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DR. W. F. SHEPPARD ON 
then 
V 0,0 Vo, 1 Vo, 2 • • • Vo, l 
Vi,o Vi.i Vi, 2 • ■ • Vi,Z 
V2,0 V2,l V2,2 • • • V2, Z 
Vz.o Vz.i Vi, 2 • • • Vz, Z 
H p ? = cofactor of i lpq in H = H ?iP ,.(10) 
r, p , g =Z p JZ, .( 11 ) 
£p, q = H p . 9 /H,.(12) 
HZ = 1.(13) 
(v.) The assumption that there is a conjugate set implies (cf. § 2 (iii.)) that 
Z is not = 0 . It follows from (13) that H is not = 0 . It also follows (see 
Appendix I., § 4 ( b)) that none of the principal minors of Z or of H are = 0 . 
4. Two Related Pairs of Conjugate Sets. —(i.) Suppose that there is another set 
of Z+l quantities u 0 , ?q, u 2 , ... u u connected with the d’s by the linear relations 
(»•= 0, 1, 2, ... i) 
K = (n) S,+ (r t ) S, + (r 2 ) S. 2 +... + (r t ) S, .(14) 
Then, by the condition of conjugacy of the d’s and the <r’s, 
( r t ) = m.p.e. of u T and a t . . 
(15) 
Let the set conjugate to u 0 , u u u 2 , ... u t be y 0 , y 1} y 2 , ... y b Then there are linear 
relations between the y s and the us and between the cr’s and the <f s, and therefore also, 
by (14), between the y s and the <r’s. To find the <r s in terms of the y s, we write (15) 
in the form 
(r t ) = m.p.e. of a t and u r ; 
and we see that (t = 0, 1 , 2 , ... /) 
— (0«) y 0 + (l«) yi+ (2 1 ) y 2 +... + ( l t ) y t .(16) 
(ii.) Similarly, if the expression for the S’s in terms of the us is (s = 0 , 1 , 2 , ... I) 
S s = {s 0 } u 0 + {sj} u, + {s 2 } u 2 +...+ {^} u b .(17) 
where, by (14), 
(n) { 0 «} +(r 0 {l t } +... + (r t ) = 0 (r ^ t) or 1 (r = t ),. . . (18) 
M (0 t ) + {sj} (l t ) +...+ {s z } (l t ) = 0 (r 5 * t) or 1 (r = t ),. . . (19) 
{$«} = m.p.e. of S, and y t , .(20) 
Vt = {0«} o- 0 + {L} 0-1 + {2 £ } cr 2 + ... + {/(} a l .(21) 
then 
