200 THEORY OF COLLINEATIONS. 
Multiplying the first equation by x, the second by y, and 
the third by z, and adding we get 
me MB WO | Mm Wy a WM | ay) az 10 
AS A Hl ES Bel BS 8 | arches, 
A” BY CY KAY Al! BY C! kA" A! BC” WC" | (45) 
or Cn Fe 0 | 
ace @ ean ae ae) =AG+Uy+ v2. (45’) 
| A” Bu Cl’ ki(2, AU +p B!4+4C") 
Equation (45) shows that the form of the function 22+ 
uy+ rz is invariant under the normal form of T. Its value 
is the same at a pair of corresponding points (a, y,z) and 
(%,,Y,,%,), and is therefore invariant under T at each of 
the invariant points of T. The function vanishes at all points 
of a certain line of the plane which can only be one of the 
sides of the invariant triangle of T. 
234. k-Relations in Subgroups of G,. The group G,(AA’) 
is defined by two sets of relations R, and R,. Suppose 
the invariant point determined by AR, is (A,B,C), then 
that determined by R, must be either A’ or A”; let us sup- 
pose it is A’. Since (A, B,C) is an invariant point of the 
group, we have the k-relation kk,’ =kk'k,k,/ ; since (A’,B’,C’) 
/ — ki key! 
° : 2 ko ames 
is also invariant, we have ,; =,.,,.- Combining these two 
v2" “ny” 
relations we get 
Hea — lok An tesa lee (46) 
Thus for the group G,( AK) we have a double k-relation. 
In like manner we can determine the k-relations in all the 
other groups of the list of Art. 209. The results are as fol- 
lows: The groups G,(A), G,(l), and G,(Al) each have a 
single k-relation, viz., k,k,’/=kk’k,k,; the groups G,(A1), 
G,(AA’), G,(ll’), G,(AA‘l), and G,(AA’A”’) each have the 
double k-relations, viz., k, = kk, and k,’ = k'k/’. 
It should be noted that each group whose invariant figure 
has at least one lineal element has the double k-relations. 
