1082:
20:
1857:
1493:
2014:
60:
1252:
566:
1488:{\displaystyle {\begin{aligned}L&=4{\sqrt {2}}\,c\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}=4{\sqrt {2}}\,c\,\operatorname {arcsl} 1\\&={\frac {\Gamma (1/4)^{2}}{\sqrt {\pi }}}\,c={\frac {2\pi }{\operatorname {M} (1,1/{\sqrt {2}})}}c\approx 7{.}416\cdot c\end{aligned}}}
45:
1237:
702:
827:
400:
1847:
1257:
2249:
405:
889:
1151:
1540:
967:
1018:
1081:
1681:
1651:
2284:
1777:
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1729:
1705:
1624:
1600:
1576:
1516:
581:
1063:
2188:
2312:
713:
325:
of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of
Bernoulli.
561:{\displaystyle {\begin{aligned}\left(x^{2}+y^{2}\right)^{2}&=a^{2}\left(x^{2}-y^{2}\right)\\&=2c^{2}\left(x^{2}-y^{2}\right)\end{aligned}}}
2387:
1865:
1782:
2487:
2433:
974:
1543:
2347:
2193:
1086:
1076:
1111:
389:
1232:{\displaystyle \operatorname {arcsl} x{\stackrel {\text{def}}{{}={}}}\int _{0}^{x}{\frac {dt}{\sqrt {1-t^{4}}}}}
2559:
2373:
1128:
842:
2554:
348:, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a
90:
1124:
2522:
2549:
1525:
1025:
314:
2323:
Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.
904:
2337:
2533:
2342:
2332:
573:
302:
1089:
relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.
2404:
2090:
341:
329:
2527:
1850:
1659:
1629:
697:{\displaystyle x={\frac {a\cos t}{1+\sin ^{2}t}};\qquad y={\frac {a\sin t\cos t}{1+\sin ^{2}t}}}
2254:
1758:
1734:
1710:
1686:
1605:
1581:
1557:
982:
2505:
2483:
2454:
2429:
2383:
1864:
The following theorem about angles occurring in the lemniscate is due to German mathematician
1102:
1098:
834:
349:
345:
340:
centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a
86:
63:
2379:
1501:
2479:
2472:
1989:
1138:
1120:
1033:
2161:
294:
283:
122:
2289:
270:
44:
1519:
1116:
19:
2543:
896:
1856:
2352:
2013:
318:
279:
822:{\displaystyle x=a{\frac {t+t^{3}}{1+t^{4}}};\qquad y=a{\frac {t-t^{3}}{1+t^{4}}}}
59:
2508:
1106:
202:
49:
1094:
290:
261:
2513:
2155:
2094:
333:
106:
53:
1654:
392:
306:
265:
194:
154:
138:
2523:"Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive
298:
337:
170:
1109:(largely unpublished at the time, but allusions in the notes to his
2409:
2148:
2012:
1855:
1080:
278:
for "decorated with hanging ribbons". It is a special case of the
275:
43:
18:
2375:
How round is your circle? Where
Engineering and Mathematics Meet
2058:
The lemniscate is symmetric to the midpoint of the line segment
1101:, as was discovered in the eighteenth century. Around 1800, the
1842:{\displaystyle |{\widehat {\rm {APM}}}-{\widehat {\rm {BPM}}}|}
2040:
and as well to the perpendicular bisector of the line segment
2403:
Lemmermeyer, Franz (2011). "Parametrizing
Algebraic Curves".
1868:, who described it 1843 in his dissertation on lemniscates.
2022:
The lemniscate is symmetric to the line connecting its foci
1916:
is any point on the lemniscate outside the line connecting
360:
The equations can be stated in terms of the focal distance
2106:
are perpendicular, and each of them forms an angle of
1123:. For this reason the case of elliptic functions with
1119:
are of a very special form, being proportional to the
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2257:
2196:
2164:
1785:
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1713:
1689:
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1608:
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1560:
1528:
1504:
1255:
1154:
1036:
985:
907:
845:
716:
584:
403:
2244:{\displaystyle {3 \over a^{2}}{\sqrt {x^{2}+y^{2}}}}
321:, by contrast, is the locus of points for which the
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2278:
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2182:
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883:
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560:
1982:is one third of the triangle's exterior angle at
1860:relation between angles at Bernoulli's lemniscate
368:of a lemniscate. These parameters are related as
2372:Bryant, John; Sangwin, Christopher J. (2008),
2017:The inversion of hyperbola yields a lemniscate
2151:tangent to its inner equator is a lemniscate.
