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terminating in T penetrates the cutting plane; if therefore we make s'L equal to Ns", it is plain that L 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'.

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By a similar construction, we may determine any number of points in the curve Q'LXR', which will be the section required.

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The curve required may be obtained still more simply by merely finding the perpendiculars NS"H,and describing the curve through L, S", м, &c. without determining the corresponding points in 'LXR.

It is evident that the plane meeting the base at right angles in r'Yq' must also meet the upper division of the conic surface, and produce another section equal and similar to q'zR. The curve determined by this construction is an hyperbola.

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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; p' 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'x produced, take KP" equal to the altitude or axis of the cone, and Let p" is the vertical projection of the summit of the cone. the cutting plane be parallel to the ground line, and meet the

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base in the horizontal trace EF, which will consequently be

parallel to AB : in KP" produced if necessary, take KY" a fourth

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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.

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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 KP"; 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".

Q

To construct the curve of intersection draw any radius r'q; from a draw am at right angles to AB, and join P", then 'Q, and r' are the horizontal and vertical traces of the slant side passing through a. 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. II. find 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

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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 KP"; 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.

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1 0.000000| 26 |1.414973|| 51|1.707570|| 76 |1.880814 2 0.301030|| 27 |1.431364|| 52|1.716003 77 1.886491 3 0.477121|| 28|1.447158|| 53 |1,724276|| 78 |1.892095 4 0.602060 29 1.462398 54 1.732394|| 79 1 897627 5 0.698970|| 30 |1.477121|| 55 |1.740363|| 80 |1.903090 6 0.778151 311.491362 56 1.748188 81 1,908485 7 0.845098 32 1.505150|| 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 831,919078 9 0.954243 34 1.531479 59 1.770852|| 84 1.924279 10 1.000000 351,544068 60 1.778151 851.929419 11 1.041393 361,556303|| 61 |1.785330|| 86 1.934498 12 1.079181|| 371,568202|| 62 | 1.792392 871 939519 13|1.113943|| 38 1.579784 631.799341 881.944483 14 1.146128 39 1,591065 64 1.806180 89 1.949390 15 1.176091| 40 1.602069|| 65 1.812913 90 1.954243 16 1.204120 41 1.612784|| 66 1.819544 911.959041 17 1.230449 42 1.623249| 67 |1.826075|| 92 1.963788 181,255273 43 1.653468 68 1.832509 93 1.968483 19 1.278754|| 44 1.643453 69 1.838849|| 94 | 1.973128 20 1.301030 45 1.653213 70 1.845098 95 1.977724 214322219|| 46 |1.662758 71 1.851258 96 1.982271 22 1,342423 47 1.672098 72 1.857333 97 1.986772 231 361728|| 48 |1.681241|| 73 |1.863323 98 1.991226 24 1,380211 49 1.690196|| 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061| 100 | 2.000000

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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.

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2 3 45 67 718 89
100 0000000434 0868 1301 1734 2166|25983029 3461|3891
4321 4750 5181 5609 6038 6466 68947321 7748 8174
8600 9026 9451 9876. 800.72411471570 1993 2415
103012837 3259 3680 4100 4521 4940 5360 577961976616
104 7033 74517868828487009116 95329947. 361.775
105 021189 1603 2016 24282841 3252 36644075 4486 4896
106
5306 57156125 6533 69427350 77578164 85718978
107 9384 9789. 195. 60010041408181222162619 3021
108 033424 3826 4227 46285029 5430 5830 6230 6629 7028
109 7426 7825 8223 8620 90179414 9811. 207. 602.998
1100413931787 2182257629693362 37554148 4540 4932
111 5323 5714 6105 6495 6885 7275|76648053 8442 8830
112
921896069993. 380. 766115315381924 2309 2694
113053078 3463 3846 4230 4613 4996 53785760 6142 6524
114 6905 72867666 8046 8426 8805 918595639942. 320
115060698 1075 1452 1829 2206 2582 2958 3333 3709 4083
116 4458 4832 52065580 5953 6326 66997071 74437815
117 8186 8557 8928 9298 9668.. 38. 407. 776 1145 1514
118071882 225026172985 3352 3718 4085 4451 48165182
119 5547 5912 6276 664070047368 7731 8094 8457 8819
120 9181 9543 9904. 266. 626.987 1347 1707 2067 2426
121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004
122 6360 6716 7071 74267781 8136 84908845 91989552
123 9905.258.611.96313151667 2018 23702721 3071
124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562
125 69107257 7604 7951 8298 8644 89909335 9681 1026
126100371 0715 1059 1403 1747 2091 2434 2777 3119 3462
127 38044146 4487 4828 5169 5510 58516191 6531 687)
128 7210 7549 7888 8227 8565 8903 9241 95799916. 253
129 1105900926 1263 1599 1934 2270 2605 2940 3275 3609
130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940
131 7271 7603 7934 8265 8595 8926 925695869915 0245
132120574 0903 1231 1560 1888 2216 2544 2871 3198 3525
133
134

3852 4178 4504 4830 5156 5481 580661316456|6781
7105 7429 7753 8076 8399 8722 9045 93689690.. 12
135 130334 0655 097712981619 1939 2260 2580 2900 3219
136 3539 8858 4177 4496 4814 5133 5451 5769 6086 6403
137 6721 7037 73547671 7987 8303 861889349249 9564
138 9879. 194. 508. 8221136 1450 1763 2076 2389 2702
139 143015 3327 3630 39514263 4574 4885 51965507 5818
140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911
9219 9527 9835. 142. 449. 756 1063 1370 1676 1982
142152288 2594 2900 3205 3510 3815 4120 4424 4728 5032
143 5336 5640 5943 6246 6549 6852 71547457 7759 8061
144 8362 8664 8965 92669567 9868. 1681. 469. 7691068
145 161368 1667 1967 2266|2564 2863 3161 3461 3758 4055
146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022
147 73177613 7908 8203 8497 8792 9086 9380 96749968
148 170262 0555 0848 11411434 1724 2019 2311|2603|2895
149 3186 3478 3769 4060 435146414932 5222|55125802)

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