III. 



DETERMINATION OF THE DISTANCES OF CORRE- 

 SPONDING IMAGE-POINTS FROM THE AXIS IN 

 THE CASE OF ANY GIVEN INCLINATION OF THE 

 RAYS. 



As already intimated, the assumptions upon which the formula 

 of Gauss are founded are entirely at variance with the actual 

 conditions of the formation of the image by means of systems of 

 lenses of high power. In Microscopes we have to deal with very 

 large angles of aperture of the image-forming pencils ; the theory 

 of Gauss assumes these angles to be so small that their cubes may 

 be neglected. 



Nevertheless, the conclusions which may be drawn by the aid of 

 these formulae afford a means of making an approximately correct, 

 and, in many cases also, practically useful diagram of the path of 

 the rays, and observation shows that the position of the cardinal 

 points determined analytically does not differ very much from that 

 found by measurement. For objects, therefore, whose marginal 

 points are not at a great distance from the axis, the calculation 

 of the corresponding magnitude of the image for any combination 

 of refracting surfaces is fairly applicable, i.e., the ratio between 

 image-magnitude and object is about the same as that between 

 the posterior and anterior focal lengths. A construction made 

 after the analogy of Fig. 4 gives, therefore, for any system 

 not merely the distances in the direction of the axis (abscissae), 

 but also the distances in the plane of the image (ordinates). 

 Trigonometrically expressed, the latter, measured from the axis, 

 are equal to the tangent of the angle at which the rays diverge 

 from the direction of the axis. If we denote this angle by B, and 

 the posterior focal distance, as above, by p*, the linear distance 

 of the image-point is consequently equal to p* X tan B. 



A more accurate analytical determination of these distances, 

 especially with reference to rays more obliquely inclined, leads, 

 however, to the result, that in the above formula the sine of the 



