THE FLATNESS OF THE FIELD OF VIEW. 77 



that is, if </> denotes the angle P^c P, and the radius of the focal 

 surface is taken as unity. From this results 



p r 2r 



Pi = v + - = r + - 



cos cos 



consequently 



_/ I i I**.'/ 4 , ,-. 



* _ PiJ \ cos 0/ 3?- 2 4- cos 



ft ' p +f ~ 2r " = ~2 ' i _i_ 2 c d>" 



r + 4- 3r ^ 



cos 



The distances p* and j9 1 * are therefore, respectively, - - and 



2 



3r 2 + cos 



-9 -, , 9 -^ ; consequently the distances of the image-points 



^ J -* ~~\ COS </) 



from the centre of curvature, which we will call d* and df, are -r 



2 



,3 2 -}- cos 



and r . - - - . r. By an evident reduction we obtain 



2 1 + 2 cos 



1 4 cos d> 



1 + 2 cos0 ' 



^ 4 



d* : d,* = 1 : 



1 + 2 



T , ^ 4 cos 



and hence d* : d* = 1 : 



cos 



If the two image-points were situated in a plane at right angles 

 to the axis of the Microscope, the ratio of their distances from the 



point c would obviously be given by 1 : - . "We will now con- 



cos 



sider whether the second term is greater in the first or in the 

 second expression. If we take /3 = 1 cos </>, there results 



4 - cos 1 3 + 3 



1 + 2 cos ' cos 3-2/3*3-3/3' 



In this latter form the two values represent two fractions, in which 

 the numerators are greater than the denominators. The numerator 

 and denominator of the first fraction are also greater than those of 



Q i Q O 



the second by /3, consequently -^ -~ <. - ^ . The image- 



O ~ 2kj O Okj 



surface is therefore curved in this case also, and its convex side is 

 directed upwards. Under the given conditions this curvature is 

 little more than zero ; for if, for instance, we take < = 4, and 