8:
2424:Eymard, Pierre; Lafon, Jean-Pierre (2004).
23:A lemniscate of Bernoulli and its two foci
1105:inverting those integrals were studied by
2534:Coup d'œil sur la lemniscate de Bernoulli
2408:
2296:
2291:
2256:
2251:. The maximum curvature, which occurs at
2233:
2220:
2214:
2206:
2197:
2195:
2163:
1970:. Now the interior angle of the triangle
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1760:
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1683:can also be defined as the set of points
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1634:
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1610:
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1607:
1586:
1585:
1583:
1562:
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1559:
1527:
1503:
1467:
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1411:
1404:
1390:
1378:
1366:
1346:
1342:
1335:
1320:
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1220:
1199:
1193:
1188:
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1153:
1035:
1004:
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844:
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792:
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757:
739:
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618:
591:
583:
543:
530:
515:
487:
474:
459:
442:
431:
418:
404:
402:
1755:, together with the locus of the points
884:{\displaystyle r^{2}=a^{2}\cos 2\theta }
58:
2364:
2147:The planar cross-section of a standard
2076:The area enclosed by the lemniscate is
260:. The curve has a shape similar to the
234:from each other as the locus of points
2449:Alexander Ostermann, Gerhard Wanner:
1992:). In addition the interior angle at
395:is (up to translation and rotation):
7:
1895:is the midpoint of the line segment
1866:Gerhard Christoph Hermann Vechtmann
1097:of arcs of the lemniscate leads to
305:of points for which the sum of the
48:The lemniscate of Bernoulli is the
1824:
1821:
1818:
1801:
1798:
1795:
1764:
1740:
1716:
1692:
1668:
1665:
1638:
1635:
1611:
1587:
1563:
1535:{\displaystyle \operatorname {M} }
1529:
1505:
1422:
1369:
328:This curve can be obtained as the
14:
2474:A catalog of special plane curves
2428:. American Mathematical Society.
2100:The two tangents at the midpoint
1071:Arc length and elliptic functions
1889:are the foci of the lemniscate,
962:{\displaystyle |z-c||z+c|=c^{2}}
2478:. Dover Publications. pp.
1998:is twice the interior angle at
1946:intersects the line connecting
769:
636:
293:was first described in 1694 by
2378:, Princeton University Press,
2273:
2258:
2177:
2165:
1835:
1787:
1452:
1428:
1387:
1372:
1242:the formula of the arc length
975:two-center bipolar coordinates
942:
928:
923:
909:
205:defined from two given points
1:
2348:Lemniscatic elliptic function
1077:Lemniscate elliptic functions
1145:Using the elliptic integral
708:A rational parametrization:
2480:4–5, 121–123, 145, 151, 184
2470:J. Dennis Lawrence (1972).
1112:Disquisitiones Arithmeticae
2576:
1676:{\displaystyle {\rm {AB}}}
1646:{\displaystyle {\rm {AB}}}
1554:Given two distinct points
1087:lemniscate sine and cosine
1074:
1026:rational polar coordinates
2528:"Lemniscate of Bernoulli"
2279:{\displaystyle (\pm a,0)}
2126:with the line connecting
1772:{\displaystyle {\rm {P}}}
1748:{\displaystyle {\rm {M}}}
1724:{\displaystyle {\rm {B}}}
1700:{\displaystyle {\rm {A}}}
1653:. Then the lemniscate of
1619:{\displaystyle {\rm {M}}}
1595:{\displaystyle {\rm {B}}}
1571:{\displaystyle {\rm {A}}}
1544:arithmetic–geometric mean
1093:The determination of the
1013:{\displaystyle rr'=c^{2}}
268:symbol. Its name is from
89:and their equivalents in
2451:Geometry by Its History.
297:as a modification of an
1511:{\displaystyle \Gamma }
199:lemniscate of Bernoulli
187:Lemniscate of Bernoulli
91:rectangular coordinates
2308:
2280:
2245:
2184:
2089:The lemniscate is the
2018:
1861:
1849:is a right angle (cf.
