PKOFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 2G9 
2(/i —A') = (?n 1 ), r - r'= {m — m') {ixv — 2) . (270) 
connect the niimher of apparent double points (A) and the rank (?•) of a curve with 
those of its complementary in the intersection of two surfaces of orders p. and v. But 
we know the characteristics of the plane curve (269) to be 
m; = Ml, r/ = Ml (Ml - 1 ), A/ = 0 .(271); 
and hence we find the characteristics of its complementary (t’65), 
= Ml^ = 2Mi2 (Ml - 1 ), 2 A = Mp (Mi - 1 )’ . . (272); 
and these in turn give the characteristics for the curve rn\ 
= r'=^,{t,~2) + S,; 2h'= ^, {t, - t, + 1) - . (273), 
and, finally, for the original curve (260) we have 
.ju — V 3 _ ■>>* . A. — OV 3 _ qv V _J_V _ o/V2 _ V\. 
Ill: — _1 ^2 ) ' -1 U -^3 “'3/ ’ 
2 A = (^i^ - No)' - (2Ni3 - 3NiNo + N 3 ) + (Ni® - No) . (274), 
where Ni, No and N 3 are the sum, the sum of the products in pairs, and the product of 
the three quantities Mi, Mo and M 3 . 
As examples, for the twisted cubic 
[/!'//:'/«] = 0 .(275), 
Mj = Mo = 1, Mg = 0, and Ni = 2, No = 1, -3 = 0, so that in = 3, r = 4, A = 1. 
For the curve 
[MMfS = 0.(276), 
Ni = 3, No = 3, Ng = 1 ; and m = 6, 7 ’ = 16, A = 7. 
These numbers admit of course of simple verification.'^ 
65. In like manner proceeding one step further we calculate the characteristics of 
the curve common to the five surfaces obtained by equating to zero the coefficients in 
the identity 
Qi(Q3Q3QiQ ) + Q3(Q3Q‘iQoQi) + Q3(QtQ3QiQ2) + Qi(Q5Q3Q3Q3) + Q5(QiQ3Q3Qt) 
= 0 . . (277) 
to he 
in = NMiMo, V = NMiNMiMg + NMiMoMg - 2 NM 1 M 0 . . . (278); 
this curve being the complementary of (260) for the fourth and fifth surfaces. 
The curve common to the five surfaces may be conveniently designated by the 
equation in double brackets 
* The expression for the rank of a carve, ‘ Modern Higher Algebra,’ Art. 284, seems to require 
modification. 
