FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
309 
broke the circuit ; the lower observer then broke circuit ; and both prepared to 
observe Coincidences of the pendulums. On examining' the interruptions of signals, 
no difficulty was found in confronting the corresponding observations. In one 
instance only (between Swings 16 and 17) was the comparison totally lost, in conse- 
quence of one of the needles sticking fast. The observation of the time of the start 
of the needle is not very delicate; and this part of the operation would be made 
much more exact, by causing each clock to register its own seconds upon a common 
revolving barrel, in the manner of the American transits. 
26. The observations on each side which had no corresponding observations on the 
other side being struck out, the means of all the times of the remaining observations 
in each group were taken, and the difference of means was formed. As a check, the 
differences of the times by the two clocks for each individual signal were formed, and 
the mean of the differences was taken. In this manner there were formed, for every 
group, mean corresponding times of Shelton (the upper clock) and Earnshaw (the 
lower clock) ; the differences of these, from group to group, gave the corresponding 
changes of clock-indications by the two clocks during the same period of absolute 
. , , . ^ Earnshaw’s change of indication , . . 
time ; and the quotient ; — ; ^ ^ — gave the apparent ratio of 
^ bHELTON s change ol indication ^ ‘ 
rates. The logarithm of this ratio was formed, by means of Callet’s logarithms, to 
eight decimals. 
27 . A very cursory examination of these ratios showed that there was a consider- 
able personal equation in the observation of the galvanometer-signals. Though (from 
the nature of the combination to be hereafter described) this scaicely produces an 
appreciable effect on the ultimate result, I thought it desirable to ascertain approxi- 
mately its magnitude, and to apply the corresponding correction. I proceeded as 
follows: — In every instance in which the signal observations at the beginning and at 
the end of a Swing were made by different observers, I compared the logarithms of 
apparent rates of during that swing with the logarithms for the preceding 
and following swings (in three instances, however, with only the preceding swing) ; 
and formed the excess of the logarithm of the intermediate swing above the mean of 
the preceding and following logarithms. I'his numerical excess may be compared 
with a symbolical formula, in which the symbols represent, not the actual error in 
time committed by each observer, but the logarithm (in 8-figure units) of the influ- 
ence of his error on rates deduced from comparisons at 4-hour intervals. Twenty- 
seven equations were thus formed. By the method of minimum squares, these were 
reduced to six, of which (from the nature of the case) one was unnecessary, their sum 
being identically equal to zero. Assuming D (Mr. Dunkjn’s error) =0, the others 
are found by solution of the equations. Though the equations are not very favour- 
able, they suffice for giving very good corrections, and they show in particular that 
Mr. SiMMONDS recorded his times too early by nearly half a second. The following 
