Karl Pearson 
29 
Hence from (xvi) : 
iVeos /3 t sin @ t cos fi s sin /3, 
t 
stN 2 sin 2 /?,, sin 2 fft 
or 
R ptP = — tan&tan/3 f (xxi), 
and agrees entirely with the value of r (s used in my Phil. Mag. paper just 
referred to. 
In other words the determinant R and its minors R ss and R st will take precisely 
the same forms as in that paper provided we 
(i) replace Eq. (xi) of that paper by our result from (xix) above, i.e. 
and 
N 
r) s = cot 2 p s = ~ ~ - 1 (xxn), 
p s s 2 
(ii) remember that we must reduce the total number of our variants 
Po > Pi > • ■ • Pti 
by the two relations S(p s ) = m, 8(sp s ) = n (xxiii), 
where m is the total number of houses and n of cases. That is to say our total 
number of frequency groups, which are not wholly dependent but are correlated, is 
u + l- 2 = u-l. 
We select as our dependent variates, which are fixed as soon as the others are 
kuown, p 0 and p u . Thus the value of the auxiliary determinant J, p. 162 of my 
Phil. Mag. memoir, is : 
J 
- Vl 1 1 ... 1 
1 -7) 2 1 ... 1 
1 1 - Vs ... 1 
1 1 1 ... ~Vu-l 
if x = (1 + Vl ) (i + v .j (i +%)...( i + ^-0. 
we find as in that memoir : 
1 1 1 
+ Vi 1 + V". 
X 
1 1 
1 + Vu-i 
.(xxiv), 
..(xxv), 
1 + Vs-l 1 + Vs+l 1 + Vu~i 
.(xxvi). 
