[ 299 ']' 
earneflly wl£hed, -and endeavoured, to accomplifla 
that bufinefs ; my aim, being to afcertain, by means 
of fiich arcs, as above-mentioned, the himit of the 
difference between the Hyperbolic Arc and its Tan- 
gent, , vvhilft the point of contadt is fuppofed to be 
carried to an infinite diftance from the vertex of the 
curve, .feeing that, by the help of that Limits the 
computation would be rendered pradlicable in the 
cafe wherein, without fuch help, the before-men- 
tioned Theorems fail. And having fucceeded to my 
fatisfadtion, 1 prefume, the refult of my endeavours, 
which this Paper contains,, will not be unacceptable 
to the Royal Society, 
I. 
Suppofe the curve ADEF (Tab. XII. fig. i.) to 
be a conic Hyperbola, whofe femi-tranfverfe axis AC 
is = w, and femi-conjugate = 
Let CP, perpendicular to the tangent DP, be 
called p j and put f Then (as 
is well known) will DP — AD be = the fluent of ‘ 
- ^ - - j p and 2 being each =- to m when 
2f z — 
A D is := 0 . 
2 . 
Suppofe the curve adefg (fig. 2.) to be a qua- 
drant of an Ellipjts, whofe femi-tranfverfe axis eg is 
itz slfTp and femi-conjugate ac zzz n. Let 
Q^q 2 C t 
n. 
