39 ® Mr, Brougham's general Theorems , &c. 
I shall throw together, in a few corollaries, the most remark- 
able things that have occurred to me concerning this curve. 
Corollary 1. The subtangent of this curve is 
Corollary 2. In order to draw a tangent to it, from a given 
point without it ; from this point as pole, with radius equal to 
d , and the curve's axis as directrix, describe a conchoid of Nico- 
medes : to its intersections with the given curve draw straight 
lines from the given point ; these will touch the curve. 
Corollary 3. This curve may be described (organically) by 
drawing one end of a given flexible line or thread along a 
straight line, whilst the other end is urged by a weight towards 
the same straight line. It is, consequently, the curve of traction 
to a straight line. 
Corollary 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 = P M and x = AP') y 
~ s/ d* ~ x z + dxhA. which may be used with 
ease, by changing the hyperbolic into the tabular logarithm. 
Thus then, the common logarithmic has its subtangent con- 
stant ; the conic parabola, its subnormal ; the circle, its normal; 
and the curve which I have described in this proposition, its 
tangent. 
Edinburgh, 1797. 
