10 ROYAL SOCIETY OF CANADA 



For values of t between 12't and 24'', the graduations of both t and z 

 would fall beyond A and increase the size of the figure: this is avoided 

 by changing the sign of M in (10). "We have then two graduations for 

 sidereal time on Au, and two graduations for bearing on AB ; the second 

 graduations are printed in red to distinguish them. Plate I shows a 

 specimen of the abacus. 



Abacus of the Altitude of the Pole Star. 



The altitude, A, of a star in terms of the latitude, hour angle and 

 polar distance is given b}^ the formula : 



sin h = sin L cos P -|- cos L sin P cos t 

 Let: 



h = L + X 

 then : 



sin Ij cos .'■ + cos L sin x = sin L cos P + cos Jj sin P cos t 



P and X are very small. Developing this expression in terms of the 

 powers of P and x, and discarding the terms which contain powers 

 above the second, we find : 



X =■ P cos t — -17- tan i sin* t 



As before, we adopt a mean value, X^^, for tan Jj.^ Allowing 0'.75 

 for refraction, we may write : 



H = h + 0'.75 



E.xpressing M, L, and P in minutes, we have : 



L -\- P cos t — -^ tan L„ sin' t sin 1' — 7f + O'.TS = 



Putting : 



(1-1) -^= L 



^'-'^J —J — = P cos t — — ^^ tan Lfj sin^ t sin 1' 



the equation becomes : 



(I'O -^— + —^ — H-\- 0'.75 = 



h '2 



The scale of m, (14j, is a regular scale of modulus l^, properly 

 selected, for one minute of latitude. It is laid out on Au, but instead of 

 measuring multiples of ^j, and numbering them in minutes of latitude, 



1 The mean value causing the least maximum error in the altitude is the 

 mean of the extreme values of tan L; it corresponds to Lg = 52°59''. 'the maxi- 

 mum error for t = &>■ or IS/i and for township or 84 is Q' Alb. 



