108 Prof. Encke ow the Calculation 



where the logarithms in [ ] are common logarithms. If we 

 assume as a first hypothesis that the quantities ^ are equal to 

 the quantities E, we have 



« = i {(;'4 -v-^ -ip^- px)} = - 28°, 



and hence by the equations : 



tang /3 = tang (45° + ? ) tang a 

 tang y = tang (45 + ^j) tang a. 



we find |3 (about) = + 36°, y = + 90°. We take, therefore, 



from the table for 



/3-a = 64°, i> (|3— «) = 6-066460 

 7-(3 = 54 , rj/ (y-jS) = 6-041780, 



and we obtain these two equations : 



k = r4.-400056'lfc^ + 0-42422 

 L -J sm 2y 



k = r6-22097'4"l Z^^ + 0-16628. 

 L J sm 2« 



Putting in these formulag first a = —28°, we obtain for k 

 1-14393 and 0-81289; and as the equations prove that the 

 diminution of the negative a will tend to equalize the values 

 of k, we readily get by making u = —24° these values : 

 0-94506 and 0-94945. 

 If we now correct the functions \I/(/3— «) &c., or rather seek 

 for every a, /3, y, the accurate values, we shall find from the 

 two first equations a = —24° 30' 31"-4, k = 0-94453. 



The above third equation gives with this value k = 1-01467. 

 This difference is so trifling that one might almost correct it 

 by the time. It would only be required to put 1823-50 in- 

 stead of 1823-27, which would perhaps be allowable. In 

 order, however, to represent most nearly the observations 

 actually adopted, I made a second trial by assuming §^ = 4-74, 

 and obtained by a new calculation of (1 2 4) (13 4) (1 3 4), 



^, ^1, ?2, for these values : a = —25° 44'-5 k = 0-99502 



by the two first equations, and k = 0-99098 by the third 



equation. It was now to be concluded with certainty that 



for §4 = 4*746 all the values would so nearly agree together 



as to allow a perfect agreement to be effected by correcting 



the times. The new definitive values were accordingly : 



(0 12) = + 10-01579 (023)=+ 2-79683 



(0 ) 3) = — 17-43508 (0 2 4) = +1-25416 



(014) =-18-62013 (0 3 4) =+3-01634 whence 



(12 3) = 30-24770 (12 4) = 29-89008 

 (13 4)= 4-20139 (2 3 4)= 4-55901 

 (12 3 4) = 34-44909 



