ON THE THEORY. OF NUMBERS. 783 
cases p=7n+2, p=7n-+4, will be found to coincide with: the formule given 
by Eisenstein. 
There can be no doubt that the principles of this method are capable of 
many other applications; but nothing has as yet been added to these researches 
of Eisenstein. 
123. Applications of Continued Fractions to the Theory of Quadratic Forms, 
—Representations of a number by quadratic forms are in certain cases dedu- 
cible from the development of its square root in a continued fraction. If A 
is any number not a square, sae ME the (n+1)th complete quotient in the 
development of A,“ the convergent fraction immediately receding that 
Pp yP 
complete quotient, so That P?n—AGn=(—1)"D,, the form (¢’,, —pnr, A), of 
which the determinant is (—1)"D,, is either properly or improperly primitive, 
and belongs in either case to the principal genus ofits order. If we investi- 
gate the transformation by which this form is reduced to the simplest form in 
its class, we shall obtain, by an operation exempt from all tentative processes, 
a representation of A by thatsimplestform. The following proposition, how- 
ever, supplies a method by which, when gq, is uneven, and (q*n, —Pn, A) 
belongs to the principal class of properly primitive forms, or when q, is even, 
and (39°, —Pn, 2A) belongs to the principal class of improperly primitive 
forms, we can frequently infer from the development of »/ A itself the solu- 
tion of the equations 
~(—1)D,Y°=A, 2X°+2XY+ abhest Td at bh: 
“ Tf (a, 6, c), (a’, b’, c’) are two primitive forms of the determinants D 
and D’, whose joint invariant ac'—2bb' + ca’ is Eps and if m and m’ are the 
greatest common divisors of a, 2b, ¢; a’, 2b’, c’; m*D' and m"”D are respect- 
ively capable of primitive representation by the duplicates of (a, 6, c) and 
a, b',¢ c’). ? 
“Thus if (a’, 8, c’) is properly primitive and ambiguous, D can be repre- 
sented primitively by (1, 0, —D’); if (a, 0’ 0) is improperly primitive and 
. For (a,6,c)and (a’,6’,c’) 
let us take (1, 0, —A) and (q,, —pa, ‘hel whose joint invariant is zero, and 
of which the first is properly primitive ; while the.second is properly or im- 
properly primitive according as q, is uneven or even, and has for its duplicate 
in the former case (q°,, —pn, A), in the latter 2 x (4q°,, —pn, 2A): so that 
itis ambiguous in both cases alike. Further, let us represent by (e,, —8,, €s—1) 
Ps Ps—-1 
Ys Ys—1 
ambiguous, 
the form into which (¢,,, —pp; nA) is transformed by ; we infer, 
from the property of the invariants, the equations 
(—1)**'D =e, €,_1—0",, es_1 D, — 28, J,—e, Ds_1=0. 
Let us first suppose that n is uneven, so that (—1)"D, is a negative deter- 
minant which we shall call —A; since 
In(In& —2P nity + Indy”) =(Gne pny) + Ay’, 
it is evident that when q,v°—2p,cy+qnAy’* attains its minimum value, 
; is a convergent to a not, we may add, the last convergent, if the last 
integral quotient in the development of ms is unity. If therefore (¢,, —pns 
n 
