LETTERS TO THE EDITOR 175 



really is able to test the graduating ability of the Gram series as 

 adequately as the more powerful, although far more complicated, 

 "error critique" of Thiele. From Pearson's derivation it appears that 

 his test is not able to take care of elementary errors beyond the first or 

 second order, while it is necessary to consider the formation of ele- 

 mentary errors of the third order in the third approximation of the 

 Gram series. In some work I have been doing in the way of con- 

 struction of compound mortality curves, I have at least found that the 

 Pearson test is inadequate, if actually not misleading, because it 

 apparently fails to measure the effect of the elementary errors of higher 

 order which enter into the formation of such compound mortality 

 curves. 



There exists, however, a very simple relationship between the 

 coefficients cs and Ce, in the third approximation. We have, namely, 

 with a fair approach to exactitude, the simple relation: c% = ^cs'. It 

 is therefore not necessary to calculate the semi-invariants or moments 

 of higher orders than those of the fourth order, since we shall have 



F{X) = Coifoiz) -j- C,,<p-i{z) + CiiPi{z) -f ^f3-<p6(2) 



as a third approximation. 



As an illustration of the above formula, we may select the expansion 

 of the point binomial (0.1 -f 0.9) i"". We have here, according to the 

 formulas on pages 263-264 of my "Mathematical Theory of Proba- 

 bilities": 



5 = 100, p = 0.1, 5 = 0.9 

 and 



X, = if = 5^ = 10, cr=^g^ = ?,, C3= -0.0444, C4 = 0.0021 



and 



C6 = W = 0.0010, 

 or 



(0.1 + 0.9)10" = i[^^(2) _ 0.0445.^3(2) + 0.0021.^4(2) + 0.0010.^6(2)], 

 where 



.^0(2) =-^e-^^--2 



z = {x - 10) : 3. 



A comparison between the above approximation and the true 

 expansion of the point binomial (0.1 + 0.9)"*" to 4 decimals is given in 

 the following table. 



