264 On 'Best' Valves of Constants in Frequency Distributions 
a number of equations of type 
/w/ (In, 
= 0 
•(I). 
These equations will generally be far too involved to be directly solved. Accord- 
ingly we proceed thus : We suppose that the values of the frequency constants 
given by the method of moments are good starting-points, and we put, if /denote 
the moment value of a frequency constant, / = / + A/". Accordingly if there be 
a number fi,/^, ...fg of independent frequency constants, we shall have a series 
of equations to find Afi, A/2, ... A/, of the type 
d^rig 
2 
+ .. 
+ s 
df.df^ 
d'^n. 
diig dn, 
IK ¥2 
A/2 
dvg dn 
A/. 
.(2 a), 
where a square bracket round the differential coefficients signifies that the frequency 
constants /^ , /g, . . ./, therein are to be given their moment values fx, ft, ■■■fg- These 
values are of course also to be used in n^. 
Since S{Tis)= N, it is clear that 
S 
dn^ 
W2 
s 
d^iis 
dfxdf^ 
= etc. 
0. 
Accordingly the above equations may be reduced to the type 
0 = S 
— ?i. 
Ml. 
+ s 
+ s 
d'^n, 
d'^Yi 
2n, 
+ 
dng\^ 
dn^ dn^ 
A/i 
A/, 
2n. 
dfidfg] L#i 'dfg_ 
dn^ dn. 
A/:.. 
It might reasonably be anticipated that terms involving the product of A/ 
and (TOj^ — ng^)jn^ could be neglected in the first place and accordingly that we 
should have as approximate type 
IS 
dn. 
dfi) } 
A/i 
g J '"s^ dn^ dn, 
\n,^ L^/i df,_ 
A/. 
+ 
S 
dn, dn. 
A/, 
.(2 6), 
but this approximation has not in every case numerically justified itself, and thus 
it cannot be invariably used as more than a reasonable starting-off point. 
(2) Fit of a Normal Curve. 
N 
Differentiating 
1U = 
V2 
e 2 0-2 
dx, 
