112 
MATHEMATICS: D. N. LEHMER 
Continuing this process, if the original set had no common factor other 
than unity, we must arrive at a set in which the first number is unity. For 
it is clear that an + \ < an except when hn is divisible by an in which case 
an + i= an. Further an + 2 < an except when both hn and Cn are divisible 
by an, in which case an + 2 = an + i = an, and so on. If now the original 
set had the greatest common divisor unity so will also the set a„, hn, Cn, . . . 
kn. In, and so not all the numbers hn, Cn, - - . K, L can be divisible by an. After 
a number of steps in the process at most equal to w — 1 an a\ must appear 
which is less than an and not zero. In the same way another set must appear 
in which the first number is less than ax and so on. This process must then 
lead to a set in which the first number is unity. By taking one more step a 
set is then obtained in which the numbers are all zeros except the last which 
is unity. 
We proceed now to reverse the above process, and taking the numbers 
ai, . . . Kii a2, ^2, ' . . K2: . . . as given we will show how to reconstruct 
the original set ai, hi, Ci, . . . h, h from them. We construct first a deter- 
minant of order m in which the element of the prinicipal diagonal are all 
units, and all the other elements are zero. Using the first set of numbers 
Oil, . . . Ki, which we will call the first 'partial quotient set' we construct 
what we will call the ' first determinant' as follows : The top row of the above 
determinant is erased and another row is added at the bottom which has for 
its elements 1, ai, (3i, . . . ki. This bottom row we will call the first 'con- 
vergent set' and for uniformity of notation we write it Ai, Bi, Ci, . . . Ki, Li. 
It is seen that the value of the first determinant is unity. 
Using the second partial quotient set, 0:2, ^2, - - - K2. we obtain from the 
first determinant a second determinant by erasing the top row and adding 
for the bottom row the second convergent set A2, B2, C2, . . . K2, L2, the ele- 
ments of which are obtained by adding the columns of the first determinant 
after multiplying the first row by 1, the second by a2, the third by (32, etc. and 
the bottom row by K2. In the same way we get the third determinant and the 
third convergent set, using the third partial quotient set as, jSs, . . . /C3. The 
nth convergent set, which is the bottom row of the nth determinant is related 
to the preceding sets by the recursion formulae: 
with similar formulae for the B„, Cn, etc. It is clear from the way the de- 
terminants are derived from each other and from the original determinant 
that the value of each is unity. 
We state now the remarkable theorem that the last convergent set is iden- 
tical with the original set ai, hi, Ci . . . h, h, from which the successive partial 
quotient sets were derived. This theorem comes out of the general theory of 
continued fractions, of multiplicity m, but without any appeal to that theory 
Professor Frank Irwin has derived it very simply from the following equations 
which are easily established by complete induction: 
