1904-5.] 
Dr Muir on General Determinants. 
921 
Towards the close of the paper (p. 44), his geometrical work 
having led him to use the identity 
’P + cl" ft" + a"'ft'")(y'S' + y'T + y '"S'") - (a S' + a'S" + a"'S'")(ft'y' + ft"y" + ft'"y'") 
= (o'Y" - a'YW'S'" - ft'"S") + (ay" - a"'y')(ft'S"' - ft'"S') 
+ (ay" - <x'y')(ft'S" - ft"S') , 
he devotes the last two pages to proving a generalisation of it, 
— a generalisation not so wide as that of Binet and Cauchy, but 
interesting because of the way in which it is arrived at. Putting 
r x o 
Vo 
*0 • • 
f ^ 
Vo 
Co • • 
• • T 0 ' 
x i 
Vi 
*1 * • 
• • h 
fii 
Vi 
Ci • • 
• • T 1 
D = det. < 
x 2 
>h 
*2 • • 
. . t 2 
k A= det.< 
^2 
V2 
C 2 • • 
• • r 2 > 
, x n 
Vn 
^ n • • 
.. t nJ 
M 
Vn 
Cn • • 
• • T n j 
and N = det.. 
cT* 
O 
lo.i 
*0.2 * 
• • • Jo,.' 
h,o 
^1,1 
^1,2 • 
• • • Ji.» 
< h.o 
l 2> 1 
l“2 , 2 • 
• • • I'i.n 
\.ln,0 
ln,l 
, 2 • 
• • • *».» , 
where 
^o.o = x o^o 4“ VoVo 4“ 2 oCo + ■ • • + t 0 r () , 
k.i = 4" VoV\ 4- ^oCi + • • ■ + ^o T i , 
l n , 0 = X n£() 4 “ VnVo 4 * Z n £ 0 + * * • + t n T Q , 
In, l ~ t 4" 4 * " ‘ 4" t n T-^ , 
ln,n ~ X nftn "t" ynVn 4“ % n £ n ' * * 4“ t'rJ’n ? 
and where therefore 
DA =N, 
he differentiates both members of this identity with respect to 
. . • t n , obtaining 
0D 
, 0N , 0N 
0N 
A;r- 
— Co g? 4- Ci 07 4- 
, n 
0D 
0 N 0 N 
0 N 
~ ^° dl ^ 0/ 4- • • 
4" Vn 0 £ 
n 
0 D 
_ 0N 0B T 
0 N 
3*» 
°34r» +Ti ^ + " 
' ’ 4- r n , 
, n 
