90 Prof. Encke on Hadley's Sextant. 



s = 2«, and make the abbreviations which the nature of the 

 case allows, we shall have for the correction of s this equation; 



As =— 2 iani^ \s{r- + sec hs{l cos {^s — ^) — icoi I sf}. (B) 

 The comparison of this formula with Bohnenberger's rules 

 will show, as miglit have been expected, a perfect agreement. 

 For Z: = / = we have by (B) As = — e'-tang \ s (Bohnen- 

 berger, p. 123). For i = and / = 0, we have by (A) 



, ,„ cos a' — sin (a— (3)' ., 



«'— « = k- ^-—- ; consequently, 



2cos/3cos(s — j3) ,, 2 cos /32 ,, . ^ -, ,„ 



A s = : — ^^ ir = k^ + sm 2 p h 



sin i tang s 



The last term being constant for all angles, does not come 

 into consideration, because it is the same in determining the 

 index error. The formula then agrees with Bohnenberger's, 

 p. 132. In the case treated by Bohnenberger, § 88, it is to 

 be observed that if k = I, and / constant, and if besides the 

 line of collimation of the telescope is to be in the plane perpen- 

 dicular to both mirrors, i is variable with the angle. For every 

 point P' of a small circle gives with the fixed point p' another 

 great circle, in every one of which A' is to be situated. If i is 

 determined agreeably to this condition, we shall obtain 



cos (^is- r=) . f 



tanff t = —-; tang / 



•-' cos -^s " 



By this equation the last term in (B) disappears altogethei*, 

 and the correction becomes 



As = — 2P tang ^s (Bohnenberger, p. 129.) 

 The formula might, therefore, likewise be written thus : 

 Let ?'• be the elevation above the plane of the sextant of the 

 point in which the great circle passing through the poles of 

 both mirrors intersects the vertical plane of the sextant in which 

 the object seen by direct vision is situated ; and we have. 



As = — (^•— e'^) tang | s — 2 tang is. P. (C) 



It is apparent from this formula, that if the errors had not 

 been considered at the same time, but each singly, and their 

 effects had been added together, the only difference would 

 have been that /' would have been assumed = 0. The angle (3 

 which we have introduced is, indeed, not immediately used in 

 measuring angles. But besides its use in the formulae for cor- 

 rection, it is used in some applications of the instrument ; so 

 that it is worth while to ascertain it for every sextant. Thus 

 we can determine by it the limit of the angles which can be 

 measured by the sextant. All reflexion ceases, when the great 

 mirror is inclined to the small one under an angle of 90° — /3. 

 The limit of measurable angles is thei'efore = 180^ — 2/3, and 

 for this reason /3 is in all sextants of nearly the same magnitude. 



It 



