PREDICTION OP SOLAR ECLIPSES. 



157 



But by (3) 



cos z'=sin <p' sin o' + cos <p' cos Ô' cos t ; 

 hence 



Up cos z dxp' ' = cos t dy — sin d' sin t dx ; 



fJp cos z dL = sin t tan <p' dy + {cos <p' cos o' + sin tp sin d' cos t) 



wo y 



Substitute the values of dx and dij from (35) in the above expressions, 

 and let // and H be determined by the formulae 



dx 



h sin H=sinN' 

 h cos H=cos N' sin 



r) 



•(36) 



then, we get, finally 



D 



•(37) 



± cos z (p d(p') = + -jj- h sin {H + 1) 



± cos z dL = — jj— [h cos (H + 1) tan <p' + cos o' cos N'] 



where dq>=p(l +e*cos*<p)a<p', neglecting the term of the fourth order in e. 

 log (l + e 2 cos 2 <p) may be found from the table III as the argument 

 of <p. 



This approximate method is not accurate, when cos z is very small, 

 that is, near the extreme points of the curve. These extreme points 

 may, however, be determined from formula} exactly similar to those 

 for penumbra (§10), of course putting in the value of J), proper for 

 the total eclipse. 



