LIMITS OF OPTICAL CAPACITY OP THE MICEOSCOPE. 421 



This gives the brightness with which the surface of image 

 included within the outline of the illuminating cone shines, 

 independent of the direction which dS and dS' have in relation to 

 the axis, and of their distances from the surface of the curve (of lens.) 



From this image fdS') we might pass on to consider a second, dS", 

 and so forth. It is obvious that between each following image and 

 dS a similar equation would arise. 



If we suppose the object and the image to lie in the same 

 medium, then the Irightness of the optical image produced hy rays 

 which incline at very slight angles to the axis and perpendicular will 

 always he eq^ual to (i.e., neither more nor less than) the Irightness of 

 the ohject, except in so far as loss of light hy reflection and ahsorption 

 may occur. 



But this law should hold good without limitation of divergence- 

 angle. For if it were possible to throw an image of any 

 bright point sending forth its light according to the conditions 

 above expressed, (namely, of rays circumscribed by a diaphragm 

 aperture) which image should shine with greater intensity than the 

 rule above given admits ; then we could cause this bundle of rays 

 to pass on as parallel rays through a plane end-surface into the air, 

 and to fall into the eye of an observer ; and in such case it would 

 happen that an object would be seen more brightly illuminated 

 through an optical instrument than it was before, — a thing contrary 

 to all experience, whatever kind of transparent refracting material 

 be used. I^ow, if this were possible with light it would also be 

 true of heat, as might be shewn by application of similar 

 reasoning ; and then the law of equal radiation of bodies possessing 

 equal temperature would be impugned. 



But the equation which premised very slight divergence-angles 

 of incident rays may be more precisely formulated, and so express 

 the same result in the case of wide divergence-angles. 



A more precise expression of the law of divergence-angles. 



In equation (5) it is a matter of indifference whether we substitute 

 for a its sine or tangent or similar functions which for indefinitely 

 small a would be its equivalent. If we assume larger divergence- 



