CONDITION OF REDUCIBILITY. 185 
¢,=8,=0 and we have another reducible group. Each of 
these two matrices (or their equivalents) viz. : 
ia; Bb) O Ov 
M, S/o. b 0, and M,= > bz ¢2|| 
| a3 bs C3 a3 b; C3 | 
gives us the largest possible reducible group in G,. Every 
other reducible group in G, will be contained within one of 
these. Hence in seeking fora general criterion of reducibility 
we need only consider these two. In the first case c, and in 
the second a, may be made unity without loss of generality. 
The matrix M, shows at once that the line « = 0 is trans- 
formed into itself by every collineation of the group G;,; 
hence the transform of G,, viz.: G,’ =S‘G,S, isa group which 
transforms into itself some linear function as l’a + m/y+ n’z. 
But the necessary and sufficient condition for the invariance 
of this linear function under T is the set of linear relations 
Va, + ma. + n/a; =V, 
Ub, + m’'bs + nbz =m’, (23 ) 
Ve, + me+n'ce; =n. 
The matrix M, shows that the point (0, 0, 7) is transformed 
into itself by every collineation of the group G,; hence the 
transform of G,, viz. G’,, leaves invariant some point whose 
coordinates are (/, m, 1). The necessary and sufficient con- 
dition for the invariance under T of the ratios of the three 
numbers /, m, 7 is the set of linear relations 
la, +mb,+nec, =/, 
lar +mb2+ne.=m, (22) 
la; + mb; + 2c3 =n. 
Hence for these two reducible groups G,’ and G,’, and there- 
fore for all reducible groups with a smaller number of par- 
ameters, a necessary condition of reducibility is the existence 
of a set of linear relations in the elements of the rows or 
columns of M. We have now established the following: 
THEOREM 17. A necessary and sufficient condition that a sub- 
group of Gs be reducible is that at least one set of its defining rela- 
tions. 2, be linear in the elements of M. 
