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71
(∆−r)·(∆‵−rv)d∆‵−r(∆‵−r)dv√∆‵2−2r(∆‵−r)v
=m√∆2−2r(∆−r)v¯¯¯¯¯¯¯¯¯¯·d∆‵−m(∆‵−r)·r(∆−r)dv√∆2−2r(∆−r)v.
then making v=0, ∆‵=∆′,
(∆−r)·∆′d∆‵−r(∆′−r)dv∆′
=m∆d∆‵−m(∆′−r)·(∆−r)rdv∆;
that is, (∆−r){d∆‵−r(∆′−r)∆‵dv
m∆d∆‵−mr(∆′−r)(∆−r)∆dv,
or {(m−1)∆+r}d∆‵=(m∆−1∆′)r·(∆−r)(∆′−r)dv;
∴ (d∆‵dv)=r·(∆−r)(∆′−r)m−1‾‾‾‾‾∆+r(m∆−1∆′)
=(∆′−r)2·(m∆−1∆′), for ∆′−r=(∆−r)rm−1‾‾‾‾‾∆+r.
The aberration is therefore (∆′−r)2(m∆−1∆′)v.
When the incident rays are parallel, or 1∆=0, this reduces to
−(∆′−r)2∆′v, that is, −(F−r)2Fv.
98. In general, the aberration is positive or negative, that is, Av is greater or less than the ultimate value, according as m∆−1∆‵ is positive or negative.
