3057:
188:
31:
2921:
and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an
1513:
2324:
1032:
we would, by changing the location of the origin, convert to the usual form of the equation of a centered trochoid. Note the change of independent variable at this point to make it clear that we are no longer using the default polar coordinate parameterization
2463:
143:
around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called
2181:
627:
727:
509:
1263:
919:
1314:
825:
1179:
2552:
1992:
2192:
2884:
1026:
2335:
1564:
2057:
292:
2828:
2778:
2065:
1930:
356:
2919:
391:
1063:
955:
1684:
1861:
520:
1649:
2018:
1752:
1613:
2737:
1717:
1306:
1825:
1097:
2705:
2661:
2641:
2621:
2601:
2581:
1881:
1799:
1779:
633:
399:
3036:
1615:, the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above, so the graph of this equation has an
3075:
1190:
1508:{\displaystyle z=b\left(e^{it}+e^{2it}\right)=be^{3it \over 2}\left(e^{it \over 2}+e^{-it \over 2}\right)=2be^{3it \over 2}\cos {t \over 2},}
840:
738:
1105:
165:
Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be
2478:
2999:
2668:
2319:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -\arccos {b \over a}\right)+{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}
1935:
3121:
158:
is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a
3130:
3150:
2960:
2839:
2458:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -2\arccos {b \over a}\right)+3b{\sqrt {a^{2}-b^{2}}}.}
963:
1883:
approaches 0, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.
1524:
3041:
173:
2023:
3145:
2020:
this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles
2928:
2176:{\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\arccos {b \over a}-{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}
181:
253:
2792:
2742:
1894:
1578:
308:
2895:
361:
298:
3125:
1036:
3027:
622:{\displaystyle x=(b+a\cos \theta )\cos \theta ={a \over 2}+b\cos \theta +{a \over 2}\cos 2\theta ,}
1654:
927:
2786:
1834:
1828:
1570:
232:
159:
3031:
2995:
2991:
2984:
2955:
2469:
1269:
244:
224:
209:
145:
39:
1625:
2950:
1997:
1722:
1687:
1592:
216:. However, some insightful investigations regarding them had been undertaken earlier by the
132:
128:
87:
3112:
2710:
305:(thus introducing a point at the origin which in some cases is spurious), and substituting
1693:
1282:
177:
1804:
1076:
3094:
2937:
2690:
2646:
2626:
2606:
2586:
2566:
1866:
1784:
1764:
722:{\displaystyle y=(b+a\cos \theta )\sin \theta =b\sin \theta +{a \over 2}\sin 2\theta ;}
166:
2687:
is a limaçon. In fact, the pedal with respect to the origin of the circle with radius
3139:
831:
213:
1863:, the cusp expands to an inner loop, and the curve crosses itself at the origin. As
231:
contains specific geometric methods for producing limaçons. The curve was named by
3103:
2680:
514:
Applying the parametric form of the polar to
Cartesian conversion, we also have
220:
149:
1574:
504:{\displaystyle \left(x^{2}+y^{2}-ax\right)^{2}=b^{2}\left(x^{2}+y^{2}\right).}
196:
3014:
1755:
30:
3098:
1273:
192:
154:
74:
2667:
1258:{\displaystyle r^{1 \over 2}=(2b)^{1 \over 2}\cos {\frac {\theta }{2}},}
217:
140:
17:
3015:
Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web
Resource.
2684:
2672:
1616:
914:{\displaystyle z={a \over 2}+be^{i\theta }+{a \over 2}e^{2i\theta }.}
136:
820:{\displaystyle z=x+iy=(b+a\cos \theta )(\cos \theta +i\sin \theta )}
208:
The earliest formal research on limaçons is generally attributed to
187:
1174:{\displaystyle r=b(1+\cos \theta )=2b\cos ^{2}{\frac {\theta }{2}}}
2666:
186:
29:
2547:{\displaystyle 4(a+b)E\left({{2{\sqrt {ab}}} \over a+b}\right).}
2931:
of a circle with respect to a point on the circle is a limaçon.
1827:, the curve becomes a cardioid, and the indentation becomes a
243:
The equation (up to translation and rotation) of a limaçon in
3107:
2892:
which is the equation of a conic section with eccentricity
102:
2623:. Then the envelope of those circles whose center lies on
108:
3116:
2922:
ellipse then the corresponding limaçon will have no loop.
235:
when he used it as an example for finding tangent lines.
