Chapter XXVIII: Appendix: , No. IV

TABLE OF THE ELEMENTS OF CATADIOPTRIC ZONES SUITED FOR LIGHTS OF THE FIRST ORDER IN THE SYSTEM OF AUGUSTIN FRESNEL.

NOTE.--The Six co-ordinates of the Generating Triangle of each Zone
in these Tables have the Focus of the Principal Lenses of the System
for their origin; it being considered more convenient, in executing
the necessary protractions, preparatory to the construction of the
grinding apparatus, at once to refer the whole of the grinding
machinery to the axis of the apparatus. To prevent the appearance of
any inconsistency, however, it is proper to mention, that the radiant
points of the series of Zones, do not exactly coincide with the Focus
of the Lenses nor with each other; and that to avoid the parallax
which the distance of the radiant points from the origin of the
co-ordinates would occasion, it is necessary to make some corrections
upon the linear dimensions, so as to find the line corresponding
to the angles θ, ξ, and the distance Δ. In the Hyperpyral series,
which stands _above_ the flame, the Zones have the radiant point
10 millimetres above the Focus of the Lenses, and each _y_ of this
series, therefore, requires a reduction of that quantity; while the
_x_ remains unchanged. In the Hypopyral series, which stands _below_
the flame, the Focus or radiant points of each Zone, varies its place
in the flame, moving upwards as the Zone is lower, so that the line
joining the Zones and the Foci, revolves as a radius vector round a
point between them. In this way, the _x_’s remain unaltered; but the
_y_’s will be lengthened successively by the addition of 10, 14, 19,
25, 32, and 40 millimetres. As these Tables contain the dimensions of
Zones which are intended as an addition to the apparatus of FRESNEL,
I have adopted the metric scale, so as to render them at once
applicable to the existing protractions of that system. It is only
necessary to add, that the conversion of millimetres into imperial
inches, is easily effected by adding the log. millimetres to the log.
̅2·59516, the sum being the log. of the equivalent of the first term
in imperial inches.

+-------+-----++--------+--------++---------+--------+--------++
| | || θ | ξ || BCA, | ABC, | BAC, ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || | || | | ||
| | || Incli- | || | | ||
| | || nation | || Obtuse | | ||
| | || of | Inci- || Angle | | ||
| | || Ray FC | dence || of the | | ||
| | || to the | of Ray || Gener- |Angle of|Angle of||
| | ||Vertical| FC on || ating | Gener- | Gener- ||
| | || Axis | Side || Triangle| ating | ating ||
| | || of the | BC of || of the |Triangle|Triangle||
| | || System.| Zone. || Zone. |of Zone.|of Zone.||
| | No. || | || | | ||
| | of || (Fig. | (Fig. || (Fig. | (Fig. | (Fig. ||
| |ZONE.|| 114.) | 114.) || 114.) | 114.) | 114.) ||
| +-----++--------+--------++---------+--------+--------++
| | || ° ′ ″| ° ′ ″|| ° ′ ″| ° ′ ″| ° ′ ″||
| | 1 ||60 45 38|44 06 09||117 26 40|31 48 10|30 45 10||
| | 2 ||56 55 38|41 28 10||116 00 42|32 32 01|31 27 17||
| | 3 ||53 05 38|38 48 29||114 31 20|33 17 30|32 11 10||
| | 4 ||49 15 38|36 07 09||112 58 40|34 04 36|32 56 44||
|HYPER- | 5 ||45 25 38|33 24 19||111 23 00|34 53 07|33 43 53||
| PYRAL | 6 ||41 35 38|30 40 05||109 44 32|35 42 57|34 32 31||
|SERIES | 7 ||37 45 38|27 54 34||108 03 30|36 34 01|35 22 29||
| OF | 8 ||33 55 38|25 07 52||106 20 06|37 26 11|36 13 43||
|ZONES. | 9 ||30 05 38|22 20 07||104 34 36|38 19 23|37 06 01||
| | 10 ||26 15 38|19 31 25||102 47 12|39 13 27|37 59 21||
| | 11 ||22 25 38|16 41 54||100 58 10|40 07 14|38 54 36||
| | 12 ||18 42 02|13 56 26|| 99 10 50|40 59 53|39 49 17||
| | 13 ||15 06 03|11 16 04|| 97 26 07|41 51 03|40 42 49||
+=======+=====++========+========++=========+========+========++
| | || ° ′ ″| ° ′ ″|| ° ′ ″| ° ′ ″| ° ′ ″||
| | 1 ||59 49 16|43 27 36||117 05 56|31 58 37|30 55 27||
| HYPO- | 2 ||55 48 55|40 42 01||115 35 07|32 44 43|31 40 09||
| PYRAL | 3 ||51 48 33|37 54 32||114 00 31|33 28 16|32 27 13||
|SERIES.| 4 ||47 51 11|35 07 32||112 23 53|34 20 23|33 15 44||
| | 5 ||43 59 40|32 23 06||110 46 32|35 08 23|34 05 05||
| | 6 ||40 16 34|29 43 19||109 10 04|35 55 37|34 54 18||
+-------+-----++--------+--------++---------+--------+--------++

