300 On the Systematic Fitting of Curves 
From (xxxiv) and (xxx) we have: 
^f2siuh?i 6 cosh n 6sinh?i) , 
0(2 - Ho = i 5 f- Z (XXXVIII). 
or : 
Eliminating G between (xxxvii) and (xxxviii) and writing ^ for (oj — a^)l(oL^ — oio), 
a constant to be found from the moments of the mortality table, we have after 
some reductions : 
12/^^ + 3/3n + 30• 
Or, substituting for the hyperbolic tangent : 
,n ^ + 2) + 3 (/g + 4) + 3 (/3 + 10) + 30 
^ (/3 - 2) _ 3 (;8 - 4) «^ + 3 (/3 - 10) n + 30 ^' 
This equation will give n to any degree of approximation required. Then (xxxviii) 
gives G, and (xxxii) gives S, and (xxx) K, whence all the constants of the solution 
can be found. 
To solve (xl) an approximate value of n = is easily found ; for c = e^'"' has 
been found from previous experience to have a logarithm very nearly '04. Start- 
ing from this value of ?io successive approximations can be obtained by Newton's 
method. Thus, put iio + li in (xl) and neglect A'-*, we find if e-"- = NID, where N 
stands for numerator and D for denominator : 

fdN^ 
\ dn J 
Writing 
Fi = 2n«+3/3/i2-|-30?i (xlii), 
= /3n^ + 12^2 + 3/3n + 30 (xliii), 
we have 
F 4- F 
'"==t~t; 
D=F2-Fi, iV = F2+Fi 
d^^dY^_d_Y^ dN _dY^ dY, 
dn dn dn ' dn dn dn 
where 
dY dV 
~ = 6/1^ + 6/3n + 30, ^ = B^n^ + 24h + 3/3 (xlvi). 