1843:
1773:
1749:
1725:
1701:
1677:
1647:
1620:
1596:
1572:
1536:
1512:
1489:
1233:
1125:complex multiplication
1090:
1059:
1058:{\displaystyle Q=2s-1}
1014:
963:
885:
823:
698:
562:
269:
190:
56:
41:
2309:
2281:
2246:
2185:
2183:{\displaystyle (x,y)}
2016:
1940:of the lemniscate in
1859:
1844:
1774:
1750:
1726:
1702:
1678:
1648:
1621:
1597:
1573:
1537:
1513:
1490:
1234:
1084:
1060:
1015:
964:
886:
824:
699:
563:
350:crossed parallelogram
336:, with the inversion
309:to each of two fixed
62:
47:
22:
16:Plane algebraic curve
2453:Springer, 2012, pp.
2338:Lemniscate of Gerono
2290:
2255:
2194:
2162:
1783:
1759:
1735:
1711:
1687:
1660:
1630:
1606:
1582:
1558:
1526:
1502:
1253:
1152:
1034:
983:
905:
895:Its equation in the
843:
714:
582:
401:
2343:Lemniscate constant
2333:Lemniscate of Booth
2307:{\displaystyle 3/a}
1853:and its converse).
1626:be the midpoint of
1298:
1198:
574:parametric equation
2506:Weisstein, Eric W.
2304:
2276:
2241:
2180:
2019:
2009:Further properties
1862:
1839:
1769:
1745:
1721:
1697:
1673:
1643:
1616:
1592:
1568:
1532:
1508:
1485:
1483:
1284:
1229:
1184:
1103:elliptic functions
1099:elliptic integrals
1091:
1055:
1010:
959:
881:
819:
694:
558:
556:
364:or the half-width
342:mechanical linkage
282:and is a rational
191:
64:Sinusoidal spirals
57:
42:
2389:978-0-691-13118-4
2239:
2212:
1831:
1808:
1456:
1450:
1402:
1401:
1340:
1327:
1326:
1278:
1227:
1226:
1181:
1179:
1142:in some sources.
1121:Gaussian integers
835:polar coordinates
817:
764:
692:
631:
330:inverse transform
87:polar coordinates
52:of a rectangular
2567:
2519:
2518:
2493:
2477:
2457:
2447:
2441:
2439:
2421:
2415:
2414:
2412:
2400:
2394:
2392:
2369:
2313:
2311:
2310:
2305:
2300:
2285:
2283:
2282:
2277:
2250:
2248:
2247:
2242:
2240:
2238:
2237:
2225:
2224:
2215:
2213:
2211:
2210:
2198:
2189:
2187:
2186:
2181:
2143:
2134:
2125:
2124:
2122:
2121:
2118:
2115:
2114:
2105:
2091:circle inversion
2085:
2072:
2054:
2039:
2030:
2003:
1997:
1990:angle trisection
1987:
1981:
1975:
1969:
1963:
1954:
1945:
1939:
1933:
1924:
1915:
1909:
1894:
1888:
1879:
1848:
1846:
1845:
1840:
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1804:
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1706:
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1703:
1698:
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1695:
1682:
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1679:
1674:
1672:
1671:
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1650:
1649:
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1642:
1641:
1625:
1623:
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1599:
1598:
1593:
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1590:
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1539:
1538:
1533:
1517:
1515:
1514:
1509:
1494:
1492:
1491:
1486:
1484:
1471:
1457:
1455:
1451:
1446:
1444:
1420:
1412:
1403:
1397:
1396:
1395:
1394:
1382:
1367:
1359:
1341:
1336:
1328:
1325:
1324:
1309:
1308:
1300:
1297:
1292:
1279:
1274:
1246:can be given as
1245:
1238:
1236:
1235:
1230:
1228:
1225:
1224:
1209:
1208:
1200:
1197:
1192:
1183:
1182:
1180:
1177:
1175:
1174:
1169:
1166:
1139:lemniscatic case
1134:
1133:
1064:
1062:
1061:
1056:
1019:
1017:
1016:
1011:
1009:
1008:
996:
968:
966:
965:
960:
958:
957:
945:
931:
926:
912:
890:
888:
887:
882:
868:
867:
855:
854:
828:
826:
825:
820:
818:
816:
815:
814:
798:
797:
796:
780:
765:
763:
762:
761:
745:
744:
743:
727:
703:
701:
700:
695:
693:
691:
684:
683:
667:
644:
632:
630:
623:
622:
606:
592:
567:
565:
564:
559:
557:
553:
549:
548:
547:
535:
534:
520:
519:
501:
497:
493:
492:
491:
479:
478:
464:
463:
447:
446:
441:
437:
436:
435:
423:
422:
383:
382:
381:
367:
363:
259:
239:
233:
222:
213:
184:
177:
168:
161:
152:
145:
136:
129:
120:
113:
104:
97:
84:
40:
31:
2575:
2574:
2570:
2569:
2568:
2566:
2565:
2564:
2560:Spiric sections
2540:
2539:
2504:
2503:
2500:
2490:
2469:
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2461:
2460:
2448:
2444:
2436:
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2422:
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2402:
2401:
2397:
2390:
2371:
2370:
2366:
2361:
2329:
2321:
2288:
2287:
2286:, is therefore
2253:
2252:
2229:
2216:
2202:
2192:
2191:
2160:
2159:
2142:
2136:
2133:
2127:
2119:
2116:
2112:
2111:
2110:
2108:
2107:
2101:
2097:and vice versa.
