1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
229 
a v ^1> C 1 
^1^2 
^1> 
^1 ) ^i» 
a 2>/^2’72 
a 3’/^3’y3 
a i»A>yi 
a 3^3’73 
^2 a l 
^2’ C 2 
a 2 , & 2 , c 2 
a 2 a 2 
a 2’ ^2’ C 2 
a 2» ^2» C 2 
a 2>/^2>y2 
“jp/Ws 
a i>/^i>yi 
a A73 
a 3 a l 
a 3’ ^8’ C 3 
a 3'> ^3> C 3 
a ‘d a 2 
^ 3 ’ C 3 
a 3 ’ ^3> C 3 
a 2’/^2’y2 
a 3^3^73 
a vftv7i 
a 3'P3’73 
Oh a, 
13 
tto.a, 
3 3 
a v \,c x 
«1> ^1’ C 1 
a iA’yi 
a 2’p2’72 
^2 
®2> ^2> C 2 
a i»A>yi 
a 2’^2»y2 
CJg, Co 
® 3 > ^ 3 » C 3 
a nft 5 yi 
a 2?^2’y2 
jSTow by either of the interchanges 
/ a i j ^2 » a 3 J a l » a 2 ’ a 3\ /®1 » ^2 > a 8 > a l > a 2 > a 3\ 
> ^ 2 » ^3 » A f @2 > /^r ' C 1 f C 2 ’ C S i Tl ) 72 J y 8 / 
the first columns of this, — and the first columns only, — would be 
affected, the a’s and a’s becoming &*s and /3’s respectively in the one 
case, and c’s and y’s in the other ; and as neither interchange could 
affect the left-hand side of our identity, we should consequently 
note that thus three different expressions would be at once obtained 
for |a 1 & 2 c 3 | . |a-,jS 2 y 3 | . Adding these together, and combining the 
nine determinants of the sum in sets of three by means of the 
addition-theorem (xlvil), we should have finally 
^1J C 1 
a\,\,c x 
a i^nyi 
a 2’Pv72 
a 3^3->73 
®2» ^2» ^2 
^2’ ^2» ^2 
a 2'> ^2> C 2 
a i^i»yi 
a 2^2^72 
a 3’^3’73 
Co 
^35 ^35 ^3 
®3’ K ^3 
a vPv7i 
a 2$2->72 
a 3^3’73 
from which it is only necessary to delete the common factor 3. 
