AT THE OBSERVATORY AT PARAMATTA. 



73 



nation. D is here the difference between the eastern and western altitudes 

 combined together; so that • is a constant factor : and a table of the reduc- 

 tions X may be constructed with the sole entry of the half interval, whereof the 

 azimuth is a function, and given opposite to it in the table of azimuths and 

 altitudes ; for hour-angle and half interval are here equivalent. A formula 

 expressed in terms of the half interval only, would probably be rather com- 

 plicate. 



sin i (? — z) sin i (? + z) , . , 



,■ I / , f\-,„^-„x > which serves in ge- 



sin -J (t + /; sm ^ sin ft ' o 



This formula results from sin x 



neral to find the change in time corresponding to that of altitude, and recipro- 

 cally ; if we suppose t = 0, this formula becomes the well-known one for 

 finding the hour-angle. 



sin 



1 ^ _ / sin|(Z+g)sin^(Z-z) 

 ^ V sin ip sin S 



used when no corresponding altitudes can be had. But by combined alti- 

 tudes the effects of any unknown error of the instrument are avoided. 2 x 

 applied to the times on either side of the meridian reduces the combined 

 altitudes to equal altitudes. 



Method of finding the Sun's Distance from the Equinox. 

 Suppose a and a' the sun's distance upon the equator from the equinox cor- 

 responding to the declinations S and h' observed with the mural circle, then 

 by the known formula for finding the equinoctial point, 



1 / )\ 1 / I r\ sin (8 — S*) 



tan ^ (a - «') = tan i (a -f a) ^i^i^ + i') 



is the obliquity of the ecliptic eliminated. But this is no advantage, as the 

 obliquity is better known than the declination. Suppose x the increase of M 

 corresponding to an increase a of declination, and j?' the increase of M corre- 

 sponding to an increase a' of the obliquity », then is 



a cot u) -, , a' tan JR. 



X = ^^ A> and X = —. 



cos iK tan CO 



if the JR. is not too near 90° or 270°.* 



• Demonstration : tan (J + a) cot w = sin (a + x) 



f tan J + tan a 1 

 cot w I i^tan^.tano / = «'" « COS ;>; + COS a sin x 



but tan S tan a =: o and cos x = I 

 therefore cot co tan J + cot co tan a = sin a 4- cos a sin x 



subtract cot a; tan J = sin a 



remains cot u> tan a = cos a sin x. 

 The other formula can be demonstrated in a similar manner. 



MDCCCXXIX. L 



