46 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735. 



c, s, D, and e, and those that revolve about d and e, serve only to guide those 

 that revolve about c ands; so that a and h, the intersections of that which 

 revolves about d, with those that revolve about e and s, be carried over the 

 fixed lines ab and af; and m the intersection of that which revolves about k 

 with that which revolves about c, be carried over a third fixed line bp; then the 

 intersection p of those that revolve about c and s, will, in the mean time, de- 

 scribe a conic section, and not a curve of a higher order. The conic section 

 degenerates into right lines, when cp and sp coincide at the same time with the 

 line OS, that joins the poles c and s, as in the preceding description ; which 

 coincides again with what is demonstrated in the abovementioned ingenious 

 paper. 



After this it is shown generally, that though the poles and lines revolving 

 about them be increased to any number, and the fixed lines over which such 

 intersections, as we described in the last two cases, are supposed to be carried, 

 be equally increased, the locus of the point p will never be higher than a conic 

 section; that is, let a polygon of any number of sides have all its angles, one 

 only excepted, carried over fixed right lines, and let each of its sides produced, 

 pass through a given point or pole, and that one angle, which we excepted, will 

 either describe a straight line, or conic section. 



Thus, if a hexagonal figure lqrpmn, fig. 8, have all its angles, excepting p, 

 carried respectively over the fixed right lines Aa, sb, og, nh, kIc; then the 

 point p in the mean time will describe a conic section, or a right line. The 

 locus of p is a right line when cp and sp coincide together with the line cs. 

 All these things are demonstrated geometrically. 



V. After this, angles are substituted instead of right lines revolving about 

 these poles; and it is still demonstrated geometrically, that the locus of p is a 

 conic section or right line. 



Suppose that there are four poles c, s, d, and e, fig. 9, about which the in- 

 variable angles pca, psk, rdm, meu revolve; and that a, m, and r, the inter- 

 sections of the legs cq and eq, of em and dm, and of dr and sr, are carried 

 over the fixed right lines Aa, sb, Gg, respectively; then the locus of p is a conic 

 section, when cp and sp do not coincide at once with the line cs; but is a right 

 line when cp and sp coincide at the same time with cs; and never a curve of a 

 higher order. 



VI. Having demonstrated this, which seems a remarkable property of the 

 conic sections, or lines of the second order; it proceeds to substitute curve 

 lines instead of right lines in these descriptions, as is always done in the treatise 

 concerning the description of lines, and to determine the dimensions of the 

 locus of p, and to show how to draw tangents to it to determine its asymptotes. 



