ACTION OF A CONVEX LENS. 



745 



with its centre at m; ^>o = the oblique incident ray the mZ is the perpendicular <) = the 



3 



angle of incidence. The refractive index given is ; the object is to find the direction of the 



A 



refracted way. From o as centre describe a circle with a radius of any length ; from a draw a 

 perpendicular, ab to m Z\ then a b is the sine of the angle of incidence, i. Divide the line a b 

 into three equal parts, and prolong it to the extent of two of these parts, viz. , to p. Draw the 

 line p parallel to m Z. The line joining o to n is the direction of the refracted ray. On mak- 

 ing a line, n s, perpendicular to m Z, n s = b p. Further, n s = sine < ) = r. So that a b : s n 



Q 



or : bp) = B : 2 or sin. i : sin. r= . 



A 



Optical cardinal point of a simple collecting system. Two refractive media (fig. 527, L and 



E 



Fig. 527. 



G), which are separated from each other by a spherical surface (a, b) form a simple collecting 

 system. It is easy to estimate the construction of an incident ray coming from the first 

 medium (L) and falling obliquely upon the surface (a, b) separating the two media, as well as 

 to ascertain its direction in the second medium, G, and also from the position of a luminous 

 point in the first medium, to estimate the position of the corresponding focal point in the 

 second medium. The factors required to be known are the following : L (fig. 527) is the 

 first, and G the second medium, a, b = the spherical surface whose centre is m. Of course, all 

 the radii drawn from m to a b (m x, m n) are perpendiculars, so that all rays falling in the 

 direction of the radii must pass unrefracted through m. All rays of this sort are called rays or 

 lines of direction ; m, as the point of intersection of all these, is called the nodal point. The 

 line which connects m with the vertex of the spherical surface, x, and which is prolonged in 

 both directions, is called the optic axis, Q. A plane (E, F) in x, perpendicular to Q, is 

 called the principal plane, and in it x is the principal point. The following facts have been 

 ascertained : (1) All rays (a to a 5 ), which in the first medium are parallel with each other and 

 with the optic axis, and fall upon a b, are so refracted in the second medium that they are all 

 again united in one point [pj) of the second medium. This is called the second principal focus. 

 A plane in this point perpendicular to Q is called the second focal plane (C D). (2) All rays 

 (c to c 2 ), which in the first medium are parallel to each other, but not parallel to Q, reunite 

 in a point of the second focal plane (r), where the non-refracted directive ray (c x , m r) meets 

 this. (In this case, the angle formed by the rays c to c 2 with C Q must be very small. ) The 

 propositions 1 and 2 of course may be reversed ; the divergent rays proceeding from p towards 

 a b pass into the first medium parallel to each other, and also with the axis C Q (a to a 5 ) ; and 

 the rays proceeding from r pass into the first medium parallel to each other, but not parallel 

 to the axis Q (as c to c 2 ). (3) All rays, 'which in the second medium are parallel to each 

 other (b to b 5 ) and with the axis Q, reunite in a point in the first medium (p) called the first 

 focal point ; of course the converse of this is true. A plane in this point perpendicular to O Q 

 is called the first focal plane (A, B). The radius of the refractive surface (m, x) is equal to the 

 difference of the distance of both focal points (p and^) from the principal focus (x) ; thus m x = 

 p 1 x-p x. From these comparatively simple propositions it is easy to determine the following 

 points : 



1. The construction of the refracted ray. Let A be the first (fig. 528) ; B, the second 

 medium ; c d, the spherical surface separating the two ; a b, the optical axis ; k, the nodal 

 point ; p, the first and p x the second principal focus ; C, D, the second focal plane. Suppose 

 x y to represent the direction of the incident ray, what is the construction of the refracted ray 



