396 PHILOSOPHICAL TRANSACTIONS. [aNNO 1740. 



of the axis r : let h p, the given distance of the point in the axis h, from the 

 focal point p, be called d; and then if the point h has been assumed, so that 



the said given quantity, or distance, d, is greater than ~^ , but less than 

 -^^, the refracting superficies g h i, that shall perform what was required, will 

 be part of a concave sphere, whose radius is =.— ~ l"-— " . ^^^ jjg ^^g jjg. 



r ' ' m(—(f/, — m)d ' 



tance of the given point h, from r, the point to which all the rays will tend, 

 after refraction at the said concave surface, (whose radius being found, as above, 

 we call v) will be = ,..1^"_ ■ • Lastly, upon the point r thus obtained, 

 as a centre, with an interval a little less than h b, describe the circumference 

 K L M, and the figure g h i m l k will denote the section of a double concave 

 lens, which, placed at the given point in the axis h, (taken nevertheless within 

 the limits above mentioned) will collect all sorts of rays proceeding from the 

 speculum, into one and the same focus, or point of the axis r, as was required; 

 for the surface g h i, which first receives those rays, will refract the most re- 

 frangible sort converging to the point p, and also the least refrangible converg- 

 ing towards q, so that both sorts, after such refraction, will concur in the point 

 B ; but the rays tending to r, it is manifest, will suffer no refraction at their 

 emergence from the superficies k l M, because r is the centre thereof, by con- 

 struction ; which point r, where a perfect image of an object infinitely dis- 

 tant will be formed, we call the focus of the telescope, to distinguish it from 

 the point p, which we have before called the focus of the speculum. 



In this manner a lens, (or instead thereof a triangular prism with two of its 

 sides ground concave, and the third plain, if that be found as practicable) may 

 be formed and situated, so as to correct the errors of the speculum arising from 

 the different refrangibility of the rays of light. But in order to render this kind 

 of telescopes absolutely perfect in their construction, the errors also that result 

 from the spherical figure, must be rectified ; and with regard to this, we assert, 

 that it is possible to assume a point in the axis, between the focus of the spe- 

 culum and its vertex, (as we have taken the point h, in the following ex- 

 ample, see fig. 4,) at which, if a refracting superficies, or lens, be constituted, 

 according to the method already delivered, it will not only correct the errors 

 occasioned by the unequal refraction of the rays of light, but also rectify such 

 as proceed from the spherical figure of this speculum, to a much greater degree 

 of exactness than is requisite for any physical purpose, meaning always the 

 errors of those rays which respect the axis. Now to find or determine this 

 point, aflxjrds a problem not easy to be solved ; and we recommend it, as worthy 

 of the consideration of geometricians. 



