ANALYTIC CONDITIONS. 57 
Let these functions be transformed by 7; the transformed 
forms f,’ are 
(la; +mar+na;) x+ (lbi+mbs.+nb;) y + (lei+mer+nes)2 , 
fi! : (lay+maz+nas) w+ (di +mbs +nbs) y/ + (ler +mertnes)z , (20') 
; (la, +mar+na;) “x”+ (1b,+mb.+nb;) y+ (ley +mes+ne3) 2”. 
Wesee here that the coefficients of each of the transformed 
forms reproduce in the elements of the columns of M_ the 
original functions. 
Again let r=2; the six functions of ourcomplete family are 
as follows : 
la?+my2+nz2+2puy+2quz+2ryz2, 
lol? + my!?+n2z!2+ 2pa'y! +2qa' 2+ 2ry'2’, 
a. bal omy? 22 op allyl + 2quell2 + 2ryl 2", 2] 
trae laa’ +myy!+nz2'+p (ry! +aly) +q(a2'+a'z) +7 (y2’+y’'Z), ( ) 
laa’ +myy!+nz2"+p (cy + avy) +¢q (v2 + 2"2) +r (yz"+y'2), 
La!a!/ +m yly" +nz2" +p (aly! + x!’y’) +q (a!2!! +-25!’2') +r (y!2" ao yz). 
It is easy to verify in this case also that the coefficients of 
each of the transformed forms reproduce in the elements of 
the columns of M’ the original forms. 
In like manner we can write down a complete family of au- 
tomorphie forms of any degree 7 and verify the fundamental 
properties of the family. 
190. The Hffect of the Transformations U and V. We 
must now return to the consideration of the equations ?; = l; 
among the elements of M, which define a system of collinea- 
tions within G, having the first group property ; and we shall 
apply the results of Theorem 9 to these equations. 
Let the three sets of cogredient variables be respectively 
the elements of the three rows of the matrix M,; let 9;(4, 8, 
ete.,) be a complete family of automorphic forms in the ele- 
ments of the rows of M,and let ~; be transformed by U. 
Since the matrix of U is the conjugate of the matrix of M, 
the coefficients of each of the transformed forms +; will repro- 
duce in the elements of the rows of M the original forms in 
the rows of M,. Equating corresponding coefficients of the 
original and transformed forms we have a set of equations 
$;(a,, b,, ete., ) =l; which satisfies both conditions of Art. 185. 
We may let our three sets of cogredient variables be re- 
