ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
113 
If the positions of the four cardinal points are known for any two 
lenses separately, then, when the lenses are placed at any given distance 
apart, the positions can be found for the four cardinal points of tho 
combination. The geometrical construction is very simple, and is 
illustrated in fig. 12. Let 7q h., be the two principal points of a lens, 
and / its principal focus for light passing through it to the right. Let 
h\ h' 2 be those of a second lens, and let /' be its principal focus for 
light passing the other way. It is required to find the position of tho 
principal points and of the principal focus of the equivalent lens. 
Consider any ray-path a b parallel to the axis. Light travelling along 
from a will, after passing the principal planes of the first lens, turn 
towards /. Similarly, light passing the other way from b, after passing 
the second lens, will turn towards /. These paths cross at o. Join o 7i 2 
and o li 1 ; and draw 7q x parallel to o h 2 ; and li\ y parallel to o h\. The 
planes x II x and y H 2 drawn through x and y will bo the desired princi- 
pal planes. And the resultant focus F is found by considering the ray 
which starts from a and passes through o towards /, and remembering 
that, as it passes through the second lens, it will be shifted forward 
through the distance between the planes 1l\ h' 2 , and turned as though 
it came from y. A little consideration will show that if the two lenses 
were close together the width Hj H 2 will be the sum of tho widths h l h 2 
and li\ h' 2 ; whilst if the two lenses are moved wider apart Hj and H 2 
will come nearer together, and may even cross past one another. If tho 
lenses are placed at a distance apart equal to the sum of their focal 
lengths, H 1 and H 2 will not only have crossed planes, but will have 
separated to an infinite distance apart. 
The formula} for calculating the resultant focal length and resultant 
width between the principal points for a combination of two lenses at a 
distance apart, are as follows : — 
resultant / = ; 
/ + f» ~ a 
resultant w - w, -f- w 2 — 2 ; 
/i +/ a - a 
1892. 
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