100 ON THE FORMULA FOR CALCULATING 



circumstances — similarly |3 Ursae Minoris and 3 Cephei, the two stars in 

 former editions of the Nautical Almanack nearest the Pole, have 1' 23* 

 and 1' 12^'' for their stationary periods. 



It must, however, be remarked, that the hypothesis, on which that 

 enquiry is conducted, is not rigorous ; for it is therein taken for granted 

 that the same vertical circle will pass through the upper and lower 

 positions of the star at equal lapses of time before and after the Maxi- 

 mum, an assumption which, though perfectly admissible for the end 

 therein proposed, will not bear to be much extended : as for instance, 

 suppose it were required to determine what would be the effect on the 

 Azimuth, if instead of the precise instant of the Maximum, the observa- 

 tion were made at any time before or after that phenomenon. 



To this end let PSZ be the polar triangle 

 right angled at S, and let be the place of a 

 star before or after arriving at >S' — Draw the 

 Arcs of great Circles PS', SS', ZS', and then 

 since PSS' is Isosceles, if a perpendicular were 

 drawn from P on SS' it would divide that side, 

 and also the angle at P into two equal and 

 similar parts, so that if h P, h Z denote the 

 variations of the hour angle and Azimuth, we 

 have 



1st. Tan P>S'*S' = Cot 1 5 P Sec 

 2d. Sin h SS' - Sin ^ h P Sin PS 



Hence, because of the right angle at S, we have sin ZSS' = cos PSS' 

 and cos ZSS' — i sin PSS' and therefore the general equation becomes 

 (vide Woodhouse's Trigon. page 157 — 3d edition). 



