PREDICTION OF SOLAR ECLIPSES. 



147 



6. To find the rising and setting limits of the eclipse. 



By these limits we mean the curves upon which are situated all 

 those points of the earth's surface where the eclipse begins or ends 

 with the sun in the horizon. 



This gives sin z'=\. Now, let it be required to find the place 

 where this condition is satisfied at a given time. Since $=IIp sin v, 

 r)=11p cos r, we have (4) transformed into the form 



D sin a—x — lip sin v ; 



D cos a = y — [J[) cos v. 



Let 



msinM=x ) 



\ (12) 



m cos M=y ) 



then from the equations 



D sin a—m sinM — lip sin v ; 

 D cos a=vi cos If— Tip cos v ; 

 we deduce, by adding their square«, 



Z) 2 =m 2 -2ra Up cos (M — v)+ fly ; 



. 2 M-v , ,., x jß-{m-IIpf 



2 sin- — t — = l — cos(M—v)= , ,, — &- . 



2 2wi Up 



If, then, we put ^ = 31— v, we have 



„•„ l ;_-. / {D + m-l/p)(D-m+//p)~ ) 

 sm-X-±J~ — iSmjS [ ( i 3) 



v = M±X J 



in which -r- X may always be taken to be less than 90°, but the double 

 sign gives two points satisfying the given condition. For the first 

 approximation, assume p to be equal to the radius corresponding to a 

 mean latitude of 45°. With the value of v thus found, we have, for 

 finding latitude and longitude of the required point, the formulae — 