96:
2900:
2026:
1938:
930:
3071:, 2nd edition, page 708, John Wiley & Sons, 1984.
2898:
2842:
2795:
2745:
2713:
2693:
2649:
2629:
2609:
2589:
2569:
2481:
2338:
2195:
2068:
2000:
1987:{\textstyle \left(b^{2}+{{a^{2}} \over 2}\right)\pi }
1897:
1869:
1837:
1807:
1787:
1767:
1725:
1696:
1657:
1628:
1595:
1527:
1317:
1308:, the centered trochoid form of the equation becomes
1285:
1193:
1108:
1079:
1039:
966:
843:
741:
636:
523:
402:
364:
311:
256:
229:
Underweysung der
Messung (Instruction in Measurement)
111:
105:
3084:, Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.
1801:, the indentation becomes more pronounced until, at
1651:, the area bounded by the curve is convex, and when
93:
99:
90:
3117:ENCYCLOPÉDIE DES FORMES MATHÉMATIQUES REMARQUABLES
2983:
2913:
2878:
2822:
2772:
2731:
2699:
2655:
2635:
2615:
2595:
2575:
2546:
2457:
2318:
2175:
2051:
2012:
1986:
1924:
1875:
1855:
1819:
1793:
1773:
1746:
1711:
1678:
1643:
1607:
1558:
1507:
1300:
1257:
1173:
1091:
1057:
1020:
949:
913:
819:
721:
621:
503:
385:
350:
286:
2468:The circumference of the limaçon is given by a
830:yields this parameterization as a curve in the
1686:, the curve has an indentation bounded by two
2879:{\displaystyle r={1 \over {b+a\cos \theta }}}
2470:complete elliptic integral of the second kind
8:
1021:{\displaystyle z=be^{it}+{a \over 2}e^{2it}}
3131:"Limacon of Pascal" on PlanetPTC (Mathcad)
2977:
2975:
1559:{\displaystyle r=2b\cos {\theta \over 3}}
3126:Visual Dictionary of Special Plane Curves
2899:
2897:
2854:
2849:
2841:
2794:
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2648:
2628:
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2162:
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2143:
2130:
2117:
2094:
2089:
2087:
2078:
2067:
2059:, the area enclosed by the inner loop is
2039:
2025:
1999:
1964:
1959:
1957:
1948:
1937:
1896:
1868:
1836:
1806:
1786:
1766:
1724:
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1656:
1627:
1594:
1546:
1526:
1492:
1467:
1430:
1407:
1379:
1352:
1336:
1316:
1284:
1242:
1225:
1198:
1192:
1161:
1152:
1107:
1078:
1038:
1006:
992:
980:
965:
934:
929:
896:
882:
870:
850:
842:
740:
694:
635:
594:
566:
522:
487:
474:
459:
446:
426:
413:
401:
363:
342:
329:
316:
310:
255:
3062:The Two-Year College Mathematics Journal
2186:the area enclosed by the outer loop is
2052:{\textstyle \pi \pm \arccos {b \over a}}
3037:MacTutor History of Mathematics Archive
2971:
191:Three limaçons: dimpled, with cusp (a
7:
3056:Jane Grossman and Michael Grossman.
1272:family of curves. This curve is the
924:If we were to shift horizontally by
3099:The MacTutor History of Mathematics
2789:with respect to the unit circle of
2329:and the area between the loops is
1573:family of curves. This curve is a
287:{\displaystyle r=b+a\cos \theta .}
25:
2986:A catalog of special plane curves
2823:{\displaystyle r=b+a\cos \theta }
2773:{\displaystyle r=b+a\cos \theta }
1925:{\displaystyle r=b+a\cos \theta }
1891:The area enclosed by the limaçon
351:{\displaystyle r^{2}=x^{2}+y^{2}}
2603:be a circle whose center is not
86:
2990:. Dover Publications. pp.
2936:A particular special case of a
2914:{\displaystyle {\tfrac {a}{b}}}
386:{\displaystyle r\cos \theta =x}
2726:
2714:
2497:
2485:
1741:
1726:
1577:, and is sometimes called the
1222:
1212:
1136:
1118:
1058:{\displaystyle \theta =\arg z}
814:
787:
784:
763:
664:
643:
551:
530:
195:), and looped. Not shown: the
148:; more specifically, they are
1:
950:{\textstyle -{\frac {1}{2}}a}
3064:, January 1982, pages 52–55.
