INVERTEBRATA, ORYPTOGAMIA, MICROSCOPY, ETC. 511 



tism whicli has just been defined, becomes a matter of vital importance. 

 When the error in the convergence is considerable and atiects the 

 whole a2)erture, the defects arising from ordinary aberration, curvature 

 of the field, and other causes, sink into insignificance —the image of a 

 flat object appears then not merely as a curved surface, but rather as 

 the apex of a cone viewed from the axis. 



In the case of large angles of aj^erture the second condition of 

 aplanatism cannot be so completely satisfied, either theoretically or 

 practically, but that evident traces of the error in divergence are left 

 in the image even when the best means of construction are employed. 

 Microscopists have given to this the very unsuitable name of " curva- 

 ture " or " unevenness of the field of view," by which it is commonly 

 known. It may, however, bo easily shown by experiment that the 

 defects in the image thus described increase in amount, not with the 

 second, but with the first power of the distance from the axis, and 

 therefore in the main have nothing to do with the actual curvature of 

 the image surface. 



By the simjile experiment about to be described, the characteristic 

 relation of convergence of rays at aplanatic points may be observed, 

 and the fact that it persists universally becomes confirmed in 

 a striking manner. This experiment is founded on the contrast 

 between the aplanatic points and the orthoscopic points of the lens- 

 system, and is deduced from the following considerations. 



If an optical system is to produce a correct image of an object 

 extended in a plane, the principal rays proceeding from points of the 

 object and crossing at a point on the axis, and the corresponding 

 principal rays of the image-forming pencils which cross in the con- 

 jugate point on the axis and proceed to points of the image, must 

 maintain a constant ratio between the tangents of their angles of in- 

 clination. It is only when a lens-system satisfies this condition for a 

 pair of conjugate points on the axis (as o. g. a properly constructed 

 eye-piece should do for tho place of the objective-opening and the 

 point of vision conjugate to it) that it is orthoscopic, and can produce 

 images which are orthogonal and free from distortion when tho object 

 or the image, or both, are eliown under a large angle. But 

 aplanatic points, by virtue of tho condition of aplanatism, are in 

 antagonism with this property of orthoscoi)ic points, and consequently 

 an aplanatic system, from its peculiar relation of convergence, must 

 give a distortion of the images, wliich can he determined beforehand, 

 when wo produce tho image of a plane distant from the aplanatic 

 point wliile the principal rays of tho image-forming pencils cross in 

 this aplanatic point. 



It will be cnougli if we determine the form which parallel lines 

 assume under these circumstances, or vice versa, find what curves 

 will appear as straight lines in tho resultant imago. 



By a simido calculation it is found that a set of parallel lines 

 in a phvne perpendicular to the optical axis, will, with an ai)lanatic 

 Bysteni, form a number of ellipses liaving tho same major axis, but dif- 

 ferent niinor axes (tho line at infinity forming a circumscribing semi- 

 circle).^ 0)1 tho other hand, a sot of liypcrbohis (of deterininiito 



