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if we choose, treat this collection of rays as a pencil proceeding from 

 that point. Hence the same beam of light may be decomposed into 

 pencils in an infinite variety of ways ; and yet, since we regard'it as 

 the same collection of rays, we may study its properties as a beam 

 independently of the particular way in which we conceive it analysed 

 into pencils. 



Now in any instrument the incident and emergent beams are com- 

 posed of the same light, and therefore every ray in the incident beam 

 has a corresponding ray in the emergent beam. We do not know 

 their path within the instrument, but before incidence and after 

 emergence they are straight lines, and therefore any two points serve 

 to determine the direction of each. 



Let us suppose the instrument such that it forms an accurate 

 image of a plane object in a given position. Then every ray which 

 passes through a given point of the object before incidence passes 

 through the corresponding point of the image after emergence, and 

 this determines one point of the emergent ray. If at any other 

 distance from the instrument a plane object has an accurate image, 

 then there will be two other corresponding points given in the inci- 

 dent and emergent rays. Hence if we know the points in which an 

 incident ray meets the planes of the two objects, we may find the 

 incident ray by joining the points of the two images corresponding 

 to them. 



It was then shown, that if the image of a plane object be distinct, 

 flat, and similar to the object for two different distances of the object, 

 the image of any other plane object perpendicular to the axis will be 

 distinct, fiat, and similar to the object. 



When the object is at an infinite distance, the plane of its image 

 is the principal focal plane, and the point where it cuts the axis is 

 the principal focus. The line joining any point in the object to the 

 corresponding point of the image cuts the axis at a fixed point called 

 the focal centre. The distance of the principal focus from the focal 

 centre is called the principal focal length, or simply the focal length. 



There are two principal foci, &c. formed by incident parallel rays 

 passing in opposite directions through the instrument. If we sup- 

 pose light always to pass in the same direction through the instru- 

 ment, then the focus of incident rays when the emergent rays are 

 parallel is the first principal focus, and the focus of emergent rays 

 when the incident rays are parallel is the second principal focus. 

 Corresponding to these we have first and second focal centres and 

 focal lengths. 



Now let Qi be the focus of incident rays, P! the foot of the per- 

 pendicular from Qj on the axis, Q 2 the focus of emergent rays, P 2 the 

 foot of the corresponding perpendicular, FjF^ the first and second 

 principal foci, A^a the first and second focal centres, then 



P i F L= P 1 Q 1= F 2 P, 

 A,F, P 2 Q 2 F 2 A 2 ' 



lines being positive when measured in the direction of the light. 



