AT THE OBSERVATORY AT PARAMATTA. 
73 
nation. D is here the difference between the eastern and western altitudes 
combined together ; so that gm ^ 
sin f D 
2 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. 
This formula results from sin x 
sin ^ (g — z) sin ^ + z ) 
sin \ (r + t) sin <p sin 8 
, which serves in ge- 
neral to find the change in time corresponding to that of altitude, and recipro- 
cally ; if we suppose r = 0, this formula becomes the well-known one for 
finding the hour-angle. 
= Jastara 
sin 
sin <p sin 8 
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 ci and h' observed with the mural circle, then 
by the known formula for finding the equinoctial point, 
tan i 0 - a') = tan £(« + «) sin(8 ~ + 8') 
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 At 
corresponding to an increase a of declination, and x 1 the increase of At corre- 
sponding to an increase a 1 of the obliquity a>, then is 
acotu , , a' tan At „ 
x = — — and x = — if the At is not too near 90 or 270 .* 
cos in. tan w 
* Demonstration : tan (8 + a ) cot w = sin (a + x) 
f tan 5 -I- tan a 1 
COt W tang, tana / = Sln a C0S * + C0S a Sln * 
but tan $ tan a = o and cos x = 1 
therefore cot co tan S + cot w tan a = sin a + cos a sin x 
subtract cot w tan 8 = sin a 
remains cot w tan a — cos a. sin x. 
The other formula can be demonstrated in a similar manner. 
MDCCCXXIX. L 
