166 KANSAS UNIVERSITY QUARTERLY. 
For which, in general terms: 
Each a (n—a) n(n—a) 
n (n—a) a a? 
a a a ee n n(n+a) 
n n Or eye. a(n-+a) 
(n--a) Be n He 
If, now, the fractional part of OA be measured outward from O, 
and the angle measured upward from the initial line, as is the cus- 
tomary way, instead of as assumed, the table becomes: 
ya (2a—n) a n 
n Tata ae) Ges (n—a) 
n ket n (2n—a) 
(2n—a) n (n—a) ‘5 
a (a—n) en (an) 
(a+n) wien n saree 
(a—n) (a—2n) (a—n) ae 
a (a—n) ees) n 
(n—a) 
a a ; 
As either the th or the ~———‘th part will serve to show the 
n 
E a Nig here: : 
desired —th portion of the angle, since it is immaterial whether 
n 
we measure it upward or downward, we have the fact that for any 
n n 
—th or the 
= ———.th 
(na) (2n—a) 
part of the primary radius may be used to determine it. 
If the 
f a ; 
fraction, as the— thespart, either the 
th part be used, measured outward from O, then the 
reGewias 
(na) 
desired —th part will be found by measuring downward from A; 
if the th part, the fraction of the angle will be found meas- 
n 
(2n—a) 
uring up from B. 
IV. Further interesting relations may be discovered by setting 
the instrument to one arc and striking across another, the radu of 
the two arcs being known, the subdivision of the angle, thus deter- 
mined, to be found in terms of the assumed radii. 
We have the following cases, Fig. 9, Z LOD being any assumed 
angle to be subdivided: (1) By turning the instrument through 
