356 Proceedings of Boy al Society of Edinburgh. [sess. 
mentioned columns the proper multiples of columns 1, 2, 3, we 
find for the eliminant ab'c" times the determinant 
where 
/ 
V 
r 
f 
r 
V 
h" 
9 
h 
h 
f 
f 
f' 
( 5 ) 
(71) (72) (73) (74) (75) (76) (77) 
(7l) = b\2c"h + H')+/G' 
(■ 72) = c'\2b'g + G")+/" H " 
(73) = c"(2af' +F") + g"K" 
(74) = aide'll + H) + #F 
(75) = a(2&y' + G) + fcF 
(76) = 5'(2c//" + r) + ^G' 
(77) = A + 4 ab'c” +/F + g' G' + h" H" . 
Rejecting, as before, the special factor ab'c", the eliminant of 
(1) is (5). Multiplying now rows 1, 2, 3, 4, 5, 6 of (5) by 
G - 2b' g" , H - 2c 'h', H' - 2 c"k , F' - 2 af", F" - 2 af, G" - 2b' g , 
adding to row 7 and dividing by 4 we get 
/ 
a h 
f g a 
v r g’ 
g' h' V 
o g" h" 
/" c" . . . h" 
- b'fg" -c"h'f r- c "g'h -af"g' ~ah"f -b'gh" 
where 
A=A + A '-f' g "h-f"gh\ 
9 
h 
h' 
f 
r 
g" 
x 
( 6 ) 
This form of the eliminant is equivalent to that given by Muir 
in the last volume of the Proceedings, p. 233 (e). In the special 
cases considered in § 6, all the elements of the last row of (6) 
except the last vanish. The last element of the last row of (6) 
reduces to either A or A', and its co-factor reduces to either 
