2171:
2081:
202:
768:
2957:
29:
1732:
2512:
2236:. The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.
2017:
All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an
540:
950:
1292:
185:
originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the
1505:
1520:
2324:
2012:
763:{\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}
343:
816:
456:
999:
and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.
2251:, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver.
1166:
1076:
1908:
1839:
2329:
1924:
1188:
2579:
1376:
1727:{\displaystyle H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.}
2507:{\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}}
1919:
247:
2975:
945:{\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}
372:
2224:
In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via
2882:
2665:
1107:
1014:
2796:
2620:
2651:
1287:{\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}
2610:
1856:
1781:
2271:
2527:
1845:
1500:{\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}
972:
790:
2318:. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:
2034:, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin
106:
2295:
is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a
2232:, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense
2039:
167:
139:
69:
65:
2961:
2898:
2824:
2135:
2125:
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2903:
2630:
2159:
2101:
2050:
2007:{\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}}
1307:
1177:
163:
143:
2727:
2765:
2757:
2194:
2147:
2061:
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190:
187:
1850:(usually simply eccentricity, as above) is often used instead of flattening. It is defined by:
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2292:
2170:
2155:
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85:
1737:
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.
109:
is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its
2832:
2812:
2749:
2625:
2595:
2590:
2228:
when they orbit a massive body in a close orbit. The most extreme example is
Jupiter's moon
2206:
2182:
The prolate spheroid is the approximate shape of the ball in several sports, such as in the
2129:
2065:
2035:
996:
469:
is the distance from centre to pole along the symmetry axis. There are two possible cases:
2057:
2043:
991:
can be used to calculate the surface area of a prolate spheroid and vice versa. However,
2969:
2828:
2080:
2785:
2255:
2218:
2027:
1511:
338:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}
2987:
2769:
2600:
2266:
elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as
2210:
2193:
of the Solar System approximate prolate spheroids in shape, though they are actually
353:
81:
2705:
2682:
2248:
2202:
2139:
2117:
1775:, is the ratio of the equatorial-polar length difference to the equatorial length:
1747:
532:
170:
model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the
2026:
The most common shapes for the density distribution of protons and neutrons in an
2018:
ellipsoidal definition considers its own values to be exact in the form it gives.
2740:
Greenburg, John L. (1995). "Isaac Newton and the
Problem of the Earth's Shape".
2244:
2143:
155:
124:
201:
2296:
2263:
2183:
2175:
2121:
2089:
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vector). Deformed nuclear shapes occur as a result of the competition between
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151:
110:
102:
92:
2927:
2286:
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2229:
2031:
1316:
77:
229:
The equation of a tri-axial ellipsoid centred at the origin with semi-axes
2956:
2277:
Many submarines have a shape which can be described as prolate spheroid.
2267:
2259:
2761:
2753:
2605:
2109:
2085:
976:
794:
171:
159:
135:
73:
62:
451:{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}
28:
2240:
2214:
2198:
2113:
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2097:
147:
128:
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162:
the Earth is often approximated by an oblate spheroid, known as the
2837:
2190:
2169:
2151:
2079:
2728:
Introduction à la Géodésie et au geopositionnement par satellites
2158:. Earth's diverse cartographic and geodetic systems are based on
2615:
2225:
2150:'s theories for their interpretation, reasoned that Jupiter and
2968:
2815:(October 1973), "The Distance to the Crab Nebula and NP 0532",
982:
These formulas are identical in the sense that the formula for
2899:"What Do a Submarine, a Rocket and a Football Have in Common?"
154:
in the direction of its axis of rotation. For that reason, in
1761:, is the ratio of the polar to equatorial lengths, while the
1161:{\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}
1071:{\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}
205:
The assignment of semi-axes on a spheroid. It is oblate if
2096:
The oblate spheroid is the approximate shape of rotating
127:. If the generating ellipse is a circle, the result is a
2306:, the moments of inertia along these principal axes are
955:
The prolate spheroid is generated by rotation about the
2817:
1913:
The relations between eccentricity and flattening are:
773:
The oblate spheroid is generated by rotation about the
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2414:
2349:
1112:
1019:
889:
707:
20:
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1922:
1859:
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1191:
1110:
1017:
819:
543:
375:
250:
2239:
The term is also used to describe the shape of some
1903:{\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}
1834:{\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}
76:
about one of its principal axes; in other words, an
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2006:
1902:
1833:
1726:
1499:
1286:
1160:
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944:
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450:
337:
2574:{\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}
174:and 6,356.752 km (3,949.903 mi) at the
150:) is not quite a sphere, but instead is slightly
2787:The Story of Civilization: The Age of Louis XIV
1008:The volume inside a spheroid (of any kind) is
465:is the equatorial radius of the spheroid, and
2072:is also a shape of archaeological artifacts.
