388 Proceedings of the Boy at Society 
The cube root of 7 is given by the progression 
- 2 0 U 66658 100986738 ^ 3 , 
0 ^ 23’ 34846’ 52791621 ’ “ V7 
formed by the multipliers 1 , - 3 , 1515. 
And, in general, the ratio sub-triplicate of that expressed by any 
two integer numbers K and L, is shown to be the asymptote of the 
progression formed by the multipliers 
r = (L-K)6; q = 3(L-K)^; = 3L2 -f 21KL-P 3K^ ; 
from the initials 
+ (L-K)-^ 0 K+2L 
-(L-K)-s’ 0’ 2K4-L ■ 
The asymptote of every progression of this kind is shown to be 
the root of a cubic equation ; while, for every cubic equation with 
integer coefficients, such progressions may be found. Thus the 
corresponding theorem in quadratics, which was thought to exclude 
periodicity from all higher equations, becomes only one case of a 
general law. 
In addition to the progression for the ratio of the long diagonal 
to the side of a regular heptagon, given in a former paper, that having 
the multipliers 1 , 4 , 3 is given thus, 
1 0 2 7 29 117 474 o 
0 ’ 0 ’ 1 ’ 3 ’ 13’ 52 ’ 211 ’ 
and for the corresponding ratio in the case of the enneagon, 
- 1 0 3 26 216 1791 o 
0 ’ 0 ’ 1 ’ 9 ’ 75 ’ 622 ’ 
having the multipliers 1 , - 6 , 9. 
It is also shown that terms placed at equal intervals in such pro- 
gressions form separate progressions of the same kind; this law 
applying also to the ordinary periodic continued fractions approxi- 
mating to the roots of quadratics 
