26 DEVILLR ON DETERMINATION OF TIME BY TExiNSITS. 



Expression of p 

 Let 



lu triaugle S'PM, we have ; — 



tan ^=sin (J + x) l:in n 



which, by neglecting the powers of p aucl n above the second, becomes 



(5) p=n sin (J + .1;) 



Developing sin (J + '■) and neglecting the powers of the small arcs above the 

 second — 



(6) p=:7i sin J + nx cos J 



lu triangle 3IPS, we have, neglecting the powers of p and x above the second — 



x^p tan a 

 Substituting this value of a; in (6) — 



p=^n sill A ■{■ np eus A (an ô 



or 



n- 

 (*7) p^^n sill J + i, sin 2 J tan S 



A table of the values of ;y has been calculated with this formula and published in 

 the " Manual of Dominion Lauds Surveys." The value of (V used was 88" 41' — . For any 

 other value of a', p, as found in the table, must be multij)lied by 



cos Ô' 

 cos 88° 41' 



The arguments of the table are d, the declination of the time star and J, the difference 

 of right ascension of the two stars. 



When the stars are not observed at the same time, let 2" and T be the times of obser- 

 vation of Polaris and the time star respectively, then : — 



A=^a-a')-{T-T') 



The time t, given by (3) or (4), is to be subtracted from or added to the time star's 

 right ascension, according as J is less or more than 12 hours. 



The table of p affords a ready way of finding an approximate value of the azimuth : 

 multiplied by the secant of the pole's altitude, it gives at once the azimuth at the time of 

 observation 



Az^p sec <l> 



The Manual of Dominion Land Surveys prescribes that the azimuths of lines are to 

 be obtained by means of time observations of Polaris and the method described above 

 was devised to enable surveyors to rate their chronometers for such observations. For 

 that purpose, the accuracy of the formula and table are ample : in ordinary circum- 

 stances, the error resulting from their use will seldom exceed one quarter of a second. 



