330 On the Measurement of the Progress of an Eclipse 



t = the time before or after the instant of calculating 



the longitude of ]) 

 T = time for which the above elements (from the Naut. 



Ahnanac) are calculated 

 T'= mean time of the observation. 

 To determine the relative orbit of the D we have 



X = A' y=X' A' + a\' = b 

 by which latter equations we may determine a and b ; but we 

 need not recur to these equations, observing that when 



/ =0 X =A ?/= X 



t=l x=A+"a' j/ = A + >'' 



/=-^ x=A + ^A'3/ = A-i-iX' / \ .,.vit 



whence, the general expres- 

 sion x= A + /A' y=.K + tx'y 

 which gives us die value 

 of X and y in time, and en- 

 ables us to determine the 

 place of the D at each in- 

 stant. 



To determine the distance of the centre of the moon and 

 cone of umbra at any moment, we have (calling the radius of 

 the cone of umbra ^) 



D' = j:'+y = (^/+V)^' + 2(AA, + ^\)^ + AVx' 

 OD=D T)+Db-q=e D^=(e + f + ^i')' 



(, + g_^')^ = (A;-l-A,)^^ + 2(AA+XX^)/+A»+A^ 



To find the value of g we have 



TT oo' TT' tang rf 



ST= 



SS'= 



TR= 



TT' 



Xxagp 

 JT:JR::TT': RR' 



TT' , jrp . . TT' (tang </— tang;)) ^ ,prjv 

 t«ng ;» ■ ■ ■ tang p 



tang p tang p' 



JS, JT: : SS'-. TT' JS-JT: JT : : SS'-TT': TT' 



TT' 



JT= 



I 



tang d — tang />. 



JR = 



