sin. A' =: sin. A sin. z. 

 sia.p = sin. h sin. x cos. S. 

 sin, k, COS. a sin. (Z? — ^ 

 = cosT? (■5) 



Also, 



sin. /i. sin. z, sin. <$. 



sm. 3r = 



sm. Z>. 

 sin. A. sin. a. sin. P. 



~ sin. Z). (^) 



There is yet a further use of the arc ^ : in observing polar dis- 

 tances with the equatorial, after reading off, the inclination of the 

 polar axis must be ascertained by the declination level. This is 

 seldom found correct, and the difference between it and the sup- 

 plement of latitude must be applied as Index error. In general the 

 level has been used only on the meridian, but this is obviously inac- 

 curate ; for observations made in other hour circles are affected by 

 any irregularity in the pivots of the polar axis or any flexure which 

 it may sustain. The framing of the Armagh Instrument (a plate of 

 which is given in Dr. Rees's Cyclopasdia) unites stiffness and light- 

 ness in the highest degree, yet it bends a little ; which becomes ap- 

 parent in observation from thePD microscopes being supported on the 

 Equator instead of points 90S from it. Theory gives me that this 

 flexure varies as the cosine of P, but as one of the cones which com- 

 pose the polar axis is not as much drawn by its screws as the other 

 three (which I find by heating each separately, while a lucid point is 

 bisected by the PD wire^ the origin of the angle is thrown to the 

 westward, and the correction to be applied to the reading is 



= — ll"Xcos.(P''— 30°.) 



This of course is eliminated by reversed observations, and in 

 other cases may be computed ; but the axis-error is more satisfae- 

 torily determined by observing the level on the same hour circle 



VOL. XV. c 



