Karl Pearson and J. F. Tocher 
173 
that the distributions of quantities like are given with sufficient accuracy by 
the normal curve then the probability of the whole result observed is given by 
1 ^id's-P,a's)^~ 
n = Product 
1 {ds -PsO-bY 
e 2 a\p,q. 
F...(xii), 
where V is the continued product of the difJerentials of the dl/s and the d'sS, 
and we require to make this a maximum for the variation of all quantities like p^. 
In other words we have u equations found by differentiating the above expression 
for Pi, P2, ■■■ Ps' ■■■ Pn (of course = 1 — p^) to determine the best values of these 
quantities. Taking the logarithmic differential of the above product with regard 
to Ps and equating it to zero, we find since Sq^ = - Sp, : 
0 = 
Hence : 
Ps 
Psqs 
2\ a,p,^q,^ 
a'sPs^qs^ 
iPs 
d, + d's ,1 qs-Ps { {ds - Ps^sf ^ {d's - Ps^' 
-7^ + 
'P' a, + a's ' 2 (a, + a',) q,p 
Now assuming to a first approximation p^ 
a 
d, + d' 
2p,qX..\xm). 
approximation p^ 
d, + d\ 
^d, d\\^ 
J we find to a second 
Ps 
Calling as before 
we have 
as + a'A 2 f^_ds + d^^ ■ 
\ \ a, + a J/ 
Ps=- 
ds + d\ 
of 
1 - 2 
IQs 
dg -\- d' „ — So 
Hence we may write p^ 
this amounts to altering the deaths by 
ds + d' 
= 1 
2 ^^') 
+ 
which is usually a very small number, or we may write 
Ps = 
+ d\ 
2 
where the factor 
/,= 1-(1-1Q/) 
will only differ slightly from unity. 
ds + d', a, + a\ 
L 2 
d, + d,' , a, + a' . 
