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CHAP. IX.
ABERRATION IN REFRACTION AT SPHERICAL SURFACES.
97. The question here is precisely similar to those we have met before, namely, to determine the difference between the ultimate value of the focal distance for refracted rays, and the value it has for a ray inclined at a sensible though small angle to the axis.
To begin with a single surface. Let v, (Fig. 95.) be the intersection of the refracted ray and the axis, every thing else as before.
Let Av=∆‵.
Referring to the beginning of last Chapter, we find that in strictness
m=QEQR·RvEv, or QE·Rv=m·RQ·Ev.
| Now QR2= | EQ2+ER2+2EQ·ERcosAER |
| = | (∆−r)2+r2+2(∆−r)r·cosθ |
| = | ∆2−2∆r+2r2+2∆rcosθ−2r2cosθ |
| = | ∆2−2r(∆−r)versinθ. |
Similarly, Rv2=∆‵2−2r(∆‵−r)versinθ.
Then putting v for versinθ,
∆−r√∆‵−2r(∆‵−r)v
(1).
=m(∆‵−r)√∆2−2r(∆−r)v
Then proceeding as in Chap. iii., we have
∆‵=∆′+(d∆‵dv)·v.
To obtain the value d∆‵dv, we must differentiate the equation (1), which gives
