NO. 6.] INTRODUCTION. CHRONOMETERS. XXIX 



square of the ellipticity, a (1 Q sin 2 u) it follows that A C = a (1 Q sin 2 u) 

 (1 a)r = a [1 Q sin 2 u (1 a)b], when 6 is the Satellite's radius expres- 

 sed as a fraction of a; from observations in recent years this is sufficiently 

 well known. Consequently 



, s . sin y ,., 



cos (y u) =-z~f4 75 v^ =5 1 (*) 



/? [1 (1 a)b Q sm 2 w] 



Here all quantities, except u, are known as soon as a convenient choice 

 of the fraction a is made. The equation gives 2 values of y u, one for D, 

 the other, with contrary sign, for R. The angle i* is considered as negative 

 on the south side of AD. 



It is seen from the same figure that if the breadth h of the segment 

 outside the shadow is measured along the elliptic normal through C, and 

 is the angle between this normal and A C, where, with the same accuracy 

 as before, 



tg = Q sin 2% ... (2) 



then the angle between the normal and the direction of relative motion is 

 90-\-u y-f-, and consequently, if dt is the increment of time and dh = 

 k.dt, 



k = tv.sin (yuQ ... (3) 



The same equation holds good for reappearance, where h increases with 

 the time, for then y u and k are negative. 



The quantity of light received from the Satellite at a given moment may 

 be supposed to be proportional to the apparent size of the illuminated seg-. 

 ment. As the dimensions of the Satellites are between Vas and I /IA of the 

 dimensions of the shadow-section, the curvature of the small part of the con- 

 tour intercepted by the satellite during the observations (which can easily be 

 taken into consideration) has been neglected in the following, because its 

 small influence is very nearly constant for each Satellite and will not disturb 

 the final results. The most extreme eclipses of Sat. IV, where observations 

 of the same D or R made with different instruments may extend over several 

 minutes, the Satellite almost grazing the shadow, must be left out of con- 

 sideration as unfit for our purpose. 



The segment 2 being thus considered as an ordinary segment of a circ e 

 it can be expressed as a function of the breadth h by the series 



