24 PROCEEDINGS OF THE AMERICAN ACADEMY 



of the star, or the disks will touch each other. They correspond to 

 the first and last contacts of an eclipse. The orbit is projected into 

 an ellipse whose major axis, a, equals the true distance of the centres, 

 and whose minor axis, b, equals the distance at the time of greatest 

 obscuration. When ?' = 0.74G, 6=1 — 0.74G = 0.254. For other 

 values of r, b must be determined from a computation of the area 

 eclipsed, by successive approximations, until such a value is found as 

 will reduce the light to 0.416. If x and y are the co-ordinates of the 

 point in the orbit reached by the satellite at the time of first contact, 

 by the properties of the ellipse x = a sin w, and y =^ b cos w. The 

 square of the distance of the centres, or D-, may be written 



2)2 = (1 -\- rY = (x^ -\- If) = a~ sin- w -\- b' cos^ lo 

 = a" — (a- — b'^) cos^ iv. 



Since w = 23^.0, 



(1 -f r)2 = 0.153 a'- + 0.847 b'\ 



Substituting the proper values of r and b, a may be deduced. The 



cosine of the inclination, ^, of the oi'bit will equal -. The three lines 



of Table IX. give the values of a, b, and i computed by these formu- 

 las for the minimum value of r = 0.764, for r = 1.000, and for 

 r = 2.000. There is no maximum value of r, which may be indefi- 

 nitely large. Let li be any large value of r, and \et a = Ji -\- A, 

 b=. E -\-B, and D^ R -\-d ; substituting these values in the formula, 

 D'^^a- sin^ iv -\- b'- cos'-^ ?p, the terms containing E- cancel each other, 

 and we have 2 R d^ 2 RA sin'^ w -\- 2 RB cos'- w, omitting the terms 

 not containing R, since when R is very large they may be neglected. 

 Dividing both sides hj 2 R gives d = A sin'^ w -\- B cos^ lo. When 

 IV = 23°, d must equal 1, and when w =: 0°, B will equal — 0.132, 

 since the arc of the large circle becomes sensibly a straight line, and 

 the segment whose versed sine is 1.000 — 0.132 has an area of 0.416, 

 or the minimum area of the uneclipsed portion. From these values, 

 we may deduce A = 7.300. The two axes, therefore, become 

 R — 0.132 and R-\-7.S00.' The inclination in this case contin- 

 ually diminishes as R increases, and would equal zero if Ji became 

 infinite. 



The residuals which will be deduced below at first led to the 

 belief that the phenomenon might be that of an annular eclipse. 

 This case has therefore been included to show the change effected in 

 the variation of the light, although the residuals are not materially 

 reduced. If the eclipse is annular, the value of r must be 0.764. 



