150 PHILOSOPHICAL TEANSACTIONS. [aNNO 1771. 



XXXVL A Disquisition concerning certain Fluents, which are Assignable by 

 the Arcs of the Conic Sections; where are tnvesligated some New and Useful 

 Theorems for Computing such Fluents. By John Landen, F. R. S. p. 2g8. 



Mr. Mac Laurin, in his Treatise of Fluxions, has given sundry very elegant 

 theorems for computing the fluents of certain fluxions by means of elliptic and 

 hyperbolic arcs; and Mr. D'Alembert, in the Memoirs of the Berlin Academy, 

 has made some improvement on what had been before written on that subject^ 

 But some of the theorems given by those gentlemen being in part expressed by 

 the difference between an arc of an hyperbola and its tangent, and such 

 difference being not directly attainable, when such arc and its tangent both 

 become infinite, as they will do when the whole fluent is wanted, though such 

 fluent be at the same time finite ; those theorems therefore in that case fail, a 

 computation thereby being then impracticable, without some further help. w 



The supplying that defect Mr. L. considered as a point of some importance in 

 geometry, and therefore he earnestly wished, and endeavoured, to accomplish 

 that business ; his aim being to ascertain, by means of such arcs as above- 

 mentioned, the limit of the difference between the hyperbolic arc and its tangent, 

 while the point of contact is supposed to be carried to an infinite distance from 

 the vertex of the curve, seeing that, by the help of that limit, the computation 

 would be rendered practicable in the case wherein, without such help, the 

 before-mentioned theorems fail. And having succeeded to his satisfaction, he 

 presumes the result of his endeavours, which this paper contains, will not be 

 unacceptable to the Royal Society. 



1 . Suppose the curve adep, pi. 4, fig. 7, to be a conic hyperbola, whose 

 semi-transverse axis ac is = m, and semi-conjugate = n. Let cp, perpendi- 

 cular to the tangent dp, be called j& ; andput/= "' ~" , z = ^. Then, as is well 



known, will dp — ad be = the fluent of ■. — — ^— , p and z being each = 



to m when ad is = O. 



2. Suppose the curve adefg fig. 8, to be a quadrant of an ellipsis, whose semi- 

 transverse axis eg is = ^/m^ -f- n% and semi-conjugate ac = n. Let ct be per- 

 pendicular to the tangent dt, and let the abscissa cp be = n v^ -. Then will the 



z 

 m' 



said tangent dt be = m -/ -^— — ; and its fluxion will be found =a mn'z -"hz y, 



_ " + mz 



V«» -\-mz a/«» + 2jz — z^ 



3. In th expression 



!/9 



-J let —-y be supposed = z. Then will ^— ^ be = y, and the 



(a + byy X (c + rfy)'' a + by 



proposed expression will be = (^z- ^T-'? x (rf '^zV+T^"- 



- <•-* X z 