2077:
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2011:
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1878:
1872:
1851:Thales' theorem
1781:
1780:
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1301:
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1251:
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1150:
1149:
1131:
1129:
1117:period lattices
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1073:
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989:
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379:
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369:
365:
361:
358:
344:in the form of
301:, which is the
295:Jakob Bernoulli
284:algebraic curve
254:
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2555:Quartic curves
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2499:
2498:External links
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1520:gamma function
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1196:
1191:
1187:
1172:
1163:
1160:
1157:
1136:is called the
1075:Main article:
1072:
1069:
1068:
1067:
1066:
1065:
1054:
1051:
1048:
1045:
1042:
1039:
1022:
1021:
1020:
1007:
1003:
999:
995:
992:
988:
971:
970:
969:
956:
952:
948:
944:
940:
937:
934:
930:
925:
921:
918:
915:
911:
893:
892:
891:
880:
877:
874:
871:
866:
862:
858:
853:
849:
831:
830:
829:
813:
809:
805:
802:
795:
791:
787:
784:
778:
775:
772:
768:
760:
756:
752:
749:
742:
738:
734:
731:
725:
722:
719:
706:
705:
704:
690:
687:
682:
678:
674:
671:
666:
663:
660:
657:
654:
651:
648:
642:
639:
635:
629:
626:
621:
617:
613:
610:
605:
602:
599:
596:
590:
587:
570:
569:
568:
552:
546:
542:
538:
533:
529:
524:
518:
514:
510:
507:
504:
502:
500:
496:
490:
486:
482:
477:
473:
468:
462:
458:
454:
451:
449:
445:
440:
434:
430:
426:
421:
417:
412:
407:
406:
357:
354:
346:Watt's linkage
252:
245:
227:, at distance
219:
210:
174:
158:
142:
126:
110:
105:: Equilateral
94:
37:
28:
15:
13:
10:
9:
6:
4:
3:
2:
2572:
2561:
2558:
2556:
2553:
2551:
2548:
2547:
2545:
2535:
2532:
2530:at MathCurve.
2529:
2526:
2524:
2521:
2516:
2515:
2510:
2507:
2502:
2501:
2497:
2491:
2489:0-486-60288-5
2485:
2481:
2476:
2475:
2468:
2467:
2463:
2456:
2452:
2446:
2443:
2437:
2435:0-8218-3246-8
2431:
2427:
2426:The Number Pi
2420:
2417:
2411:
2406:
2399:
2396:
2391:
2385:
2381:
2377:
2376:
2368:
2365:
2358:
2354:
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2349:
2346:
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2341:
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2336:
2334:
2331:
2330:
2326:
2324:
2318:
2301:
2297:
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2234:
2230:
2226:
2221:
2217:
2207:
2203:
2199:
2174:
2171:
2168:
2157:
2153:
2150:
2146:
2139:
2130:
2104:
2099:
2096:
2092:
2088:
2084:
2080:
2075:
2068:
2062:
2057:
2050:
2044:
2035:
2026:
2021:
2020:
2015:
2008:
2002:
1996:
1991:
1986:
1980:
1974:
1968:
1959:
1950:
1944:
1938:
1934:. The normal
1929:
1920:
1914:
1905:
1899:
1893:
1884:
1875:
1871:
1870:
1869:
1867:
1858:
1854:
1852:
1828:
1811:
1805:
1656:
1549:
1547:
1545:
1521:
1478:
1475:
1472:
1468:
1464:
1461:
1458:
1447:
1441:
1437:
1434:
1431:
1425:
1417:
1414:
1408:
1405:
1398:
1391:
1383:
1379:
1375:
1363:
1361:
1353:
1350:
1347:
1343:
1337:
1332:
1329:
1321:
1317:
1313:
1310:
1305:
1302:
1294:
1289:
1285:
1281:
1275:
1270:
1267:
