200 PROCEEDINGS OP THE AMERICAN ACADEMY 



by putting 



a! = a arc V -\- tan l, cos^ t, — (85) 



(86) 

 (87) 



a" = u' arc 1" + i cot C ^"^ (88) 



by putting 



The factor, arc I", is introduced in equations (85) and (88), because 

 in (82) a is expressed in seconds of arc, while a' and a" must be 

 expressed in parts of radius. Bessel lias given a table of the values 

 of «', which may be found in his " Astronomische Untersuciiungcn," 

 Vol. I. pp. 198 and 199; and also in Cliauvenet's "Spherical and 

 Practical Astronomy," Vol. II. pp. 572 and 573 ; but it must be noted 

 that our a' is Bessel's a". From the differences of consecutive values 



da' 



of a' eiven in that table, the values of — have been obtained, and 

 ^ ' dC 



then the values of «", given below, were computed by means of equa- 

 tion (88). 



The quantities in the following table correspond to a state of the 

 atmosphere such that the barometer would stand at 29.597 inches, its 

 attached thermometer at 32° F., and the external thermometer at 

 48°.75 F. The first and fourth columns of the table contain the 

 argument. The second and fifth columns contain the values of log. a", 

 to the argument true zenith distance, computed by means of equation 

 (88) ; but it must be carefully noted that, in using these logarithms, 

 their characteristics must be diminished by 10. The third and sixth 

 columns contain the values of dr, to the argument true zenith distance, 

 computed by means of equation (80) ; and the seventh column, 

 headed dr', contains the values of dr to the argument apparent zenith 

 distance, for all cases where they differ sensibly from the values given 

 in the sixth column. 



Strictly speaking, a", dr, and dr' are functions, not only of the 

 zenith distance, but also of the density of the atmosphere ; but so 

 long as the temperature of the latter remains between -j- 30° and -j- 

 75° F., and its pressure between 29 and 30 inches of mercury, the 



