382 Proceedings of Eoyal Society of Edinburgh. [sess. 
QR 
2AA' 
. 
- 2C'E 
-AB 
. 
. 
RP 
2BB' 
-2A'P 
-BC 
2CC' 
. 
PQ 
-CA 
, 
-2B'Q 
. 
. 
A 
. 
. 
Q 
-2C' 
B R . . -2 A! 
G ... V . -2B' ^ 
-QRB' -RPC' -PQA' -CAA' - ABB' -BCG' ABC + PQR + 4 A'B'C' . j 
Here the elements 2AA', 2BB', 2CC' can easily he rendered zero, 
and after them the elements - QRB', - RPC', - PQA' in the same 
columns. When this has been done the first three columns will 
resemble the first three columns of the original determinant, so 
that the simplification made upon that determinant can be repeated, 1 
with the effect of reducing the order by as much as before. The 9 
result then will be 
2ARB' 
2APA' 
cr 
- 2EPC' - 4AA'B' 
cr 
2BPC' 
2BQB' 
- 2PQA' - 4BB'C' 
2CRC' 
or 
2CQA' 
- 2QEB' - 4CC'A' 
- 2CAA' - 4RB'C' - 2 ABB' - 4PC'A' - 2BCC' - 4QA'B' o- + 16 A'B'C' 
where o- stands for ABC + PQR. How this is exactly the form of 
the eliminant found in § 15— a result not to be surprised at if we 
consider that the process now gone through is closely allied to the 
process of eliminating the six variables y^, y\ z^x from 
the ten equations of § 19, 
