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But when p is greater than 1, put n = vf, and in this Oase the correct equation of the fuents is

. sin. (n . hyp. log. ) This equation shows us that we shall have y = 0 as often as 7. hyp.log-becomes equal to any complete number of semi. circumferences : if therefore x = 3.1416, and n = any aumder in the series 1, 2, 3, 4, 5, &c. we can have y = 0 only when n. hyp. log -- nt, from which we have

t=t.ee supposing hyp. log. e = 1, and therefore

1 which shows the relation between the number of vibrations s and the time r in which they are performed.

Hence it is manifest, that the times or durations of the seperal successive vibrations constitute a series in geometrical progression,

NT

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LOGARITHMS

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1 0.000000 261 414973 51|1.70757 761.880814 2 10.301030 27 1.431364 || 521 7:6003 771 886491 3 0.477121|| 28 1.447 158 53 1•724276 781.892095 4 0.602060 29 1.462398 54 1732394 79 1.897627 5 0.698970 30 1.477121 55 1.740363 80 | 1.903090 6 0.778151 31 1.49 1362 | 56 748188 81 1 908485 7 0.845098 || 32 1:505150 || 57 1.755875 821.913814 8 0.903090 || 331.518514|| 58 1.763428 831.919078 9 0.954243 34 1 531479 59 1.770852 84 1.924279 10 | 1.000000 351.544068 | 601-778151 85 | 1.929419 111•041393 || 36 1.556303 61 785330 861 934498 12 1.079181 371 568202 || 62 1.792392 871 939519 131.113943 38 1.579784 || 63 1.799341 88|1.944483 141.146128 39 1.591065 64 1.806180 891.949390 15 1.17 609 1 || 40 1.602060 65 1.812913 901.954243 16 1.204120 || 411.612784 66 1.819 544 911.959041 17 1•230449 || 42 |1 623249 67 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 941.973128 201•301030 || 45 1.653213 ||701.845098 95 | 1.977724 21 1•322219 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 1.863323 98 | 1.991226 241 1.380211 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 0'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 Logarithm in the second colum,

2

N. 0

3 4 5 6 7 819 100 00000010434|08681301|1734 21662598 3029/34+, 113891 101 4321 475 5181560916038 64666894732 i 7748.8174 102 860090269459876. 300.724 1147 1570|1993 2415 103 0128373259|3684|41004521 49405360 57746197 66 16 104 703374517868 82848700.9116 4532 9947). 361). 775 105 02118916032016 2428 2841 3252 3664 4075 4486 4896 106 5306 57 15 612565336942 73507757 81 64 857 18978) 107 9384/978..195. 600 1004 140811812122162619 3021 108 033424 38204227 4628|5029 5430 58306230 6629 7028 109 74267825 8223 862090 17 94149811. 207.602.998 110 04:393 17872 1 82 2576 2969 3362 3755 4 1 484540 4932 111 5323 57 14610564956085 727 5 7 66480538442 8830) 112 9218960619993. 807661153 1538 1924|2309 2694 113053078 3463 3846 4230 4613 4996 5378 57606.426524 114

6905728617666 80468426 8805 9185 95639942. 3201 115 060678 1075 1452 1829 -206 258229583333 3709 4083 116

4458 48325206 5580 5953 6326|6699707174437815 117

818685578928 92989668..38). 407. 77611451514 118 071882 22502617 2985 3352 37184085 4451 48165182 119

5547 59126276 6640700473687731 8094 8457 8819 120

9181 95439904. 266. 626 987 1347 17072067 2426 121 | 032785 3144 3503 38614219 457 6 4934 529156476004 122 6360 671670717426778181368490 88459198.9552 123 9905. 258.611.96313151667 2018 2370127213071 124 093422 377241224471 4820 51695518 5866 6215 6562) 125

69107257700479518298 8644 89909335968. 026 126 1003710715 105914031747 20912434 2777|3119 3462 127

3804 41464487 4828 5169 5510 585 6.9165316871 128

721075497888 8227 85658903924195799916 253 129 / 1105900926.263 1599 1934 227026052940-2753609 130

