46 



But from ((j) we 

 becomes 



MEMOIltS OF THE NATIONAL ACADEMY OF SCIENCES. 

 0' 



that <pi = /Jo 



whence, dividing by the value of <?, the magnifying power 



as a perfectly general expression. 



For the telescope proper, po = 1 and 



P" .D 



1 also, since the length of the iustruineiit is neg- 



igible with respect to the distance of the object. In this case the expression reduces to -^ , or, 



if we make fl° equal to the diameter of the objective, it is eipial to the quotient derived by divid- 

 ng the diameter of the objective by the diameter of its image produced by the ocular, an old rule 

 first given, I believe, by Ramsdeu. 



For the microscope the magnifying power is defined somewhat differently from our definition 

 above, namely, as the ratio of the apparent size to the size as seen at a distance of ten inches, - 



which is T^ times as great as ours. Thus modified the expression becomes 



"fflOi d ''■ ' 



since po becomes the same as the index of refraction of the " immersion fluid " employed. Not only 

 this, but we have also ,, y ^^ *^^ tangent of one-half the so-called " angular aperture," whence the 

 true aperture equals 



n sin < tg ^--, > 



It has long been known that the complete optical power of a telescope — that is, both its mag*^ 

 nifying power and resolving power — could be determined by two linear measurements, the a" and 

 (("lOf our discussion, but it has, perhaps, n;>t bean suspected before that with three linear measure- 

 ments it is possible to determine both the magnifying power and resolving power of a microscope. 



To illustrate the application of these formulas, we may quote the determinations of the optical 

 constants of a Zeiss microscope employed with ocular No. 2, shortest tube, and three different ob- 

 jectives. The objectives were, a Zeiss J, of which his catalogue gives magnification 52 and numer- 

 ical aperture 0.20, a 4""' objective made by the writer, and a iV°' Wales water immersion. The 

 measures were made by placing a glass scale upon the table of the microscope and bringing the 

 objective into contact with it, the number of divisions of the scale visible above the ocular, and 

 also the absolute length of its image was then recorded, the former length being «" and the latter 

 al\. Then the tube was raised a measured distance {(I) until the scale was in focus. In the table 

 the measures are in millimeters. M is the calculated magnification and A the aperture. 



It should be noted that, in order to determine .If with precision, the ratio of a" to a",, near the 

 axis, should be taken, since the surface a" of Fig. 4 is not plane, but a portion of a wave surface. 

 For small apertures this distinction is insignificant. 



To investigate the resolving power of an optical apparatus used in conjunction with the eye 

 we may proceed as follows: 



Suppose that we have a plane area at a distance of 10 inches fi-om the eye divided into regular 

 spaces, and that we observe it through a hole of diameter «"i. The smallest angular values of the 



