288 On the Systematic Fitting of Curves 
By actual solution of tliese equations I find* : 
415?;i — Ifibij + 55»3 — d>u 
72 
755»i - 493?t, + 191»3 - 33», 
48 
a. e- = — 
72, 
05V 
119».i-97?i, + 47»3-9n, 
6 
125/ii — 115?)2 + 65?i3 — 15?i4 
12 
74, 
= 75 
.(xv). 
These values of 7 can be easily calculated. Now let hje = p. Then if we take h 
equal unity as usual, 1/p will measure the fraction e is of h. Of course very 
usually p = 1, but this is not necessary ; thus in certain diseases the frequency of 
cases in each of the first five years of life is often recorded, but later only in five- 
year periods. Making these substitutions we may write our final results for the 
moments : 
720 "^ 30240 
' - ' _ JL _ Phi 4. p*y± _ 
12 120 "^3024 
p-72 
+ 
P'74 
240 126 1440 
+ 
48"' 288 "^3168 
,5,7, 31 
P 72 , P'74 
80 ^ 528 
(xvi). 
(xvi) and (xv) form the solution of the problem. It is of course sometimes more 
convenient to use (xi) directly. In any individual case we must first find the i'"s — 
generally only v^' to v^' are needful — about one end of the range. Then we 
calculate the 7's and p and so determine the /Lt"s by (xvi). Then by (ix) we 
find the values of the fi's transferred to the centroid. Of course the process 
will be much simplified if there be high contact at both ends, for then all the 
7's may be put zero. The methods here indicated seem of such importance 
that it is desirable to fully illustrate them in various special examples, each of 
which has been selected with a view of illustrating some peculiar point or 
difficulty. My first two examples are illustrations of the fitting of skew frequency 
curves ; my third deals with the fitting of sine curves when only a portion of 
* The rfader must remember that iij, Hj, "3 and 11^ are not the total frequencies in the first four 
elements, but the proportional frequencies. 
