394 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I9. 



order, both those that have double points, and those that have none: but in 

 this short specimen of our method, these must be omitted. 



7 . Let the Hnes and angles remain as in prop. 3 : but let the concourse of 

 the lines MT, nk (fig. 12) be now drawn along the indefinite line agi; then 

 the concourse of the legs MR and nl will describe a line of the 5th order, 

 having a quadruple point in s. I have also other methods of describing curves 

 of the 5th order, which have a triple or double point, or two double points, or 

 none but simple points: but these above may suffice to show the simplicity and 

 universality of the method. But it must be observed, that in particular and 

 simpler circumstances of the lines and angles, sometimes a curve line will pass 

 into another of an order inferior to that explained in the proposition. Indeed 

 all the propositions afford particular methods of describing some curves of every 

 inferior order. 



8. General Prop. Let there be taken at pleasure any number n of right 

 lines any how situated in the same plane, as bn, er, ft : also let any number 

 m of other lines be taken at pleasure, as dm, gl, hk, &c: and let the angles 

 CNR, nrt, rtg, &c, as also the angles sml, mlk, lkq, &c, be invariable, 

 while the angular points n, r, t, m, l, k move along the indefinite lines 

 BN, ER, FT, DM, GL, HK ; and let the concourse of the legs Ta and kg be 

 drawn along the indefinite hue aq : to find the order of the curve which shall 

 be described by the concourse of the leg sm with any one of the lines cn, nr, 

 RT, TQ, &c, for instance, with the line rt. 



In the series of lines cn, nr, rt, tq, &c, let s denote the number of the 

 line rt, by the concourse of which with sm the curve is to be described, from 

 the line cn inclusively ; which in this case is the third, or j = 3 : then will the 

 curve be of an order expressed by the number sm -\- s -\- n -\- I . Hence in 

 the case denoted by the figure, since j = m = n = 3, the curve will be of 

 the l6th order. 



In these descriptions we have only postulated that right lines and angles 

 should be given. But often the more complex curves are easier described by 

 means of simpler ones; and I have investigated propositions of this kind not 

 less universal than these. But I omit them at present, with the demonstra- 

 tions of these, as too prolix, though I may publish them hereafter. 



An Account of an ancient Roman Inscription lately found at Caerleon upon 

 Usk, By Mr. Wm, Rice, Rector of that Place. With some Conjectures 

 thereon, by the Rev. Dr. John Harris, S.T.P. and R. S. S. N° 359, P- 945. 



A person being at plough in a close near the bank of the river Usk, which 

 the ancients called Isca, (which glides by us about a quarter of a mile off from 

 Caerleon, and in sight of the town) came across a stone, and finding letters 



