VOL. XVII.] PHILOSOPHICAL TKANS ACTIONS. 5^5 



of the lens be rejected as not sensible, the rule will be much shorter, viz. 

 ■ ", — =: /, or in glass -^ — — :; — —- — = / ; all the terms wherein 



dr + df - pr f ~" ° dr + df — 2 r f -^ ' 



t is found being omitted, as equal to nothing. In this case, if d be so small, 

 as that 2 r p exceed dr -\- dp, then will it be — f, or the focus will be negative, 

 which shows that the beams after both refractions still proceed diverging. 



To bring this to the other cases, as of converging beams, or of concave 

 glasses, the rule is ever composed of the same terms, only changing the signs 

 of + and — ; for the distance of the point of concourse of converging beams, 

 from the point B, or the first surface of the lens, I call a negative distance or 



— d ; and the radius of a concave lens I call a negative radius or — r, if it be 

 the first surface, and — p, if it be the second surface. Let then converging 

 beams fall on a double convex of glass, and the theorem will stand thus 



— J J — -- — = 4- /■; which shows that in this case the focus is always 



affirmative. 



If the lens were a meniscus of glass, exposed to diverging beams, the rule is 



— J — -—; — ;'-— — =f; which is affirmative when 'Irp is less than dr — do. 



— dr + af + 2rf'' ^ f 

 Otherwise negative : but in the case of converging beams falling on the same 



+ 2 c/ r p . 1 1 . 



meniscus, it will be -— -j y — — - — =/; and it will be -I- /, while dp — dr 



+ a r — a f + 2 r f • j ' r 



is less than 'irp, but if it be greater than 2rp, it will always be found nei 



gative or — f. If the lens be double concave, the focus of converging beams 



is negative, where it was affirmative in the case of diverging beams on a double 



convex, viz. -— r— ;-t :; — = Jl which is affirmative only when 2 r a 



exceeds d r •}- d p : but diverging beams passing a double concave have always 

 - . —2drf 



a negative focus, viz. — -j — --; — --- — = — j- 



The theorems for converging beams are principally of use to determine the 

 focus resulting from any sort of lens placed in a telescope, between the focus 

 of the object-glass and the glass itself; the distance between the said focus of 

 the object-glass and the interposed lens being made = — d. 



I here suppose my reader acquainted with the rules of analytical multiplication 

 and division, as that -1- multiplied by + makes the product +, + by — makes 

 — , and — by — makes -j- ; so dividing + by -f makes the quote -f-, -(-by — 

 makes—, and — by — makes -{■ ; which will be necessary to be understood 

 in the preceding examples. 



In case the beams are parellel, as coming from an infinite distance, which is 

 supposed in the case of telescopes ; then will d be supposed infinite, and in the 



4 G 2 



