Fundamental Propositions in Optics 471 



A nearly similar investigation in terms of the corpuscular 

 theory is to be found in Thomson and Tait's ' Natural Philo- 

 sophy' (1867), §§ 326, 327, where the authors emphasize 

 strongly the optical importance of their conclusions. " The 

 most obvious optical application of this remarkable result is, 

 that in the use of any optical apparatus whatever, if the eye 

 and the object be interchanged without altering the position 

 of the instrument, the magnifying power is unaltered." And 

 again, " Let the points 0, 0' be the optic centres of the eyes 

 of two persons looking at each other through any set of 

 lenses, prisms, or transparent media arranged in any way 

 between them. If their pupils are of equal sizes in reality, 

 they will be seen as similar ellipses of equal apparent dimen- 

 sions by the two observers/'' 



It will be remarked that in general the " magnifying 

 power " varies in different directions, a circle being seen as an 

 ellipse. This phrase and that of "apparent distance" may 

 be understood in an extended sense. Thus the apparent dis- 

 tance may be taken to mean the distance at which the object, 

 if seen direct, would present the same angular area as when 

 seen through the lenses &c. 



The demonstration of the law of apparent distance upon 

 the basis of Hamilton's characteristic function, is of course 

 satisfactory, and perhaps indispensable to a complete investi- 

 gation ; but it is not unimportant to remark that the essential 

 part of the law is really included in the vastly more general 

 reciprocal theorem, established in the first instance by v. Helm- 

 holtz for vibrations in a uniform gaseous medium*, and 

 capable of extension to all vibrating systems, even though 

 subject to dissipative forces f. 



Let ^P denote the intensity of a radiant source at 0, yjr the 

 corresponding amplitude of luminous vibration at 0'. By 

 the doctrine of rays, the energy transmitted across any section 

 of a pencil is the same, and thus if a t be the area of the cross 

 sections at 0' of a thin pencil of rays from whose angular 

 magnitude is there a> ly we have 



where H is an absolute constant. 



In like manner if <r 2 be the area at of a pencil starting 

 from 0' with angular opening g> 2 , 



o- 2 ^ 2 =Hft> 2 ^ /2 , 



* Theorie der Luftschwingungen in Rohren mit offenen Enden. Crelle, 

 Bd. lvii., 1860. 



t Proc. Math. Soc. June 1873; ' Theory of Sound,' vol. i. §§ 107, 108, 

 109. 



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