318 foote: visibility equation 



Equation (2) may be solved for g (s) by using the reciprocal 

 integral (3) derived by Pincherle, 2 



9 



(s) = ^e sl e p ' /l dt . (3) 



the integral starting at — c° below the axis of reals, encircling 

 all singularities in the finite complex plane and ending at — » ■ 

 above the axis of reals. On performing this integration and 

 expressing g (s) in terms of V (X) one obtains: 



r "- b — f bc^ 



fTi \ a 2 I \n \ n-\ 



_i 

 where R = A e " cf 1 = constant. 



If c 2 =14,350 and the values of a and b obtained by Ives and 

 Kingsbury are substituted in (4) the final visibility equation 

 takes the form. 



y W = R.-^x«£(— ?Y-L_ (5) 



^ \ X / \n \ n-\ 



The series in (5) is of the form Xx n / \n \ n — 1 where x is over 

 100,000. The ordinary methods of making such a series more 

 rapidly convergent, such as Euler's transformation, etc., do not 

 appear to assist materially in the present instance, and it is 

 not evident that (5) can be expressed in closed form. Accord- 

 ingly unless (5) can be put in a rapidly convergent form its main 

 interest is in the method of its derivation, this being the first 

 example in which a visibility equation has been derived from a 

 luminosity equation. The ordinary procedure is the converse 

 of this. 



2 Mem. Accad. Sci. 1st. Bologna, IV, vol. 8. 



