420 
to twice ND ; then the time of descent in the arc AN is to the time 
of descent in BN as OD is to ON. 
By treating the arc BN in the same way, we can obtain a third 
arc, and so on, thus forming a series of arcs decreasing more and 
more rapidly as we proceed. 
Let the angle NO A he denoted by 4A 0 , NOB by 4A X , the next 
angle of the series by 4A 2 , and so on ; then we have, for any two 
consecutive terms of the progression, 
(tan A n) 2 = sin 2A« + i, and 
time in 4A n = (sec A w ) 2 x time in 4A ra + 1 
Now the time of descent in a small circular arc approaches to the 
time in a cycloidal one, and hence, 
time in 4A 0 = {sec A 0 x sec A t x sec A 2 x etc.} 2 x 
time in cycloid. 
As an example of the ease and rapidity of the calculation, I 
subjoin the work for an angle of 160° from the nadir line, which 
shows that the time of oscillation in an arc of 320° is rather more 
than double of that in a minute arc. 
A n 
2 Log sec A n 
2 Log tan A n 
2 Aft-f-i 
0 / // 
40 00 00-000 
22 22 40-669 
4 52 47-180 
12 31-721 
1-493 
•2314920670 
680053042 
31540158 
57684 
0 
9-8476270604 
9-2292047378 
7*8626575152 
5-1607141918 
o / ir 
44 45 21-339 
9 45 34-359 
25 03-441 
2-986 
•3026571554 
= log. 2*00750740. 
The progression may be put in another form, thus — let t Qi t v 
f 2 , &c., be the tangents of the angles A 0 , A p A 2 , &c. ; then, 
3 1 - a /(1 — t » 4 ) _ the 
” +1 1+a/(1 -t/) 2 + 2\/(l-f„ 4 )-i„ 4 ’ 
coefficient of correction being 
(1 + £ 0 2 ) (1 + ^ 2 ) (,1 + ^2 2 ) , & c * 
