772 REPORT—18638. 
Similarly we may investigate the conditions for the ambiguity of a class. 
In order that a class should be ambiguous, the nodal system representing it 
must be symmetrically equivalent to itself. If therefore there is but one 
reduced parallelogram, that parallelogram must be symmetrically equivalent 
to itself, 7. e. it must be either a rectangle or a rhombus. When there are 
two reduced parallelograms, we have seen that they are symmetrically equi- 
valent to one another; and when there are three, they are each of them 
rhombs. We thus obtain the conclusion that if (a, 6, c) is the reduced form 
of an ambiguous class, either b=0, or a=c, or a=2b (Art. 94), 
121. Application of Formule relating to the Division of the Circle to the 
Theory of Quadratic Forms.—We have already referred to the trigonometrical 
solutions of the equation T?—DU*=1 (Art. 96, ix.) and to the connexion 
existing between them, and the number of classes of quadratic forms of 
determinant D (Art. 104.) 
If p isa prime of the form 3xn+1 or 4n+1, the coefficients of the cubic, 
or biquadratic, equation of the periods depend on the values of the indetermi- 
nates in the equation 4p =a” +37’, or p=" + y’ (Art. 43). Thus in these two 
cases, if, for any given value of p, we calculate the equation of the periods, we 
obtain, by a direct though tedious process, the values of the indeterminates in 
certain simple quadratic decompositions of 4p or p. But the theory of the 
division of the circle supplies a method equally direct and of more general 
application for the investigation of such decompositions in certain cases. The 
principles of this method were discovered by Gauss, who deduced from them 
the first of the three following theorems :—- 
“Tf p=4n+1=2°+y’, 
TI2n 
w= ee mod p; «==1, mod 4; 
Ti2n .T2n - 
Cake lan d ».” 
Y= Iln . in ae 
(Gauss, Theor. Res. Biq. Comm. prima, art. 23.) 
“Tf p=3n+1, 4dp=a’+dy’, 
Nl2n 
Tn. In’ 
y = 0, mod 3.” 
(Jacobi, Crelle, vol. ii. p. 69 ; Stern, 2b. vol. vii. p. 104, vol. ix. p. 198, vol. xviii. 
p. 375; Clausen, 7b. vol. vil. p. 140.) 
Tf p=8n4+1=2a°+2y/’, 
Ay Hn 
C= 3. 
IIn . T4n 
(Jacobi, Crelle, vol. xxx. p. 168; Stern, 7b. vol. xxxii. p. 89.) 
In all these formulz the absolute value of « is evidently < 3p; so that x 
is determined without ambiguity as the minimum residue for the modulus p 
of the binomial coefficient. And the combination of the two congruences 
satisfied by w gives rise in each case to a remarkable property of the coefli- 
cient: thus, from the two congruences satisfied by in the first theorem, we 
infer that “if pis a prime of the form 4x+1, the minimum residue of 
rales for the modulus p is of the form 4m-+1.” 
To show, by an example, how these formule are obtained, we shall consider 
the last of them in particular. Resuming the notation of Art. 30, let 6 be a 
p-l 
primitive root of the equation #?—!—1=0; and let "=05 =w,(?"=w'=i, 
mod p; #=1, mod 3; 
LSS — 
,mod p; w=1, mod 4.” 
