1 



8 PHILOSOPHICAL TRANSACTIONS. [anNO J735. 



one of which b is to be triple, fig. 6. Join the points bf, fh, he, and pro- 

 ducing these three lines ; and through the points eg, gb, let the lines egd, 

 BGV be drawn, of which let egd cut bf in d, and the other line bgv cut fh in 

 V. Then having joined v and d, and produced vd to meet he in a, draw the 

 line dABR through the points a, b. Then from the points b, e let the lines 

 Has, EPQ be inflected to the given point q, of which let the first Bas meet fh 

 produced in s ; and through the points a, s having drawn as, meeting eq in p, 

 let DPO be produced through p and d, and meet Bas in o : and note the point o. 

 In like manner, from the same b, e, to another given point t, let the lines 

 BTS, EpT (supply the figure) be inflected, of which let bts meet fh in s ; and 

 having drawn as cutting EpT in p, draw npz through p and d, meeting bts in 

 z, and mark the point z. And thus let lines be drawn from the same b, e, to 

 the other given points m, l, and drawing lines from a and d as before, let the 

 points so found be marked x, y. Then through the 4 points thus found, 

 o, z, X, Y, and the given point b, let a conic section be described (see prop. 3, 

 exercit.), cutting fh in the points i, k, and the line dAB in b, r. Through the 

 points A, I draw the line ax, cutting the conic section in i and c ; join the points 

 K, R, and let this line kr be produced. Now about the 5 points a, b, c, d, e, 

 as poles, let as many lines, as, bs, cn, do, Ea, revolve, of which three as, bs, 

 ON meet each other in n, s, o ; and let the intersections n and s, of the lines 

 as, on, and as, bs, be drawn along kr and fhk, while the line dpo passes 

 through the pole d, and the intersection o of the lines bs, cn, and cut the line 

 as in p ; and through p and the pole e, let EPa be produced to cut bs in a : 

 then this intersection a will describe a line of the 4th order, passing through 

 the 9 given points, b, e, f, g, h, l, m, t, a, one of which b will be triple. 



By a method not much unlike this, a line of the 4th order may be described 

 through 8 given points, 3 of which are double; as also a line of the same order 

 through 1 1 given points, 2 of which are double ; with many other cases of that 

 kind. 



As to the number of points which determine a line of any order, Mr. B. says, 

 that if n denote the number of the dimensions of a line ; then tr + 1 will be 

 the number of points through which the line may be described. For instance, 

 a line of the 2d order through 5 points, one of the 3d order through 10, of 

 the 4th order through 17, of the 5th order through 2d points. And hence is 

 deduced, that if a line of the 11th order have an « — 1 multiple point, it may 

 be described through 2n + 1 points. For instance, a line of the 3d order, 

 with a double point, (viz. n — 1 = 2) through 7 points, and a line of the 4th 

 order with a triple point, through Q, &c. And generally, if p, q, r, &c. denote 

 multiplex points, the number of which is m, a curve can be described through 



