282 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
1 , 0 , *' 
0 , 1 , * 
0 , 0 , 1 ; 
( 33 ) 
for if one of them could be so transformed into another, these two would contain as a 
part the same form <p, and the values of Q, Q' in the primitive contravariant of the one 
would be congruous, for the modulus M, to the values of Q, Q! in the primitive contra- 
variant of the other; the two forms would thus be derived from the same solution of the 
same congruence (32). Again, the primitive representations of the forms (<p) by the 
forms (/) are equal in number to the positive transformations of the forms (/) into the 
forms (/''). For every positive transformation of a form of (/) into a form of (/) 
supplies a primitive representation of some form of (<p) by that form of ( / ) ; and these 
representations are all different, because the same form /cannot be transformed into two 
of the forms (/'), or twice into one of them, by positive substitutions of which the first 
two columns are the same ; otherwise one of the forms (/') could be transformed into 
another by a substitution of the type (33), or else one of those forms would have an 
automorphic of that type, whereas no substitution of the type (33), in which z and it 
are different from zero, can be an automorphic of any ternary form. There are therefore 
at least as many different primitive representations of the forms (<p) by the forms (/), as 
there are positive transformations of the forms (/) into the forms (/'). And there are 
no more ; for if 
a , /3 
ct , j3' 
a", P" 
is a given primitive representation of <p by/*, let y, y', y" be numbers which render the 
determinant of the substitution 
a , (3 , y 
P', y' 
P", y" 
(34) 
equal to + 1 ; and let / be the form, containing <p as a part, into which / is transformed 
by the substitution (34). The coefficient of z 2 in the primitive contravariant of / is 
M, and if the coefficients of 2 yz, 2 xz in that contravariant are Q n Q[, these numbers 
supply a solution of the congruence (32). Let /' be that form of (/') which is deduced 
from this solution ; then/ is equivalent to/', and is transformed into it by a substitution 
of the type (33), in which % = Therefore/ is transformed into/' by 
the substitution 
a , p , y -f/t'a +x/3 
P', +*/3' 
a", p", S'+xW+zP", 
