104 



ON THE FORMULA FOR CALCULATING 



It will now be in place to explain how the above formula can be 

 simplified, to which purpose let S', S^'' be two positions of a circumpolar 

 star at equal lapses of time before and after 

 the maximum, so that the angles at P, viz. 

 SPS', SPS''^ may be equal to each other ; then 

 if the diagonal *S' S''^ of the Quadrilateral formed 

 by the pole and the three positions of the star 

 be drawn, it will intersect the other diagonal 

 PS Sit right angles in <r and the two vertical 

 circles ZS', ZS^' (produced in the former case) 

 will intersect the great circle PS obliquely in 

 two points a', a" making two very small tri- 

 angles S' G s' , 8" a each equal and similar 

 to the other and right angled at a. Now in 

 A ZS (s which is right angled at S, if we denote 

 as before by ^ the variation or SZa', we have 



1st sin SZ zz cot 8 Z. tan Sa'; or tan S a' sin SZ. tan h Z 

 S a' — sin SZ. ^ Z zz sin P. cos X. d Z (a) 



2d cos ff' — sin ^ Z. cos S Z 



.J- ff'j = cos SZ.hZ; and a' - -—cosSZ. h Z 



(in ^S'tra') 

 (3) 



3d /. S'(^' or Z. S' z= cos SZ. IZ 



And since tan a a' ~ sin S' a. tan S' 



Therefore a a' — S> s. cos SZ. h Z 

 Again in A PS' g right angled at a we have 

 1st cosh P =. cot PS', tan Pa; but tan P a ~ tan (PS—Sa) 



ta?i PS — - tan S ff 



" 1 -j- tan PS. tan S a 



So- 



=z COS h p. tan PS 



tan PS 



COS' PS 



— COS h P. tan PS 



