[171 
172] 
51 
7—2 
/3 2 + 7 2 ), 
; factor 
172. 
NOTE ON THE LOGIC OF CHARACTERISTICS. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 257—259.] 
;here is The conditions in order that an equation of the sixth degree 
{a, b, c, d, e, f g), (a;, y) 6 = 0 
may have five of its roots equal are 
A — ae — 4 bd + 8c 2 = 0, 
respect B = af-3be + 2cd = 0, 
quatlon G = bf-4<ce + 3d> = 0, 
D = ag — 9ce + 8 d 2 = 0, 
E = bg — 3 c/+ 2 de = 0, 
F = eg — 4 df+ 3e 2 = 0, 
equivalent of course to four relations between the coefficients: among the connections 
of these equations are 
fA - eB~ bF+ cE= 0, 
(3e 2 - 2df) A - 2deB + ecD - AF - 2cdE + (3c 2 - 2bd) F = 0. 
The system is one of the tenth order. To verify this, I write first 
(A, B, C, F) = (A, B, C, F, cE) = (A, B, C, F, c) + (A, B, C, F, E), 
i.e. the equations A = 0, B = 0, (7=0, F= 0 imply (by the first of the connectives) the 
additional equation cE = 0, viz. the system A = 0, B = 0, G= 0, F= 0, cE = 0, or what 
is the same thing, one of the systems A = 0, B = 0, (7 = 0, F = 0, c = 0 and A = 0, 
B = 0, (7=0, F= 0, E = 0.