of the Vertical on Latitude Observations. 199 



groups of stars whose mean right ascensions and declinations are 

 ajSj, a 2 8 2 , &c. We have 



<f> +A 1 (j) = S 1 + z 1 + L 1 , 



cf) + A.,cj) = 8 2 + z 2 + L 2) 



where L lt L 2 are the deviations of the plumb line at the sidereal 

 times a lt a 2 ; <£„ denotes the mean latitude, and A^, A 2 (/> the 

 variation of latitude. In Kiistner's method of determining the 

 aberration constant from latitude observations with a zenith 

 telescope, four groups of stars are generally chosen at intervals 

 of 6 hours, and two consecutive groups are observed on every 

 observing night. The variation of latitude is eliminated by sub- 

 tracting the equations -given by the two groups, and we get four 

 equations of the form 



0=8 1 -8 2 + z 1 -z 2 + L 1 -L 2 , 



or putting in the term depending on the correction to the adopted 

 constant of aberration, 



== S 1 — S 2 + z x — # 2 + («i — a 2 ) Ak + L x — L 2 . 



For most observatories in the northern hemisphere, we can 

 neglect the second term in equation (1) for L, since cos 2</> is less 

 than i. The star groups are separated by half the period of the 

 first term, so that L 1 — L 2 has its maximum value, which is 



3 Ma? . 



t i=r^> ^ sin l(p cos 2 b cos 2h, 



4 ED i T 



where h is the moon's hour-angle at the sidereal time a x . The 



maximum value of this is ^ nrf^, which is about or 0"'0l7, 



and the whole range of the deviation of the plumb line in the 

 observed quantities is 0""034<. This is a small quantity, but its 

 determination does not seem impossible. The probable error of 

 one determination of the latitude may be taken as + //- 12 : each 

 group of stars contains as a rule 10 pairs of stars, and accordingly 

 the probable error of the difference of two groups is 



± 0"-12 Y-.^ = ± // -054. 

 V 10 



The problem then is to determine the amplitude of an oscillation 

 of known period from observations of which the probable error is 

 about three times the quantity sought. 



Besides the moon's tide-raising force there is also the sun's, 

 which is nearly half that of the moon, and the effects of the two 

 are superposed. But since, in Kiistner's method, the observations 

 are made symmetrically about midnight, i.e. one group is observed 



14—2 