3943 4277 4611 4944/5278 561 15943 6276 6608 6940

7271 76037934 8265 8595 8926925695869915). 245) 132 12057409031231 1560 18882216 25442873198 3525 133 3852 41784504 4830 515615481580616131645616781 134

7105 7429 7753 807 83998722190459368 9690 135 130334 0655 0977 12981619|19392260 2580 290013219 136 35393858 4177 4496/4814151335451576960866403 137 672170377354 767 17987830, 861889349449/9564 138 9879. 194 508. 822 1136.450|17632070 2389 2702 139 1430)5 3327 363 395 i 42634574 4805 519655075818 140 6128 6438 6748 7058173677676 1985 8294 86038911 141 921995279835 1421. 449.756106313701676 1982 142152288 2594 2900 320535103815 4:20 4424 4728 5032 143 5336 5640 5943 6246654968527 : 547457 7759/8061 144 8362 8664/8965 926695679868. 168. 469.769 1068 145 161368.667 1967 2266 2564 2863316 3464-758 4055 146 4353 4650 4947 5244 5541 58386:34 643016726 7022 47 7317|7613790882038497 879290869380 9674 9968 148 170262 0555 0848 1141 1434 1726 2019/231112603 2895 149 3186)347813769 4060143511464114932522215512 58021

131

121 N. 0 1 1 | 2 | 3 4 5 6 | 7 | 8 9 1 50 17609 1 6381 6670 6959 7248/7536 7825 81138401 8689 151 89779264955298391. 126. 413 699 9801272|1558 152 18184421292415 2700 298532703555 3839 4123 4407 153

4691 4975 5259 5542 5825 61086391 667 4/69567239 154

7521780380848366 8647 89289209 9490/9771..51 155 7903320612089211711451 1730/2010 2289 2567 2846 156 3125 34033681 3959 4237 4514 4792 5069|5346 5623 157 5899 6176 6453/6729 700571817556.7832 8107 8382 158 8657 8932]920694819755.. 29 303 577 850 1124 159 2013971670 194322162488 27613033 330535773848 160

4120439 1 4663 4934 5204 547 5 57466016 62866556 161 68267096736576347804 8173 84418710897919247 162 9515 9783.51 319. 586. 8531121138811654 1921 163 212188 2454 272029863252 55181378340494314 4579 164 48445109 5373563815902 6166 6430 6694 6957 7221 165 74847747 8010 8273853687969060193239585 9846 1662201080370 063108921153 1414 1675 193621962456 167 2716 297632363496 3755 40154274 4533/47925051 168 5309 5568 58266084|6342166001685871157372 7630 169 78878144 8400 8657 891391701942696829938. 193 170 230449|07040960112151470/1724 1979 2234 2488 2742 171 2996 32503504 3757 4011 4264 4517 47705023]5276 172 5528 578 1 6033 6285/6537678970411729217544 7795 173

8046829718548 8799 904992999550 9800 501. 300 174 240549 0799 10481297 15461795 2044229325412790 175 3038/328613534|3782 403014277 4525 4772 5019 5266 176 5513 5759 6006/6252 6499674516991 7237 74827728 177 79738219 84648709895419198 9443 96879932176 178250420106641090811511395 163818812125 2368 2610 179 285330963338 358013822 4064 4306 4548 4790 5031 180 327315514 5755 5996 6237 647716718695871987439 181 767917918 81588398 8637 887791169355/95949833 182 | 260071051010548 0787 1025 1263 1501 1739/1976 2214 183 245126882925 3162 33993636 387341094346 4582 184 4818 5054 5290 5525 5761 5996 6232 64676702 6937 185 7172740676417875 8110 8344 8580881290469279 186 9513974699801 213). 446. 679. 912 1144 1377 1609 187 | 271842/2074 2306/2538 2770/3001132333464 3696 3927 188 41584389 4620 4850 5081 5311 5542 5772 6002 6232 189 6462 669269217151 7380 7609 7838 806782968525 190 87548982/92119439 9667 9895. 123. 3511. 578. 806 1912810331261/1488 1715 1942 2169/2396 2622 2849 3075 192 3301 352737533979 1 205 44314656488251075332 193 5557 5782 600715232 6456 6691169057130173547578 194 7802 8026 824984738696 8920914393669589 9812 195290035 02570480 07020925 1 147 1369 1591 1813 2034. 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197

