ON THE THEORY OF NUMBERS. 785 
and the second negative ; Goepel shows that —8,6,,, << A. For we find 
Ge0—1 + Ys—18,=(—1)*(9, Ps1—P,Jo—-1)> 148, + e—1€, = (—1)*+(9,, P,— 9.) 
1. € 
Ys €s—1 + Ys—1 es i 
Ys Ost+Yo—1 &s Ve Poti vi 
whence 
Ys €s—1 +9,_1 9, > Hs(Ye 8, +901 €,) 3 
or multiplying by e,, 
Aq, => —6d, Os41 Ye Eg Os41 Ys—15 
that is, A> —0, 0,41, because 6,4, is negative. 
Thus if A=1, we have necessarily 8,=0; whence e,=e,_1=1, D,-,=D,, 
A=J*,+D’,. If A=2, we have either 
(1) 5.=0, €._1=2, e=1, D,2)=22D,, A=J*,+2D*, ; 
or (2) d:=0, e1=1, e=2, D,=2D,-,, A=J?,4+2D*,_;; 
or (3) é=1, Os41= —1, pe=2, e=1, €s—1=€s41=3 ; 
D,1=3D,—23,, A=(J,—D,)?+2D*,=(J.41—D,)’ + 2D*,. 
If A=3, we have either 
(1) m0, =3, c=, D2, 330. A =3",-- ols 
or (2) —9, e¢1=1, ¢=3,) DoD, A=’ Poly: 
or (3) O=1, b41=—2, #,=38, ¢,=1, e_1:=4, D,1=4D,—23, 
A=(J,—D,)?+3D*, ; 
or (4) 6=2, o4:=—1, B,=3, €,=1, €41:=4, Dp4;=4D,—25,41, 
A=(J.4:—D,)?+3D*, ; 
or (5) 6=1, d41=—1, p=2, €=1, es 1=641=4, 
3 D,1=4D,—23,, A=(J,—D,)?+3D?,; 
or (6) 6,=1, ds41=—1, He=1, ¢,=es-1=€041=2; D,1=D,—J,=J.4i 
D.41=D,—Js4i1=I,, A=J?,—D,J,+D? =3,4:—D, Joa +D? 
the last case occurring always and only when q, is even. If A=7, andif we 
suppose qg,, even, so that (¢,, —pn, dnA) is improperly primitive, we shall cer- 
tainly arrive at a form (e,, —ds, e,_)), in which 8,=+1, and either e,=2, 
€,—1=4, or vice versd e,_,;=2, es=4; so that there are four cases 
(1,2) +J,=2D,_.—D,, A=J?,¢I,D,14+2D%-1, 
(3, 4) +J,=2D,—D,_,, A=J*,FI,D,+2D%_1. 
Let us next suppose that n is even, so that (—1)"D,,= A is a positive deter- 
minant. Then it is evident d, 6,... are all positive, for 
Inos= ed (GnPs—P ne) UnPs—1 —Pr Qo) + AG,95— 1? 
of which both parts are positive. Again, the numbers €,, €, +++ forma conti- 
nually decreasing series ; for ¢n¢s=(4nPs—PnYs)°—Ag’s; of which the positive 
part continually decreases, and the negative increases in absolute magnitude. 
But e,=q,, and e, = —Agq,,; there exists, therefore, a term e,_, which is posi- 
tive, while the following term e, is negative; whence 3°,=A+e,e,_1<A. 
1863. 3 
