Elkanok Pairman and Karl Pearson 
233 
e.g. deaths in infancy, disease incidence in infancy, wages, incomes or house-valna- 
tions, we have details for the ends of the range on different subranges to those for 
the bulk of the curve. These modihed subranges will be termed /;„ and h^,, and 
when either of them is less than It, we shall get more accurate corrective terms if 
they are- used instead of the frequencies on subranges h. At the same time it must 
be remembered that in calculating the value of the chordal area terms, suthcient 
of these //j and hp subrange frequencies must be clubbed together to give sub- 
frequencies on ranges h. 
Let the frequencies on the first five subranges or h from x = x^ be A^n/, Nn.!, 
Nthj, Nn^', Nn^', so that «/, n-J, /(/, Vr' are proportional frequencies, then 
a., a 
1 ! 2 
3 ! ^ 4 ! 
«5 
5"! 
or 
(^1 «2 «3 
1 ! 2 ! 3 ! 
4 ! ^ 5 ! 
Similarly 
— Ill - = 
^2 + ^2^ + ^2«+"i2M^'2 
^ 1!2!3!4!5! 
- - - n., = 3 + 3- + 3-^ + 4-, 3^ + o' 
— — lU — II, = 
Fi^+ri^ +3^? +4^^^ 
ft'.-. 
5"! 
r!^+2v5' + 3-!-5-'+4l'5' + ,5-,-^ 
Solving these equations we find 
«1 = 
_ 1 
0 0 
{137n/- 
163h./ + 137«;/- 
637;,; + r2»,;j,\ 
a.,= 
1 
1 2 
f 45)V- 
109/?/+ 105«,; - 
a-. - 
_ 1 
4 
54h.,' + (347;,;/ — 
34«;+ 7/;/;, 
((-4 = 
{ 3»/- 
11/;,/+ 15«;/- 
9»;+ 2/;,'|, 
«s = 
! «/- 
47// + GihJ - 
4"/+ ih'\.) 
Similarly we find for the b coefficients 
^'i = + ik 1 137?i'^, - 1637/p_i + 137h,'^j_.j — (i3/i'^,_;j + 1 '2ii'p_^] 
6, = - [ 4<5,i'p - 109/i'p_i + 105?i.p_o - 5l7;/^,_3 + 10n\,_,} 
h:s = + l{ I7n'p— 5in'p^i+ 64?i'^,_., — 34//^,_3 + 7ii'p^^} 
64=- j Sn'p- 11 + loii'p_.,- 9/i'^,_3+ 2h'^,_,; 
n'p- 4/i,'^,_i + 
67i'^j_o 
4/1^-.! + 
.(iv) 
where Nn'p-4, Nn'p_.i, Nn'p^^^ Nn'p^-^, Nn'p are the five successive frequencies ad- 
jacent to the terminal of the subranges hp or li as the case may be. 
Biometrika xii • 16 
