LIV. On the Effect of a Change of Polar Distance on the Re- 

 duction to the Meridian of a Zenith Distance observed out of 

 the Meridian. By A CORRESPONDENT. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 



T TROUBLE you with a few lines on the effect of a change 

 -- of polar distance on the reduction to the meridian of a 

 zenith distance observed out of the meridian. Retaining the 

 symbols used in a paper in your last Number, where A and Z 

 are understood to refer to the time of meridian transit; let 

 us suppose that for the hour angle P we have the polar di- 

 stance = A +& A, and the zenith distance Z + #. The cor- 

 rection x to be applied to the zenith distance Z + .r, in order 

 to obtain Z, the zenith distance on the meridian is to be found 

 from these two equations : 



sin L cos A + cos L sin A = cos Z 



sinLcos(A +8 A) +cosL . sin (A + 8A) cos P = cos (Z + x) 



Considering the square of 8 A and oc? as evanescent, we shall 

 find 



cos L. sin A . , cos L 2 sin 2 A n n i 



x = - TJ - 2 sin \ P 2 -- ^^f - cotang Z2sin^P* 

 sm Z sm 2 Z 



sin LA 



The part 8 A is owing to the change of polar distance, which 

 from A +8 A at the time P of the observation, is become A 

 at the time of the body's transit over the meridian, and 



8 A - : rj 2 sin i P 2 is the increase of the first term 

 sin LA 



of the value of x by the substitution of A + 8 A for A, as that 



term belongs to a polar distance A ; whereas the real polar 



distance belonging to the hour angle P was A 



The equation may be thus written : 





cosL. s in(A+SA) _ 2 ^ p _ 



_ 

 sm Z sm Z 1 



The correction of x depending on 8 A is hardly ever re- 

 quired ; but the preceding formula shows how easy it is to 

 take it into account. 



LV. Ob- 



