88 Prof. Encke on Hadley's Sextant. 



in which form it agrees with the result deduced from the plane 

 triangles. 



Agreeably to the notation here given, we have 

 6' = 2«', c' = 2p'A', b'l = 2p' F 

 B"= Qp'A'— Q/ZP'; and from the triangles 

 Qp'A' and Q^ P' we have these equations: 



cos I c' = sin /?: . sin i + cos /c cos i cos /3 

 sin \ c' . sin Qp' A' = cos i sin /3 

 sin h d . cos Qp' A' = cos k sin i — sin /c cos « cos /3 



cos ^ i" = sin k sin / + cos k . cos / . cos a 

 sin i i" sin Qp' P' = cos / . sin a 

 sin ^ i" cos Qy P' = cos ^ . sin /— sin k cos I . cos a. 

 From the latter three we obtain 



cos b" — 1—2 sin a^'. cos P—2 (sin Z cos /i;— sin k cos Z . cos a)' 

 = cos 2a + 2 sin a^sin^* — 2 (sin/cos/t — sin/uCosZ.cosa)" 

 and from the second, third, fifth, and sixth, we derive 



C cos i . cos k sin / . sin j3 

 sin \ c' . sin h b" sin B" = < + cos / cos I sin k sin (a— /3) 



(_ — cos k . cos I sin z sin a 

 hence, having this equation 



cos 2a.' — cos Z»" + 2 sin ^ c' '^ sin i i''- sin B"', 

 we derive this strictly exact formula : 



sin («' — a) sin (a'-f «) = 

 [— cos r- (tang Z sin a)'^ (A) 



4- cos P . cos Z-' (tang Z— tang Z- cos a)^ 

 — cos/^cosPcosZ-(tgZsin/3-f- tg/^rsin (a— /3) — tang ?' sin«)®} 



Tlie quantities i and Z' are by their nature constant as long as 

 the sextant is not changed. The quantity /, however, may 

 change with the angle. Its evanescence depends on two cir- 

 cumstances: 1st, that the axis of rotation is vertical; and 2dly, 

 that the I'eflecting plane is parallel to the axis of rotation. If 

 the first condition is fulfilled without the second, the pole de- 

 scribes a small circle parallel to the plane of the sextant, and I 

 is constant. If the second condition is complied with without 

 the first, the pole describes a great circle inclined to the plane, 

 and I is changeable with the angle. If neither of these condi- 

 tions is complied M'ith, the pole describes a small circle in- 

 clined to the plane, and I is likewise variable; we will as- 

 sume the latter most general case,and denote by y the distance 

 from the pole of the plane of the sextant, of the point in which 

 the axis of rotation produced upwards intersects the sphere, 

 and by u the angle at the pole, between the arcs y and Qp 

 (counted in the order of the divisions) : and lastly, by S the in- 

 clination of the plane of the mirror to the axis of rotation (po- 

 sitive 



