Vol. IX] WOODWORTH— OPTICS OF THE MICROSCOPE 179 



astigmatism is contrary to these fundamental conceptions. It 

 is perhaps one of the most remarkable facts in the whole history 

 of optics that the hypothetical conoids of Sturm should have 

 been accepted by physicists and used by all practical computers 

 of optical instruments in the face of the recognized fact that 

 nothing corresponding to them can be obtained experimentally. 

 When it is fully realized that the theory based on this con- 

 ception also violates the fundamental laws of focus formation, 

 the study of these interesting mathematical forms will be 

 removed from optics to their proper place in geometry and in 

 their place rational methods of measuring the astigmatism 

 will prevail. 



The laws of focus formation as regards object points on 

 the optical axis are correctly interpreted even in elementary 

 treatises, but it is not so well understood that precisely the 

 same laws apply to oblique refraction. It is not difficult to 

 state them in such general terms that their universal applica- 

 bility is at once evident. 



Law I. All possible foci of an object point lie on the line 

 through that point normal to the spherical refracting surface. 



This law has often been stated in the form "the focus of a 

 point is a line" without defining the position of the line. The 

 law is true of all refractions of a point source of light through 

 a single spherical surface. The equation of this line may be 

 given in either of the following forms : 



(1) 



(2) 



the angles e and e' in each case being measured irom the 

 normal to the surface. 



The pair of aplanatic points are those in which c=r — .c'=r-^ 

 « = a' and e' = x in which case the ratios of cosines be- 

 come equal to unity. The common aplanatic point is where 

 c, c', a and x' are all zero and 6 = 6', These are the only finite 



