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373:, yeast converts sugars into carbon dioxide and alcohol. By monitoring the decline in SG over time the brewer obtains information about the health and progress of the fermentation and determines that it is complete when gravity stops declining. If the fermentation is finished, the specific gravity is called the final gravity (abbreviated FG). For example, for a typical strength beer, original gravity (abbreviated OG) could be 1.050 and FG could be 1.010.
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usually 20 °C (68 °F) for both sample and water but in some parts of the world different temperatures may be used and there are hydrometers sold calibrated to, for example, 16 °C (60 °F). It is important, where any conversion to °P is involved, that the proper pair of temperatures be used for the conversion table or formula being employed. The current
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With typical values of 1.050 and 1.010 for, respectively, OG and FG this simplified formula gives an ABV of 5.31% as opposed to 5.23% for the more accurate formula. Formulas for alcohol similar to this last simple one abound in the brewing literature and are very popular among home brewers. Formulas
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Points can be used in the ADF and RDF formulas. Thus a beer with OG 1.050 which fermented to 1.010 would be said to have attenuated 100 × (50 − 10)/50 = 80%. Points can also be used in the SG versions of the alcohol formulas. It is simply necessary to multiply by 1000 as points are 1000 times (SG −
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quantitative analysis but can also be done by evaporation in a water bath. If the residue is made back up to the original volume of beer which was subject to the evaporation process, the specific gravity of that reconstituted beer measured and converted to Plato using the tables and formulas in the
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Specific gravity is the ratio of the density of a sample (of any substance) to the density of water. The ratio depends on the temperature and pressure of both the sample and water. The pressure is always considered (in brewing) to be 1 standard atmosphere (1,013.25 hPa) and the temperature is
1226:
This formula is fine for those who wish to go to the trouble to compute TE (whose real value lies in determining attenuation) which is only a small fraction of brewers. Others want a simpler, quicker route to determining alcoholic strength. This lies in
Tabarie's Principle which states that the
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The difference between the original gravity of the wort and the final gravity of the beer is an indication of how much sugar has been turned into alcohol. The bigger the difference, the greater the amount of alcohol present and hence the stronger the beer. This is why strong beers are sometimes
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The amount of extract which was not converted to yeast biomass, carbon dioxide or ethanol can be estimated by removing the alcohol from beer which has been degassed and clarified by filtration or other means. This is often done as part of a distillation in which the alcohol is collected for
841:
Knowing the amount of extract in 100 grams (3.5 oz) of wort before fermentation and the number of grams of extract in 100 grams (3.5 oz) of beer at its completion, the amount alcohol (in grams) formed during the fermentation can be determined. The formula follows, attributed to
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depression of specific gravity in beer to which ethanol is added is the same as the depression of water to which an equal amount of alcohol (on a w/w basis) has been added. Use of
Tabarie's principle lets us calculate the true extract of a beer with apparent extract
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By considering the original gravity, the brewer or vintner obtains an indication as to the probable ultimate alcoholic content of their product. The OE (Original
Extract) is often referred to as the "size" of the beer and is, in Europe, often printed on the label as
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However, the above approximation has increasingly larger error for increasing values of specific gravity and deviates e.g. by 0.67°P when SG = 1.080. A much more accurate (mean average error less than 0.02°P) conversion can be made using the following formula:
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such as this one make it possible to mark hydrometers with "potential alcohol" scales based on the assumption that the FG will be close to 1 which is more likely to be the case in wine making than in brewing and it is to vintners that these are usually sold.
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referred to as high gravity beers, and "session" or "small" beers are called low gravity beers, even though in theory the final gravity of a strong beer might be lower than that of a session beer because of the greater amount of alcohol present.
