MATHEMATICS: D. N. LEHMER 363 
The characteristic cubic was obtained from the last three convergents at 
the end of the first period. If we form a cubic in the same way from the 
last three convergents at the end of the second period we get what we shall 
call the second characteristic cubic. Similarly for those of higher order. The 
following theorem holds: 
Theorem. V. The roots of the characteristic cubic of order A are the \th powers 
of the roots of the first characteristic cubic. 
The proof is again by induction. 
By a process similar to that by which Theorem IV was obtained we have 
also derived the following: 
Theorem: VI. If the characteristic cubic has one root pi whose modulus is 
greater than the modulus of either of the other two then the fractions B n / A „, C n / A n and 
C„B n approach for n— > °° limits which are cubic irrationalities and are con- 
nected with the irrational number pi by linear fractional transformations. If the 
characteristic cubic has two imaginary roots whose common modulus is greater 
than the absolute value of the real root then these fractions do not approach any 
limit. 
The actual equations connecting pi with ai= Lim (B n /A n ) and cr 2 — Tim 
(C n /A n ) and c 3 = Lim (C n /B n ), when these limits exist are a little difficult to 
obtain, but turn out as follows: 
(7i = ft Pl + iA_2 - A k - 2 B k )/(A kPl + Ak^Bk - A k B k -{) 
(7 2 = (C k _ ipi +C k -2 A k _i — C k -i A k -2)/ {A k _ipi-\- A k C k _i — A k _i C k ); 
(7 3 = (C k _ 2 pi + C k -iB k _ 2 — B k -iCk-2)/ (B k - 2 pi + B k C k - 2 —B k _ 2 C k ). 
The cubic equations satisfied by a h a 2 and 03 may be obtained from these 
relations, but the algebraic work involved is tedious. It turns out that all 
the coefficients of the cubic in 01 are divisible by the determinant of the trans- 
formation connecting it with <t\. It follows that the discriminant of the cubic 
in (7i is equal to the discriminant of the characteristic cubic multiplied by the 
square of the determinant of the transformation. Similar results hold for 
(7 2 and 0-3. The actual cubic equation satisfied by u\ is 
Eal + Fu\ + G01 + II = 0 
where, E. F. G. H have the following values (we write for shortness A, B, C, 
iorA k , B k , C k ; A^Bi. & for A k „ h B k -i, Ck-!; andy4 2 , B 2 , C 2 for^_ 2 , B k - 2 , 
C*- 2 ): 
E = A 2 Ci + AAiBx - AA\C - A\B, 
F = A1BB1 + A^BC - 2AiA 2 B + AAA + AA 2 B 1 - AA 2 C - AB\ +AB x C 
-2ABd + A 2 C 2 , 
G = -A\B + A,BB X + A 2 BC + A X BB 2 - 2ABC 2 - BB X C + B 2 d + AA 2 B 2 
- 2AB1B2 + AB 2 C, 
H = A2BB2 - BB 2 C + B 2 C 2 - AB\. 
(The equation has been freed from the factor A X B 2 — AB (A 2 — Bi) — A 2 B 2 
which is the determinant of the substitution connecting u\ and pi) 