1679:{\displaystyle a<b<2a}
169:-shaped, or it may be oval.
36:r = 2 + cos(π – θ)
34:Construction of the limaçon
2982:J. Dennis Lawrence (1972).
2671:Limaçon — pedal curve of a
1856:{\displaystyle 0<b<a}
3167:
2961:List of periodic functions
1569:making it a member of the
1518:or, in polar coordinates,
1268:making it a member of the
1781:is decreased relative to
297:This can be converted to
3042:University of St Andrews
2558:Relation to other curves
1099:, the polar equation is
131:formed by the path of a
1644:{\displaystyle b>2a}
2915:
2880:
2824:
2774:
2733:
2701:
2675:
2657:
2643:and that pass through
2637:
2617:
2597:
2577:
2548:
2459:
2320:
2177:
2053:
2014:
2013:{\displaystyle b<a}
1988:
1926:
1877:
1857:
1821:
1795:
1775:
1748:
1747:{\displaystyle (-a,0)}
1713:
1680:
1645:
1609:
1608:{\displaystyle b>a}
1560:
1509:
1302:
1259:
1175:
1093:
1059:
1022:
951:
915:
821:
723:
623:
505:
387:
352:
288:
200:
70:
27:Type of roulette curve
3058:"Dimple or no dimple"
2916:
2881:
2825:
2775:
2734:
2732:{\displaystyle (a,0)}
2702:
2670:
2658:
2638:
2618:
2598:
2578:
2549:
2460:
2321:
2178:
2054:
2015:
1989:
1927:
1878:
1858:
1822:
1796:
1776:
1749:
1714:
1681:
1646:
1610:
1561:
1510:
1303:
1260:
1176:
1094:
1060:
1023:
952:
916:
822:
724:
624:
506:
388:
353:
299:Cartesian coordinates
289:
190:
33:
3082:A Survey of Geometry
3028:Robertson, Edmund F.
2896:
2840:
2793:
2743:
2711:
2691:
2647:
2627:
2607:
2587:
2567:
2479:
2336:
2193:
2066:
2024:
1998:
1936:
1895:
1867:
1835:
1805:
1785:
1765:
1723:
1712:{\displaystyle b=2a}
1694:
1655:
1626:
1593:
1525:
1315:
1301:{\displaystyle a=2b}
1283:
1279:In the special case
1191:
1106:
1077:
1073:In the special case
1037:
964:
928:
841:
739:
634:
521:
400:
362:
309:
254:
3108:Mathematical curves
3077:pp. 725 – 726.
3026:O'Connor, John J.;
2739:has polar equation
1820:{\displaystyle b=a}
1619:or isolated point.
1092:{\displaystyle a=b}
2911:
2909:
2876:
2820:
2770:
2729:
2697:
2676:
2653:
2633:
2613:
2593:
2573:
2544:
2455:
2316:
2173:
2049:
2010:
1984:
1922:
1873:
1853:
1817:
1791:
1771:
1744:
1709:
1676:
1641:
1605:
1579:limaçon trisectrix
1556:
1505:
1298:
1255:
1171:
1089:
1055:
1018:
947:
911:
817:
719:
619:
501:
383:
348:
301:by multiplying by
284:
233:Gilles de Roberval
201:
146:centered trochoids
127:, is defined as a
119:, also known as a
71:
3151:Roulettes (curve)
3122:Limacon of Pascal
3113:Limaçon of Pascal
3095:Limacon of Pascal
2956:Centered trochoid
2908:
2874:
2700:{\displaystyle b}
2656:{\displaystyle P}
2636:{\displaystyle C}
2616:{\displaystyle P}
2596:{\displaystyle C}
2576:{\displaystyle P}
2535:
2521:
2450:
2409:
2374:
2311:
2281:
2263:
2231:
2168:
2138:
2125:
2104:
2047:
1974:
1876:{\displaystyle b}
1794:{\displaystyle a}
1774:{\displaystyle b}
1688:inflection points
1554:
1500:
1483:
1446:
1420:
1395:
1270:sinusoidal spiral
1250:
1233:
1206:
1169:
1000:
942:
890:
858:
702:
602:
574:
245:polar coordinates
139:when that circle
121:limaçon of Pascal
40:polar coordinates
16:(Redirected from
3158:
3045:
3044:
3032:"Cartesian Oval"
3023:
3017:
3012:
3006:
3005:
2989:
2979:
2920:
2918:
2917:
2912:
2910:
2901:
2885:
2883:
2882:
2877:
2875:
2873:
2850:
2829:
2827:
2826:
2821:
2779:
2777:
2776:
2771:
2738:
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2735:
2730:
2706:
2704:
2703:
2698:
2662:
2660:
2659:
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2639:
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2619:
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2579:
2574:
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1777:
1772:
1754:is a point of 0
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3136:
3135:
3091:
3074:Howard Anton.
