IN THE HIGHER GEOMETRY. 23 



y 



V a * w* 4- a x h. 1. j 7 > a nna l equation to the 



' a -*~ V * _ y* 



curve as required. Q. E. i. 



I shall throw together, in a few corollaries, the most re- 

 markable things that have occurred to me concerning this 

 curve.* 



Corol. 1. The subtangent of this curve is V ( 2 ?/ 2 )- 



Cord. 2. In order to draw a tangent to it, from a given 

 point without it ; from this point as pole, with radiiis equal 

 to a, arid the curve's axis as directrix, describe a concoid of 

 Nicomedes: to its intersections with the given curve draw 

 straight lines from the given point; these will touch the 

 curve. 



Corol. 3. This curve may be described, organically, by 

 drawing one end of a given flexible line or thread along a 

 straight line, while the other end is urged by a weight to- 

 wards the same straight line. It is consequently the curve of 

 traction to a straight line. 



Corol. 4. In order to describe this curve from its equation ; 

 change the one given above, by transferring the axes of its 

 co-ordinates: it becomes (y being = I>'M and x = AP'), y = 



QC 



>J (a* x?) -4- a x h. 1. r ; which may be used 



+ v (* x*) 



with ease, by changing the hyberbolic into the tabxilar 

 logarithm. Thus, then, the common logarithmic has its sub- 

 tangent constant; the conic parabola, its subnormal; the 

 circle, its normal ; and the curve which I have described in 

 this proposition, its tangent. f 



* There are other properties of this curve noted in Tract V. of this 

 volume. 



t This Tract was printed in Phil. Trans, for 1798, part 2. The fluxional 

 notation lias alone been altered to the differential. The schol., p. 17, is 

 subject to doubt from the leminscata and other similar curves. See Note I. 

 at end of this volume. The subject of Porisms is treated of in Note II. 



