222 
222] 
on Lambert’s theorem for elliptic motion. 
565 
the 
so that 
e cos u + e cos u = 
(p + o- - 2k), 
u — u' = 2 tan -1 
e sin u — e sin u 
2A 
(p + o') 2 
, _ 4 (er — p) A 
(p + o') 2 / 
pcr 
4A (p + cr — 2k) 
= sm _1 —— 
Vpo- (p + o- — 2&) (p + cr) 2 V per 
which is 
if 
, , • • / X 4 (p + er — 2k) A 4 (cr — p) A 
w — u —(e sin u — e sin u) = sm -1 —A r/ 
(p + cr) 2 Vper (p + cr) 2 Vper 
= c/> — </>' — (sin 0 — sin </>'), 
1 - cos 0 ^ (FM + №+ ilfA) = ^-^y 2 {p (er +p) + &(<r - p) + 2A}, 
1 — cos <fi' = — (Ailf + AA—if A) = —yj {p (cr + p) + & (cr — p) — 2A}. 
In fact we then have also 
1 + COS (f) 
1 + cos </>' 
(J+ay i 0- ( a + P) ~ k O ~P) ~ 2A \> 
(7~+ cr) 2 O + P) - k O - P) + 2A }> 
and thence 
sin | + ^ (Vp (p + cr — &) + V/kcr), sin | 0' = (Vp (p + cr — A;) — V&er), 
cos \ c/> = (Vcr (p + cr — &) — V&p), cos | 9' = (Vcr (p + cr — &) + V&p), 
whence 
sm 
sin 
* = - 24 >+- 
(cr — p) VA 
^ = (7T^) 2 { V/5<r(p + tT “ 2A; ) 
V. 
per 
and therefore 
• , • ,, 4(cr —p)VA 
sm <p — sm <p = r 
sin | (0 - f) = 
(p + cr) 2 Vper ’ 
2JcA 1 ., N p + cr-2k 
, COS £(4>-<£) = - : , 
(p + ff ) Vper 
P + cr 
and therefore 
„:„/i j/x 4 (p + cr — 2&) A 
0> + -)V^ ’ 
which verifies the formula.