564 
ON LAMBERTS THEOREM FOR ELLIPTIC MOTION. 
[222 
Writing FQ — p, QF = a, and as before the exterior angle of inclination = 20, the 
actual expressions for the various lines of the figure are easily found to be 
%(p + o-) , = a, 
GF = Gf= |- Vp 2 + <r 2 + 2pa cos 20, = a, 
GQ = ^ Vp 2 + o' 2 — 2pa cos 20, = a', 
CR = Vpcr , = b‘, 
where CR (not shown in the figure) denotes the semi-diameter conjugate to GQ. 
I _ e 2 = ^P°L s ; n 2 a 
1 e ( p + a .y sm °> 
cos^- P + LT^ cosg.^7 2 *, 
sin F = 
2 ae 
a sin 20 
2ae 
. n a sin 20 . „ pa sin 20 
, Sin Q= — , , sm G =' - 7 , 
2a 4a ae 
where jP, (7, Q, denote the angles of the triangle FCQ, respectively, 
EG = ^°~ cos 0, and therefore EH = — 
p + o’ p + <r 
QQ = -J C —2a', 
p + cr 
and, if for shortness A = *Jkpa (p + a — k), then 
{EM, EN) = (A + ka cos 0), 
p + c 
so that 
(FM, FN) = - {p (a + p) + k (a — p) ± 2A cos 0], 
p -\- cr 
2A 
GM=GN = l(EM + EN) = -=±- t 
p + cr 
i(FM+FN) = 
p + cr 
{p(a + p) + k(cr-p)}, 
and moreover 
(ecos u, e cos u) = K 0 " — p)(p + cr — 2k) + 4Acos 0], 
(e sin u, e sin u') = —-— i ± + 2 (p + a - 2k) Vpa cos , 
(p + tr) 2 l v pa )