220 On the General Laws of Brightness of Images. 



we understand by A 2 the projection of this surface on a 

 normal section of the beam. 



When A-i is the image of A 2 the " theorem on apparent 

 size " requires modification, for the centre of A l will no longer 

 receive one ray from each point of A 2 , but will receive all its 

 rays from one point only, which we shall regard as the centre 

 of A 2 . Let a section orthogonal to all the rays which go from 

 the centre of A 2 to the centre of A 1 be taken anywhere in the 

 intermediate region, but not so as to coincide with a surface 

 at which there is an abrupt change of direction in any of the 

 rays. For instance, if the image is formed by a lens, the 

 section may be taken either in the substance of the lens or in 

 the external medium, but must not coincide with a face of the 

 lens. Divide this section into parts comparable in size with 

 A-i or A 2 , and let A 3 denote the area of one of these parts. 

 Also let ii z denote the index of refraction at the centre of A 3 . 

 Since A 1 is the image of A 2 , the rays which go from the 

 perimeter of A 2 to a fixed point of A 3 will go on to the peri- 

 meter of A x ; thus A 3 and A x will subtend the same solid angle 

 as seen from A 3 , call it co 3 . Hence, by applying (1) first to 

 A 1 and A 3 and then to A 3 and A 2 , we have 



(Ji^A 1 o3 l — fi 5 2 A z co 3 = /A 2 2 A 2 fi> 2 , . . . (14) 



&>! and g> 2 denoting the solid angles formed at Aj and A 3 by 

 rays which traverse the perimeter of A 3 . 



Equations (14) show that the ratio of co 1 to <w 2 is the same 

 for all the portions such as A 3 into which the intermediate 

 section is divided. Let £l x H 2 be the whole solid angles at 

 Ai A 2 formed by the aggregate of all the rays which go from 

 A 2 to A v Ilj is the sum of all the partial angles co 1} and fl 2 

 is the sum of all the partial angles a> 2 ; hence we have 



/^AAl = fi 2 2 A 2 n 2 , (15) 



a result which is similar in form to (1). 



The rays which go from A 2 to A x diverge from A 2 in the 

 solid angle I2 2 , and converge to Aj in the solid angle fl v 

 Let Q be the whole light sent from A 2 on the way to A 1; and 

 JcQ, the portion of it which actually reaches A 1} then the 

 intrinsic brightness I of the real image A x will be 



, kQ kA 2 n 2 Q _,//M 2 Q 



•£)s3s-*(2K a* 



Ai^li A l n 1 A 2 fl 2 \yti 2 / A 2 I2 2 \/j, 2 . 

 as in (13). When the pupil of the eye is placed at the image, 

 the brightness of the field of view will be identical with this 

 intrinsic brightness of the image, if the pupil is filled. The 

 apparent brightness of A 2 as seen from A x will then be un- 

 affected by the circumstance that A x is the image of A 2 . 