+-------+-----++-------+-------+-------++
| | || BA, | BC, | AC, ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ++
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || Re- | | ||
| | || flect-| | ||
| | || ing | | ||
| | ||Side of| | ||
| | ||Zone in| Inner | Outer ||
| | || Milli-| Re- | Re- ||
| | || metres| fract-| fract-||
| | || being | ing | ing ||
| | || Chord |Side of|Side of||
| | || of |Zone in|Zone in||
| | ||the Arc| Milli-| Milli-||
| | ||_b_ A. |metres.|metres.||
| | No. || | | ||
| | of || (Fig. | (Fig. | (Fig. ||
| |ZONE.|| 115.) | 115.) | 115.) ||
| +-----++-------+-------+-------++
| | || | | ||
| | 1 ||160·331| 92·379| 95·209||
| | 2 ||155·430| 90·249| 93·011||
| | 3 ||151·551| 88·729| 91·434||
| | 4 ||148·580| 87·768| 90·424||
|HYPER- | 5 ||146·444| 87·332| 89·947||
| PYRAL | 6 ||145·087| 87·403| 89·986||
|SERIES | 7 ||144·481| 87·977| 90·536||
| OF | 8 ||144·609| 89·060| 91·604||
|ZONES. | 9 ||145·476| 90·671| 93·209||
| | 10 ||147·089| 92·843| 95·384||
| | 11 ||145·361| 93·000| 95·413||
| | 12 ||143·362| 93·000| 95·271||
| | 13 ||141·379| 93·000| 95·127||
+=======+=====++=======+=======+=======++
| | || | | ||
| | 1 ||159·369| 92·000| 94·806||
| HYPO- | 2 ||158·051| 92·000| 94·783||
| PYRAL | 3 ||156·592| 92·000| 94·719||
|SERIES.| 4 ||155·083| 92·000| 94·619||
| | 5 ||153·489| 92·000| 94·488||
| | 6 ||151·863| 92·000| 94·334||
+-------+-----++-------+-------+-------++

+-------+-----++--------------------------------------------------++
| | || Co-ordinates of the Apices of the Generating ||
| | || Triangle, in Millimetres, having the Axis of the ||
| | || System cut by the horizontal plane of the focus ||
| | ||of the Annular Lenses, for its origin. (Fig. 116.)||
| | ++--------+-------+--------+-------+--------+-------++
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | || | | | | | ||
| | No. || | | | | | ||
| | of || | | | | | ||
| |ZONE.|| A_{_y_}|A_{_x_}| B_{_y_}|B_{_x_}| C_{_y_}|C_{_x_}||
| +-----++--------+-------+--------+-------+--------+-------++
| | || | | | | | ||
| | 1 || 593·369|986·260| 551·481|831·497| 525·000|920·000||
| | 2 || 663·063|957·410| 617·423|808·832| 593·369|895·816||
| | 3 || 734·313|926·908| 684·958|783·619| 663·063|869·605||
| | 4 || 807·357|894·222| 754·267|755·451| 734·313|840·920||
|HYPER- | 5 || 882·445|858·858| 825·547|723·921| 807·357|809·337||
| PYRAL | 6 || 959·847|820·325| 899·011|688·608| 882·445|774·426||
|SERIES | 7 ||1039·852|778·108| 974·899|649·051| 959·847|735·730||
| OF | 8 ||1122·784|731·645|1053·471|604·730|1039·852|692·742||
|ZONES. | 9 ||1209·000|680·322|1135·025|555·059|1122·784|644·900||
| | 10 ||1298·900|623·433|1219·892|499·354|1209·000|591·556||
| | 11 ||1390·290|559·378|1308·183|439·428|1288·900|531·963||
| | 12 ||1482·755|490·169|1398·007|374·538|1390·290|467·217||
| | 13 ||1576·048|415·991|1488·972|304·611|1482·755|397·403||
+=======+=====++========+=======+========+=======+========+=======++
| | || | | | | | ||
| | 1 || 593·815|985·212| 550·915|831·725| 525·000|920·000||
| HYPO- | 2 || 682·731|981·810| 634·862|831·183| 610·872|920·000||
| PYRAL | 3 || 779·464|978·917| 726·831|830·694| 704·730|920·000||
|SERIES.| 4 || 885·046|974·441| 827·926|830·261| 807·657|920·000||
| | 5 ||1000·663|970·608| 939·385|829·882| 920·871|920·000||
| | 6 ||1127·665|966·771|1062·591|829·557|1045·740|920·000||
+-------+-----++--------+-------+--------+-------+--------+-------++

