VOL. XH.j PHILOSOPHICAL TRANSACTIONS. 393 



xyy = bx- + ex -{• d, will, as they say, give 2 liyperbolo-parabolical figures, 

 equally distant on each side the diameter a b. See the 57th figure in Newton's 

 Treatise, and this is his 53d species, and Stirling's 57th. 



1. If the equation bx^ — cj; + (/= O, have 2 equal roots, both with the sign 

 -j- ; the equation xyy ^ boe^ — ex + d, will, as they say, give 1 hyperbolo-para- 

 bolical curves, crossing each other at the point t in the diameter. See fig. 58 

 in Newton ; and this is his 45th species, and Stirling's 58th. 



3. But if the equation bx"^ -\- ex -\- d=. o have 1 possible unequal negative 

 roots Ap and At, the curve given by the equation xyy = + bx'^ -{- ex -\- d, will 

 consist of 2 hyperbolo-parabolical parts, as also of an oval on the contrary side 

 the asymptote or principal absciss, as fig. 1, pi. 9. And this is one of the 

 species omitted by Sir Isaac and Mr. Stirling, which is really the 59th 

 species. 



4. Also if the equation bx' -^ ex -^ d := O, have two equal negative roots Ap 

 and At; the curve given by the equation xi/y := + bx^ -\- ex + d, will consist of 

 2 hyperbolo-parabolical parts, and also of a conjugate point on the contrary 

 side the asymptote or principal ordinate, as figS2. And this is the other spe- 

 cies of these curves omitted by Sir Isaac and Mr. Stirling, which is really the 

 60th species. 



5. If the roots of the equation bx'^ — ex -\- d = O, be real, and unequal, hav- 

 ing both the sign -f-; the curve given by the equation xyy = bx"^ — ex -{- d, will, 

 as they say, consist of a conchoidaThyperbola and a parabola, on the same side 

 the asymptote or principal ordinate. See fig. 59 in Newton ; and this is really 

 the 6 1st species. 



6. If the roots of the equation bx^ + cx — d= O, have contrary signs, the 

 equation xyy = bar' + ex — d, will, as they say, give a conchoidal hyperbola 

 with a parabola on the contrary side the asymptote or principal ordinate. See 

 fig. 60 in Newton ; and this is really the 62d species. 



^n Acemint, by Mr. Harris, of several Alterations and Contrivances about the 

 Terrestrial Globe, to render it, as he thought, more commodious and useful in 

 Practice. N° 456, p. 321. 



A New Method of improving and perfecting Catodioptrical Telescopes, by form- 

 ing the Speculums of Glass, instead of Metal. By Caleb Smith. N*" 456, 

 p. 326. 



The imperfections of telescopes are attributed to two causes ; viz. the unfit- 

 ness of the spherical figure to which the glasses are usually ground, and the 

 different refrangibility of the rays of light. 



VOL. VIII. 3 E 



