418 The Inverfe Method 
NOTE, 
(Referred to in page 375) 
Let VA BD (Fig. 8.) be a traje€tory defcribed by a 
body round a centre of force C, Y an apfe, and A the next 
following one; V being at a greater diftance from the 
centrethan 4. It is evident, that if the body were pro- 
je&ted from A, at right angles to C A, and with the 
fame velocity it had when it arrived there, it would 
accurately defcribe the arch 4V, and have the fame velo- 
city at V that it firft begun with, For during the time of 
moving over any particle of the curve, the force a&ing 
upon the body, and the dire€lion of the force are the {ame 
in both cafes; hence the conclufion is clear, But if the 
body, inftead of moving towards V, be projeé&ed in a 
contrary dire&ion, at right angles to AC, and with the 
fame velocity, an arch A B, equal and fimilar to AV, will 
be defcribed; B being an apfe, and CB=CV, Hence in 
the above equation y can have but two different values ; 
but as thefe may lie in oppofite direétions, two may be 
pofitive, and two equal tothem and negative, The other 
roots, if any, muft either be impoffible, or relate to fuch 
parts of the algebraical curve as have their concavity 
turned from the centre of force, or fuch parts as are 
feparated from that part in which the body moves; that 
is, it cannot be a curve of continued curvature, as that 
mujt be in which the body moves. 
The fame conclufion may be deduced immediately from 
the nature of the equation found above for determining 
the 