+-------+-----++--------------------------++
| | || Inclination of the Sides ||
| | ||of the Generating Triangle||
| | || to the Vertical Axis of ||
| | || the System. (Fig. 117.) ||
| | ||--------+--------+--------++
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || | | ||
| | || Incli- | Incli- | Incli- ||
| | No. || nation | nation | nation ||
| | of || of | of | of ||
| |ZONE.|| AB. | BC. | AC. ||
| +-----++--------+--------+--------++
| | || ° ′ ″| ° ′ ″| ° ′ ″||
| | 1 ||74 51 19|73 20 31|44 06 09||
| | 2 ||72 55 27|74 32 32|41 28 10||
| | 3 ||70 59 39|75 42 51|38 48 29||
| | 4 ||69 03 53|76 51 31|36 07 09||
|HYPER- | 5 ||67 08 12|77 58 41|33 24 19||
| PYRAL | 6 ||65 12 36|79 04 27|30 40 05||
|SERIES | 7 ||63 17 03|80 08 56|27 54 34||
| OF | 8 ||61 21 35|81 12 14|25 07 52||
|ZONES. | 9 ||59 26 08|82 14 29|22 20 07||
| | 10 ||57 30 46|83 15 47|19 31 25||
| | 11 ||55 36 30|84 16 16|16 41 54||
| | 12 ||53 45 43|85 14 24|13 56 26||
| | 13 ||51 58 53|86 10 02|11 16 04||
+=======+=====++========+========+========++
| | || ° ′ ″| ° ′ ″| ° ′ ″||
| | 1 ||74 23 03|73 38 20|43 27 36||
| HYPO- | 2 ||72 22 10|74 53 06|40 42 01||
| PYRAL | 3 ||70 21 45|76 05 59|37 54 32||
|SERIES.| 4 ||68 23 16|77 16 21|35 07 32||
| | 5 ||66 28 11|78 23 26|32 23 06||
| | 6 ||64 37 37|79 26 45|29 43 19||
+-------+-----++--------+--------+--------++

