28 THEORY OF THE MICROSCOPE. 



therefore 



p _ jyo . Mr _ 2dr 



~ 



_ 



d - Qr ~ Qr - d ' 



f = 

 J 



- d 



The principal points of a bi-convex (or a bi-concave) lens with 

 equal curvatures lie, therefore, symmetrically on both sides of the 

 optical centre ; the distances from the two lens- vertices are equal 

 to each other. In a sphere, where d = 2r, the principal points lie 

 in the centre. If d is greater than 2?% as 'in cylindrical lenses, the 

 second principal point will be in front of the first. 



We will now give determinate values to the quantities r and d, 

 in order to apply the formulae we have obtained to the three 

 double-lenses of our objective. For greater convenience, .let / 

 and /' be, in each case, the focal lengths of the flint- and crown- 

 glass lenses, d Q and d' their thickness, and c, i, c, % their principal 

 points ; further, let </>, </>', <" be the focal lengths of the first, second, 

 and third double-lenses, and j, 7, E, /', E", r their principal 

 points ; and let N Q , N l , N 2 . . . . N* be the successive lens-vertices. 

 In the combination of the flint- and crown-glass lenses, therefore, 

 t' ( = e i) will be equal to the distance of the point c from the 



surface of contact, w = -- and u = -^. As the calcula- 



J J 



tion is easy, it will be sufficient to point out the operation to be 

 performed for the first double-lens only, and, for the others, simply 

 to compare the results obtained in a similar manner. 



First double-lens. Let r = 1 , d = - , d' 1 ; 



. ^ 



then 



^ T + - d = N Q + * X I = A T + jg , i = X 1 , 



,o _ . f, - _ __ 



i "IT " 3'' ' rf - 6r 



5 Xr 



