Fundamental Propositions in Optics. 475 



(involving in some degree a mental judgment) is, of course, 

 not to be expected. For the microscope, the purely physical 

 question of the distribution of light in the ultimate image of 

 a mathematical point cannot be definitely solved without some 

 assumption as to the manner in which the light is radiated in 

 different directions. Even if the radiation were uniformly 

 distributed, it does not follow that the light emerging from 

 the eye-piece consists of ordinary plane waves, equally intense 

 over the area of section. It would seem, indeed, that such a 

 uniform distribution of rays is inconsistent with good defini- 

 tion, except at the very centre of the field of the microscope. 

 For, in accordance with principles already discussed, it im- 

 plies at all points an equal magnifying power, which however 

 is to be reckoned always in relation to an object supposed to 

 be perpendicular to the initial direction of the ray. But in- 

 asmuch as the extreme rays of the pencil start in a direction 

 oblique to the axis, it is evident that in relation to a given 

 external object different parts of the system have different 

 magnifying powers. In order that the efficient magnifying 

 power (in the radial direction) may be alike for the whole of 

 the emergent pencil, the rays must be concentrated towards 

 the outer parts according to a law which will be obvious. In 

 this way the resolving power might perhaps come to be a 

 little greater than that estimated by v. Helmholtz, the concen- 

 tration of rays towards the circumference playing somewhat 

 the same part as a central stop. 



Much more might be said upon this subject, but probably 

 without results of practical importance. My main purpose 

 has been to emphasize fundamental optical principles which 

 have met with strange neglect, and to show how much 

 excellent work had been done in this direction by some early 

 writers. 



April 18. 



P.S. Reference should have been made above to an inter- 

 esting paper of Clausius*, in which the author develops very 

 fully the analytical theory of radiation, as based upon Hamil- 

 ton's function. One general theorem, previously established 

 with rather less generality by v. Helmholtz, may here be 

 noticed. The angles of the cones formed by the pencil of rays 

 at an object and at its image (supposed to be ' astigmatic) 

 stand simply in the inverse ratio of the areas of the corre- 

 sponding elements of object and image. 



In terms of apparent distance the argument may be put 

 thus. Consider a cross section of the pencil at any interme- 



* Pogg. Ann. t. cxxi. p. 1 (1864). 