+-------+-----++----------------------------------------++
| | || ||
| | || ||
| | || AB, Reflecting Surfaces (convex). ||
| | || (Fig. 115.) ||
| | ++-------+-------+-------+-------+--------++
| | || | | Verti-| | ||
| | || | Hori- | cal | | ||
| | || | zontal| dis- | | ||
| | || | dis- | tance | | ||
| | || | tance | of | | ||
| | || | of | centre| | ||
| | || | centre| of | | ||
| | || | of | curva-| | ||
| | || | curva-| ture X| | ||
| | || | ture |_below_| | ||
| | || | X from| the | | ||
| | || Radius| the |_outer_| |Inclina-||
| | || of | axis | arris | In- | tion ||
| | || Curva-| of the| of the| clina-| of the ||
| | ||ture in| System| _Zone_|tion of| Outer ||
| | || Milli-| in |at A in|the Two| Radius ||
| | No. ||metres,| Milli-| Milli-| Radii | in A ||
| | of || XA or | metres| metres| in A | to the ||
| |ZONE.|| X _b_.| = OX. | = OA. | and B.| Vertex.||
| +-----++-------+-------+-------+-------+--------++
| | || | | |° ′ ″| ° ′ ″||
| | 1 ||8750·19|3194·76|8466·88|1 02 59|14 38 11||
| | 2 ||8253·90|3306·80|7909·94|1 04 45|16 32 11||
| | 3 ||7850·30|3411·75|7446·67|1 06 22|18 27 11||
| | 4 ||7527·08|3514·22|7056·39|1 07 51|20 22 11||
|HYPER- | 5 ||7275·23|3617·56|6730·66|1 09 12|22 17 11||
| PYRAL | 6 ||7082·17|3723·81|6459·63|1 10 25|24 12 11||
|SERIES | 7 ||6944·09|3835·23|6234·93|1 11 31|26 07 11||
| OF | 8 ||6857·03|3954·67|6052·36|1 12 30|28 02 11||
|ZONES. | 9 ||6817·87|4081·70|5902·53|1 13 20|29 57 11||
| | 10 ||6824·46|4226·67|5795·68|1 14 05|31 52 11||
| | 11 ||6880·52|4385·63|5718·52|1 12 37|33 47 11||
| | 12 ||6980·89|4558·83|5672·64|1 10 36|35 38 59||
| | 13 ||7123·05|4747·24|5654·92|1 08 15|37 26 59||
+=======+=====++=======+=======+=======+=======+========++
| | || | | |° ′ ″| ° ′ ″||
| | 1 ||8674·74|3243·48|8375·64|1 03 09|15 05 22||
| HYPO- | 2 ||8419·60|3456·44|8047·72|1 04 31|17 05 32||
| PYRAL | 3 ||8278·82|3685·41|7823·68|1 05 01|19 05 13||
|SERIES.| 4 ||8252·95|3941·89|7701·00|1 04 36|21 04 24||
| | 5 ||8335·86|4228·06|7673·05|1 03 18|23 00 10||
| | 6 ||8522·64|4549·97|7732·80|1 01 15|24 51 43||
+-------+-----++-------+-------+-------+-------+--------++

+-------+-----++----------------------------------------++
| | || ||
| | || ||
| | || ||
| | ||AC, Outer Refracting Surfaces (concave).||
| | ++-------+-------+-------+-------+--------++
| | || | | Verti-| | ||
| | || | | cal | | ||
| | || | Hori- | dis- | | ||
| | || | zontal| tance | | ||
| | || | dis- | of | | ||
| | || | tance | centre| | ||
| | || | of | of | | ||
| | || | centre| curva-| | ||
| | || | of | ture | | ||
| | || | curva-|_above_| | ||
| | || | ture | the | | In- ||
| | || | from | outer | | clina- ||
| | || | the | arris | In- | tion ||
| | || Radius| axis | of the| clina-| of the ||
| | || of | of the| _Zone_| tion | Outer ||
| | || Curva-| System| at | of the| Radius ||
| | No. ||ture in| in | A in | Radii | at A ||
| | of || Milli-| Milli-| Milli-| in A | to the ||
| |ZONE.||metres.|metres.|metres.| and C.| Vertex.||
| +-----++-------+-------+-------+-------+--------++
| | || | | |° ′ ″| ° ′ ″||
| | 1 ||4000·00|3825·31|2817·77|1 21 50|45 12 56||
| | 2 || ... |3923·65|2683·55|1 19 56|47 51 52||
| | 3 || ... |4015·06|2542·31|1 18 34|50 32 14||
| | 4 || ... |4098·54|2394·23|1 17 42|53 14 00||
|HYPER- | 5 || ... |4173·08|2239·64|1 17 18|55 57 02||
| PYRAL | 6 || ... |4237·70|2078·82|1 17 20|58 41 15||
|SERIES | 7 || ... |4291·45|1912·18|1 17 48|61 26 32||
| OF | 8 || ... |4333·30|1740·12|1 18 44|64 12 46||
|ZONES. | 9 || ... |4362·26|1563·10|1 20 06|66 59 50||
| | 10 || ... |4377·24|1381·63|1 21 58|69 47 36||
| | 11 || ... |4376·72|1194·94|1 22 00|72 37 06||
| | 12 || ... |4360·70|1009·41|1 21 12|75 22 58||
| | 13 || ... |4329·31| 828·19|1 21 46|78 03 02||
+=======+=====++=======+=======+=======+=======+========++
| | || | | |° ′ ″| ° ′ ″||
| | 1 ||4000·00|3855·83|2785·60|1 21 28|45 51 40||
| HYPO- | 2 || ... |3983·21|2644·16|1 21 28|48 37 15||
| PYRAL | 3 || ... |4104·84|2494·82|1 21 24|51 24 46||
|SERIES.| 4 || ... |4218·56|2340·02|1 21 20|54 11 48||
| | 5 || ... |4322·95|2182·17|1 21 12|56 56 18||
| | 6 || ... |4416·91|2023·99|1 21 04|59 36 09||
+-------+-----++-------+-------+-------+-------+--------++

