332 On the Measurement of the Progress of an Eclipse 

 The tinie will be given by the equation 



_ —A A,- XX, + T 



v+v- 



in which t indicates the middle of the eclipse. 



If we would have the beginning and the end of the eclipse, 

 we make e = 2d', and the time of each will be given by the 

 equations 



._ — AA,-XX, + ^(AA, + XX,)^+(a;- -f X'')[(r + gdQ'^-Aa-X^] "t- T 



taking the sign — for the beginning and + for the end. 

 The duration of the eclipse will be given by 

 1 -n— V(AA, + xx,)^ + (A/^ + x/0[(t + s^jQ^-A":^ 



i^- v^r^ . 



If we desire the end of the immersion and the beginning of 

 the emersion, we make £ = o, and the tunes will be given by 

 the equation 



_ — AA,— XX,+ V(AA, + XX,)^ + (A/^ + XZ-'JCt^-A'J-X'^) + T 



a;^+V 

 being after the middle of the eclipse if we take the sign + of 

 the radical and before if we take — . 



Let I represent the duration of the total immersion of the 

 ]) , /. e. the time during which she remains completely invisi- 

 ble, and we have 



IT — \/ (^^/ + ^^/)- + i^:' + *■ ^)(t^-a^-x^) 



2 ^ - A, + X,^ 



Lastly: we have the hour to which any enlightened part e 

 corresponds, by making s equal to tliis part, and deducting 

 the corresponding value of t. 



In all these equations we use or re}Teat nearly the same lo- 

 garithms, which very much expedites the calculation. 



Let us suppose that we have measured the chord of distance 

 between the two horns of the moon, which seems to me to 

 admit of more precision ; we have only to make the following 

 additions in the original expressions for the elements, 



c — \ distance of the horns of ]) 



t' = mean time of the observation of c. 



T = 1^ {p + p' —d)-= radius of the section of the conus 

 umbra. 



a=A/ + A /3 = X/ + A j/5= «x a = -^ = !■ 



tang 2 = a sin z = /3 cos i — a 



7/= §x'+ai/ x—axf-^^y' {Biot, Geom. Anal. Nu. 77.) 



