198 PROCEEDINGS OF THE AMERICAN ACADEMY 



Assuming the well-known relations, 



d (cos 6,) . „ ._„. 



rf(tanei) 



de^ 



= sec^ dj^ (74) 



dividing (73) by (74), multiplying the quotient by (72), and writing 

 6 for d^, we have, with sufficient accuracy, 



d (cos By) Tsin^ 6 cos 61 d^r 



dc ~L Ix ■ J ^ ^ '' 



The vertical distance between the centres of the ordinates i/y and i/^ 

 being very approximately s cos d, we now have 



^1 ^v fsin^ e cos- e~l . d-r ,_„. 



s (cos d, + cos 6,) = - I J s^- - (76) 



and therefore 



, r <? a sin2 e COS- d~\ o d-r /nn\ 



2/1 + ^2= [cos' ^ J « ^ (77) 



The value of jw is 0.99974. Consequently, it may be taken equal to 

 unity without appreciable error, and then 



yi + ^2 = cos^ s' - (78) 



As 1/y and y., are any pair of opposite ordinates of the curve which 

 defines the outline of the apparent sun ; and as dr is the distance ik, 

 or, in other words, the ordinate of the centre of gravity of the appar- 

 ent sun ; we have, by reasoning identical with that employed in deduc- 

 ing equation (63), 



^ir 9 ^=^ — i'"' 

 dr = s'- :i^,.- 2: cos* d (79) 



in which m is the number of points on the sun's limb at which ordi- 

 nates have been computed. If we assume these points to be equidis- 

 tant upon the apparent sun, and take m = 12, as in the measurements 

 of the photographs, then 



c/2r 



dr = 0.375 s^ — (80) 



nC* 



