J. C. Watson on the Elements of the Orbit of a Comet. 219 
The codrdinates “ the first place of the comet referred to the third 
place of the earth a 
&, = cos 8 cos4-+-g cos G, 
eh mene cos # sin 4+4-¢ sin G, 
z,—=4 sin 8, 
vive vt is the dincies of the comet from the earth at the first ob- 
Wear ws us iow put 
2,=D cos B cos L, 
y¥,—D cosB sin L, 
z,—D sin 
and we shall have 
D cos B cos (L— G)=4 cos 8 cos (4—G)+-9, 
at cos B sin (L—G)=4 cos sin (4—G), (2) 
oB = sin ?. 
If we represent 43 @ the angle at the third place of the earth between 
the first and third places of the comet, we obtain 
cos p=cos B cos 8” cos (4’’—L)-+sin B sin 8”. 
Let us now put 
nm sin m=sin 6", (3) 
: m COS M= COS Bu cos (4’’—-L), 
and we shall have 
os p=n cos (B— m). (4) 
Let * be ob chord * the orbit of the comet between the first and third 
Places, and e ge 
*2—D24.4"2 —2D4" cos g, 
a *2—(d’’—D cos g)?-+-D? sin? g, 5 
where 4” is the distance of the comet from the earth corresponding to 
the > observation. 
and w’’ represent the angles at the earth between the sun and 
comet, at the first and third ert Sener e shall have 
cos ahead bas cos ati 9", ( ) 
Then, if we denote by 7 and “dg the distances of the comet from the sun, 
at the times ¢ and t’’, we obta 
2 Re +R? sin? y, = 
raid, —R,cone Je Rt, sn sin? y” (7) 
Let us now put 
Rs sin @ ae 2 COs an 
= cos ==¢, 
R’ein y” Nan", B’cos y=", 
and equations fe and (7) become 
OR erp | 
r = (a —e )2--U% (8) 
aN (ae EHO? 
