Barrow, Isaac
,
Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur
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<
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<
s
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xml:space
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">XXV. </
s
>
<
s
xml:id
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echoid-s13582
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xml:space
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">Nè ſpeculatio præſens, _ob bujuſmodi complures metbodos Cy-_
<
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_clometricas indies promulgatas_, aſpernanda videatur, adjungemus con-
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ſectarium unum vel alterum, quibus fortè ſolis hæc paucula meruerant
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<
note
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="
right
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xlink:label
="
note-0279-01
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xlink:href
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note-0279-01a
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xml:space
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">Fig. 148.</
note
>
impendi; </
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<
s
xml:id
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echoid-s13583
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xml:space
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">à quibus nempe _Maxima, Minimaque_ ſui generis innume-
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ra determinantur.</
s
>
<
s
xml:id
="
echoid-s13584
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xml:space
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"/>
</
p
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<
p
>
<
s
xml:id
="
echoid-s13585
"
xml:space
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">Sit _Semicirculus_ ABZ, cujus centrum C; </
s
>
<
s
xml:id
="
echoid-s13586
"
xml:space
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preserve
">ſitque _ſegmentum
<
unsure
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_
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ADB; </
s
>
<
s
xml:id
="
echoid-s13587
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13588
"
xml:space
="
preserve
">huic adſcripta _paraboliformis_ AFB, cujus exponens {_u_/_m_};
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</
s
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<
s
xml:id
="
echoid-s13589
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xml:space
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">ſit item AD = {_m_ - 2 _n_/_m_ - _n_} CA; </
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>
<
s
xml:id
="
echoid-s13590
"
xml:space
="
preserve
">_paraboliform@s_ autem _parameter_ (hoc
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eſt recta, cujus aliqua poteſtas in poteſtatem ſegmenti axis, ſeu AD,
<
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ducta conficit _poteſtatem_ ordinatæ, ceu D B) nominetur _p_; </
s
>
<
s
xml:id
="
echoid-s13591
"
xml:space
="
preserve
">erit _p_ in ſuo
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genere _maximum_.</
s
>
<
s
xml:id
="
echoid-s13592
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xml:space
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"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s13593
"
xml:space
="
preserve
">Nam utcunque ducatur GE ad DB parallela, & </
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>
<
s
xml:id
="
echoid-s13594
"
xml:space
="
preserve
">ad GE poſita
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concipiatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter_ di-
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catur _q_. </
s
>
<
s
xml:id
="
echoid-s13595
"
xml:space
="
preserve
">quum ergò _paraboliformis_ AFB _circulum_ extrorſum contin-
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gat, erit GF &</
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<
s
xml:id
="
echoid-s13596
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xml:space
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">gt; </
s
>
<
s
xml:id
="
echoid-s13597
"
xml:space
="
preserve
">GE; </
s
>
<
s
xml:id
="
echoid-s13598
"
xml:space
="
preserve
">adeóque GF {_m_/ } &</
s
>
<
s
xml:id
="
echoid-s13599
"
xml:space
="
preserve
">gt; </
s
>
<
s
xml:id
="
echoid-s13600
"
xml:space
="
preserve
">GE {_m_/ }; </
s
>
<
s
xml:id
="
echoid-s13601
"
xml:space
="
preserve
">hoc eſt _p_ {_m_ - _n_/ } x
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AG {_n_/ } &</
s
>
<
s
xml:id
="
echoid-s13602
"
xml:space
="
preserve
">gt; </
s
>
<
s
xml:id
="
echoid-s13603
"
xml:space
="
preserve
">q {_m_ - _n_/ } x AG {_n_/ }; </
s
>
<
s
xml:id
="
echoid-s13604
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xml:space
="
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">quare _p_ &</
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<
s
xml:id
="
echoid-s13605
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xml:space
="
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">gt; </
s
>
<
s
xml:id
="
echoid-s13606
"
xml:space
="
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">_q_.</
s
>
<
s
xml:id
="
echoid-s13607
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s13608
"
xml:space
="
preserve
">Notandum eſt eſſe _p_ {2 _m_ - 2 _n_/ } = ZD
<
emph
style
="
sub
">_m_</
emph
>
x AD {_m_ - 2 _n_/ }. </
s
>
<
s
xml:id
="
echoid-s13609
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13610
"
xml:space
="
preserve
">q {2 _m_ - 2 _n_/ }
<
lb
/>
= ZG {_m_/ } x AG {_m_ - 2 _n_/ }. </
s
>
<
s
xml:id
="
echoid-s13611
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xml:space
="
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">unde ZD {_m_/ } x AD {_m_ - 2 _n_/ } &</
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>
<
s
xml:id
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echoid-s13612
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xml:space
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">gt; </
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<
s
xml:id
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echoid-s13613
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xml:space
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">ZG {_m_/ } x AG {_m_ - 2 _n_/ }.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s13614
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xml:space
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">quare ZD {_m_/ } x AD {_m_ - 2 _n_/ } eſt maximum.</
s
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<
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echoid-s13615
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</
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<
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">Exemp. </
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<
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">1. </
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<
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">Sit _n_ = 1, & </
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<
s
xml:id
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echoid-s13619
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xml:space
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">_m_ = 3. </
s
>
<
s
xml:id
="
echoid-s13620
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xml:space
