302 On the Systematic Fitting of Curves 
This value is correct to the last figure or we have 
11 = 2-807,343,873. 
Hence by using the exponential theorem : 
g»l _ g2 y g-807 g'000,343,873 
= 16-565,858,706,268. 
Similarly 
= -060,365,] 17,060. 
Thus 
sinh?2- 8-252,746,794,593, 
cosh 71 = 8-313,111,911,675. 
Hence we determine from (xxxviii) : 
(? = - -064,875,005,350, 
and from (xxxii) : 
^ = -•002,866,074,767. 
Finally from (xxx) 
Z = 3-888,100,258. 
Calculating : c = e""'' we have : 
c = 1098,096,393,273, 
which I believe is true to the last figure. 
The value of c as found by Messrs King and Hardy is : 
0 = 1 095,612,204. 
The difference is partly due to difference of range, partly due to method of 
calculation. 
Thus finally we obtain for L^, the logarithm of the number of survivors of 
age ao + X years : 
= 3-888,100,258 -xx -002,866,074,767 
- -064,875,005,350 (1-098,096,393,273)=^. 
In comparing our formula with others of a like kind, it must be remembered 
that our x is measured from 55 years as origin. For use it may be noted that the 
reciprocal of c is 
- = -910,666,865,065, 
c 
which will be wanted when x is negative. 
Clearly c and - are wanted to many places of figures as they have to be raised 
c 
to high powers. The values of Lx for x— — 30 to + 30 were found by repeated 
multiplication with a Brunsviga, so that in no part of the work has a table of 
logarithms been used. 
I give here a table of the observations and calculated values and add Messrs 
King and Hardy's results*, asking the reader, however, to remember that these 
* Text-hook of Institute of Actuaries, Part ii. p. 88. 
