VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 45 



about the poles c and s, so that the angle acp bears always the same invari- 

 able proportion to asp, suppose that of m to n. In the line cs, take the point 

 T, so that ST may be to ct in that same proportion of ot to n; then this point 

 T will be an invariable point; sin(;e cs is to ct. as ot — n to n. Draw tp, and 

 constitute the angle spn, equal to cpt, so that pn and pt may lie contrary 

 ways from sp and cp, and pn shall be a tangent of the curve described by 

 the motion of the point p. Several other theorems of this kind are subjoined 

 here. 



IV. After these, lines or angles are supposed to revolve about three or more 

 poles, and the dimensions of the curves with their tangents and asymptotes are 

 determined. Suppose in the first place, that the three poles are c, s, and d, 

 fig. 6, and that lines or rulers ck, sq, qdr, revolve about these poles. The 

 line which revolves about d, serves only to guide the motion of the other two, 

 so that its intersection with each of them being carried over a fixed right line, 

 their intersection with each other describes the locus, which is shown to be a 

 conic section. The intersection of qdr with sq, is supposed to be carried over 

 the fixed right line af ; the intersection of the same auR with cr, is supposed 

 to be carried over the fixed right line ae ; and in the mean time, the intersec- 

 tion of the right lines sa, cr, that revolve about the poles s and c, describes a 

 conic section. 



This conic section passes through the poles c and s; and if you produce dc 

 and DS, till they meet with Aa and ar in f and e, it will also pass through f 

 and e: it also passes always through a the intersection of the fixed lines qf 

 and ER; from which this easy method follows, for drawing a conic section 

 through five given points. Suppose that these five given points are a, f, c, s, 

 and e: join four of them by the lines af, fc, ae, es, and produce two of these 

 FC, es, till they meet, and by their intersection give the point d. Suppose 

 infinite right lines to revolve about this point d, and the points c and s, two 

 of those that were given, and let the intersections of the line revolving about 

 D, with those that revolve about c and s, be carried over the given right lines 

 AE, af; then the intersection of those that revolve about c and s with each 

 other, will, in the mean time, describe a conic section, that shall pass through 

 the five given points a, f, c, s, and e. 



It is then shown, that when c, s, and d are taken in the saine right line, the 

 point p describes a right line, fig. 7> as also when c, s, and a are in the same 

 right line; which also follows from what is demonstrated in that very ingenious 

 paper concerning Pappus's porisms, communicated by Mr. Simson, professor 

 of mathematics at Glasgow, published in the Phil. Trans. N" 377. 



In the next place it is shown, that if four right lines revolve about four poles 



