786 REPORT—1863. 
Thus if A=2, we shall have 6,=1, e,.)=1, es=—1, 2J,=D,+D,;-1, 
A=(J,+D,)’—2D?,=(J,+D,-1)?>—2D*,_1. If A=38, we shall have either 
(1) j=1, 6.=1, e,=—2, 2J,=D,+2D,_1, A=(J,+D,_1)?—3D’5_1; or 
(2) d.=1, é;1=2, e-=—l, 2J,=2D,+D,-1, A=(J,+D,)?—3D*,. 
If a is the integral number immediately inferior to »/ A, the period of inte- 
gral quotients in the development of s/ A is of the type 
Fis Pao+++ Pk—1s b, Pk-1s Pk—29 +++ i> 2a; 
and it is sometimes possible to assign @ priori the value of D;, the denomina- 
tor of the complete quotient corresponding to 6; for that denominator is always 
a divisor of 2A, and is besides <2,/A. Thus if A is a prime, D,=1 or 2; 
if 3A is a prime, D,=1, 2, or 4. Hence if A or 3A is a prime of the form 
4n+1, (—1)*D,=—1; for the equations «7—Ay’=+2, = +4 are impos 
sible on the supposition that w and y are relatively prime, and the equation 
«’—Ay’=1 is inadmissible, because } is not the last quotient of a period. 
Similarly if A or 3A is a prime of the form 4m+3, (—1)*D,=2 ote 
according as the prime is of the form 8m+7 or 8m+3; if ZA is a prime of 
the form 4m+8, (—1)'D,= +3 or —3, according as the prime is of the form 
12m+11 or 12m+7; and, in general, if \ and = are each of them primes of 
the form 4n+3, and if 2\< A, (—1)*D,=A, or —A, according as X is or 
is not a quadratic residue of 2 We thus obtain a direct method for the repre- 
sentation of primes of the forms 4m+1, 8m+3, 8m+7, or the doubles of 
such primes, by the forms «*+y’, a°+2y, #?—2y*: when d is a prime of 
the form 12m-+ 7, the developments of / and 2 a/ ; will give represen- 
tations of 3A by the forms a°—ay+y’, «°+3y?: when d is a prime of one of 
the forms, 28m +11, 28m+15, 28m+23, the development of sth will give 
a representation of 7 by the form w?—awy+2y’, &e. 
The theorem relating to primes of the form 4n+1 is very celebrated ; it 
was established independently by Gauss and Legendre, and it no doubt sug- 
gested the researches of Goepel in his doctoral dissertation ‘De quibusdam 
eequationibus indeterminatis secundi Gradis’ (Crelle, vol. xlv. pp. 1-13). 
Goepel confined his investigation to the case D,=2, though his method, 
which in the main is that here described, is of a much more general cha- 
racter. The theorems relating to the case A=—3 were first given by M. 
Stern, who employs Goepel’s method with very little modification (Crelle, 
vol. lili. pp. 87-98). A paper by M. Hermite, which appeared in Crelle’s 
Journal (vol. xlv. p. 191) prior to the republication there of Goepel’s disserta- 
tion, contains a method (see pp. 211-213) which is very similar to that of 
Goepel, but which does not connect itself so readily with the common theory 
of continued fractions. In these researches of M. Hermite the invariant 
ac’ —2bb'+<a’c appears explicitly ; which is not the case in Goepel’s paper. 
Berra). 