3053:
3051:Further reading
3048:
3025:
3024:
3020:
3013:
3009:
3002:
2981:
2980:
2973:
2969:
2947:
2894:
2893:
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2709:
2708:
2689:
2688:
2645:
2644:
2625:
2624:
2605:
2604:
2585:
2584:
2583:be a point and
2565:
2564:
2560:
2524:
2503:
2477:
2476:
2440:
2427:
2385:
2381:
2360:
2344:
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2201:
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2196:
2191:
2190:
2158:
2145:
2090:
2074:
2073:
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2063:
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2021:
1996:
1995:
1960:
1944:
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1934:
1933:
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1892:
1889:
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1148:
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1103:
1075:
1074:
1071:
1035:
1034:
1002:
976:
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178:algebraic curve
176:rational plane
172:A limaçon is a
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3089:External links
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3067:Howard Anton.
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225:Albrecht Dürer
210:Étienne Pascal
205:
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129:roulette curve
125:Pascal's Snail
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3080:Howard Eves.
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1069:Special cases
1068:
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832:complex plane
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247:has the form
246:
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215:
214:Blaise Pascal
211:
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84:
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51:
47:
41:
32:
19:
3081:
3068:
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3035:
3021:
3010:
2985:
2467:
2328:
2185:
1890:
1760:
1719:, the point
1621:
1588:
1568:
1517:
1278:
1267:
1183:
1072:
1031:
923:
829:
731:
513:
302:
296:
242:
228:
212:, father of
207:
171:
164:
153:
150:epitrochoids
124:
120:
82:
78:
72:
49:
45:
42:' origin at
2707:and center
1887:Measurement
221:Renaissance
135:fixed to a
3140:Categories
2967:References
1575:trisectrix
393:to obtain
227:. Dürer's
174:bicircular
2871:θ
2868:
2818:θ
2815:
2768:θ
2765:
2438:−
2399:
2390:−
2387:π
2299:−
2253:
2247:−
2244:π
2156:−
2128:−
2115:
2037:
2031:±
2028:π
1982:π
1920:θ
1917:
1756:curvature
1730:−
1549:θ
1544:
1490:
1434:−
1245:θ
1240:
1164:θ
1159:
1134:θ
1131:
1050:
1041:θ
932:−
904:θ
875:θ
812:θ
809:
797:θ
794:
782:θ
779:
714:θ
708:
689:θ
686:
674:θ
671:
662:θ
659:
614:θ
608:
589:θ
586:
561:θ
558:
549:θ
546:
433:−
375:θ
372:
279:θ
276:
239:Equations
3069:Calculus
2951:Roulette
2945:See also
2929:conchoid
1274:cardioid
957:, i.e.,
193:cardioid
155:cardioid
75:geometry
3104:Limaçon
2992:113–118
2787:inverse
1994:. When
223:artist
204:History
199:limaçon
83:limacon
79:limaçon
66:
54:
18:Limacon
2998:
2685:circle
2673:circle
2396:arccos
2250:arccos
2112:arccos
2034:arccos
1831:. For
1617:acnode
218:German
197:convex
182:degree
152:. The
137:circle
2683:of a
2681:pedal
1690:. At
1622:When
1589:When
167:heart
141:rolls
133:point
52:) = (
38:with
2996:ISBN
2927:The
2785:The
2563:Let
2005:<
1848:<
1842:<
1829:cusp
1668:<
1662:<
1633:>
1600:>
1585:Form
1571:rose
1184:or
358:and
160:cusp
77:, a
68:, 0)
3124:at
3115:at
3106:at
3097:at
2865:cos
2812:cos
2762:cos
1932:is
1914:cos
1761:As
1541:cos
1487:cos
1237:cos
1150:cos
1128:cos
1047:arg
806:sin
791:cos
776:cos
705:sin
683:sin
668:sin
656:cos
605:cos
583:cos
555:cos
543:cos
369:cos
273:cos
184:4.
180:of
123:or
81:or
73:In
3142::
3060:,
3040:,
3034:,
3030:,
2994:.
