126 A NEW METHOD OF DETERMINING 
4 (c+ 6) = (2) + | Gr) — Gx) | cos 45° 
4(¢— 7) = | (4) — Gs) ] cos 22°.5 + [ 3) — G4) | cos 67°.5 
A (e+ 6) = (8) —| Gr) — Ger) | cos 45° 
A (¢—6) = | (4) — Gs) |sin 22°.5 — | (3) — Gs) | sim 67°.5 
4 (s:+ 8) = [(4) + G's) | sin 22°.5 + [ GA) + Gs) | sin 67°.5 
L(a—s) = [ (27) ap (Gh | cos 45° + (45 
4 (s,+8;) = | (4) + Gs) | cos 22°.5 — | (2,) + Gs) | cos 67°.5 
A(s,—s;) = |G?p) + Gér))| cos 45°— Gs) 
The values of ¢,, s, must satisfy the equation 
(c) (s) 
in OF A; = 4+ 6,c08g + & cos 2g + ete. 
+ s,sing + s,sin 2g + ete. 
. Oo 0 (7) . 
2 answering to? in b,, and x being any one of the numbers, from 0 to 15 inclusive, 
23 
(¢) 
into which the cireumference is divided. We use ¢, s, as abbreviated forms of C,,,, 
(s) (c) (ec) (s) 
C;,,, ete. Having found the values of ¢,, s, from the 16 different values of A), A,, i, 
(Cc) (s) (ec) (8) a > (4 
Aly, Aly 5 6 9 lay 4b, lootiin ioe m (4) and war ( “)s we have the values of these func- 
tions given by the equation 
a (c) (s 
n (s) c = 
(3) = $33 (C= S,,) eos [GF ») g 1B" F £35 (C,,4 S,,) sin[ GF») gE | 
The values of the most important quantities from the eccentric anomaly £ to ¢,, 
s,, needed in the expansion of w (“) and «a> Cr are given in the following tables, 
first for uw (5) , and then for wa? fe 3 when not common to both. 
