1 5 ß SHIN HIRAYAMA; 



Since x'—ç'=n'sinN', 



y —-rf = n cos N', 



we have 



*=*£ < 34 > 



15. To find the northern and southern limits of total eclipse. 



We may compute this curve by the previous method by simply 

 putting in I) the proper value for the total or annular eclipse. But 

 it is more convenient to deduce these points from the previously 

 computed curve of central eclipse. The co-ordinates of the points on 

 the central curve corresponding to the time T being £=x y=y, those 

 for a point on the limiting curve may be denoted by x + dx and y + dy, 

 because the two limiting curves of total or annular eclipse lie very 

 near to the central curve. These being substituted for £ and r { in the 

 equations 



Ç=x^£Dcos N' ; 

 y=y+D sin N' ; 



we have 



dx= T-D cos N' ) 



(35) 



dy=±DsinN I 



Let us now differentiate the following equations 



-tj—=cos f sin t ; -A— — sin <p r cos o'—cos ip' sin o' cos t. 



Then, since we have dt = dL, 



jj— = — sin <p' sin t df' + cos <p' cos t dL ; 



dy _ 

 Up- 



{cos ip cos o +sin tp' sin o' cos t) dtp' + cos <p' sin a' sin t dL. 



