1 terminating in r penetrates the cutting plane; if therefore we make S't equal to Ns", it is plain that I will be the position on the horizontal plane of the point denoted by s', s", by the revolution of the cutting plane about the intersection R'e'. By a similar construction, we may determine any number of points in the curve Q'LXR', which will be the section required. The curve required may be obtained still more simply by merely, finding the perpendiculars ns"H, and describing the curve through L, S", M, &c. without determining the corresponding points in e'LXR'. It is evident that the plane meeting the base at right angles in P'YQ' must also meet the upper division of the conic surface, and produce another section equal and similar to R'R'. The curve, determined by this construction is an hyperbola. PROBLEM VI. To construct the intersection of a conic surface by a plane parallel to one of the slant sides of the cone. Let AB be the ground line; r' the centre of the cone's base, which is supposed to be coincident with the horizontal plane; and let the base EFK touch the ground line in K: in P'K produced, take Kr" equal to the altitude or axis of the cone, and P" is the vertical projection of the summit of the cone. the cutting plane be parallel to the ground line, and meet the Let 1 base in the horizontal trace EF, which will consequently be parallel to AB : in Kr" produced if necessary, take Kr" a fourth - 1 proportional to the three straight lines KP', KX, KP", and the straight line G"Y"H" parallel to AB, will be the vertical trace of the cutting plane. The angle which the slant side passing through z makes with the horizontal plane is evidently the acute angle at the base of a right-angled plane triangle of which the base is zr', and perpendicular equal to Kr"; and the angle which the cutting plane makes with the horizontal plane is also the acute angle at the base of a right-angled triangle of which the base is xx and perpendicular KY"; and since these two triangles are in the same plane and have the bases and altitudes proportionals, it is plain that the acute angles at their bases are equal, and that the slant side passing through z is parallel to the plane of which the traces are EF, G"H". 11 To construct the curve of intersection draw any radius r'e; from a draw am at right angles to AB, and join PM, then F'Q, and r"m are the horizontal and vertical traces of the slant side passing through e. Find by prob. 2. chap. Iv the horizontal and vertical projections s' and s" of the point in which this slant side meets the cutting plane; and by prob. 12. chap. 1. find I the position on the horizontal plane of the point of which s' and s" are the projections by the rotation of the cutting plane about the intersection EF, and I is a point in the required curve. In a similar manner we may proceed in determining any number of points in the required curve FIVE. The ordinates NO, uw are obtained by the construction given in prob 2, chap. iv. for the slant sides passing through the extremities of the diameter Dr parallel to the ground line AB. The vertex v is found by taking ka equal to KY", and P'D equal to Kr"; then drawing ax and KD, we have the position c' of the vertex of the curve on the horizontal plane; and therefore making xv equal to xc, the point v will be the vertex of the curve. It is obvious that the curve FIVE is a parabola. : ريع ! 1 : 1 1 0.000000261.41497351 1.707570761.880814 20.301030 27 1.431364 52 1.716003 77 1.886491 30.477121 28 1.447158 53 1.72427678 1.892095 4 0.602060 29 1.462398 54 1.732394 791897627 50.698970 30 1.47712155 1.74036380 1.903090 6 0.778151 31 1.49136256 1.748188 81 1.908485 70.845098 321.505150 57 1.755875 82 1.913814 8.0.903090 33 1.518514 58 1.763428 83 1,919078 9 0.954243 34 1.531479 59 1.770852 84 1.924279 10 1.000000 35 1.544068 60 1.778151 85 1.929419 11 1.041393 36 1.556303 61 1.78533086 1.934498 12 1.07918137 1.568202 62 1.792392 87 1.939519 131.113943 38 1.579784 63 1.799341 88 1.944485 14 1.146128 39 1.591065 64 1.806180 89 1.949390 15 1.17609140 1.602060 65 1.812913 90 1.954243 16 1.204120 41 1.6/2784 66 1.819544 91 1.959041 17 1.230449 42 1.62324967 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 20 1.301030 45 1.65321370 1.845098 95 1.977724 211322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 48 1.681241 73 1.86332398 1.991226 241.380211 49 1.690196 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061|100|2.000000 N. B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are now introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the corresponding natural number in the first column stands in the next lower line, and its annexed first two figures of the Logarithms in the second column. N. 0 4321 4750 5181 5609 6038 6466 689473217748 8174 102 8600 9026 9451 9876.300.724 1147 1570 19932415 1030128373259 3680 4100 4521 4940 53605779 6197 6616 104 703374517868 8284 8700911695329947.361.775 105021189 1603 2016 2428 28413252 3664 4075 4486 4896 106 5306 5715 6125 6533 6942 7350 77578164 85718978 107 93849789.195.600 1004 1408 1812221626193021 108 0334243826 4227 4628-5029 5430 58306230 6629 7028 109 7425 782582238620901794149811.207.602.998 110041393 1787 2182 257629693362 37554148 4540 4932 111 5323571461056495 6885 72757664805384428830 112 92189606 9993.380.766 1153 1538 1924 2309 2694 1130530783463 3846 4230 4613 499653785760 6142 6524 114 6905 728676668046 8426 8805 918595639942.320 115060698 1075 1452 1829 2206 2582 295833333709 4083 116 4458 4832 52065580595363266699 707174437815 117 818685578928 92989668..38.407.7761145 1514 118071882 2250 2617 2985 3352 3718 4085 445148165182 5547 5912 6276 6640 70047368 7731-809484578819 91819543 9904.266.626.987 1347 1707 2067 2426 1210827853144 3503 38614219 4576 4934 5291 5647 6004 122 6360 6716 707174267781 8136 8490884591989552 123 9905.258.611.963 1315 1667 2018 2370 2721 3071 124093422 3772 4122 44714820 5169 5518586662156562 125 69107257 7604 7951 8298 8644 8990933596811026 126 100371/0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5169 5510 58516191 6531687) ! : 1 72107549 7888 8227 8565 8903 924195799916.253 129 1105900926 1263 1599 1934 2270 2605 2940 32753609 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 131 727176037934 8265 859589269256958699150245 132 120574 0903 1231 1560 1888 2216 2544 28713198 3525 13338524178 4504 4830 51565481/5806613164566781 134 71057429 7753 8076 8399 8722 9045 9368 9690..12 135 1303340655 0977 1298 1619 1939 2260 2580 2900 3219 136 35393858 4177 4496 4814 5133 5451 5769 6086 6403 137 6721/7037 735476717987 83038618893492499564 138 9879.194.508.8221136 1450 1763 2076 2389 2702 139143015 3327 3630 39514263 457448855196 5507 5818 140 6128 6438 6748 7058 7367 767679858294 86038911 141 92199527 9835.142.449.756 1063 1370 1676 1982 142152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 53365640 5943 62466549 6852 7154 7457 7759 8061 144 8362/8664 8965926695679868.168.469.769 1068 145161368 1667 1967 2266 2564 286331613461 37584055 4353 4650 4947524455415838 6134 6430 6726 7022 73177613 7908 82038497 87929086 9380 96749968 1481702620555 0848 1141 1434 1724 2019 2311 2603 2895 149 3186/3478 3769 40604351/4641 4932 5222 5512 5802 |