2060:. Rotating equilibrium spheroids include the
1178:Radius of the Earth § Radii of curvature
8:
2783:Durant, Will; Durant, Ariel (28 July 1997).
2703:A derivation of this result may be found at
2680:A derivation of this result may be found at
2706:"Prolate Spheroid - from Wolfram MathWorld"
2291:For a spheroid having uniform density, the
2683:"Oblate Spheroid - from Wolfram MathWorld"
2116:. Saturn is the most oblate planet in the
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959:-axis of an ellipse with semi-major axis
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777:-axis of an ellipse with semi-major axis
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251:
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2877:(6 ed.). Elsevier Health Sciences.
2875:Introduction to Vascular Ultrasonography
200:
22:Spheroids with vertical rotational axes
2873:John Pellerito, Joseph F Polak (2012).
2708:. Mathworld.wolfram.com. 7 October 2003
2642:
1193:
2287:Ellipsoid § Dynamical properties
1101:is the polar diameter, the volume is
241:aligned along the coordinate axes is
35:
16:Surface formed by rotating an ellipse
7:
2154:are oblate spheroids owing to their
91:If the ellipse is rotated about its
2521:is the mass of the body defined as
2088:is a slight oblate spheroid with a
1182:Let a spheroid be parameterized as
166:, instead of a sphere. The current
2852:"Nuclear fission - Fission theory"
14:
2932:MathWorld--A Wolfram Web Resource
2955:
2112:, and the quickly spinning star
1091:is the equatorial diameter, and
348:The equation of a spheroid with
27:
1751:of an oblate spheroid/ellipse,
971:may again be identified as the
895:
887:
713:
705:
134:Due to the combined effects of
2621:Prolate spheroidal coordinates
1539:
1527:
1395:
1383:
1278:
1215:
1209:
1197:
1:
2611:Oblate spheroidal coordinates
2726:Brial P., Shaalan C.(2009),
2146:'s pendulum experiments and
2162:, all of which are oblate.
2042:repulsion between protons,
2022:Occurrence and applications
3015:
2284:
1175:
2742:History of Exact Sciences
2270:may be measured by their
789:may be identified as the
26:
2813:Trimble, Virginia Louise
2748:(4). Springer: 371–391.
2650:Torge, Wolfgang (2001).
2056:Spheroids are common in
800:A prolate spheroid with
521:An oblate spheroid with
2976:Encyclopædia Britannica
2856:Encyclopedia Britannica
2685:. Mathworld.wolfram.com
1366:. Then, the spheroid's
55:ellipsoid of revolution
2979:(11th ed.). 1911.
2575:
2508:
2179:
2093:
2008:
1904:
1835:
1728:
1501:
1288:
1162:
1072:
946:
764:
452:
339:
226:
215:(left) and prolate if
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2509:
2173:
2083:
2009:
1905:
1836:
1729:
1502:
1289:
1163:
1073:
947:
765:
508:reduces to a sphere.
453:
340:
204:
168:World Geodetic System
2964:at Wikimedia Commons
2528:
2325:
2281:Dynamical properties
2160:reference ellipsoids
2126:planetary flattening
1920:
1857:
1782:
1521:
1377:
1189:
1108:
1015:
963:and semi-minor axis
817:
781:and semi-minor axis
541:
373:
356:is given by setting
248:
59:rotational ellipsoid
2926:Weisstein, Eric W.
2904:Scientific American
2829:1973PASP...85..579T
2631:Translation of axes
2272:long and short axes
2195:triaxial ellipsoids
2104:, including Earth,
1308:parametric latitude
164:reference ellipsoid
144:figure of the Earth
119:, flattened like a
113:, the result is an
101:, elongated like a
53:, also known as an
23:
2754:10.1007/BF00374704
2571:
2547:
2504:
2502:
2474:
2423:
2358:
2180:
2148:Christiaan Huygens
2094:
2062:Maclaurin spheroid
2004:
2002:
1900:
1831:
1724:
1497:
1368:Gaussian curvature
1284:
1158:
1121:
1068:
1028:
942:
893:
760:
711:
494:: prolate spheroid
448:
335:
227:
191:geopotential model
95:, the result is a
21:
2960:Media related to
2907:. 8 November 2010
2658:Walter de Gruyter
2546:
2473:
2422:
2357:
2302:, and minor axes
2293:moment of inertia
2166:Prolate spheroids
2156:centrifugal force
1998:
1958:
1898:
1896:
1826:
1807:
1719:
1715:
1492:
1120:
1027:
937:
892:
873:
827:
810:has surface area
755:
710:
699:
663:
607:
551:
482:: oblate spheroid
440:
413:
327:
300:
273:
107:American football
86:circular symmetry
84:. A spheroid has
47:
46:
3006:
2980:
2972:
2970:"Spheroid"
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2701:
2695:
2694:
2692:
2690:
2678:
2672:
2671:
2656:(3rd ed.).
