784 REPORT—1863. 
YnA) is properly primitive and of the principal class, we shall have, for some 
value of s, e,=1; whence 
D,a= —20,5,+ e.—1D,; and A=J*,+D,Ds-1 =(J,—é,D,)°+AD*,. 
If (ns —Pns QnA) is improperly primitive, and of the principal class of its 
order, we shall have for some value of s, e,-=2, D,;.=—6,J,++4es_;D,, 
24=2(J.— fail D,) +2(I.— — D,)D.+ An 
We may therefore enunciate the theorem: “If?” is an inferior convergent 
Gn 
to / A, and A=q*,A—p*,; when (qa; —Pas YnA) is of the principal class of 
forms of determinant —A, A is of the form X°+ AY’, and Y is the denomi- 
nator of a complete quotient in the development of/ A; when (¢n, —Pn» YnA) 
is of the principal class of improperly primitive forms of determinant —A, 
A is of the form 2X?4+2XY-+ sl 3 Y’, and Y is the denominator of a com- 
plete quotient in the development of / A.” 
When (¢,, —Pn; ga) is ambiguous and properly primitive, but of some 
other class than the principal class, we must distinguish between two cases, 
that in which the reduced form equivalent to (¢;, —px, YnA) is itself an am- 
biguous form, and that in which it is of the type (a, 6, a). In the former 
case we shall arrive at a form (e,, —6,, ¢s—1), In which e,, being the least 
number which can be represented by (qn, —Pns YnA), 18 a divisor of 2¢,, and 
consequently of D, and 2A; and we shall find 
3 2 
ie (1-2, =) +a De 
Es Es 
In the latter case we shall, in the series of forms (¢., —ds, es—1), arrive at a 
sequence of one or other of the three types: (1), (2[a—d]|, —[a—b], a), 
(a, a—b, 2fa—b]); (2), (a, —[a—4], 2fa—b]), (a, b, a); (3), (a, —b, @), 
(2[a—b], a—b, a); 7. e. we shall arrive at a form in which e, is the least 
number but one, which can be represented by (Gn; —Pn; YnA), and is a divisor 
of D, and 2A; we shall then find 
ut | 
(al) A=(,—]D)'442"3 
2 
(2) A=(Jeu1+4D,)?+ ant 
(3) A=(e+3D) +80" 
Similar results may be enunciated for the case in which (q,, —Pny QnA) is 
improperly primitive and ambiguous, but not of the principal class. 
In applying the preceding formule to particular cases, the following 
theorem of Goepel’s is very useful. Since 
i Yup" s— 2PrP As + AG? ae 6,=%,,PsPs—1 —Pn( P.qs-1 +Ps-14.) = AGn eT e— ls 
we find, if u, is the integral quotient immediately succeeding A , that 
& 
ds41==0,—p,e, Hence 6, 6,... form a continually decreasing series. But 
é Bs s 1 2 y 
¢, =Pn 1s positive, and 6,=—Ag,_, is negative; there exists, therefore, a 
pair of consecutive terms 6, and 6,4, of which the former is positive, or zero, 
