450 
GALBRAITH’S ASTRONOMICAL OBSERVATIONS. 
the horary distance from the meridian, when 
the observation is made, extend to 30 minutes 
of time; though, no doubt ibis error is dimi- 
nished when combined with observations 
made near the meridian. Again, when the 
latitude is 40 **, the declination 20 °, and the 
zenith distance also 20^, the same formula to 
three turns gives results incorrect to about 
half a second in excess, while the first two 
turns, or those commonly used, give an error 
of about 4 in defect. Lastly, when the lati'ude 
is so hi:^h as fifty degrees, the declination 
still 200 . and the zenith distance 30 «. Delam- 
bre’s formula to these turns gives, at 30 
minutes distance From the meridian, correct 
results ; while two turns give a small error of 
about half a second in defect. Assuming dif- 
ferent numbers somewhat analogous but with 
similar relations, the same conclusion would 
follow. It may, therefore, be concluded that 
when the zenith distance in mean latitudes 
amounts to about 30 ®, two terms of Delam- 
bre’s formula, or their results in tables, are 
sufficiently correct for practical purposes at a 
horary distance from the meridian of about 
30 minutes, and then the calculation for the 
mean of a considerable number of repetitions 
is comparatively simple. 
Instead of Delambre’s formula, or tables 
derived from it, some practical astronomers 
recommend a table given by the late Dr. 
Thomas Young, consisting of natural versed 
sines, which are nothing more than the first 
part of Delambre’s table in a less conve- 
nient form, and requiring the additional trou- 
ble of employing a constant log within to 
convert them into Delambre’s numbers in every 
operation, without any equivalent advantage 
in any respect over the other method;* in the 
words of Dr. Pearson, “ Dr. Young having 
simplified (complicated he should have said) 
the preceding formula by omitting the second 
term,” &c. Now it has already been shown 
that the second term cannot be admitted un- 
less the zenith distance be considerable, not 
less than 20 °, or 30 °. at 30 minutes from the 
meridian, or the object to be observed be a 
circumpolar star not very distant from the 
pole, in mean latitudes, and of this any ob- 
server may easily satisfy himself. 
If, for example, at London circummeridian 
observations be made extending to 24 minutes 
from the meridian, (the extent to which Dr. 
Young’s table has been carried, in a tract 
published by Messrs. Troughton and Simms,'' 
to determine the obliquity of the ecliptic at 
the summer solstice, the first two terms of 
Delambre’s formula would be sufficient, 
though Dr. Young’s table, recommended by 
Dr. Pearson, and more lately approved by 
Mr. Simms, woixld, at 24 minutes from the 
meridian, give results erroneous to about 7", 
a quantity quite inadmissible, though this 
problem is just such a one as is, under the 
given cii-cumstances, suited to the smaller 
class of altitude and azimuth circles, gene- 
rally in the hands of astronomical students, 
and repeating circles previously alluded to. 
♦ The author of these remarks has endea 
voured to remedy this elsewhere. 
If, however, the horary distance from the 
meridian be, under such circumstances, re- 
stricted to 12 minutes of time, which will 
admit of a sufficiently extensive number of 
repetitions useful to exterminate casual errors 
of observation, reading; and dividing ; two 
terms of Delambre’s formula will be fully 
adequate for the purpose, while the error 
arising from the use of Dr- Young’s table 
will not exceed half a second. 
With regard to the most eligible size of an 
instrument, it is difficult to come at an accu- 
rate conclusion. That must, in a great degree, 
be regulated by the purposes for which it is 
intended- 1 am strongly inclined to think, 
however, that circles of moderate size, and of 
the most simple yet substantial construction, 
are the most likely to give satisfaction. Very 
large mural circles that do not revolve in azi. 
muth, especially when employed to make 
observations on the sun, are liable to sulFer 
unequal expansions from heat on that side next 
the sun, being acted on powerfully if not 
shaded, which it is difficult to do completely, 
while the opposite side is slightly affected 
by its position in the shade of the other, and it 
is doubtful, in my opinion, whether a consi- 
derable number of microscopes except under 
|)eculiar circumstances will correct the errors 
arising from this cause. On the other hand, 
a much smaller instrument revolving in azi- 
muth, and by that means having its different 
sides, though as much shaded as possible, 
exposed partially in succession to the sun will 
expand much more equally, and when the 
mean of three or four verniers or microscopes 
read at each observation, which may be repeat- 
ed two or three times in pairs of double observa- 
tions, within the j)roper time near the meri- 
dian ; on the principles of the theory of proba- 
bilities, the errorsarising from all the different 
causes affecting the accuracy of the results 
must, in a great degree, destroy each other. 
Though this conclusion is the most proba- 
ble in reference to a steady well constructed 
instrument, yet it must be received under 
certain qualifications, since too much praise 
has doubtless been lavished on the omnipo- 
tence of Borda’s repeating circle, especially 
by foreigners. M. Biot, afier explaining the 
principles of the repeating circle, says, ‘‘ Let 
us examine, now, in what respect the repeated 
multiplication of the angle proves advantage 
ous. It would have none, if the divisions cut 
upon the circle were mathematically exact, and 
if the observer could dii’ect the interst-ctions 
of the cross wires in his telescope perfectly cor- 
rect, for, in that case, one observation would 
give the zenith distance exact. But as these 
conditions cannot be accomplished in practice, 
the repetition of the angles supplies the defect 
by compensations. With regard to the error 
of the divisions, it is clear, that the arcs mea- 
sured, follow without interruption upon the 
limb, in such a manner that the print of 
the limb, which is the termination of the 
previous observation, becomes the origin of 
the succeeding. From this it follows,” says 
M. Biot, “ that the sum of the observations, 
or the whole arc passed over by the verniers, 
comprehends no intermediate error, but the errors 