2647:
2626:Rotation of axes
2596:Equatorial bulge
2591:Ellipsoidal dome
2580:
2578:
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2377:
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2350:
2317:
2313:
2309:
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2301:
2130:equatorial bulge
2124:of 0.09796. See
2102:celestial bodies
2076:Oblate spheroids
2066:Jacobi ellipsoid
2058:3D cell cultures
2036:angular momentum
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1303:reduced latitude
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98:prolate spheroid
31:
24:
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2660:. p. 104.
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2289:
2283:
2197:. Examples are
2168:
2142:, working from
2078:
2044:surface tension
2040:electromagnetic
2024:
2001:
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188:Earth's gravity
116:oblate spheroid
80:with two equal
17:
12:
11:
5:
3012:
3010:
3002:
3001:
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2951:
2950:External links
2948:
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2838:10.1086/129507
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2279:
2201:'s satellites
2167:
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2077:
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2028:atomic nucleus
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2015:
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1995:
1991:
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461:The semi-axis
459:
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198:
195:
82:semi-diameters
45:
44:
39:
33:
32:
15:
13:
10:
9:
6:
4:
3:
2:
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3000:
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2933:
2929:
2922:
2919:
2906:
2905:
2900:
2894:
2891:
2886:
2884:9781455737666
2880:
2876:
2870:
2865:
2862:
2857:
2853:
2847:
2844:
2839:
2834:
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2826:
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2808:
2805:
2800:
2794:
2791:. MJF Books.
2789:
2788:
2779:
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2771:
2767:
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2751:
2747:
2743:
2736:
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2700:
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2684:
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2667:9783110170726
2663:
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2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2601:Great ellipse
2599:
2597:
2594:
2592:
2589:
2588:
2584:
2568:
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2256:atomic nuclei
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2250:
2249:Fresnel zones
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2165:
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2161:
2157:
2153:
2149:
2145:
2141:
2137:
2136:Enlightenment
2133:
2132:for details.
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2099:
2091:
2087:
2082:
2075:
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2059:
2054:
2052:
2051:shell effects
2049:
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2041:
2037:
2033:
2029:
2021:
2019:
1993:
1989:
1985:
1982:
1977:
1974:
1971:
1969:
1964:
1953:
1949:
1945:
1942:
1939:
1934:
1932:
1927:
1916:
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1914:
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1770:
1767:(also called
1766:
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1114:
1104:
1103:
1102:
1099:
1095:
1089:
1085:
1065:
1062:
1057:
1053:
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1046:
1043:
1038:
1034:
1030:
1024:
1021:
1011:
1010:
1009:
1003:
1001:
998:
995:then becomes
986:
980:
978:
974:
967:; therefore,
939:
932:
928:
922:
918:
912:
909:
906:
901:
897:
883:
879:
875:
869:
866:
862:
857:
854:
850:
844:
840:
836:
