Fan. 4, 1883 | 
thermometers, the bulbs of which are bent nearly at right 
angles to the stem, and are inserted into small holes in 
the tubes. In order to protect the tubes as far as possible 
from changes of temperature they are wrapped round 
with several thicknesses of cloth, and are further inclosed 
in a wooden box, out of which the two ends of the tube 
just project. During the measurement of a base line each 
rod is supported on two trestles, at one-fourth and three- 
fourths of its length, provided with screw arrangements 
giving slow motions laterally and in elevation. The rods 
are not, however, accurately levelled, and a correction 
has to be made for dislevelment. To measure the small 
angle of inclination each rod is fitted with avery sensitive 
level. One end of the level works on trunnions, the 
other is connected to a micrometer screw by means of 
which the level can be raised or lowered. The bed of 
the level is attached to the top of the box, but in such a 
manner that it can be adjusted truly parallel to the tube. 
The value of each micrometer division was determined 
by means of the meridian circle in the observatory at 
Christiana. It will be seen from the above that, as the 
measurement of a base line proceeds, the following read- 
ings are required for each rod: (1) the contact lever ; (2) 
the thermometers; (3) the micrometer for inclination. 
These readings were taken and booked independently by 
two observers. Both base lines were measured twice, 
once in each direction. 
Before and after the measurement of each base line 
each rod was compared with a standard rod, the exact 
length of which was known, namely : 
= 1727'96641 (I + 0000011476 (¢ — 16°'25)) + 0.00058 
expressed in Paris lines* based on Bessel’s toise, ¢ being 
expressed in degrees Centigrade. It was found that the 
rods were slightly diminished in length during the mea- 
surement of a base line (on an average 0'o05 lines) owing 
to abrasion. An allowance was made for this diminution 
in length. The apparatus with which these comparisons 
were made consists of a massive cast-iron beam, turned 
up at both ends, and carrying two supports fitted with 
rollers upon which the rod to be measured rests. 
One end of this beam is fitted with a fixed steel stud, 
against which the contact lever of the rod under com- 
parison bears ; the other end carries a sliding scale, con- 
nected with a contact lever, and read by means of a 
micrometer microscope. A set of readings consisted in 
first measuring the standard rod, then each of the four 
measuring rods in succession, and lastly the standard rod 
again ; the temperature of each rod was carefully noted. 
For a complete comparison twelve such sets of readings 
were taken. 
The time occupied in measuring the Egeberg base was 
18 days, and the observations for each measuring rod 
occupied 4 minutes ; the Rindenleret base was measured 
more rapidly, namely, 2} minutes per rod, due to the site 
being more level. 
A considerable portion of Part I. is taken up in con- 
sidering the errors to which the measurements of these 
base lines are liable, in estimating the allowances to be 
made to correct these errors, and in computing the 
probable errors of the final results. These errors are due: 
(1) to errors of observation in the actual measurement of 
the base lines ; (2) to the error in the adopted length of 
the measuring rods. 
Firstly, the errors to which the actual measurement of 
a base line is liable are as follows :— 
A slight uncertainty attaches to the micrometer read- 
ings of the levels measuring the inclination of the rods. 
The probable error is computed to be 
Egeberg base 
+ 0°350 lines 
Rindenleret base 
=EyOnTS3inny 
The errors due to the contact levers are next con- 
t A Paris line is defined by 1 Paris line =;4, toise, hence 1 Paris line 
=0'088813 English inch. 
NATURE 
225 
2S ee eet ee 
sidered. It is shown that the error caused by the small 
uncertainty in the value of a degree of the scale over 
which the long arm of the lever moves, is too small to be 
taken into account, but the error caused by uncertainties 
in reading the scale is of sensible amount, and is com- 
puted to be 
Egeberg base 
: + 0°015 lines 
Rindenleret base 
£o014 ,, 
Further, the surface of the steel studs, at the end of the 
rods, is a portion of a sphere whose radius is con- 
siderably less than the length of a rod. Hence an error 
will occur each time a contact lever does not touch at the 
centre of the stud, that is if it makes an eccentric contact, 
and although every care was taken to obtain accurate 
contacts, it is considered that a correction of the following 
amounts should be made— 
Egeberg base 
— 0°351 + 0'175 lines 
Rindenleret base... 
— 0°314 £0°157 ,, 
The next source of error is that due to errors in align- 
ment, these errors will always be negative, and are due 
to the uncertainty in placing the rods in the line given by 
the directing theodolite. This error is computed to 
amount to 
Egeberg base 
¢ ; — 0'294 + o'ror lines 
Rindenleret base... 
— 0'262 £o0'090 ,, 
The computed variation of length of the rods due 
to alterations in temperature is vitiated by several 
errors. In the first place, the coefficient of expansion 
of the rods, as determined by Prof. Lindhagen, is 
affected by the small uncertainty, o‘o00000015. Further, 
the correction for expansion is computed on the sup- 
position that the thermometers do actually indicate 
the mean temperature of the rods at the time of taking 
the readings ; but this is an assumption, and in fact it is 
estimated that the temperature indicated by the thermo- 
meters is the temperature the rod had 20°0 + 59 minutes 
before taking the reading. This estimate is arrived at as 
follows : It will be remembered that each base line was 
measured twice ; the difference between the two measure- 
ments is due to the various errors under consideration, 
and its probable value can therefore be computed ; this 
computed value will contain, as an unknown, the time of 
which an estimate is required. Hence, by equating the 
computed difference to the actual difference the time can 
be found. The total error in the allowance made for 
expansion is found to be 
Egeberg base... 
+ 0085 + 0°52 
Rindenleret base st 
+ 0'071 + 0'250A 
where q = 20:0 + 5'9 minutes. 
Secondly, the errors due to the uncertainty in the 
accepted length of the rods are considered under four 
heads, namely : (1) the error in the length of the standard 
rod ; (2) the error due to the bending of the beam of the 
comparing apparatus (some experiments were made to 
obtain data for the calculation of this error) ; (3) the error 
in comparing the rods with the standard; (4) the error 
due to the assumption that the diminution in length of 
the rods by abrasion is proportional to the length of time 
in use. The probable error of the accepted length of a 
rod during the measurement of the Egeberg base is com- 
puted to be + o’o008r lines, and during the measurement 
of the Rindenleret base + 0'00071 lines. 
Finally, the base lines had to be reduced to the sea- 
level; data had been obtained for this purpose by means 
of spirit-levelling operations. The reduction in the length 
of the base lines due to this cause is 
Egeberg base = — 33°89 lines 
Rindenleret base so “ae = 0°852.,, 
Applying all these various corrections to the measured 
lengths of the base lines the final results are as follows: 
