154 On the Theories of Multiple and Partial Contingency 
Multiply by and add the series, and we find 
- A'2 = AiA'i + A2A2 + ... + (xxxii). 
Multiply by and add the series, and we find, by aid of (x), 
- A'l = Ai + A2 cos (12) + A3 cos (13) + ... + Ag cos (1^). 
Similarly : 
A2 = Ai cos (21) 
+ A3 cos (23) + ... + Ag cos (2g), 
Kt = Aj cos {t\) + A2 cos (i2) 
+ A( + ... + Ag COS {tq), 
- Ag = Ai cos (gl) + A2 cos (^2) + ... + ... + ... + A,. 
These are q equations to find A^, A2, ... A,, and we can then substitute in (xxxii) 
to find the required P^. 
Now consider the determinant 
R 
1, 
K2, 
. K, 
1, 
cos (12), 
cos (13), . 
. cos (Iq) 
A2, 
cos (21), 
1, 
cos (23), . 
. cos (2^) 
cos (31), 
cos (32), 
1, 
. cos (Sg) 
Kg, 
cos {ql), 
cos {q2), 
cos (^3), . 
1 
..(xxxiii), 
and let us call the first row and the first column the 0 row and 0 column, then 
clearly 
— A( = — i?o'/-^00 J 
where R^t is the minor of the sth row and tth column, and 
Aji^oi + K2R02 + ••. + KfRot + ••■ + KgR^g 
- A^ 
00 
l-K^=RIR,„ 
or A2 = 1 - P/i2o(, (xxxiv). 
If we call A the minor > have: 
Rot = - AiAii - K2^2t - •■• - KtAtt - ■■■■ 
Thus A2 = (a,2 + 2S {liji, (xxxv). 
From this we deduce that the probability of a sample which gives = Xo^ 
with q linear conditions must be obtained from 
P = 
J 0 
.(xxxvi) 
dx 
