354 Proceedings of Royal Society of Edinburgh. [sess. 
ffv+ fy 
g'x+fy 
g"x +f"y + 2 c'z 
The Jacobian of these quadrics is 
2 ax+ hy + gz hx +fz 
h'y + g'z h'x + 2b'y + fz 
h”y + g"z h"x + f"z 
Expanding and rejecting the factor 2 we get 
- aEx z - b'G'y* - c"E!’z 3 
+ b'(2c"h + H ')fz + e"(2 b'g + G ")yz* 
+ c"(2af + Y")z 2 x + a(2d'ti + K)zx 2 . . (2) 
+ a(2b'g" + G)x 2 y + b\2af + Y)xy 2 
+ ( A + 4 ab f c")xyz 
where A is the determinant 
f 9 h 
f 9 h' 
9" h' 
and E, G, etc., are the co-factors of f g , etc., in A . 
4. Eliminating x 2 , y 2 , z 2 , yz , za;, xy dialytically from (1) and the 
three differentials of (2), we obtain for the eliminant of (1) the 
determinant 
a • . / 9 h 
V . / g' h f 
f 9" h" 
-3 aP b\2af + r) c\2af + F ; ') A + 4a&V' 2a(2c"h' + H.) 2a(2%" + G) 
a(2&V' + G) - 3&'G' c\2b'g + G") 2&'(2c"ft + H') A + 4 a&'c" 2&'(2a/ # + F) 
a(2c'7i' + H) 7/(2 c'7h-H') - 3c"H" 2c"(2&V + G") 2c\2af + F") A +4 cd/d' 
Taking out the factors a, b\ c” from columns 1, 2, 3 and then 
reducing to zero all the elements common to rows 1, 2, 3 and 
columns 4, 5, 6 by subtracting from columns 4, 5, 6 the proper 
multiples of columns 1, 2, 3, we find for the eliminant 4 s ab'c" times 
the determinant 
a(b'c" -/'/") +/E a(c"h' -f f g f ) + pE a(b'g" - h'f) + h¥ 
b\c"h -fg") +/G' b\c"a - g"g) + g’G } b\af - gh") + KG' (3) 
c"Q/g - h'f) + /" H" c\af - g'h) + g" H" c"(ab’ - hh') + h" H" 
5. If A, B', C"‘ are the co-factors of a, b\ c" in A ' where 
A' = 
h 
V 
r 
9 
f 
c" 
