VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 7 



HD, to meet each other in b. Then through a and the points p, M, draw the 

 right lines apms, amhs, cutting the right line kr in n and n, and the right line 

 HK in s, s ; tlirough these points s, s, to the point b, draw bs, bs ; and through 

 D, to the points p, m, draw the lines dpo, dmt, meeting bs, bs, in o and t. 

 Draw ON, xn, produced to meet in c. Then about the points a, b, c, d, as 

 poles, let there be revolved the lines as, bo, co, do, of which the three as, bo, 

 CO intersect in s, n, o ; and let the two s, n move along the lines hk, kr, while 

 the line do always passes through the remaining point o, and cuts ans in p ; 

 then this intersection p, of the right lines as, do, will describe a line of the 3d 

 order, passing through the ^ given points, a, d, h, k, m, p, r, and doubly 

 through the given point a. 



Lines of the 3d order also are more generally, but less commodiously, de« 

 scribed after this manner, which also comprehends the former. About 5 given 

 points. A, b, c, d, e, fig. 4, as poles, let as many right lines ans, bos, cno, 

 dpo, EPS, revolve, of which the three ans, bos, cno intersect each other in 

 the points n, s, o; let the two s, n be moved along the lines dK, kr given in 

 position ; and through one s, of the two n, s, and the remaining point o, let 

 the lines eps, dpo pass, being drawn through the poles e, d, and meeting in p: 

 then this point p will describe a line of the 3d order, with a double point in the 

 pole E. 



In like manner may lines of the 4th order be described. About the 5 given 

 joined points a, b, c, d, e, fig. 5, as poles, in any plane, let as many right lines, 

 ans, bqs, cno, dpo, EPa, be moved ; of which the three ans, bqs, cno meet ' 

 in the three points s, n, o ; let the two points of intersection s, n be drawn 

 along the lines dK, rk, given in position, while the line dpo, moveable about 

 the 4th pole d, passes through the remaining point o, and cuts the line ans in 

 p ; then let the line epg, drawn from the 5th pole e, be drawn through p, and 

 be produced both ways to meet the lines sas, cno, in q and w : then will the 

 points a and w describe lines of the 4th order ; as is demonstrated by prop, ll 

 of the Exercitatio. Through the poles a, e, and b, d, let the lines aeh, bdf, 

 pass, meeting dK, given by position, in h and f ; join de ; and through the 

 poles d and a, the line ad being drawn, meeting dK in v; from which point v 

 let the line vb be drawn to the pole b, and cut the line de in g. Then the 

 figure described will pass through the 5 points b, e, g, f, h, and triply through 

 the pole B. Through the poles a, b, let there be produced the line abr, meet 

 ing the line kr, given by position, in r ; then the curve will also pass through 

 the points r, k. 



Hence is derived a method of drawing a line of the 4th order through Q given 

 points, one of which is a triple point. For let b. e. f, g, h, l, m, t, a be given, 



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