KiRSTiNE Smith 
21 
Hence 
^+iA = . 2"-! if -If {2^- . 2"'-i> '^-+2) . ^,.,n . [(2(7 + 2p - l)^ 
p+iA = {If . 2*'-i (2^ - 1)2 . ^)}^ 2" . 
which agrees with (24). 
(4) Next we have to evaluate the minors of „A necessary for calculating the 
numerators in (23). For this purpose we only need the minors pA,^,,, but to carry 
(2 
through the proof by induction pA^,^ for any values of s and r is needed. 
Q 
For 3A2 3 we directly find, 
1 -22.2 
,A, 
^"2.3 - 1) (25 + 1) {2q + 3) {2q + 5) 
1 22 . 22 
and oA, = 
^ (2^- 1) (2(7 + 3)2(2^ + 7) • 
these both agree with the following formula which will be proved by induction, 
^A,,, = (- l)'-+^iS,_,.._, . i8,_i,,._i {1^-2 . 2.-3 (^ - 3)2 {jy - 2)}2. 2<f-i) <f-2» ^n,,. 
f9 
0 
1^—1 Q 
Bn-i s-^ is the binomial coefficient r — H=t=^- and „n, the product of all the 
\s -1 \p - s 
Q 
elements of jjA, ,.. 
The relation has to be proved first for r = s, then for r = 2^ and finally for any 
combination s and r. 
For the first two proofs we u^e the relation between the minors of an ortho- 
symmetrical determinant 
A A ' ' A " " - A^ ' " 
AT;.; ~ A,. A,,,'," + A,.- A,',',," 
This is found from two relations given by Professor Pearson* by dividing one 
of them by the other. 
a 
(5) Let A be A, s' = 1 and s" = 2^ + 1, then 
9 Q <i Q ^ 
Q Q g <1 Q 
P+l^l, P+1 11+1^1, 1, JJ+1, v+1 • v+l^.f, s, 1, j)-f-l p+l^p+l, s, 1 • p+l^l, l,s, p+1 
q q+2 
■'^'^^ 39 + 1^.9811= 2)'^S-1,.5-1 1 
<I Q 
v+l^s, s, D+l, 71+1 ~ v^ss' 
q 1+1 
A —I— A 
53+l'^l,l,I'+l,»+l ~ V ^) V-^1.7>> 
q <1+1 
Pfl'^s, s, 1, j)+l — (~ 1)'' J)^S-1,S> 
* Biometrika, Vol. xi. pp. 232-3. 
(27). 
