&5'C- de Zach’s Ajlronomical Oljh'v-atlons . 
The angle x -defines therefore the pofition of the axis and 
the two anomalies required, the perihelia! diftance being 
p *= 2 ( ci± 2 y. fin. x , it will be known alfo by the angle x. 
In order to find the time the comet employs in running* its 
x J O 
anomalies, let the perihelia! diftance juft now inveftigatedp be 
equal to the radius of the earth’s orbit, the parabolic area 
fwept by the radius vector will be by the nature of the para- 
bola f PO x OM -HSOx -OM = i^T^ + 3 SO, x OM t Now 
6 
the periphery of the earth’s orbit is 7 : 22 :: gp : dip; 
therefore the whole area ~ n p>\p~~ p 2 . It is known that the 
velocity of a heavenly body moved in a circular path, is to 
the velocity in a parabolic path in the ratio s/ 2 : 1 . If the pa- 
rabolic area of the comet is divided by 2 it comes out 
2 — -.T — equal to an area that the earth defcribes in 
6^ 2 
the very fiume time*; put therefore A equal to the time of a 
fide real year, we fhall recover the analogy; the whole area of 
the earth’s orbit is to the time in which it is defcribed as the 
parabolic area is to the time con fumed in fweeplng it; therefore 
22 
p* . A ... ( 4 1>q *1 3SO ) MO > 7 A ( 4 PO s- 350 ) MO . ^ 
6^2 ‘ 72 / v' 2 
OM 
SM . fin. ahom. PSM and OS = SM . cof. anom. PSM ; let 
•the anomaly be = <?, we have OM = m fin. J, and OS =zm cof. 0 ; 
therefore PO = p — m cof. 1 . Subftituting we obtain 
7A (4^—4 m cof. £4- 3?wcof. f) m fin. £ yrgpp p 7 A (4/> — m cof. £) m fin. £ 
72 p 
2 p 1 2 
vwliqreby the time is found in parts of a fidereai year. 
I am, &c. 
SI R, 
1 