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">erit ideò _p_ {4/ } = ZD {3/ } x AD =
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ZD_q_ x BD_q_; </
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<
s
xml:id
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echoid-s13621
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xml:space
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">vel _p_
<
emph
style
="
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= ZD x BD. </
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<
s
xml:id
="
echoid-s13622
"
xml:space
="
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">Item AD =
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{1/2} CA.</
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<
s
xml:id
="
echoid-s13623
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xml:space
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"/>
</
p
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<
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<
s
xml:id
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echoid-s13624
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xml:space
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">2. </
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<
s
xml:id
="
echoid-s13625
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xml:space
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">Sit _n_ = 3, & </
s
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<
s
xml:id
="
echoid-s13626
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xml:space
="
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">_m_ = 10. </
s
>
<
s
xml:id
="
echoid-s13627
"
xml:space
="
preserve
">erit p {14/ } = ZD
<
emph
style
="
sub
">10</
emph
>
x AD
<
emph
style
="
sub
">4</
emph
>
.
<
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</
s
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<
s
xml:id
="
echoid-s13628
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xml:space
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preserve
">vel p {7/ } = ZD {5/ } x AD
<
emph
style
="
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">2</
emph
>
= ZD
<
emph
style
="
sub
">3</
emph
>
x BD
<
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style
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">4</
emph
>
. </
s
>
<
s
xml:id
="
echoid-s13629
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xml:space
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">& </
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<
s
xml:id
="
echoid-s13630
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xml:space
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">AD
<
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= {4/7} CA.</
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<
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xml:id
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echoid-s13631
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xml:space
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"/>
</
p
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<
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<
s
xml:id
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echoid-s13632
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">XXVI. </
s
>
<
s
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echoid-s13633
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xml:space
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">Sit item _hyperbola_ (æquilatera) cujus centrum C, axis
<
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ZA; </
s
>
<
s
xml:id
="
echoid-s13634
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xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s13635
"
xml:space
="
preserve
">huic inſcripta _paraboliformis_ AFB cujus expo-
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<
note
position
="
right
"
xlink:label
="
note-0279-02
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xlink:href
="
note-0279-02a
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xml:space
="
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">Fig. 149.</
note
>
nens {_n_/_m_} _parameter p_; </
s
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<
s
xml:id
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xml:space
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">ſitque AD = {2_n_ - _m_/_m_ - _n_} CA; </
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<
s
xml:id
="
echoid-s13637
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xml:space
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">erit _p_ ſui gene-
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neris maximum.</
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<
s
xml:id
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echoid-s13638
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xml:space
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</
p
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<
p
>
<
s
xml:id
="
echoid-s13639
"
xml:space
="
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">Nam utcunque ducatur EG ad BD parallela; </
s
>
<
s
xml:id
="
echoid-s13640
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s13641
"
xml:space
="
preserve
">ad EG conſtituta
<
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/>
intelligatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter q._
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s13642
"
xml:space
="
preserve
">quum ergo _paraboliformis_ AFB _hyperbolam_ introrſum contingat,
<
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erit GF {_m_/ } &</
s
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<
s
xml:id
="
echoid-s13643
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xml:space
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">lt; </
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<
s
xml:id
="
echoid-s13644
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xml:space
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">GE {_n_/ }; </
s
>
<
s
xml:id
="
echoid-s13645
"
xml:space
="
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">hoc eſt _p_ {_m_ - _n_/ } x AG
<
emph
style
="
sub
">n</
emph
>
&</
s
>
<
s
xml:id
="
echoid-s13646
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xml:space
="
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">lt; </
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>
<
s
xml:id
="
echoid-s13647
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xml:space
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">_q_ {_m_ - _n_/ } x AG
<
emph
style
="
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">n</
emph
>
; </
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<
s
xml:id
="
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xml:space
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quare _p_ &</
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<
s
xml:id
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echoid-s13649
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xml:space
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">lt; </
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<
s
xml:id
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echoid-s13650
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xml:space
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">_q_.</
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<
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