1265:
1260:
1249:
1248:
1247:
1221:
1217:
1213:
1210:
1205:
1202:
1194:
1189:
1185:
1170:
1161:
1158:
1155:
1148:
1147:
1146:
1143:
1141:
1140:
1135:
1126:
1122:
1118:
1114:
1113:
1108:
1104:
1100:
1096:
1088:
1083:
1078:
1070:
1052:
1049:
1046:
1043:
1040:
1037:
1030:
1029:
1027:
1023:
1005:
1001:
997:
993:
990:
986:
979:
978:
976:
972:
954:
950:
946:
938:
935:
932:
919:
916:
913:
901:
900:
898:
897:complex plane
894:
878:
875:
872:
869:
864:
860:
856:
851:
847:
839:
838:
836:
832:
811:
807:
803:
800:
793:
789:
785:
782:
776:
773:
770:
766:
758:
754:
750:
747:
740:
736:
732:
729:
723:
720:
717:
710:
709:
707:
688:
685:
680:
676:
672:
669:
664:
661:
658:
655:
652:
649:
646:
640:
637:
633:
627:
624:
619:
615:
611:
608:
603:
600:
597:
594:
588:
585:
578:
577:
575:
571:
550:
544:
540:
536:
531:
527:
522:
516:
512:
508:
505:
503:
494:
488:
484:
480:
475:
471:
466:
460:
456:
452:
450:
443:
438:
432:
428:
424:
419:
415:
410:
397:
396:
394:
391:
387:
386:
385:
376:
372:
355:
353:
351:
347:
343:
339:
335:
331:
326:
324:
320:
316:
312:
308:
304:
300:
296:
292:
287:
286:of degree 4.
285:
281:
277:
273:
272:
267:
263:
258:
251:
244:
238:
232:
226:
218:
209:
204:
200:
196:
188:
182:
172:
166:
156:
150:
140:
134:
124:
118:
108:
102:
92:
88:
82:
78:
74:
70:
65:
61:
55:
51:
46:
36:
27:
21:
2550:Plane curves
2512:
2509:"Lemniscate"
2473:
2450:
2445:
2425:
2419:
2398:
2374:
2367:
2353:Cassini oval
2322:
2319:Applications
2137:
2128:
2102:
2082:
2078:
2066:
2060:
2048:
2042:
2033:
2024:
2000:
1994:
1984:
1978:
1972:
1966:
1957:
1948:
1942:
1936:
1927:
1918:
1912:
1903:
1897:
1891:
1882:
1873:
1863:
1553:
1497:
1241:
1144:
1137:
1110:
1092:
374:
370:
359:
327:
322:
319:Cassini oval
311:focal points
310:
288:
280:Cassini oval
256:
249:
242:
236:
230:
224:
216:
207:
198:
192:
186:
180:
164:
148:
132:
116:
100:
80:
76:
72:
68:
34:
25:
2536:(in French)
1107:C. F. Gauss
274:, which is
271:lemniscatus
264:and to the
223:, known as
203:plane curve
50:pedal curve
2544:Categories
2464:References
1988:(see also
1779:such that
1095:arc length
291:lemniscate
2514:MathWorld
2410:1108.6219
2380:pp. 58–59
2262:±
2156:curvature
2095:hyperbola
1829:^
1812:−
1806:^
1506:Γ
1476:⋅
1462:≈
1426:
1418:π
1399:π
1370:Γ
1351:
1314:−
1286:∫
1214:−
1186:∫
1159:
1050:−
917:−
879:θ
873:
786:−
686:
662:
653:
625:
601:
537:−
481:−
390:Cartesian
356:Equations
334:hyperbola
307:distances
262:numeral 8
107:hyperbola
71:= –1 cos(
54:hyperbola
2327:See also
1655:diameter
994:′
393:equation
315:constant
240:so that
195:geometry
155:Cardioid
139:Parabola
2455:207-208
2123:
2109:
1542:is the
1518:is the
1130:√
1115:). The
378:√
323:product
299:ellipse
2486:
2440:p. 200
2432:
2386:
1602:, let
1550:Angles
1498:where
338:circle
197:, the
178:
176:
171:Circle
162:
160:
146:
144:
135:= −1/2
130:
128:
114:
112:
98:
96:
2405:arXiv
2359:Notes
2149:torus
2093:of a
1348:arcsl
1156:arcsl
572:As a
332:of a
313:is a
303:locus
289:This
276:Latin
201:is a
151:= 1/2
85:) in
2484:ISBN
2430:ISBN
2384:ISBN
2154:The
2135:and
2031:and
1955:and
1925:and
1910:and
1880:and
1578:and
1522:and
1085:The
899:is:
388:Its
317:. A
225:foci
214:and
123:Line
119:= −1
103:= −2
32:and
2190:is
2158:at
2081:= 2
1976:at
1973:OPR
1964:in
1473:416
1178:def
1127:by
1024:In
973:In
870:cos
833:In
677:sin
659:cos
650:sin
616:sin
598:cos
193:In
183:= 2
167:= 1
75:),
2546::
2511:.
