ANALYTIC CONDITIONS. 161 
195. The Second Group Property. It remains to be shown 
that the system of collineation satisfying a set of relations 
?;=l, also has the second group property, viz.: that the in- 
verse of every collineation in the set is also in the set. 
We shall begin by establishing the second group property 
for the set of linear relations given in equations (22). A 
collineation T and its inverse 7~ are as follows: 
%, =a,e+biy+¢12, Ag=Ajr,+AryitAszv, 
pee Yi=a2x+boyte.z, F-1: Ay=Bi21+ Boy: + Ban, 
2 =a," +b3y + ¢32, 2=C\9,+ Coy: + C3z. 
The coefficients of T satisfy the relations 
la,+mb,+ne,=l, 
Ke : la,+mbo+ne.=m, (@22)) 
la; +mb;+ne3;=n. 
If the coefficients of T~‘ also satisfy R,, we must have 
1A,+mA.+nA;3=Al, 
R’, :  lBitmB, +nBs=Am, (26) 
1C,+mC.+nC; = An. 
But equations R,’ may be derived directly from R, as fol- 
lows: Multiplying the first equation of R, by A,, the second 
by A,, the third by A, and adding, we get the first equation 
of R,'. In like manner by multiplying by the B’s and C’s we 
get the other equations of R,’. Hence the system of collinea- 
tions satisfying the relations R, has both group properties and 
is therefore a group. 
The same process may be applied to the set of quadratic 
relations given in (24). If we multiply the six equations of 
(24) respectively by A/, B?,C?, 2A,B,, 2A,C,, 2B,C,, and 
add, we get 
LA?+mB/+nC?+ 2pA,B,+2q4,C,+2rB,C,=A°l. 
In like manner we obtain the whole set of relations 
lAi?+mBy2+ - - - - =A?2l, 
tAo?+ - - - - - =A??m, 
lA3?+ - - - = - =A®%n 
/ 5 3 ’ 
Fee, ae mes ne (27) 
LA, A3+ - Se 5 5 SA, 
lA, A3+ = SS Se == / NAPs 
But these relations show that 7T~ satisfies equations (24). 
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