2974:^
2830:is
2679:A
2472::
1758:.
1581:.
1276:.
1065:.
834::
162:.
48:,
3004:.
2906:b
2903:a
2862:a
2859:+
2856:b
2852:1
2847:=
2844:r
2809:a
2806:+
2803:b
2800:=
2797:r
2780:.
2759:a
2756:+
2753:b
2750:=
2747:r
2727:)
2724:0
2721:,
2718:a
2715:(
2695:b
2651:P
2631:C
2611:P
2591:C
2571:P
2542:.
2538:)
2532:b
2529:+
2526:a
2519:b
2516:a
2511:2
2505:(
2501:E
2498:)
2495:b
2492:+
2489:a
2486:(
2483:4
2453:.
2446:2
2442:b
2433:2
2429:a
2423:b
2420:3
2417:+
2413:)
2407:a
2404:b
2393:2
2383:(
2378:)
2372:2
2366:2
2362:a
2355:+
2350:2
2346:b
2341:(
2314:,
2307:2
2303:b
2294:2
2290:a
2284:b
2279:2
2276:3
2271:+
2267:)
2261:a
2258:b
2240:(
2235:)
2229:2
2223:2
2219:a
2212:+
2207:2
2203:b
2198:(
2171:,
2164:2
2160:b
2151:2
2147:a
2141:b
2136:2
2133:3
2123:a
2120:b
2108:)
2102:2
2096:2
2092:a
2085:+
2080:2
2076:b
2071:(
2045:a
2042:b
2008:a
2002:b
1978:)
1972:2
1966:2
1962:a
1955:+
1950:2
1946:b
1941:(
1911:a
1908:+
1905:b
1902:=
1899:r
1871:b
1851:a
1845:b
1839:0
1815:a
1812:=
1809:b
1789:a
1769:b
1742:)
1739:0
1736:,
1733:a
1727:(
1707:a
1704:2
1701:=
1698:b
1674:a
1671:2
1665:b
1659:a
1639:a
1636:2
1630:b
1603:a
1597:b
1552:3
1538:b
1535:2
1532:=
1529:r
1503:,
1498:2
1495:t
1481:2
1477:t
1474:i
1471:3
1465:e
1461:b
1458:2
1455:=
1451:)
1444:2
1440:t
1437:i
1428:e
1424:+
1418:2
1414:t
1411:i
1405:e
1400:(
1393:2
1389:t
1386:i
1383:3
1377:e
1373:b
1370:=
1366:)
1360:t
1357:i
1354:2
1350:e
1346:+
1341:t
1338:i
1334:e
1329:(
1325:b
1322:=
1319:z
1296:b
1293:2
1290:=
1287:a
1253:,
1248:2
1231:2
1228:1
1223:)
1219:b
1216:2
1213:(
1210:=
1204:2
1201:1
1196:r
1167:2
1154:2
1146:b
1143:2
1140:=
1137:)
1125:+
1122:1
1119:(
1116:b
1113:=
1110:r
1087:b
1084:=
1081:a
1053:z
1044:=
1028:,
1014:t
1011:i
1008:2
1004:e
998:2
995:a
990:+
985:t
982:i
978:e
974:b
971:=
968:z
945:a
940:2
937:1
909:.
901:i
898:2
894:e
888:2
885:a
880:+
872:i
868:e
864:b
861:+
856:2
853:a
848:=
845:z
815:)
803:i
800:+
788:(
785:)
773:a
770:+
767:b
764:(
761:=
758:y
755:i
752:+
749:x
746:=
743:z
717:;
711:2
700:2
697:a
692:+
680:b
677:=
665:)
653:a
650:+
647:b
644:(
641:=
638:y
617:,
611:2
600:2
597:a
592:+
580:b
577:+
572:2
569:a
564:=
552:)
540:a
537:+
534:b
531:(
528:=
525:x
499:.
495:)
489:2
485:y
481:+
476:2
472:x
467:(
461:2
457:b
453:=
448:2
443:)
439:x
436:a
428:2
424:y
420:+
415:2
411:x
406:(
381:x
378:=
366:r
344:2
340:y
336:+
331:2
327:x
323:=
318:2
314:r
303:r
282:.
270:a
267:+
264:b
261:=
258:r
115:/
112:n
109:ɒ
106:s
103:ə
100:m
97:ɪ
94:l
91:ˈ
88:/
63:2
60:/
57:1
50:y
46:x
44:(
20:)
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