VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. - Q 



w'^ — p'^ — q- — r^ -\- m -\- \ points, of which there are m multiplex points. 

 Thus, a line of the 4th order, with 3 double points, may be described through 

 8 points : for n = 4, /) = 9 = r = 2, m = 3, and l() — 4 — 4 —44- 

 3+1=8. 



There is another method also of describing lines of the 4th order, not much 

 different from the former, but a little more complex. About 7 poles. A, b, c, d, 



E, F, G, (fig. 7), let there revolve as many lines, as, bs, cn, ds, en, fo, gt, one 

 of which ANS, by revolving, cuts the lines dK, kk, given by position, in the 

 points s, N ; let the lines cn, bn be drawn through one of them n, and the 

 lines BS, ds through the other s, and meet the lines cn, kn, in the points o, t, 

 describing conic sections as above; while the lines fo, gt, drawn from the poles 



F, G, pass through the same o, t, and meet in p : then the intersection p will 

 describe a line of the 4th order, with a double point in both the poles f, g. 



But not to dwell longer on these, Mr. B. now gives the following general 

 theorem. About the poles a, b, c, d, e, f, g, h, &c. (fig. 8), whose number is 

 n, let as many lines as, bs, cn, dp, eq, fw, gx, hy, &c. revolve, of which the 

 three as, bs, cn intersect each other in the points n, s, o ; let two, s, n, be 

 drawn along the lines dK, kr, given in position; while through the third o and 

 the pole D, passes the line dp, cutting as in p ; and through p and the pole e 

 draw the line Ea, cutting bs in a ; and from a through the pole f let Fa be 

 drawn, cutting as in w ; also through w and the pole g draw gw, cutting bs 

 in X ; and then through x and the pole h draw hy, meeting sa in y ; and so 

 on : then the concourse y, of the line hy, drawn from the last pole h, with 

 either of the lines as, bs, will describe a line of the n — 1 order, and have the 

 n — 2 multiplex point in the pole a or b, like as it was described by the inter- 

 section of the line as or bs. The points o, p, a, w, x, y, &c. will describe lines 

 of the 2d, 3d, 4th, 5lh, 6th, 7th, &c. order. But if all the poles a, b, c, d, e, 

 F, G, H, &c. be situated in the same right line, then those points o, p, a w x 

 Y, &c. will also describe as many right lines. 



The Newtonian description of curves is also greatly promoted by this method. 

 It is well known, that if the given angles oan, obn. revolve about the given 

 points a, b (fig. 9) and the intersection n, of the legs an, bn, be drawn along 

 the line nr, given in position ; then the concourse o, of the legs ao, bo, will 

 describe a conic section. Now let another point c be taken, about which let 

 the line ocp be moved, which shall always pass through the intersection o of 

 the legs AO, bo, and meet the other leg an of the angle a in p : then the in- 

 tersection p will describe a line of the 3d order, passing doubly through the 

 pole a. In like manner, if by the intersection of the leg bn, of the angle b, a 

 curve be describedj it will be of the same order, and have a double point in the 



VOL. VIII, C 



