58 Mr Bowdich on the Measurement of an Eclipse 
To determine the relative orbit of the }) we have, 
x — ay + h x — L y — x A -f tix —b 
x = A' y — a' A' -j- a a' — h. 
by which latter equations we may determine a and h ; but we 
need not recur to these equations, observing, see Plate IV. Fig. 2, 
that when 
t— 0 x— A y — a 
= 1 ^ = A + A' ?/ = A + A' ' 
t—\ # — A -j- |A' y — a + |a'; 
whence the general expression, x A -j- t a' y=z},-\- tx\ 
which gives us the value of x and y in time, and enables us to 
determine the place of the )) at each instant. 
To determine the distance of the centre of moon and cone of 
umbra at any moment, we have (calling the radius of the cone 
of umbra ^), see Fig. 3. 
D* — . X? -}- y 2 = (A, 2 -f A 2 ) t 2 -f 2 (a A, -f AA,) t A 2 + A 2 
0 D = D D -f- D6 — ? = s D 2 = (g + C + ^') 2 
( 8 + e — *)* = (A 2 + A 2 ) f + 2(aA, + AA y ) t + A 2 + A 2 
To find the value of we have 
TT' _ T T' tang S TT' 
tang p 
ST = 
SS' = 
TR 
tang p tang p tang p' 
JT:JR::TT':RR'; JS: JT:: SS':TT'; JS—JT: JT::SS'—TT':TT' 
TT' JT TT 7 (tang $ — tang p) . TT , j T __ TT' 
tangjp 
JR = JT 
tangy? 
tang § — tang p 
■TR 
TT' 
TT' 
tang d — tang p tang p' 
JB — ( ten g P'— ten S l ± tang p) TT' 
tang p' (tang £ «— tangy?) 
TT' (tang p — tang S + tang;/) TT' TT< . 
tang p ’ tang p' (tang £ — tang p) 
tang 
RR' 
tan g$- rp Tl = 
RR' = TT' (tang p — tang j + tang p) 
tang p f 
tang p — tang d -f tang p\ 
tang g = tang^? + tang p e — tang £ p +p' — ^ ; 
(A^-J- A y 2 ) ^-f^AA^-J-AA,) t -{- A 2 + A 2 = y 2 = (e — d' -\-p -J- p ' — 5) z 
a , o^A. + AA,) — Z + p+p' — J)* — a 2 — >? 
