90 THEORY OF COLLINEATIONS. 
When these equations are simultaneous, their resultant van- 
ishes ; thus 
po b, C, | 
|, b,—p G = (I). (9) 
|; b, C5? 
This cubic in p, designated by A(p) = 0, is called the charac- 
teristic equation of the collineation. 
There are several cases to be considered. The equation 
A(p) =0 may have three single roots ¢,, p., p;, one single root 
p, and a double root p., or a triple root p,. Moreover the 
coefficients of the three equations (2) may satisfy certain 
conditions so that these three are equivalent to only two 
equations ; or they may satisfy such conditions that the three 
are equivalent to one. If aroot of A(p) = 0, say p,, is substi- 
tuted for p in (8) and no other conditions are imposed on the 
coefficients, then the three equations (8) are equivalent to 
only two. They are satisfied by one and only one set of val- 
ues of the ratios w:y:z. (Geometrically speaking the three 
lines represented by (8) meet in a point.) If the first minors 
of (9) are all simultaneously zero, then equations (8) are 
equivalent to only one. They are satisfied by ~‘ sets of val- 
ues of the ratios «:y:z and these sets of values satisfy a 
linear relation. (Geometrically speaking the three lines (8) 
coincide and may be considered as intersecting at all points of 
a line.) 
113. Type I. Let us consider the case where A(p) = 0 has 
three single roots, p,, p2, p;. In this case the first minors of 
(9) can not all vanish; for if they do, the conditions for a 
double root are satisfied. A double root of A(p) =0 satis- 
fies not only A(p) = 0, but also its derivative A’(p) =0. But 
A’ (ep) = — (A, +A4,,+A,;), Where A,,, ete., are the first 
minors of the elements in the principal diagonal of (9). 
Hence, if the first minors of (9) all vanish, A(p) =0 has a 
double root. 
If one of these roots, as p,, be substituted for p in equations 
(8), these three equations have a common solution. Solving 
equations (8) for the ratios «:y¥:z, we thus find the coordi- 
