MATHEMATICS: D. N. LEHMER 
361 
tion, and showed that these coefficients sometimes become periodic. Bach- 
mann 3 showed later that with the coefficients determined by Jacobi's method 
periodicity will not always ensue. Berwick 4 has obtained periodic expansion 
for cubic irrationalities but his algorithm differs from Jacobi's. 
A closer study of Jacobi's expansion reveals many remarkable points of 
contact with ordinary continued fractions not hitherto observed. Instead 
of starting with a cubic irrationality and finding an expansion to fit it, we 
start with a periodic expansion and find associated with it a definite cubic 
irrationality. 
The set of three numbers (A n , B n , C n ) defined as above we call the nth con- 
vergent to the ternary continued fraction, 
(pi, qi\ p2, qi\ pz;q%\ ) 
The extension to four or more sets is obvious, and most of the theorems which 
follow hold also for quaternary and higher continued fractions. We reserve 
discussion of these, however. 
The f ollwing theorem is of fundamental importance : 
Theorem It If (A\, B\, C\) is the convergent of order X in the ternary contin- 
ued fraction (pi, qi; p2, q2,.-..), and (A\>,B\> C'\) the convergent of order X' 
in the ternary continued fraction (p\, q\; p\, q\) ) then the convergent of 
order X + X', (A'\ + \, B"\ + a', C"\ + X') ? of the ternary continued fraction 
(pi, qi; P2, 
formulae. 
■>'P\, q\; Ph q ii P 2, qi',- ■ •; P \; q\d may be obtained by the 
A'\ + \'-= A\C'\> 
B'\ + v = B K C'\> 
C'\ + \> = C A C\' ■ 
+ B X - 1 B' V 
f A X -2A\. 
+ B X -oA\ f , 
- C\ -2A'\>. 
More generally, the convergents of order X + X', X + X' — 1 and X — X' — 2 
may be obtained by the rule for the multiplication of determinants, so that 
A\, A\ _ i, A\ 
B\, B\ _ i, A\ 
C\, C A _ i, C\ . 
A\>,A\> _ i, A\> 
A'\ + x', A'\ + X' - i, A'\ + X' - 2 
B'\ + X', B'\ + x' 
X + X', C X + X' 
i, B'\ + x' - 2 
Clt 
X + A' - 2 
The proof is by induction. 
Consider now the purely periodic ternary continued fraction of period k: 
(pi, qi', p2, q2\ pky qk) 
Associated with this fraction is the following cubic equation which we shall 
call the characteristic equation of the ternary continued fraction : 
A k -2 
A k -u 
A k , 
Bk -2 
B k -i 
Bk 
P, 
Ck -2 
Ck - 1 
Ck - P 
0 
