488 Proceedhigs of the Royal Irish Academy. 



their results by means of the clock-rate (found independently of per- 

 sonal errors) to a common epoch (O*" Sid. time). The differences be- 

 tween the resulting clock-errors for this epoch are then equal to the 

 equations bet-sveen the observers (with reverse sign, according to the 

 mode of designation chosen in this paper). 



During deteiTainations of longitude, the personal equation between 

 the observers has often been eliminated, by letting them exchange 

 their stations, and begin observing again. The mean of the two result- 

 ing values for the difference of longitude is then the exact value, and 

 half their difference is equal to the personal equation. This method 

 supposes the personal error of both observers to be constant during the 

 whole operation, but as this is not always the real case, it is the safest 

 to determine the equation in one of the usual manners, best by letting 

 the observers compare themselves with both instruments used at 

 the two stations. Sometimes the equation has also been found by 

 letting the two astronomers observe simultaneously at the same place, 

 each using his own instrument ; but as the uncertainty in the deter- 

 ]nination of the instmmental errors gets a considerable influence on 

 the result, this method ought never to be used. 



But the comparison between the habits of two observers does not 

 give us any information about the absolute error of each of them, 

 which it, of course, is of far greater interest to study; and it has, 

 therefore, duiing the last twenty years, been attempted by several 

 astronomers to construct an apparatus by which the personal error of 

 an observer could be found. As we shall often, in the following pages, 

 quote results obtained by such apparatuses, it will not be superfluous to 

 give a short description of them, only entering a Kttle into details 

 respecting those with which important and trustworthy results have 

 been obtained.* 



Hartmann has described an apparatus in Gn^nert's "Ai'chiv fiir 

 ITathematik," XXXI., 1858 (also in the Astron. Xachiichten, LXY.), 

 which only allows observations by eye and ear. A centrifugal pendu- 

 lum tui'ns in one second an axis on which a small disc, cut like an 

 Archimedes-spiral, is fixed. At a certain phase of the rotation, an 

 arm, which slides on the spiral, falls down and produces audible 

 second-beats, while it, in the same moment, sets an escapement free, 

 and causes a system of wheels, which before were at rest, to begin to 

 move, and after having moved a certain part of a rotation — which, 

 when the instrument is stopped, may be conveniently measured — 

 causes an artificial star to pass behind the wire in a small telescope. 

 While tliis goes on, the observer estimates the moment between two 

 consecutive second-beats, in which the star (a steel pearl, on the cir- 

 cumference of a wheel, illuminated sideways) is bisected by the wire. 



* Prazmo"wski in "Warsaw seems to have been the first one Trho has invented 

 such an instrument, ■n'hich was very much on the same principle as the modern 

 " Time-collimators." As far as we know, no researches made with it have been 

 published.— Cosmos, T. IV. (1854), p. 44.5. 



