M. de Zach’s Ajlron&mical O.bfer.vationh» 
-finus totus = i is to cofine m :: as cotangent k is to the tan- 
'gent of an angle which I put = and 90° - k n will 
give an angle which I put = q. Lafily, the analogy 
cof. n : cof. q ; 1 fin. k •. will give the cofine of an angle ip, 
which is the required motion upon the orbit, or the angle 
comprehended between the two radius veblors m and | w. Let 
therefore ECPMND be the apparent parabolic path of a comet ; 
S the fun s center ; M and N two places of the comet, the angle 
MSN equal to its motion in longitude, or the comprehended 
angle ; P the perihelion ; it is required to find the two ano- 
malies PM, PN, that is, PSM and PSN, the perihelial dif- 
tance SP, and the time the comet employed to come from its 
perihelion P to M and N.. 
Refolutiom 
SM - m 
•SN ~ p 
MSN = 4, 
NSB ± v 
MSB =(tJ/*=*) 
Parameter— p 
In the right-angled triangleSM'R and SNVwe have 
MR- OS ±=/» fin. (iD±.v) N V = QS = ^ fin. a-; 
therefore OP- \p - m (fin. ip =*= x) and PQjp 
fin- x ; but by the nature of the parabola 
wehaveSM-AP 4 - POandSN =AP + PQ; that is 
m zz f ,p — m (fin. tpdz v) p ~ § p p fin. x 
m + m (fin ,pz±zx)z=.\ p fin . x~\ p 
m ( 1 + fin. i| [/=*=*) = ip ft (1 rp fin. x) = § p 
and i-j-fin. (d/ x) — J— 
putting into a fum 1 + fim (i[/=±= x) 
tion made 2 ±(in,j+£n. (^±j) = [—) p ; but by trigono 
metrical formulae we have fin. — fin. xp cof. .Vrfcfin. x 
■cof. ip. Subftituting this expreffion in its place we obtain, 
2 fin. x -f- fin.rjrcof.vrtfin. x cof.ip = ^ ^T~') By the fonae 
formulae we have cof . 1 ^-1- fin ,. 2 * and gpf. x = Vi-£mfx\. 
U 2 Sub- 
fill. X Z~ L~;bv 
2 u J 
P 
x 4 - — - ; reduc- 
2 m 
