$ 



they were found without difficulty, but may supply a blank in all the 

 practical treatises which I have examined. 



The form in which our President has arranged the tables of re- 

 fraction, given by him in the 12th volume of our Transactions, af- 

 fords a facility for calculating the refractions, in PD and AR, 

 which neither the French tables nor those of Dr. Young present. 

 He gives it as the difference of two quantities, each of which is 

 divisible by tan. Z D ; their other factors being, one the product of 

 the height of the barometer by a number taken from his first table, 

 the other a minute quantity whose effect is insensible till 35° from 

 the Zenith. 



To apply this we must estimate the 

 effect of a variation of Z. D, in PD and 

 AR. Let S be the observed place of 

 a star and T the true ; put ST=u, 

 SP,=D,TP= D+A,ZPT=P+n, 

 and Z S P =z S, then we have 



COS. (D + A) = COS. D COS. It — sin. D sin. u, cos. S. 



or expressing the functions of A and m in terms of their halves, and 

 omitting powers above the second 



em. - + tang. D. sin. ^= sin.'^+'ang. D. cos. S. sin. ^ 



and 



2 sin. - =: 



developing 



( / 4sin.'ff") 



.| = tang.i).^_l+N/ l+i££!ii.sin.«+ 2( 



^ (^ tang. 7). 2^ tang." D.\ 



2 sin. :t = 2 sin. !f . cos. S. - if£i-i-^ . 4 sin. « fl + &c. 

 2 2 tang. D. 2 ^ 



To find the difference of the hour angle; by tiie triangle SPT, we 

 have 



