LINEAR RELATIONS. a 
like manner we see that the points (m, q,t) and (,7,w) are 
also transformed into themselves. If the determinant 
|l m n | 
= 2 de \ ay 
}s t¢ u 
the group leaves invariant a triangle and may properly be 
designated by G,(AA’A”’). 
When the relation 7 =v is transformed by substituting for 
A,,8,, ete., their values, we may collect in a different manner 
and equate the coefficients of a,, b,, etc., to the coefficients of 
A,, B,, ete., in@; this gives us a set of nine relations in the 
elements of MV, as follows: 
la, + moa.+na,=1, PaAt+qa+ras—=p, s4+ta+uUa;=s, 
1p, +-mBo+-nBs=m, pei+-qpe--rps—aq, sfi-+tfo+uhs—t, (29) 
lyitmre+nrz=n, pPnitde2tris=r, syittye+uy3,=—u. 
If we assume the nine relations 
1A, +m4.+nA3=l, pA,+qéA2+rA3;=p, sA\+td.+uUA3=s, 
1B, + mB.+nB;=m, pB\+qB.+7rB;=q, $8: +tB.+uB;=t, ( Zo.) 
1C,+mC,+nC;=n, pli+ql:+rCz;=r, sC,+tC, +uls;=—u, 
and transform them as before we get the same nine relations 
in the elements of the columns of M. Thus we have again 
the conditions for a two-parameter group. The geometric 
invariant of this group is the set of three lines whose equa- 
tions are lxa+my+nz=0, put+qytrz=0, suttytuz=0. 
If the determinant 
l mn 
ae ie 0, 
the group leaves a triangle invariant and may be designated 
by G, (lll). 
208. Implied Linear Relations. If we have given two 
sets of linear relations of the same kind, say R, and R,,, we 
can derive from these another set of linear relations of the 
other kind R,. Let Fk, and R,, be as follows : 
la, + mb; + nce, =l, Vaytm’b,+n'q=l’, 
Ines : las+mbs+ne.=m, Fees, > Vactmb+no=m, (22) 
la; -+mb3;+nce3;=n, UVa3+m/b3+ 7 c3=n'. 
