▼ OL. LIV.l PHILOSOPHICAL TRANSACTIONS. 125 



lines pq, qr, rs, ,it, tv, &c. touching the conic section in the points p, q, k,s, t, 

 &c: then will the content /)P X ^a X rn X s* X &c. = p? X ar X R* X si X 

 Tf X &c; or, which is the same, the sum of all these ratios, pp : p^, t^q-.ar, 

 Rr : Ks, &s : st, &c, will be equal to nothing. 



Carol. Let the ellipse pansTv &c. (fig. 3) be circumscribed by any polygon 

 pqrstmv &c, whose sides touch the ellipse in the points p, a, r, s, t, v, &c: then 

 will the content J&P X ?« X 'R X ss X It X v\ &c. = p? X ar X R* X sd X Ti; 



X vw X &c. 



Carol. Draw the lines pq, qr, RS, st, &c; and for the sines of the angles wpp, 

 apq, Rar, QRr, SR.«, rst, &c. write respectively a, p, b, q, c, r, d, s, &c: then will 

 abed &c. = pqrs &c. 



And the same of polygons inscribed between the conjugate hyperbolas. 



The same is true of a polygon, of which the sum of the sides or the area is a 

 minimum, described about any oval always concave in itself, as appears by the 

 Miscell. Anal. 



Theor. 4. — Let the ellipsis PAaBRCSDTEVF &c, (fig. 4) be circumscribed by 

 the two polygons abcdef&cc, pqrstu &c. having the same number of sides; their 

 sides at, be, cd, de, ef, &c, pq, qr, rs, st, tv, &c, respectively touching the ellipsis 

 n the points A, b, c, d, e, p, &c, and p, q, r, s, t, u, &c; and let oa : a^ :: j&p 

 : p^, and Ir : bc :: qo. : a?-, .and cc : cd :: rR : rs, and ^d : oe :: is : st, and soon: 

 then will the area of the polygon abedef &c, be equal to the area of the polygon 

 pqrstv, &c. 



Carol. Two parallelograms, abed and pqrs, described about the conjugate 

 diameters (ac and bd, pr and as) (fig. 5) will be equal to each other. For in 

 this case ca = xb, bR = bc, cc = cc^, do = va, andpv = pq, qa = ar, rR = 

 iw, ss = sp; consequently aA : kb :: />p : vq, and bR : bc :: ^a : or, and so on: 

 therefore by the theorem these two parallelograms are equal ; which is the known 

 property of the ellipse. 



The same may be said of polygons in like manner described between conju- 

 gate hyperbolas. 



Theor. 5. — Let a conic section revolve about its diameter al, (fig. 6) and let 

 MAM &c. be the solid thus generated; \etpq, qr, rs, st, tv, vw, wp, &c, be lines 

 touching the solid in the respective points p, Q, R, s, t, v, w, &c: then will the 

 content pp X q<^ X tr x ss, X ti X va X. wvf X &c. = pq X ar x Rs X st X 

 TV X y^v X &c. 



Theor. 6. — Let the ellipsis apbocr &c. revolve about its diameter bd, (fig. 7) 

 and about its conjugate diameters (ac and bd, pr and as) be described the 

 circumscribing cylinders pqrs, and acbd ; these will be equal to each other. 



Let there be two solids composed of truncated cones, circumscribing the gene 



