158 THEORY OF COLLINEATIONS. 
spectively the elements of the three columns of M,,; if the 
complete family »; of automorphic forms in the elements of 
these columns be transformed by V, the coefficients of each of 
the transformed forms ¢; will reproduce in the elements of 
the columns of M, the original forms. Equating as before 
corresponding coefficients of the original and the transformed 
forms we have a set of equations ¢;(«,,«,, etce.,) =1; which 
satisfies both conditions of Art. 185. 
We are now able to restate Theorem 8 in a more precise 
form, thus: 
THEOREM 8a. ‘The set of equations ¢; =1; on the elements of the 
matrix M, which defines a system of collineations within Gs having 
the first group property, consists of a complete family of automor- 
phic forms in the elements of the rows or columns of M, each mem- 
ber of the family equated to the corresponding coefficient of the 
family. 
191. The Linear Families. If our complete family of au- 
tomorphic forms is linear in the rows of M, the equations 
or— rare 
la, +mb\+nca=l, 
J hee - laotmb,+ne,.=m, (22) 
la;+mb;+ne;=n, 
where |, m, and n, are arbitrary constants and not all zero. It 
is an easy matter to write these equations in terms of 4,, ete., 
transform them by U, equate coefficients of corresponding 
terms and get the same equations in a, b,, ete. 
A convenient method of verifying the sufficiency of these 
conditions is as follows: Write down these relations on the 
elements of M and M,, thus: 
la;+mb,;+ne,=l, la,tmj,t+n7,=l, 
la,st+mb,+nes=m, and la.+mfo+ny2=mM, 
la; +mb;+ne3=n, lo,+miz+n73=N. 
Substitute for J, m, and n, on the left-hand side of the sec- 
ond set their values from the first set and collect, we thus get 
1A, +™m8,+nC,=l, 
lAs+-mBs+-nCs=™m, (227) 
1A, +mB8;+nC,=n. 
