Diagrams illustrative of the Table, Appendix, No. IV.

Fig. 114.

References to table

Fig. 115.

References to table

Fig. 116.

References to table

Fig. 117.

References to table

 

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, BA, BC, AC, 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.) Inclination of the Sides of the Generating Triangle to the Vertical Axis of the System. (Fig. 117.) AB, Reflecting Surfaces (convex). (Fig. 115.) AC, Outer Refracting Surfaces (concave). BC, Inner Refracting Surfaces (convex). Δ,  
No.
of
Zone.		 Inclination
of Ray FC
to the
Vertical
Axis
of the
System.		 Incidence
of
Ray FC
on Side
BC of
Zone.		 Obtuse
Angle
of the
Generating
Triangle
of the
Zone.		 Angle of
Generating
Triangle
of Zone.		 Angle of
Generating
Triangle
of Zone.		 Reflecting
Side of
Zone in
Millimetres
being
Chord
of the
Arc b A.		 Inner
Refracting
Side of
Zone in
Millimetres.		 Outer
Refracting
Side of
Zone in
Millimetres.		 Ay Ax By Bx Cy Cx Inclination
of AB.		 Inclination
of BC.		 Inclination
of AC.		 Radius of
Curvature
in
Millimetres,
XA or X b.		 Horizontal
distance
of centre
of
curvature
X from
the axis
of the
System in
Millimetres
= OX.		 Vertical
distance
of centre
of
curvature
X below
the outer arris of the
Zone at A in
Millimetres
= OA.		 Inclination
of the
Two Radii
in
A and B.		 Inclination
of the
Outer
Radius
in A
to the
Vertex.		 Radius of
Curvature
in
Millimetres.		 Horizontal
distance
of centre
of
curvature
from the
axis
of the
System in
Millimetres.		 Vertical
distance
of centre
of
curvature
above
the outer
arris of the
Zone at A
in
Millimetres.		 Inclination
of the
Radii
in A
and C.		 Inclination
of the
Outer
Radius
at A to the
Vertex.		 Radius
of
Curvature
in
Millimetres.		 Horizontal
distance
of centre
of
curvature
from the
axis of
the System
in
Millimetres.		 Vertical
distance
of centre
of
curvature
below the
outer arris of the
Zone at A
in
Millimetres.		 Inclination
of the
Radii in
C and B.		 Inclination
of the
Outer
Radius
in C to
the Vertex.		 Distance
of C
from the
Focus for
the Zones,
in
Millimetres
= FC.		 No.
of
Zone.
  (Fig. 114.) (Fig. 114.) (Fig. 114.) (Fig. 114.) (Fig. 114.) (Fig. 115.) (Fig. 115.) (Fig. 115.)                                                 (Fig. 114.)  
    ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″                   ° ′ ″ ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″    
HYPER-
PYRAL
SERIES
OF
ZONES.		   -  1 60 45 38 44 06 09 117 26 40 31 48 10 30 45 10 160·331 92·379 95·209  593·369 986·260  551·481 831·497  525·000 920·000 74 51 19 73 20 31 44 06 09 8750·19 3194·76 8466·88 1 02 59 14 38 11 4000·00 3825·31 2817·77 1 21 50 45 12 56 4000·00 2021·19 3777·07 1 19 24 15 58 47 1054·34  1
 2 56 55 38 41 28 10 116 00 42 32 32 01 31 27 17 155·430 90·249 93·011  663·063 957·410  617·423 808·832  593·369 895·816 72 55 27 74 32 32 41 28 10 8253·90 3306·80 7909·94 1 04 45 16 32 11 ... 3923·65 2683·55 1 19 56 47 51 52 ... 1918·37 3797·40 1 17 34 14 48 41 1069·02  2
 3 53 05 38 38 48 29 114 31 20 33 17 30 32 11 10 151·551 88·729 91·434  734·313 926·908  684·958 783·619  663·063 869·605 70 59 39 75 42 51 38 48 29 7850·30 3411·75 7446·67 1 06 22 18 27 11 ... 4015·06 2542·31 1 18 34 50 32 14 ... 1813·60 3815·77 1 16 16 13 39 01 1087·52  3
 4 49 15 38 36 07 09 112 58 40 34 04 36 32 56 44 148·580 87·768 90·424  807·357 894·222  754·267 755·451  734·313 840·920 69 03 53 76 51 31 36 07 09 7527·08 3514·22 7056·39 1 07 51 20 22 11 ... 4098·54 2394·23 1 17 42 53 14 00 ... 1707·55 3831·95 1 15 26 12 30 46 1109·85  4
 5 45 25 38 33 24 19 111 23 00 34 53 07 33 43 53 146·444 87·332 89·947  882·445 858·858  825·547 723·921  807·357 809·337 67 08 12 77 58 41 33 24 19 7275·23 3617·56 6730·66 1 09 12 22 17 11 ... 4173·08 2239·64 1 17 18 55 57 02 ... 1599·72 3846·04 1 15 04 11 23 47 1136·14  5
 6 41 35 38 30 40 05 109 44 32 35 42 57 34 32 31 145·087 87·403 89·986  959·847 820·325  899·011 688·608  882·445 774·426 65 12 36 79 04 27 30 40 05 7082·17 3723·81 6459·63 1 10 25 24 12 11 ... 4237·70 2078·82 1 17 20 58 41 15 ... 1489·62 3858·14 1 15 08 10 17 59 1166·57  6
 7 37 45 38 27 54 34 108 03 30 36 34 01 35 22 29 144·481 87·977 90·536 1039·852 778·108  974·899 649·051  959·847 735·730 63 17 03 80 08 56 27 54 34 6944·09 3835·23 6234·93 1 11 31 26 07 11 ... 4291·45 1912·18 1 17 48 61 26 32 ... 1376·71 3868·30 1 15 36  9 13 16 1201·46  7
 8 33 55 38 25 07 52 106 20 06 37 26 11 36 13 43 144·609 89·060 91·604 1122·784 731·645 1053·471 604·730 1039·852 692·742 61 21 35 81 12 14 25 07 52 6857·03 3954·67 6052·36 1 12 30 28 02 11 ... 4333·30 1740·12 1 18 44 64 12 46 ... 1260·38 3876·59 1 16 32  8 09 30 1241·16  8
 9 30 05 38 22 20 07 104 34 36 38 19 23 37 06 01 145·476 90·671 93·209 1209·000 680·322 1135·025 555·059 1122·784 644·900 59 26 08 82 14 29 22 20 07 6817·87 4081·70 5902·53 1 13 20 29 57 11 ... 4362·26 1563·10 1 20 06 66 59 50 ... 1139·94 3883·03 1 17 56  7 06 33 1286·15  9
10 26 15 38 19 31 25 102 47 12 39 13 27 37 59 21 147·089 92·843 95·384 1298·900 623·433 1219·892 499·354 1209·000 591·556 57 30 46 83 15 47 19 31 25 6824·46 4226·67 5795·68 1 14 05 31 52 11 ... 4377·24 1381·63 1 21 58 69 47 36 ... 1014·67 3887·66 1 19 48  6 04 19 1336·99 10
11 22 25 38 16 41 54 100 58 10 40 07 14 38 54 36 145·361 93·000 95·413 1390·290 559·378 1308·183 439·428 1288·900 531·963 55 36 30 84 16 16 16 41 54 6880·52 4385·63 5718·52 1 12 37 33 47 11 ... 4376·72 1194·94 1 22 00 72 37 06 ...  884·95 3893·00 1 19 56  5 03 46 1394·36 11
12 18 42 02 13 56 26  99 10 50 40 59 53 39 49 17 143·362 93·000 95·271 1482·755 490·169 1398·007 374·538 1390·290 467·217 53 45 43 85 14 24 13 56 26 6980·89 4558·83 5672·64 1 10 36 35 38 59 ... 4360·70 1009·41 1 21 12 75 22 58 ...  752·78 3897·32 1 19 56  4 05 38 1457·22 12
13 15 06 03 11 16 04  97 26 07 41 51 03 40 42 49 141·379 93·000 95·127 1576·048 415·991 1488·972 304·611 1482·755 397·403 51 58 53 86 10 02 11 16 04 7123·05 4747·24 5654·92 1 08 15 37 26 59 ... 4329·31  828·19 1 21 46 78 03 02 ...  618·37 3901·10 1 19 56  3 10 00 1525·43 13
                                                                       
    ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″                   ° ′ ″ ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″       ° ′ ″ ° ′ ″    
HYPO-
PYRAL
SERIES.		   -  1 59 49 16 43 27 36 117 05 56 31 58 37 30 55 27 159·369 92·000 94·806  593·815 985·212  550·915 831·725  525·000 920·000 74 23 03 73 38 20 43 27 36 8674·74 3243·48 8375·64 1 03 09 15 05 22 4000·00 3855·83 2785·60 1 21 28 45 51 40 4000·00 2002·52 3781·91 1 19 04 15 42 08 ——  1
 2 55 48 55 40 42 01 115 35 07 32 44 43 31 40 09 158·051 92·000 94·783  682·731 981·810  634·862 831·183  610·872 920·000 72 22 10 74 53 06 40 42 01 8419·60 3456·44 8047·72 1 04 31 17 05 32 ... 3983·21 2644·16 1 21 28 48 37 15 ... 1918·55 3801·51 1 19 04 14 27 22 ——  2
 3 51 48 33 37 54 32 114 00 31 33 28 16 32 27 13 156·592 92·000 94·719  779·464 978·917  726·831 830·694  704·730 920·000 70 21 45 76 05 59 37 54 32 8278·82 3685·41 7823·68 1 05 01 19 05 13 ... 4104·84 2494·82 1 21 24 51 24 46 ... 1836·22 3818·93 1 19 04 13 14 29 ——  3
 4 47 51 11 35 07 32 112 23 53 34 20 23 33 15 44 155·083 92·000 94·619  885·046 974·441  827·926 830·261  807·657 920·000 68 23 16 77 16 21 35 07 32 8252·95 3941·89 7701·00 1 04 36 21 04 24 ... 4218·56 2340·02 1 21 20 54 11 48 ... 1756·33 3834·20 1 19 04 12 04 07 ——  4
 5 43 59 40 32 23 06 110 46 32 35 08 23 34 05 05 153·489 92·000 94·488 1000·663 970·608  939·385 829·882  920·871 920·000 66 28 11 78 23 26 32 23 06 8335·86 4228·06 7673·05 1 03 18 23 00 10 ... 4322·95 2182·17 1 21 12 56 56 18 ... 1679·85 3847·38 1 19 04 10 57 02 ——  5
 6 40 16 34 29 43 19 109 10 04 35 55 37 34 54 18 151·863 92·000 94·334 1127·665 966·771 1062·591 829·557 1045·740 920·000 64 37 37 79 26 45 29 43 19 8522·64 4549·97 7732·80 1 01 15 24 51 43 ... 4416·91 2023·99 1 21 04 59 36 09 ... 1607·39 3858·56 1 19 04  9 53 43 ——  6