C 527 ] 
Thus is this problem in all cafes folved either by a 
right line, or an hyperbola given in portion, which 
fhall interfe6l the projected orbit in the point fought. 
For though in each cafe the proje&ion of the planet 
has here been confidercd as within the orbit of the 
earth, the form of argumentation will be altoge- 
ther fimilar, were the projection of the planet without. 
And this is agreeable to the method, I have pur- 
fued throughout this difcourfe, where I have al- 
ways accommodated the expreffion to one fituation 
only of the terms given and fought in each article ; 
the variation neceffary for the other cafes, when one 
has been duely explained, being fufficiently obvious. 
In the 5th volume of the Commentaries of the 
Royal Academy at Peterfbourg is given an algebrai- 
cal computation for a general folution of this problem 
in the orbits of any two planets projected on the 
plane of the ecliptic ; but with this overfight of ap- 
plying to the projected orbits a propolition from Dr. 
Keil’s Agronomical Lectures, which relates to the 
real orbits (<3). 
However from the geometrical folution now given 
a calculation for affigning the point D may be formed 
without difficulty. L D M being the orbit of the 
earth, A is the focus, and R P perpendicular to the 
(tf) The demonftration of Dr. Keil’s proportion proceeds on 
the known property in the planets of having their periodic times 
in the fefquiplicate ratio of the axes of their orbits, which con- 
fines the propofition to the real orbits; for in each planet the pe- 
riodic time through the projected orbit is the fame, as through the 
real, though the axis in one be not equal to the axis of the 
other. 
Yyy 2 
axis.. 
