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Transactions of the Society. 
towards the right hand from the vertex of s". The true focus/" = 
— 1*459. Next we have to find p and q the principal points of the 
crown equi-convex. This is done in precisely the same manner, but 
s being negative makes — g s positive, and r s negative ; h being also 
negative, p is positive and q negative ; the first is therefore measured 
to the right, and the second to the left, and both points are situated 
within the lens. The true focus, which is *4747, is positive. 
We have next to combine the left-hand diverging meniscus with 
the equi-convex lens, and to find the second principal point of this 
combination, which we will call Q, and its true focus <£. Now, 
although these lenses are in physical contact, yet, optically speaking, 
they are not so ; we must therefore first determine the optical distance 
cl between the lenses ; this is the distance between the second prin- 
cipal point q" of the diverging meniscus and the first principal point 
p of the equi-convex ; this obviously is *1517 — *0673 = *0844. 
By going through another arithmetical calculation, we find that 
the second principal point Q of the combination is situated at a point 
*03748 to the right hand of q, and therefore *1142 to the left of the 
posterior surface of the equi-convex lens ; the focus of the combination 
is *6481 and is positive. 
This doublet combination must now be combined with the second 
diverging meniscus. It will be unnecessary to calculate the principal 
points p' c[ and focus f of the second meniscus, as the lens is perfectly 
symmetrical with the first lens, all we have to do is to put p' = — q’, 
f — *— p ”, and f = f" ; both ,the radii being negative, p' and / will 
also be negative, and therefore will be measured from the proper 
vertices to the left. 
The optical distance D, between the doublet and the right hand 
diverging meniscus, must now be determined, it is the distance Q p' } 
both these points lie inside the equi-convex crown lens, Q being 
*1142 and p' *0673 from its posterior surface; D therefore is 
*1142 — *0673 = *0469; it is positive because measured towards 
the right. 
The last computation, which is similar to the one preceding it, 
shows us that the posterior principal point Q' is *0798 to the left of 
q' ; now as q l is • 1353 to the left of the vertex of s' (see p" in the 
first meniscus) Q r is *2151 to the left of the posterior surface of the 
triple ; the true focus of the triple is 1*102; this is measured from 
Q', it therefore is situated * 887 from the last surface. Consequently, 
the magnifying power of the combination is 8 * 07, and its working 
distance is * 887. 
Example 12. — We will now go through the same process with the 
other triple (in Example 6, Case 4, fig. 17), consisting of two bi-con- 
vexes enclosing an equi-concave. 
There is nothing needing any special explanation with regard to 
the principal points and foci of the separate lenses, but when the two 
first are combined you will notice that the second principal point Q 