+-------+-----++----------------------------------------++-------++
| | || || Δ, ||
| | || || ||
| | || || ||
| | || BC, Inner Refracting Surfaces (convex).|| ||
| | ++-------+-------+-------+-------+--------++ ||
| | || | | | | || ||
| | || | | Verti-| | || ||
| | || | Hori- | cal | | || ||
| | || | zontal| dis- | | || ||
| | || | dis- | tance | | || Dis- ||
| | || | tance | of | | || tance ||
| | || | of | centre| | || of ||
| | || | centre| of | | || C from||
| | || | of | curva-| | || the ||
| | || | curva-| ture | | || Focus ||
| | || | ture |_below_| | In- ||for the||
| | || | from | the | | clina- || Zones,||
| | || | the | outer | In- | tion of|| in ||
| | || Radius| axis | arris | clina-| of the || Milli-||
| | || of | of the| of the| tion | Outer || metres||
| | || Curva-| System| _Zone_| of the| Radius || = FC. ||
| | No. ||ture in| in |at A in| Radii | in C || ||
| | of || Milli-| Milli-| Milli-| in C | to the || (Fig. ||
| |ZONE.||metres.|metres.|metres.| and B.| Vertex.|| 114.) ||
| +-----++-------+-------+-------+-------+--------++-------++
| | || | | |° ′ ″| ° ′ ″|| ||
| | 1 ||4000·00|2021·19|3777·07|1 19 24|15 58 47||1054·34||
| | 2 || ... |1918·37|3797·40|1 17 34|14 48 41||1069·02||
| | 3 || ... |1813·60|3815·77|1 16 16|13 39 01||1087·52||
| | 4 || ... |1707·55|3831·95|1 15 26|12 30 46||1109·85||
|HYPER- | 5 || ... |1599·72|3846·04|1 15 04|11 23 47||1136·14||
| PYRAL | 6 || ... |1489·62|3858·14|1 15 08|10 17 59||1166·57||
|SERIES | 7 || ... |1376·71|3868·30|1 15 36| 9 13 16||1201·46||
| OF | 8 || ... |1260·38|3876·59|1 16 32| 8 09 30||1241·16||
|ZONES. | 9 || ... |1139·94|3883·03|1 17 56| 7 06 33||1286·15||
| | 10 || ... |1014·67|3887·66|1 19 48| 6 04 19||1336·99||
| | 11 || ... | 884·95|3893·00|1 19 56| 5 03 46||1394·36||
| | 12 || ... | 752·78|3897·32|1 19 56| 4 05 38||1457·22||
| | 13 || ... | 618·37|3901·10|1 19 56| 3 10 00||1525·43||
+=======+=====++=======+=======+=======+=======+========++=======++
| | || | | |° ′ ″| ° ′ ″|| ||
| | 1 ||4000·00|2002·52|3781·91|1 19 04|15 42 08|| -- ||
| HYPO- | 2 || ... |1918·55|3801·51|1 19 04|14 27 22|| -- ||
| PYRAL | 3 || ... |1836·22|3818·93|1 19 04|13 14 29|| -- ||
|SERIES.| 4 || ... |1756·33|3834·20|1 19 04|12 04 07|| -- ||
| | 5 || ... |1679·85|3847·38|1 19 04|10 57 02|| -- ||
| | 6 || ... |1607·39|3858·56|1 19 04| 9 53 43|| -- ||
+-------+-----++-------+-------+-------+-------+--------++-------++