K. Pearson 
301 
Thus every part of (xli) can be readily found numerically by calculating Fj, F,, 
and their differentials, as given by (xlvi), for any value of Hq. Of course for 
accurate calculations we must go to 9 to 12 places of decimals and the ordinary 
tables of logarithms are of no service. For the numerical illustration now to be 
given a large Brunsviga calculator was used, and exponentials and reciprocals 
found so as to be true to twelve places of figures. The calculations were of course 
long and laborious, and I owe an immense amount of solid help to my former 
colleague, Mr Leslie Bram ley- Moore, for independent arithmetic and for verifica- 
tion of my own calculations. 
I selected the mortality table given in the Text-book for actuaries, but I 
limited the range I of life to the 60 years from 25 to 85 inclusive. I did this 
because the data after 85 is really sparse, because the material before 25 begins to 
diverge from Makeham's law, and lastly because as a mei'e illustration of method 
it is a sufficiently big task to calculate area and moments for a system of 61 
ordinates. Using z„ to Zg^ at equal distances I could apply Weddle's Quadrature 
Rule (see (^) of p. 275), in which I have great confidence for a fairly smooth 
curve like that given by the mortality table. The ordinates, of course, are 
z = L = log Ix for the area, wz for the first, x^z for the second, and x^z for the 
third moment, where attention must be paid to the sign of x. 
The following values were found : 
A = 221-843,235 
^/u, = - 275-103,222 
AfM,= 64,46-!<-355,986 
= _ 102,062-316,564. 
Whence 
a,= 3-697,387,250, a,= 3-581,353,110,3 
a,^- -917,010,740, = - 1-000,384,670,148. 
These lead to 
/3= -718,529,308,595. 
By a rougher quadrature process I got for /3 for the whole range from 20 to 90 
/3=: -801,086,783. 
The value of /3 as found by (xl) from the n which corresponds to Messrs King and 
Hardy's c is : 
/3 = -804,162,5, 
but their range is from 17 to 88 years. 
The next point is the solution of (xl). Working in the manner indicated with 
yS = -718,529,309, 
and calculating necessary terms to 12 places of decimals, we found the following 
series of approximations to the value of n 
2-7, 2-8, 2-807,68, 2-807,312, 2-807,346,8, 2-807,343,62, 
2-807,343,87 and 2-807,343,873. 
