W. A. Norton on the Dimensions of Donati’s Comet. 55 
(1.) For orbit of particle leaving comet at perihelion, 
; D 1 
e=—m+1; po=m.D; ee P=D: O20: cos Y= (13.) 
_ (2.) For orbit of particle leaving comet at any other point of 
its orbit, / 
eV im! sin?8 (m!+-2)-+-1; pam! sins. R; p=+2, 
R 548 . (14) 
A= ——.., 
m! +. 2 
'sin 2 @D 
cos @= et =-(16.) T=c.tang (aree ane (16.) 
(3.) For position of particle in its orbit, 
tang 0= a (are® tang“) ame es rk - (18) 
mr 
008 0 == : +1) “Pa nee ae ee 
4=0,- 9 . . . (20. yr 0nd oy oo () 
If the particle is emitted after the perihelion passage, y=0+4. 
For a particle describing a Ancabots concave toward the sun, 
€qus. (18) to (19) become 
e=m—1; or ex m' gin 28 (m'—2) 41 “on gaphead 
(23.) 
a P2 ‘“ coe P2 
Carrot 2% 3, 008 == 22s _ ) ve alban (24.) 
The true anomaly was calculated from the time, in this case, 
by an indirect method founded upon the law of areas, which 
admits of any desired degree of approximation. The calculation 
might also be made by means of Gauss’ formule. ~ : 
In the instance of a repelled particle, which describes an orbit 
oe toward the sun, the true anomaly was calculated direct] 
¥ equ. (17). In this equation, and in equation (16) a and ¢ are 
Constants for the same hyperbolic orbit, but vary from one orbit 
to another, with the initial cireumstances of motion, and the 
Supposed, intensity of the repulsive force. The laws of, their 
Variation are quite simple, so that their values having been found 
atone hyperbolic orbit they may be readily computed for any 
er, ; 
A= P=2_; coso= 
R . 
mM =D" e+1 
m’ sin 28—1 
¢ 
