176 THEORY OF COLLINEATIONS. 
fine a group G,(K’) which leaves invariant a conic K’ whose 
equation is 
L'x?+M'y?+ N’2?+ 2P'ay+2Q’/az+2R'yz=0. 
If 1’, m’, ete., are chosen so L’, M’, equal tol, m, etc., the 
conic K’ is the same as the conic K of Theorem 14. 
THEOREM 15. There is only one variety of group, viz.: Gs(K), 
defined by a set of relations quadratic in the elements of M ; this 
group may be defined in two ways, either by a set of relations quad- 
ratic in the elements of the rows or columns of M. 
212. Groups Defined by Linear and Quadratic Relations. 
A set of quadratic relations R, imposes five conditions on 
eight independent elements of 17; a set of linear relations R, 
or R, imposes two conditions on M. Evidently a quadratic 
and a linear set may be simultaneously imposed on the ele- 
ments of M and leave us still one independent parameter. 
Suppose that R, and R, are imposed simultaneously on the 
elements of M; these will define a one-parameter group 
G,( AK) leaving invariant a point and a conic. Similarly R, 
and R, define a group G,(/K) whose geometric invariant is a 
line and a conic. 
If we have given R, and R,, we can derive from these two 
sets another linear set of the kind R,. Let the set R, be as 
follows : 
WVaj+m'b+n/cq=l', 
Ress Vas+m'b.+n!/e.=m’, (22) 
Pp 
Va3;+m'b3;+n'ce3=n', 
and let R, be given by equations (24). Multiply the first 
equation of R, in turn by la,, pa., ga,;; the second equation 
of R, by pa,, ma., ra;, the third by qa,, ra,, na;; add the 
nine products and reduce by means of R,, we get 
.  (U'+pm!+ qn’) a+ (pl’+mm!'+rn’) ar+ (ql’+nm'+nn')as 
fi, i =ll!+pm'+qn’, (32) 
similarly we get 
(ll! + pm! +qn')b; + (pl! + mm! +rn’) bo + (ql’+rm!+nn') b;=(pl’+mm'+rn'), 
(ll + pm! +qn')e,+ (pl! + mm! +rn’) e2+ (ql’+rm!+nn') es= (ql +rm'+nn’). 
