C E V 



II. Caustics produced by refraction. 



1. Required the focus of a thin pemil of rays, after l,'*ingr refracted ob- 

 liquely at a plane surface. 



Let Q and q bo the foci of incident and 

 refracted rays, and 0' the /s. of inci- 

 dence and refraction, then 



. . . sin . sin 0' 



2. Required the same at a curved sur- Q- 

 face. 



Let the incident ray -*> 



the refracted ray ~ u 



/of incidence 



/ of refraction = p 



Radius of curvature of the surface at 

 the point of incidence r 



then 



u r cos c'. tan <? 



u tan <p (u + r cos ?; tan < 



(w 4- fcos 0; sin y 

 w r cos <j>' 2 . fan 



When 7< is infinite, or the incident rays are parallel, 



__ r cos &'. tan. <? _ r cos s's. sin .? 

 tan tan <^ x ~ sin (<p <p') 



When <p is a right angle, or u a tangent to the surface, 



v r cos 0'. 

 When r- is infinite, or the refracted rays parallel, 



r cos <5 tan 0' sin <?' 



W i -2 T COS <5 2 . : 



tan ^' tau 9 sin ;?' <?) 



For further information on this subject, see Coddingtan's Optics. 



CENTRAL FORCES. 



1. Of the motion of bodies in r,'/r'/,/rtr orbitf. 



Ijet V velocity of a body in a circle, R radius, P periodic time, 

 F ar.-elpratinir force, * 3.14-159 &c. then 



V* 4 r* R 



, or - - . 



IT 



48 



