PALMER: THE HYPERBOLIC SPIRAL. 163 
through M, and its intersection K, with the arc AB determines the 
fardeot LOD horas betore,, 7 MOG) “~ LOD, and as 
every point of the instrument of course moves through the same 
angle when the instrument is turned, point A moves through one- 
third Z LOD from A to K. 
Relations Between Fractions of Angle and Lntercepts of Sides.— 
The turning of the instrument thus suggests a series of interesting 
results which may be developed by setting in this way to distances 
of different numbers of times the primary radius, first along OL and 
then along OD, with the instrument in the position shown, and 
striking spiral arcs across the first arc and also across other arcs at 
chosen distances from the pole. There are four sets of these rela- 
tions: 
I. Between the intercepts on the two sides of the angle. II. 
Between fractional part of angle and length on OL, using arc AB. 
Already noticed. III. Between fractional part of angle and length 
on OD, using arc AB. IV. Between either the intercept on OL or 
that on OD aud fractional part of angle, when using one or more 
arcs other than AB, of assumed radii. 
I. In Fig. 8, turn the instrument as before, bringing the point 
at AdowntoB. Then point W bisect OA, for ZB”’OW=2/B"”OB. 
., OW==3 OB=f OA. If now we turn till the curve goes through 
F, the point V, found as W was, gives OV—1 OA. Continuing 
this, we have the series or fractional parts of the primary radius, 
. 1 1 1 a 
on OD: i, 4,4,4, 4, 4, 4, etc., etc., 
corresponding to the series: 
ceylbbigl ines octet Hage) 
WINDS". Bide tod) sD! iors 
of OB along the side OL. 
Instead of drawing the circular arcs each time, the instrument 
may be inverted, when the lengths OW, OV, etc., will be laid off 
alternately on the one side and the other of the angle. 
Continuing to settings greater than OB, when the instrument is 
set at C, where OC=2 OB, we have OX=3 OA, Likewise setting 
to OM=3 OB we have OY=# OA. For total angle C’OX=3 COA. 
Hence OX=2 OA. And total angle M”’OY=4 MOX, so OY=# 
OA. So we have for 
Arg Oy: Jy, Oy. Oy CLC, times Ob 
4, B, $, i, 3, io eles of OA: 
: a : 
And hence, in perfectly general terms, setting to (=) times the 
