VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS. 2S 



o{ the curve determined to a maximum or minimum, the first towards the con- 

 cave part of the curve, and the other towards the convex part. 



Postscript. First, it will be easy by this method to determine the tangent, 

 by operating at the convex side of the curve, as before on the concave side. 

 For let AC be the vertical tangent, and c a point in it taken at pleasure. Make 

 AC = w, CO = z, (by which symbol let all the lines be denoted, which are drawn 

 from the point c to the convex curve aeg). Then drawing mo always perpen- 

 dicular to AC, it will be CM = n — y. And since om = ^, it will be zz = nn 



— 2ni/ -\- yy — XX \ and therefore, for an extreme value of z, lyy + Ix.v — - 

 Iny = 0. In which equation, if 2x.v be expounded according to the nature of 

 the curve, we shall have the line cz determined, which in this place performs 

 the office of a subnormal. This is too clear to need any illustration by 

 examples. 



Secondly, as in the foregoing method we have found the tangents of curves, 

 by determining to extremes the lines le or co, drawn from a given point either 

 in the axis or in the vertical tangent ; thus by considering the lines qe, &c. 

 drawn from a given point in the axis beyond the vertex, the same may be per- 

 formed, and that universally. For all the lines qe are of a flowing and variable 

 nature, but the tangent qp alone, (supposing qp to touch the curve) is constant 

 and determined to one value. Therefore in this place we shall not insist on the 

 hypothesis of an extreme, but shall only consider it as a permanent quantity. 

 Let two points a, l, be assumed, and thence to the same point of the curve e 

 let two lines le, qe, be always drawn. The angle gel between the point of 

 contact p and the vertex, will always be obtuse, but on the other side of the 

 point p it will be acute ; supposing, as said before, that qp touches the curve, 

 and PL is at right angles to it. Make qa = p, al = w, ab = .r, be = y 

 and qe = z. Also ve = v, which is intercepted between the points e and v, 

 where qv falls perpendicularly from q upon le produced. Now because of the 

 obtuse angled triangle qel, we shall have this equation, zz = pp -{- 2pn — yy 



— XX -\- V yy •\- nn — Inx •\- xx y, 1v \ or instead oi V yy -}- wn — 'Inx -{- xx 

 writing f^ it will be zz = pp -\- 2pn — yy — xx -{■ Inx — 2/1', and thence 

 Q,zz =: yy — Q,x.v -{■ Ini — 2fv — 2vjf. Now if z be a constant quantity, in 

 which case qe will coincide with the tangent qp, it will be then — 2yy — 2x.v 

 -f- 2n.r = 0, the rectangle 2/1', and therefore its fluxion entirely vanishing. 

 But this is the very general equation, that was determined by the foregoing 

 method, which is deduced with the same ease from the supposition of a con- 

 stant quantity, as before from the principle of an extreme quantity. 