2482:.
2382:,
1731:,
1707:,
1546:.
1132:−1
1028::
977::
837::
576::
384:.
373:=
352:.
255:=
250:PF
243:PF
185::
169::
153::
137::
121::
93::
83:/2
79:=
73:nθ
2517:.
2492:.
2438:.
2413:.
2407::
2393:.
2314:.
2302:a
2298:/
2294:3
2274:)
2271:0
2268:,
2265:a
2259:(
2235:2
2231:y
2227:+
2222:2
2218:x
2208:2
2204:a
2200:3
2178:)
2175:y
2172:,
2169:x
2166:(
2144:.
2141:2
2138:F
2132:1
2129:F
2120:4
2117:/
2113:π
2103:O
2086:.
2083:c
2079:a
2073:.
2070:2
2067:F
2064:1
2061:F
2055:.
2052:2
2049:F
2046:1
2043:F
2037:2
2034:F
2028:1
2025:F
2004:.
2001:O
1995:P
1985:R
1979:O
1967:R
1961:2
1958:F
1952:1
1949:F
1943:P
1937:n
1931:2
1928:F
1922:1
1919:F
1913:P
1907:2
1904:F
1901:1
1898:F
1892:O
1886:2
1883:F
1877:1
1874:F
1836:|
1825:M
1822:P
1819:B
1802:M
1799:P
1796:A
1788:|
1765:P
1741:M
1717:B
1693:A
1669:B
1666:A
1639:B
1636:A
1612:M
1588:B
1564:A
1530:M
1479:c
1469:.
1465:7
1459:c
1453:)
1448:2
1442:/
1438:1
1435:,
1432:1
1429:(
1423:M
1415:2
1409:=
1406:c
1392:2
1388:)
1384:4
1380:/
1376:1
1373:(
1364:=
1354:1
1344:c
1338:2
1333:4
1330:=
1322:4
1318:t
1311:1
1306:t
1303:d
1295:1
1290:0
1282:c
1276:2
1271:4
1268:=
1261:L
1244:L
1222:4
1218:t
1211:1
1206:t
1203:d
1195:x
1190:0
1171:=
1162:x
1053:1
1047:s
1044:2
1041:=
1038:Q
1006:2
1002:c
998:=
991:r
987:r
955:2
951:c
947:=
943:|
939:c
936:+
933:z
929:|
924:|
920:c
914:z
910:|
876:2
865:2
861:a
857:=
852:2
848:r
812:4
808:t
804:+
801:1
794:3
790:t
783:t
777:a
774:=
771:y
767:;
759:4
755:t
751:+
748:1
741:3
737:t
733:+
730:t
724:a
721:=
718:x
689:t
681:2
673:+
670:1
665:t
656:t
647:a
641:=
638:y
634:;
628:t
620:2
612:+
609:1
604:t
595:a
589:=
586:x
551:)
545:2
541:y
532:2
528:x
523:(
517:2
513:c
509:2
506:=
495:)
489:2
485:y
476:2
472:x
467:(
461:2
457:a
453:=
444:2
439:)
433:2
429:y
425:+
420:2
416:x
411:(
380:2
375:c
371:a
366:a
362:c
266:∞
257:c
253:2
248:·
246:1
237:P
231:c
229:2
220:2
217:F
211:1
208:F
181:n
165:n
149:n
133:n
117:n
101:n
81:π
77:θ
69:r
66:(
38:2
35:F
29:1
26:F
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