An Upper Limit for the Value of a Determinant. 
325 
But whatever a,., A,., a,., A',., ... may be 
a^ + a,.i b,. + b'^i c,. + cji 
A,-A;i B,-B;i a-c;i 
because the left-hand member is the sum of three terms, each of which is 
the product of a complex quantity by its conjugate. Hence 
I (2a^A, - 2a;.A;.) - (Sa,.A; + I^a,A,.)i A; + A;^ + B;. + B; ^ + C; + C;.^ | = ' 
or 
(Sa; + i:<^)(SA;. -t- SA;^) > (Sff.,A, - S^A;.)^ + (Sa,A', + Sa;A,)^ (II.) 
From (I.) and (II.) it follows that for r = 1, 2, 3 
(S^i;, + S6^;.^)(SA;. + SA;-) > I /z h ; 
so that if s;, Sf. stand for Sa;. + SA;. + SA;.% we have 
and .-. 
5,5253.8,8283 > 1^1 . |M|. 
Now any reasons for 515353 being < | ft ' would be equally effective in 
showing that 8,8283 < | M i, and would thus by multiplication entail a 
result at variance with that just reached. Hence finally 
5,5053 ^ I /i I, 
as was to be proved. 
6. If /x' be what jj, becomes on writing —i for i, then 5^5^53 is evidently 
the diagonal term of the determinant got by multiplying jj. column-wise by 
jll' : and the result of the preceding paragraph is that iijjl' is not greater 
than its oiun principal diagonal term, the product-determwiant being 
obtained by column-toisc multiplication, i.e. 
(a-; + a;' + b\ + + + c\^){al -f ai^ + b\ + + c\ + cl^)(a^ -f . . .). 
Of course we could prove m similar manner that nfi' is not greater than its 
oion principal diagonal term lohen the product-determinant is obtained by 
multiplying row by roio, i.e. 
[t- + + cd -1- + 4- a^){b\ + + b\ + b\^ -h b\ ^ l)f){c\ -f c,^ -f- c> 
^cl^cf). 
7. From § 4 it is seen that the limit s\s\s\ which j /z h cannot exceed will 
actually be reached when for 7* = 1, 2, 3 
a^^d^i 
b,. + b'yi 
a^ — a'yi 
by — b'yi 
A,-A;i 
B,-B;i 
A,+A>- 
By -|- B/i 
ay + cLyi 
c, + c;i 
ay — a'yi 
; • 
Cy Cyl 
Ay-A'yl 
c,-c;^ 
A,+A;i 
6,.-f b'^i 
Cy-\~c'.,.i 
by — b'^i 
Cy — C'yi 
B, + B>- 
c,.+c;i 
a^ 
ayi 
b,. — bfi c. 
Ay + Kyi B, + B;i Gy+C\i 
0, 