833:
830:
821:
813:
812:
811:
808:
804:
798:
796:
792:
757:
750:
746:
740:
736:
730:
727:
724:
719:
715:
701:
695:
692:
689:
684:
681:
678:
672:
668:
665:
660:
655:
651:
645:
642:
637:
633:
629:
626:
623:
619:
615:
612:
609:
604:
598:
594:
590:
587:
581:
578:
574:
568:
564:
560:
557:
554:
545:
537:
536:
535:
534:
529:
525:
516:
511:
509:
506:
502:
492:
488:
484:
480:
476:
472:
471:
470:
445:
442:
435:
431:
425:
421:
415:
408:
404:
397:
393:
389:
384:
380:
369:
368:
367:
364:
360:
355:
354:symmetry axis
332:
329:
322:
318:
312:
308:
302:
295:
291:
285:
281:
275:
268:
264:
258:
254:
244:
243:
242:
223:
219:
213:
209:
203:
196:
194:
192:
189:
184:
179:
177:
173:
169:
165:
161:
157:
153:
149:
145:
141:
137:
132:
130:
126:
122:
118:
117:
112:
108:
104:
100:
99:
94:
89:
87:
83:
79:
75:
71:
67:
64:
60:
56:
52:
43:
40:
38:
34:
30:
25:
19:
2974:
2935:. Retrieved
2931:
2921:
2909:. Retrieved
2902:
2893:
2874:
2864:
2855:
2846:
2823:(507): 579,
2820:
2816:
2807:
2786:
2778:
2745:
2741:
2735:
2722:
2710:. Retrieved
2699:
2687:. Retrieved
2676:
2652:
2645:
2516:
2290:
2276:
2253:
2243:such as the
2238:
2226:tidal forces
2223:
2217:' satellite
2188:
2181:
2140:Isaac Newton
2134:
2118:Solar System
2095:
2055:
2025:
2016:
1912:
1847:eccentricity
1846:
1843:
1768:
1762:
1757:
1753:
1748:aspect ratio
1746:
1744:
1741:Aspect ratio
1736:
1509:
1361:
1338:
1306:
1302:
1296:
1181:
1097:
1093:
1087:
1083:
1080:
1007:
984:
981:
973:eccentricity
954:
806:
802:
799:
791:eccentricity
785:, therefore
772:
533:surface area
527:
523:
520:
504:
500:
498:The case of
497:
490:
486:
478:
474:
460:
362:
358:
347:
228:
221:
217:
211:
207:
182:
180:
146:(and of all
133:
115:
114:
97:
96:
90:
68:obtained by
58:
54:
50:
48:
41:
36:
18:
2245:Crab Nebula
2144:Jean Richer
2084:The planet
156:cartography
123:or a plain
2988:Categories
2928:"Spheroid"
2798:1567310192
2637:References
2297:major axis
2285:See also:
2264:lanthanide
2184:rugby ball
2176:rugby ball
2138:scientist
2122:flattening
2100:and other
2092:of 0.06487
2090:flattening
1844:The first
1769:oblateness
1764:flattening
1176:See also:
512:Properties
111:minor axis
103:rugby ball
93:major axis
2962:Spheroids
2770:121268606
2566:ρ
2550:π
2234:volcanism
2207:Enceladus
2120:, with a
2032:spherical
1986:−
1978:−
1946:−
1872:−
1816:−
1798:−
1700:β
1697:
1669:−
1619:β
1616:
1588:−
1537:λ
1531:β
1479:β
1476:
1448:−
1393:λ
1387:β
1317:longitude
1276:β
1273:
1261:λ
1258:
1252:β
1249:
1237:λ
1234:
1228:β
1225:
1207:λ
1201:β
1194:σ
1172:Curvature
1137:≈
1115:π
1047:≈
1031:π
997:imaginary
913:−
837:π
731:−
693:−
669:
646:π
630:π
613:
591:−
561:π
181:The word
152:flattened
78:ellipsoid
2999:Quadrics
2994:Surfaces
2869:Page 559
2762:41134011
2585:See also
2260:actinide
2189:Several
2070:Spheroid
2064:and the
1756: :
1510:and its
1360:−π <
225:(right).
197:Equation
183:spheroid
140:rotation
70:rotating
51:spheroid
2911:13 June
2825:Bibcode
2712:24 June
2689:24 June
2653:Geodesy
2606:Lentoid
2258:of the
2241:nebulae
2219:Miranda
2110:Jupiter
2098:planets
2086:Jupiter
2048:quantum
1364:< +π
1355:
1343:
1335:
1323:
1315:is the
1301:is the
977:ellipse
975:. (See
826:prolate
795:ellipse
793:. (See
610:arctanh
352:as the
172:Equator
160:geodesy
148:planets
136:gravity
125:M&M
74:ellipse
66:surface
63:quadric
61:, is a
42:prolate
2937:16 May
2881:
2795:
2768:
2760:
2664:
2517:where
2314:, and
2268:testis
2215:Uranus
2211:Tethys
2209:, and
2199:Saturn
2114:Altair
2106:Saturn
1341:< +
1319:, and
1297:where
1004:Volume
988:oblate
876:arcsin
550:oblate
142:, the
129:sphere
121:lentil
105:. The
37:oblate
2766:S2CID
2758:JSTOR
2730:, p.8
2616:Ovoid
2304:a = b
2203:Mimas
2191:moons
2152:Earth
1337:<
1140:0.523
891:where
805:>
709:where
526:<
489:>
477:<
220:>
210:<
176:poles
2939:2018
2913:2015
2879:ISBN
2871:in:
2793:ISBN
2714:2014
2691:2014
2662:ISBN
2262:and
2254:The
2213:and
2128:and
2046:and
2030:are
1745:The
1358:and
1050:4.19
979:.)
