i^S Cn ihe Afichant/m if the Eye. 



Prop^oH III. Prohlem. (Plate XIII. Fig. |.) 



At the vertex of a given triangle (CBA), to place a given refracting furface (B), fo 

 that the incident and refracted rays may coincide with the fides of the triangle (A B 

 und B C.) 



Let the fides be called d and e; then in the bafe take, next to d (or AB), a portion (AE) 



equ^l to ~~ , or (AD zz) — ■ ■ ■ , ..; draw a line (EB, or DB) to the vertex, and 



n d -\- m e m d -\- n e 



the furface muft be perpendicular to this line, whenever the problem is phyfically pofliblc. 



When e becomes infinite, and parallel to the bafe, take — or next to J, for the in- 



m ft 



terfedtion of the radius of curvature. 



Prqpojiim IV. Theorem. (Fig- a.) 



In oblique refraftions at fpherical furfaces, the line (AI, KL,) joining the conjugate foci 

 (A, I; K, L;) pafles through the point (G), where a perpendicular from the centre (H) 

 falls on the line (EF), bife£ting the chords (BC, BD,) cut off from the incident and re- 

 frafted rays. 



Corollary i. Let t and u be the cofines of incidence and refraflion, the radius being i, 

 and d and e the refpe£tive diftances of the foci of incident and refrained rays; then 



tn d u u 



^ I — I II. 



m du — ndt — n t t 



Corollary 2. For a plane furface, e z: -'— — — . 



m u u 



Corollary 3. For parallel rays, dzr co, and ezz 



m u^-n t 



Scholium I. It may be obferved, that the cauftic by refra£tion ftops fhort at Its cufp, 



»ot geometrically, but phyfically, the total refle6tion interfering. 



^ „ /n ,. *" " " > I f' 1 1 , b d 



Corollarv 4. Call ■ , », and , <■; then e — -. , and * — * = 



•' 711 tt — n t m u — nt a — c 



m ■; or, In words, the rectangle contained by the focal lengths of parallel rays, paffing 



and repafllng any furface in the fame lines, is equal to the reiStangle contained by thCi 



.^iffsrencef between thefe lengths and the diftances of any conjugate foci. , , ,^ 



■ ■ ■ . '" ^ "* " •<• 1. 'i « '~i. 



Corollary 5. For perpendicular rays, e = r = m -\- -j ; or, if the raaiu»bc aj 



■ % and if d and e be given to find the radius, a = 



-d—na' ^ md + ne 



Corilkry 6. For rays perpendicular and parallel, < — »w, ox ez:zma. 



Corollary 



