828 SUMMARY OF CURRENT RESEARCHES RELATING TO 



by 8000, and we get nearly 1/12 in., wliicla it is obviously absurd to put 

 as the limit of visibility in the microscopic image. 



The difference does not affect Sir H. Koscoe's argument, for the capacity 

 to see even the 1/1,000,000 of an inch vs^ould still leave us far from the point 

 •when atoms would be visible, but we call attention to his statement because, 

 coming from so high an authority as a President of the British Association, 

 it may give rise to a serious misapprehension as to the powers of the Micro- 

 scope of the present day. 



Heath's ' Geometrical Optics.' *— Measure of the Aperture of the 

 Microscope. — Dr. E. S. Heath's book is, we believe, the first English 

 treatise on optics in which aperture is dealt with. The following is the 

 author's treatment of the subject : — 



It has been shown that the brightness of an image given by a Micro- 

 scope is determined by the formula 



where X is the conventional image distance, p the radius of the pupil of the 

 eye, m the magnifying power, and a the divergence of the cone of rays 

 proceeding from the object in a medium whose refractive index is u. Thus 

 for an instrument of given magnifying power, 



I oc (m sin ay, 



and accordingly, u sin a may be taken to be the numerical measure of the 

 aperture. 



This measure of the aperture may be expressed in terms of the focal 

 length of the objective, and diameter of the pencil passing through it. The 

 diameter of the pencil as it passes through the object varies from the first 

 to the last surface. We shall suppose that the diameter is taken at the 

 back surface of the objective as the pencil emerges from it. This will be 

 so close to the second principal focus of the objective in microscopic objec- 

 tives of the ordinary type of construction, that the difference in the distance 

 may be disregarded. We shall therefore suppose that 6 is the semi-diameter 

 of the pencil at the second focal plane of the objective, and that / is the 

 focal length of the objective. Let u' be the distance of the image from the 

 second principal focus ; then, using the ordinary notation, 



Also by Helmholtz's theorem, we have 



u y8 sin a = u' /3' sin a', 

 and therefore 



It sm a = M — sm a 

 P 



u\ . , 

 = — - u sm a . 



The angle a' is always very small in Microscopes, never exceeding a 

 few degrees, and therefore u' sin a will not differ sensibly from u' tan a'. 

 But h = — u' tan a', and therefore 



u'b 



UBin a = — • 



'*■ Heath, K. S., ' A Treatise on Geometrical Optics,' xvii. aud 356 pp., figs., 8vo, 

 Cambridge, 1887, pp. 29i-6. 



