196 



PROCEEDINGS OP THE AMERICAN ACADEMY 



As the assumed artificial sun is circular in form, it has no minor 

 axis ; and therefore d may be put equal to zero, and we have finally 



2 ^ = <^ 

 e = IP 



2 ^ = ^ 



fe = — 2" COS 5 (7? — E') 



(65) 



from which the values of 8i/ and dx can be found by means of a table 

 of natural sines and cosines, and a Crelle's multiplication table ; no 

 logarithms being required. 



As it is now evident that the assumption of an elliptical outline for 

 the apparent sun is unnecessary, we proceed to determine for dr a more 

 accurate expression than equation (16). With this view, let abed, Fig. 

 5, be the outline of the true sun, and i its centre, through which the 



Fig. 5. 



vertical circle ac passes ; and let ehfd be the outline of the apparent 

 sun, and k its centre of gravity. Then, if gh is any diameter of the 



