OP DIOPTRICS AXD CATOPTRICS. 



71 



at the common surface of two mediums is the of the focus of incident rays, the sum, divided 

 quotient of their respective indices. ^ b}' the index of refraction, is equal to the 



For the indices being r and qr, if the sine of incidence sum of the reciprocals of the rudius, and of 

 from a v.cuum be ;, the sine of refraction in the first me- jj^g distance of the focUS of refracted rayS : 



dium will be 2., and this interposed medium being ter- tlie distances being considered as negative 



minated by parallel surfaces, the sine of refraction in the when the respective focl are On the COncave . 

 second medium will be the same as in the absence of the side of the surface. 



first, or -L, which is toi. as 1 to?. Let AB be the axis, and AC a f-^q 



?'' *" ray infinitely near it ; let D be 



410. Theorem. The angle of deviation the centre, and e the focus con- a 

 being given, the angles of incidence and re- jugate to A. Call BD, a, ab, 



fl-action will be equal to the angles at the '^'^^' '' '^' '"''"" °^ refraction, r. and the angle CAB, or 

 base of any triangle, of which the sides are 

 as 1 to the index of refraction, including an 

 angle equal to the angle of deviation. 



For the sines of the angles of a triangle are proportional 

 to the sides opposite to them. 



411. Definition. The point of inter- 

 section of the directions of any two or more 

 rays of light is called their focus y and the 

 focus is either actual or virtual, accordingly 

 as they meet in it, or only tend to or from it. 



412. Definition. When the divergence 

 or convergence of rays is altered by refrac- 

 tion or reflection at any surface, the foci of 



the incident and refracted or reflected rays 415. Theorem. The distances of the 

 are called conjugate to each other ; and the conjugate foci from a plane refracting sur 

 new focus is called the image of the former 



its sine, s. Then CD : CA : : Z.CAB : /_CVB, the arcs 



coincidingultimatelywiththeirsines,and z.CDBzr s, and 



a 



the angle of incidence ACF=— s+s, whence DCE=:;— 

 a r 



ACF—{d+a).. 



But sin. DCE : sin. CDE : : DE : CE, 



-<- ,and^-f — ::: — — . But since E is on the con- 



rd ra rd a e 



cave side, we must substitute — e for « to make the theorgm 



general,and 1 — l— 



ra rd a e 



1 _ 1 1 1 



c ra ^'' 



rd 



focus. 



413. Theore-M. In reflections at a plane 

 surface, the conjugate foci are at equal dis- 

 tances from the surface, and in the same 

 perpendiculai'. 



For in the triangles ABCj 

 DBC, z.ABC=DBC, since 

 EBC=:ABF=:DBF ; and z. 

 ACBrrDCB, and CB is com- 

 mon to both triangles ; there- 

 fore BD=AB, and the triangle BDF=BAF, and DF=AF. 



414. Theorem. For rays falling on a 

 spherical surface nearly in the direction of 

 the axis, the reciprocal of the radius being 

 itncreased by the reciprocal of the distance 



face are in the ratio of the sines ; and both 

 are on the same side. 



For here azz 33 , and — _ , or rd'ZZe, and both dis- 



e rd 



tances are positive, pr both negative. 



416. Definition. When the focus of 

 incident rays becomes infinitely distant, the 

 rays are parallel, and the conjugate focus of 

 such rays is called the principal focus of a. 

 surface or substance. 



417. Theorem. The principal focus of 

 a spherical reflecting surface is at the dis- 

 tance of half the radius. 



By making r= — 1 , we accommodate the general theorem 



for refraction to reflecting surfaces, and — — - + 



a a a 



