PREDICTION OF SOLAR ECLIPSES'. 



149 



For the beginning and end generally we have, therefore, 



(D+np)sinM=x ; 

 {D+IIp)cosM=y. 



Let T be the time where the conditions are satisfied, and put 



T=T +t; 



where T is the epoch of the eclipse tables, for which the values of 

 x and y are x and y . Then 



Puttin« 



x = x + x't ; 



m sin M = x ; 

 m cos M =y ; 



the above conditions become 



n sin N= x' 

 n cos N=y' 



(D+flp) sin M=m sin M +z. n. sin N ; 

 (D + IIp) cos M=m cos M Q + v. n. cos N ; 

 (D + IIp) sin (M-N)=m sin (M -N) ; 

 (D + lip) cos (M-N)= m cos (M -N)+nz 



so that, if we put M—N=0, we have 



, m sin (M —N) 



D+IJf 

 D+IIy 



?n 



cos é—^^- cos (M—N) 

 n n 



.(15) 



• (16) 



T=T +t 

 in which cos <p may be taken with either the negative or the positive 



For the two interior contacts we have- 



