QUADRATIC RELATIONS. 175 
We shall now show that the set of relations R, (24) implies 
still another set R.’. Let us subtract the square of the last 
equation of (24) from the product of the second and third and 
reduce the result ; we get 
LA,? + MA,? + NA;?+2PA,A2+2QAiAs+2RA2A3=L, 
where L, M, ete., and A,, A,, ete., are the cofactors of 1, m, 
etc., and a,, a,, ete., in the respective determinants 
Ep Gi 
Pp m 
q r 
a, bd, Cc) 
a2 bs Co}, 
a; b3 ¢3 
and 
S 
n 
In like manner we deduce the entire set of relations R,”’ as 
follows: 
BAZ eiAs + = 
LB, + MB,2 + 
B62 Bf ogi) ee 
Le Bae: 
R, 2 LAB, + - = - - 
ions ier eo VARESS eel 
(27’) 
ot wow wea 
POVNA SS 
LB,C,+ US aioe sete Se 
If the six equations of R,’’ be multiplied respectively by 
a,, 67, ete., and added, we get as in Art. 195, 
La,? + Mb,?2+ Ne\?+ 2Paib, + 2Qaie, + 2Rbic, = L. 
In like manner we deduce the entire set of relations R,’” as 
follows : 
La,? + Mb,?+ Ne? + 
Lay? + - - 
R,!": an ; eA CaS 
La\a3;+ - - - - - - 
Lasa;+ - - - - - - 
Thus we see that if we have given a complete family of 
automorphic forms quadratic in the columns of M, we can 
deduce from it three other sets of quadratic relations. In 
fact if we have given any one of the four sets R,, R,’, R,/’, or 
R,/"’, we can deduce from it the other three. The last set of 
relations R,’’’ is a complete family of automorphic forms 
quadratic in the rows of M. It follows from this that if we 
have given a complete family of automorphic forms quad- 
ratic in the rows of M, such as l’a,7-+m’b,?+ ete.; these de- 
C270) 
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