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| Now m∆−1∆′ | =m∆−1m∆−m−1mr |
| =(m−1m)1∆−m−1mr | |
| =m2−1m∆−m−1mr | |
| =m−1m(m+1∆−1r). |
This is positive, if ∆ be less than (m+1)r, and negative when A is above that value; when ∆=(m+1)r, there is no aberration. See p. 54. Note.
When r is negative, or the surface convex, the aberration is always positive.
99. We will now pass on to the aberration in a lens, (Fig. 96.)
We may consider this as consisting of two parts:
- The variation in the second focal distance arising from the aberration in the first (α).
- The additional aberration in the refraction at the second surface (β).
As to the first, we may consider the ratio of the variations as the same with that of the differentials of ∆′ and ∆″.
Now 1∆″=m∆′−m−1r′; ∴ d∆″d∆′=m∆″2∆′2.
Then since the first aberration is (∆′−r)2(m∆−1∆′)·v,
α=m∆″2∆′2(∆′−r)2(m∆−1∆′)·v.
For the second part we must alter our formula by putting
Now 1m for m, v′ for v, ∆′ for ∆, ∆″ for ∆′, r′ for v.
