154 



SHIN H1RAYAMA 



I cos d x = (1 — c) cos d' 

 lsind 1 = sind' 



sin ~B — -ry\ cos (f x sin t = sin B 

 cos B sin A = -rrj ; cos (f x cos t=cosB cos (A + d x ) 



III 



cos B cos A= cos z ; sin <p x 



cos B siii (A +o t ) 



tan<f = —, — ^ L = t—T +e 



1 — c 



(29) 



13. To find the time of the beginning and end of central eclipse upon 

 the earth. 



Here the cone of the umbra touches the earth and the central 

 eclipse is observed at sunrise and sunset, respectively, that is, cos z'=0; 

 or, by (3) and (28), 



x = Ftp sin v 



y = Up COS V 



.(30) 



This is the condition to be satisfied by x and y. 

 Let T be an arbitrarily assumed epoch ; 



T—T +~ be the required time of beginning or ending ; 



x y be the mean hourly changes of x and y. 



Then, 



x +x'z=rip sinv 

 y +y / T=IIp cosv 



Hence, putting m sin M= -rf- n sin N= -n- 



m cos M- 



lip 



n cos N= 



Up 



• (31) 



we get, as usual, 



