1890 - 91 .] Dr T. Muir on some unproved Theorems. 
75 
then any one of the twelve other determinants is expressible in the 
form 
0i(1) + 0 2 (2) + <9 3 (3). 
The above demonstration has the advantage of showing what 
0 V 6 0 ?j are in every case. 
There is, however, quite a different mode of viewing and investi- 
gating the theorem. The identities (A), (A'), (A"), (B), (B'), (B") 
may each be looked on as furnishing the result of an elimination. 
For example, having used (A) to prove that if |1234| = 0 and 
|1235| = 0 then |1245| = 0, we may manifestly view the work thus 
accomplished as the elimination of the suffix 3 from the given 
equations. The question consequently arises, May the demonstra- 
tion not be presented in the form of an ordinary process of elimina- 
tion? 
Writing the first given equation in the form 
a 1 |& 2 c 3 ^ 4 | - a 2 |5 1 c 3 c? 4 | + a 3 |& 1 c 2 c? 4 | - a 4 |& 1 c 2 c? 3 | = 0 , 
and the second in a similar manner, 
a i\h e A\ - a 2\ b i c s d b\ + a s\ b i c 2 d b\ - a 5\ b l C 2 d 5\ = 0 > 
and from these eliminating a 3 we have 
a \{ ~\ b l G 2 d ^\ b) 2 C ^ d ^ 4* \ b \ G 2 d ^\\ b 2 C ^ d ^[\ 
- a 2 { - \ b l C 3 d 4\ + W C 2 d ±\W C A\} 
+ a ^{\ b i G 2 d ^[ l^l C 2^sl} 
~ ^{\ b l C 2 d M b l C 2 d s\) =°> 
from which, on striking out the common factor | there results 
- a $i c A\ + a i\ h i c 2 d b\ “ a 5 |&iC 2 ^ 4 | = 0, 
i.e. |cq& 2 e 4 dy = 0. 
Turning now to (B), and observing that the result there ob- 
tained is the elimination of the suffixes 2, 3 from the equations 
|1234| = 0, |1235| = 0, |1236| = 0, we write the said equations in the 
form 
a il^2 C 3^4:l — %|^i c 2^4l “ ^4|^l c 2^3l = 0 j \ 
a i\ b 2 C 3 d 5\ ~ a 2^1 C 3^5l + a 3^1 C 2^5l ~ a 5^1 C 2^3l = ^ 5 
«il ~ a z\ b i c A\ + a *\ b l C 2 d 6\ ~ a Q\ b l G 2 d s\ ~ 0 » ' 