531:has
517:Area
237:and
158:and
138:and
2833:doi
2750:doi
1688:cos
1607:cos
1514:is
1467:cos
1370:is
1305:or
1270:sin
1255:sin
1246:cos
1231:cos
1222:cos
1096:= 2
1086:= 2
1081:If
797:.)
193:).
72:an
57:or
2990::
2973:.
2930:.
2901:.
2854:.
2831:,
2821:85
2819:,
2764:.
2756:.
2746:49
2744:.
2310:,
2274:.
2247:.
2230:Io
2221:.
2205:,
2186:.
2174:A
2108:,
2068:.
2053:.
1771:)
1311:,
666:ln
503:=
446:1.
366::
361:=
333:1.
233:,
178:.
131:.
88:.
49:A
2941:.
2915:.
2887:.
2858:.
2835::
2827::
2801:.
2772:.
2752::
2716:.
2693:.
2670:.
2569:.
2563:c
2558:2
2554:a
2544:3
2541:4
2535:=
2532:M
2519:M
2498:,
2494:)
2489:2
2485:a
2481:(
2477:M
2471:5
2468:2
2462:=
2458:)
2452:2
2448:b
2444:+
2439:2
2435:a
2430:(
2426:M
2420:5
2417:1
2411:=
2404:C
2397:,
2393:)
2387:2
2383:c
2379:+
2374:2
2370:a
2365:(
2361:M
2355:5
2352:1
2346:=
2339:B
2336:=
2333:A
2316:B
2312:A
2308:C
2300:c
2178:.
1994:2
1990:e
1983:1
1975:1
1972:=
1965:f
1954:2
1950:f
1943:f
1940:2
1935:=
1928:e
1892:2
1888:a
1882:2
1878:c
1869:1
1864:=
1861:e
1829:.
1824:a
1821:c
1813:1
1810:=
1805:a
1801:c
1795:a
1789:=
1786:f
1773:f
1758:a
1754:c
1722:.
1713:2
1710:3
1704:)
1692:2
1683:)
1677:2
1673:a
1664:2
1660:c
1655:(
1651:+
1646:2
1642:a
1637:(
1632:a
1629:2
1623:)
1611:2
1602:)
1596:2
1592:a
1583:2
1579:c
1574:(
1570:+
1565:2
1561:a
1557:2
1553:(
1549:c
1543:=
1540:)
1534:,
1528:(
1525:H
1495:,
1488:2
1483:)
1471:2
1462:)
1456:2
1452:a
1443:2
1439:c
1434:(
1430:+
1425:2
1421:a
1416:(
1409:2
1405:c
1399:=
1396:)
1390:,
1384:(
1381:K
1362:λ
1352:2
1349:/
1346:π
1339:β
1332:2
1329:/
1326:π
1321:−
1313:λ
1299:β
1282:,
1279:)
1267:c
1264:,
1243:a
1240:,
1219:a
1216:(
1213:=
1210:)
1204:,
1198:(
1156:.
1153:C
1148:2
1144:A
1134:C
1129:2
1125:A
1118:6
1098:c
1094:C
1088:a
1084:A
1066:.
1063:c
1058:2
1054:a
1044:c
1039:2
1035:a
1025:3
1022:4
993:e
985:S
969:e
965:a
961:c
957:z
940:.
933:2
929:c
923:2
919:a
910:1
907:=
902:2
898:e
884:)
880:e
870:e
867:a
863:c
858:+
855:1
851:(
845:2
841:a
834:2
831:=
822:S
807:a
803:c
787:e
783:c
779:a
775:z
758:.
751:2
747:a
741:2
737:c
728:1
725:=
720:2
716:e
702:)
696:e
690:1
685:e
682:+
679:1
673:(
661:e
656:2
652:c
643:+
638:2
634:a
627:2
624:=
620:)
616:e
605:e
599:2
595:e
588:1
582:+
579:1
575:(
569:2
565:a
558:2
555:=
546:S
528:a
524:c
505:c
501:a
491:a
487:c
479:a
475:c
467:c
463:a
443:=
436:2
432:c
426:2
422:z
416:+
409:2
405:a
398:2
394:y
390:+
385:2
381:x
363:b
359:a
350:z
330:=
323:2
319:c
313:2
309:z
303:+
296:2
292:b
286:2
282:y
276:+
269:2
265:a
259:2
255:x
239:c
235:b
231:a
222:a
218:c
212:a
208:c
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