The Helmholtz Theory of the Microscope. By J. W. Gordon. 433 



subject to the proviso that 6 is not to be greater than — . For the 

 tangent scale we have in like manner 



tan 6 . 



T = 



6 



(H) 



subject to the same proviso. 



So, again, connecting the sine and tangent scales directl}-, we have 

 the equation 



T=-^ (15) 



cos $ 



These magnitudes are measured upon the surfaces in which the images 

 are formed. It is obvious that the sine and tangent scales are fully- 

 comparable to one another, for they are represented by straight lines. 



Fig. 105. 



If in the arc scale we inquire for axial distances we find, of course, that 

 they coincide with the scale measurement of the^ sine scale so that the 

 sine law applies to axial distances in spherical fields. 



But we have still to investigate these relations, substituting the 

 divergence angle u for the position angle 6, and in proceeding to do so 

 it is useful to remember that as between conjugate images we may 

 determine the scale of magnification by observations made in any part 

 of the field. It is quite true, and will presently appear, that in the 

 tangent field the magnifying power varies from zone to zone, whereas 

 in the sine field it is uniform all over the area of the field and hence 

 an unsymmetrical system giving a sine image in one field and a tangent 

 image in the other, although it might be aplanatic, could not yield an 

 undistorted image. But an image in one tangent field of an object in 

 another tangent field is not impaired by the variation of magnifying 

 power, for, as the position angle 6 is of necessity the same for both 

 conjugate points, the change in apparent magnitude of the object is 

 proportional to what the image gains by the alteration of focal length, 

 and so the correctness of the projection is maintained. We must, there- 

 fore, postulate symmetry in this sense and may then assume uniform 

 magnifying power in any kind of field. 



That being granted, it is easy to see that we are at liberty to choose 



