , Q 3 ) may be found, and in the same way, as 

 many more as are desired. The detailed work of the first step is 

 indicated below: 



Substituting k = 2 in (16), 



U = M*+IL4+1{R 1 +L z -M{)t+}L&+}R } U. (17) 



Substituting in (17) the values known from (11), 



U = L 3 t+(-t 4 -2f- + 2)/3Q + M 2 . (18) 



Let L 3 be a polynomial of the form (Ast 3 -\-A2t 2 -\-Ait-\-A ) and sub- 

 stitute in (18). Upon equating coefficients of like powers of t, we 

 find that A 3 = 1/36, A 2 = 0, 4 1 = 5/36, A = 0, and M 2 -l/l2. JR a 

 is then obtained by substituting these values in (15) and Q3 from (14). 

 The results are as follows: 



L 3 = (J 3 -50/36, i? 2 =(/ 2 -5)/36, | 



M 2 = l/12, Q 3 = (P- 70/36. i 



The actual work of computing these coefficients has been performed 

 up to and including £ = 11 (Ln, M w , Rio, Qn)- These results are pre- 

 sented in the attached tables: Q n in I, L n in III, L» in IV, R n in V, 

 and M n in VI. From this information the next coefficient in the 

 series (1), Q12, can be computed by the method outlined above. 



It maybe pointed out in conclusion that the expansion of M presented 

 in Table VI is the asymptotic series obtained in Stirling's expansion 

 of T(c), as is to be expected from equations (5). This in itself con- 

 stitutes a partial check upon the determination of the coefficients. 



Additional Properties of the Curves 



At the probability P = 0.5, the difference (c — a) = 1/3, approxi- 

 mately. 4 Discrepancies are so small as not to be positively dis- 



4 This recalls the approximate rule that the median lines one-third of the distance 

 from the mean towards the mode. (Yule, Theory of Statistics, 1911, p. 121.) But 

 in the Poisson exponential the median never lies between the mean and the mode; 

 the median occurs at the first integer above or below the mean, whichever integer 

 corresponds to the c curve cutting P = 0.5 next below the mean, while the mode is 

 always at the first integer less than the mean. For the range of cases having a given 

 mode, however, the mean and the median are, on the average, greater than the mode 

 by 5 and approximately J, respectively; thus the median must line one-third of 

 the distance from the mean towards the mode in the case of the corresponding hetero- 

 geneous samplings. 



