VOL. XXX.] I'HILOSOPHICAL TRANSACTIONS. 3Q3 



respectively; and let sd = 2sa. Then if sd be less than cs, the curve de 

 scribed according to the first case, will be a parabola with a node and an oval, 

 of the 68th species of Newton's curves. But if sd = cs, the oval will vanish, 

 and the node become a cusp, and the curve described will be the Neilian or 

 semicubical parabola. And if sd be greater than cs, the curve will be a 

 punctated bell-formed parabola, of the 69th species. 



3. Let the given angles rmt, knl (fig. 7) move in such a manner, that the 

 points M and n may run over the indefinite lines bm and D;sr respectively; and 

 let the legs rm and kn always pass through the given points c and s. Now, 

 1st, if the concourse a of the legs mt and nl be drawn along the indefinite 

 line AG ; then the concourse p of the legs mr, nr, will describe a line of the 

 4th order, having 2 double points, the one in c and the other in s. But, 2dly, 

 if the concourse MR and nk (fig. 8) be drawn along the indefinite line aq; 

 then the concourse of the legs mt and nl will describe a line of the 4th order, 

 having no double point. 



4. Now in the first case of this construction, if the lines cmr, snk (fig. 7) 

 coincide together with cs; then the points c and s become simple, and the 

 curve will be of the 3d order, without a double point. For example, let bm, 

 AG, DN (fig. 9) be parallel lines, and all perpendicular to cs; and let the angles 

 RMT, KNL be right ones ; then if a curve be described according to the rule 

 of the first case, the legs cmr and snk will coincide with cs; and by this con- 

 struction may be described Newton's curves 10, II, 20, 21, 40, according to 

 the various positions of the points c and s in respect of the three lines bm. 

 Ad, dn; and all these species will be without a double point. 



5. Now lines of the 4th order, which have a triple point, may be thus con- 

 structed; Let Ad, bn, dm (fig. 10) be three lines given in position; also let 

 the angles qct, snm, nml be given and invariable; and let the points n and m 

 be carried along the lines bn and dm, so that the leg ng may always pass 

 through the given point s; also let qct revolve about c, so that the con- 

 course of the legs ck, sn may pass along a third line aq: then the concourse of 

 the legs CT, ml, will describe a line of the 4th order, having a triple point in c. 



6. I have shown how lines of the 4th order may be described, which have a 

 triple point, or two double points. Others having only one double point, may 

 be conveniently described thus: Let aq, bn, dm (fig. Jl) as before be three 

 lines given in position, and snk, sml, rct three given angles; also let the 

 points N, M, s be always in the same right line; and let the points n and m as 

 before move along the lines bn and dm: then if the concourse of the legs cr, 

 NK be drawn along the indefinite line Aa, the concourse of the legs ct, ml 

 will describe a line of the 4th order, having one double point only in c. And 

 these last two propositions give us new methods for describing lines of the 3d 



VOL. VI. 3 E 



