16 Prof. H. H. Turner. 



" Nautical Almanac Office, 



" March 26, 1898. 

 " Dear Turner, 



** You will find the deduction of the formulae (used in 

 'N.A. Circular No. 16 ' and others) in appendix to N.A. 1836, p. 117. 

 These equations will give sensible accuracy for a distance of about 

 50 miles from the point for which they are computed. In this case 

 the distance is about 150 miles. 



" I find that the most expeditious way of getting accurate results 

 in this work, is to use the Besselian Elements (page 4 of circular 

 No. 16). You can infer the time of middle of eclipse to the nearest 

 minute from the Table on p. 3 of circular, and then 10 minutes 

 work gives you the times of beginning and ending of totality with 

 sensible accuracy. 



" Of course for accurate determinations of times of first and last 

 contacts (partial phase) special computations for the approximate 

 times of these contacts would be necessary. 



" Yours very truly, 



"A. M. W. DOWNING." 



The following remarks on .the geometrical significance of these 

 formulas may not be out of place. 



The time of a contact t at a place of geocentric latitude I and 

 longitude X is given by a pair of equations of this form 



cos u = A + B sin l + C cos I cos (X-f-D) 

 = E + Fsinw + GrsmZ + HcosZcos(X + K). 

 Write these in the form 

 cos ta = A + ~L{sin Zsin Z + cos ZcosZ cos (X X )} = A4-LcosPP , 



t = E + F sin w + M {sin Z sin Z^cos Z cos Zi cos (X X^} 



= E + Fsinw + McosPPj, 



where P is the point (Z, X) and P a point whose geocentric latitude 

 and longitude are Z X given by the equations 



C tan Z = B = L sin Z , X + D = 0, 



and PP is the arc of the sphere between P and P . 



Similarly for the point P lt 



In the case of the second and third contacts, which determine 

 totality, the equations only differ in the sign of F : so that the dura- 

 tion of totality is constant when w is constant. The condition u>=o 

 gives us points for which the eclipse is just total for an instant, 

 i.e., the points on the borders of the totality belt. But from the 

 equation 



cos to = A + L cos PP 



