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XXVI. An InveJHgation of a general l^heore 7 n for finding 
the Length of any Arc of any Conic Hyperbola^ by Means 
of Lwo Elliptic Arcs, with fo 77 te other new and ufeful 
’theorems deduced therefrom^ By John Landen, F. R. S, 
Redd'e, Mar. 23, 
1775- 
1ST a joaper, which the Society did me tile 
honour to publilli in the Philofophical 
Tranfadtions for the year 1771,1 announced, that I had 
difcovered a general theorem for finding the length of 
any arc of any conic hyperbola', by means of two ellip- 
tic arcs ; and I promifed to communicate the inveftiga- 
tion of fuch theorem.. I now purpofe to perform my 
promife; and, being pleafed with the difcovery (by 
which we are enabled to bring out very elegant conciu- 
fions in many interefting enquiries, as-well mechanical 
as purely geometrical), I cannot but flatter myfelf, that 
what I am , about to communicate wdll be acceptable to 
gentlemen who are curious in fuch inquiries. • 
I . From, the theorem taken notice of in Art. i . of the 
paper I have jufi: no w mentioned, it follows, that in the hy- 
perbola AD (tab. VII. fig. I.), if the femi-tranfverfe axis 
AchG=?n-n; the femi-conjugate= 2 x mn ^^ ; and the per- ^ 
pendicular cp, from the center c upon the tangent dp, 
the difference (dp -ad) between the faid. 
tangent 
