J. H. Alexander on a New Formula for Interpolations. 21 



the epochs for interpolation required between the first and second 

 terms, become (instead of 1*1; 1*2; 1*3; 14;) 1-2; 1*4; 1*6; 

 and 1*8, respectively. These are the values then to be success- 

 ively substituted for n in the formula II, in order to obtain the 

 values of / in terms of the temperature. Taking, then, a, as 100° ; 

 6, as 112° -2 : and c, as 122° ; and calling n = 1*2, we have 



^144-3-6+2 i22-(l-44-4-8+3) . m^ l *tj±* . 100 



I # 2 



102° 632. 



When n is 1-4, then 





^ l%-4-2+2 . 122 -(i96-5-6-f3) . 112-2+ -' 96 -~—- 100 



2 ^ 



2 

 105°-168. 

 When n is 16, then 



z= 2-56-4-8+2 >122 -(2-56-6'4 + 3) . 1122 + 2 56 " 8+ - .100 



2 



2 

 107° -608. 

 And finally when n is 1-8, then 



^3-24-5-4+2 1 12 2_ (3-24-7-2 + 3). 112-2 + ?Jfz±!J. 100 

 109° 052. 



Assembling these results into a series, and letting the epochs of 

 pressure return to their actual value, we have, as under 



Pressures in > , H ^ ,3 14 j. 5 2 &c , 



atmospheres: ) 



Corresponding ) ]QQU 10 oc. 632 . 10 5 o lG3; 107°-6C8 ; 109°-952; 112°-2; 122°. &c. 

 temperatures: J ' ' ' 



It is hardly necessary to say that these interpolated results are 

 exactly the same with those that Weisbach arrives at, after a 

 more complicated process. 



2. Wallace, in his article on Interpolation, for the Edinburgh 

 Encyclopedia, has quoted the following lunar distances of Alde- 



baran, viz. 



1847. Nov. 1. Noon, 53°- 20'- 16" ; Midnight, 59°- 33'- 47" ; 



Nov. 2. " 65°' 52'- 58" ; " 72°- 18'- 10" ; 



and has then calculated the corresponding distances at the third, 

 sixth and ninth hours, respectively, of the first interval. I shall 

 compare his results with those of the present formula. 



Ranking the given terms in a natural series of equidistant in- 

 tervals of 12 hours, and reducing the angular values to seconds of 

 space for convenience, we have as follows : 



Ordinates, 1. 2. 3. 4. 



Distances, 1 920 16". 21 427". 237 1 78". 2602 90". 



