1824.] Col. Beanjoy's Astronomical Observations. 329 



a + b <p x 



vj, x = p ~ ' ' b* + be + c* (24) 



c ■ $x v ' 



a 



which coincides with Mr. Babbage's solution of r}/ 9 x = a?. And 

 because - = 1 — — it is evident that rj/ - * a; ought to be the in 



4 4 



verse of (24). Put therefore in (21) r = 1, v == — j, and of 

 course making k as above = 1 . From these data, we have 



vk\ >■ </3 . kh . 4 a, ,/ 3 



^ — = sin -~3 = - T ■r"! 8~ 3kK 8* 1 



, , cos — =cos — = — 



vk\ \ 1 ftX 4X 1 n 3 2 



cos — = ccs. — - = - cos — — cos — 



n 3 2 n 3 a 



Whence 



A+c ( , J-O 



fl H J - 1 + , £ <f> X 



i-ij» p- ' g . < 6+c > _ g -l c»*-« 



ft + c < 6-c) J'-4c+ c 2 r Z 



3 _1 — - — [ <sx b + - 



2 ( ' b + ci 3 a 



3 a 



■ <px 



which it may be easily shown is the inverse of (24), or equal to 



\J/ _1 x. 



This method of solution may, for the sake of distinction, be 

 called the algebraic. 



( To be continued'") 



Article III. 



Astronomical Observations, 1824. 

 By Col. Beaufoy, FRS. 



Bushey Heath, near Stanmore. 



Latitude 51° 37' 44-3" North. Longitude West in time 1' 20-93". 



Oct. 2. Immersion of Jupiter's second <13>> 50' SO" Mean Time at Bushey 



? 13 :>v 



satellite ( ' 3 ^2 11 Mean Time at Greenwich. 



srsion of Jupiter's first U 7 01 OS Mean Time at Bushey. 

 satellite ( IT 02 29 Mean Time at Greenwich. 



