BOOK OF MATHEMATICAL PROBLEMS. 



1G05. A ray is propagated in a medium of variable density 

 in one plane (xy) which divides the medium symmetrically : prove 

 that the projection of the radius of curvature at any point of the 

 path of the ray on the normal to the surface of equal density 

 through the point is equal to 



V () *(%) 



1606. A small pencil of parallel rays of white light, after 

 transmission in a principal plane through a prism, is received on 

 a screen whose plane is perpendicular to the direction of the 

 pencil: prove that the length of the spectrum will be propor- 

 tional to 



fa, - fQ sin t . 



cos 2 /) cos (D + L <}>) cos <f> ' 



where i is the angle of the prism ; <, <f> the angle of incidence 

 and reflexion at the first surface, and D the deviation, of the 

 mean ray. 



1607. Two prisms of equal refracting angles are placed with 

 one face of each in contact and their other faces parallel : prove 

 that the condition of achromatism for two colours is 



fy*, = J>/* 2 . 

 cos <' cos \f/' y 



where <, </>' are the angles of incidence and refraction at the first 

 surface, 



and $', \j/ are the angles of incidence and refraction at emergence. 



1608. When a ray of white light is refracted through a 

 prism in a principal plane, so that the dispersion of two given 

 colours is a minimum, 



Bin(3<fr'-2 t ) 2 m 



in <f> [jf ' 



$ being the angle of refraction at the first surface and t the 

 refracting angle. 