4466 4687 4907 5127 5347 5567 5787 6007 62261446 198

666568847104 7323 75427761 7979 8198/8416 8635 199 8853/9071/928919507972519943). 1611. 3781. 595), 813

1

N. 0

2 | 3 | 4 | 5 6 7 8 g 200 301030'1247|1 464 168118982114 2331 2547 2764 2980 201 3196 3412 3628 3844 4059 4275 449 1 4706 4921 5136 202 5351 5566|5781 59966211 6425 6639168547068 7282 203 74967710792481378351 8564 8778 899192049417 204 9630 9843..56 268.481. 693.90611181330 1542 205 311754 19662177 23892600 2812 3023 3234 3445 3656 206 3867 407842894499 4710 49205130 5340555115760 207 5970 618016390|6599 680970187227743676467854 208 80638272 84818689 88989106/93149522|97309938 2093201 46103540562|07 69|0977|1184 1391|1598 1805 2012 210 2219 2426)26332839/30463252 3458 3665 38714077 211 4282 4488 469448995105 5310 5516 5721 59 26 6131 212 63366541 6745 69501715573597563 77677972)8176 213 8380 8583 87878991/9194193989601 9805...8.211 214 330414|06170819 10221225 1427 16301832 2034 2236 215 2438 2640 2842|30443246/3447 3649 3850 4051 4253 216 4454 4655 4856 5057 5257 5458 5658 58596059 6260 217 646016660 686017060172607459 76597858|8058 8257 218 84568656 88559054|92539451 9650 9849.. 47. 246 219 340444|06420841 10391237 1435 1632 1830 2028 2225 220 2423 26202817 301413212 3409 3606 5802 3999 4196 221 4392/4589 4785 4981 5178 537415570 5766 59626157 222 3353 65496744/69397135733017525 7720 79158110 223 8305 8500 8694888990839278 947296669860 . 54 224 5502480442 06360829 1023121614101603 17961989 225 21832375 256812761129541314733393532 3724 3916 226 4108 4301/4493 46854876 5068 5260 5452|5643 5834 227 6026 6217 6408 65996790 698171727366 75547744 228 793581258316 850618696888690769260/94569646 229 9835. . 25. 215. 404). 593.783 97211611350 1539 230 361728 1917 2105 2294 2482 2671 2859|3048 32363424 231

3612 3800 39884176 43634551 4739 4926 5113,5301 232

548856751586216049 623616423661016796 69837169 233 7356754217729 7915/810182871847386598845 9030 234 92169401 9587 977219958. 143. 328.513) 698. 883 235 | 3710681253 14371622 1806 1991/2175 2360 2544 2728 23 2912 3096 3280/3464/3647|383140154198 43824565 237 4748 4932 5115 5298 5481 5664158466029)62126394

65776759/69427124730617488 767078528034 8216 239 8398 8580 876189439124 9306948796689849)..30 240 380211039205730754 09341115|1296 1476/1656 1837 241 20172197 2377 255712737 2917 3097 32773456 3636 242 9815 3995 4174 4353 4533 4712 4891 5070524915428 243 5606 5785 596461426321|64996677 6856703417212 244 73907568 7746792318101 8279/8456 8634 8811 8989 245 9166 934395209698 9875..51. 228 405 5821. 759 246 | 390935 11121288 1464 1641 1817|19932169 2345 2521 247 2697 28733048 32243400 3575 3751 3926 4101 4277 248 44524627 4802 4977 51525326 5501 5676 5850/6025 249 61991637 4/654816722 689617071172451741917 592 77 66

238

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