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table is (20 °C/20 °C) meaning that the density is measured at 20 °C (68 °F) and referenced to the density of water at 20 °C (68 °F) (i.e. 0.998203 g/cm or 0.0360624 lb/cu in). Mathematically
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559:) measure density directly and so must use a hydrometer, whose stem is bathed in air, or pycnometer weighings which are also done in air. Hydrometer readings and the ratio of pycnometer weights are influenced by air (see article
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The relationship between SG and °P can be roughly approximated using the rule-of-thumb conversion equation "brewer's points divided by four", where the "Brewing" or "Gravity points" are the thousandths of SG above 1:
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The original gravity is the specific gravity measured before fermentation. From it the analyst can compute the original extract which is the mass (grams) of sugar in 100 grams (3.5 oz) of wort (°P) by use of the
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Because ABV depends on multiplicative factors (one of which depends on the original extract and one on the final) as well as the difference between OE and AE it is impossible to come up with a formula of the form
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As an example, a wort of SG 1.050 would be said to have 1000(1.050 − 1) = 50 points, and contain 50/4 = 12.5 °P of extract. This is simply the linear approximation to the true relationship between SG and °P.
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or sometimes just as a percent. In the Czech
Republic, for example, common descriptions are "10 degree beers", "12 degree beers" which refers to the gravity in Plato of the wort before the fermentation.
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tools are available to brewers to convert between the various units of measurement and to adjust mash ingredients and schedules to meet target values. The resulting data can be exchanged via
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Most brewers and consumers are used to having alcohol content reported by volume (ABV) rather than weight. Interconversion is simple but the specific gravity of the beer must be known:
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article. The use of "apparent" here is not to be confused with the use of that term to describe specific gravity readings which have not been corrected for the effects of air.
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The drop in extract during the fermentation divided by the OE represents the percentage of sugar which has been consumed. The real degree of attenuation (RDF) is based on TE
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gives the number of grams of alcohol per 100 grams (3.5 oz) of beer i.e. the ABW. Note that the alcohol content depends not only on the diminution of extract
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654:{\displaystyle {\text{SG}}_{\text{true}}={\text{SG}}_{\text{apparent}}-{\rho _{\text{air}} \over \rho _{\text{water}}}({\text{SG}}_{\text{apparent}}-1)}
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However, the ASBC table uses apparent specific gravities, so many electronic density meters will produce the correct °P numbers automatically.
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Several different scales have been used for measuring the original gravity. For historical reasons, the brewing industry largely uses the
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Because of the near linear relationship between (SG − 1) and °P specific gravities can be used in the ADF formula as shown.
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833:. This is the number of grams of extract remaining in 100 grams (3.5 oz) of beer at the completion of fermentation.
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The final gravity is the specific gravity measured at the completion of fermentation. The apparent extract, denoted
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for details) and are called "apparent" readings. True readings are easily obtained from apparent readings by
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2395:{\displaystyle {\text{ADF}}=100{(p-m) \over p}\approx 100{({\text{OG}}-{\text{FG}}) \over ({\text{OG}}-1)}}
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by weight at 20 °C. Inserting this into the alcohol formula the result, after rearrangement, is
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This formula gives the true specific gravity i.e. based on densities. Brewers cannot (unless using a
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2097:{\displaystyle A_{v}=250f_{pm}({\text{OG}}-{\text{FG}}){{\text{SG}}_{\text{beer}} \over 0.79661}}
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The amount of extract in degrees Plato are thus approximately given by the points divided by 4:
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is a simple constant. Because of the near linear relationship between extract and (SG − 1) (see
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This is the number of cubic centimetres of ethanol in 100 cc (6 cu in) of beer.
1216:{\displaystyle f_{pn}={1 \over (2.0665-1.0665p/100)}\approx 0.48394+0.0024688p+0.00001561p^{2}}
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799:{\displaystyle n=P_{\text{recon}}{{\text{SG}}_{\text{recon}} \over {\text{SG}}_{\text{beer}}}}
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as a function of OE and AE. It is again possible to come up with a relationship of the form
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545:{\displaystyle {\text{SG}}_{\text{true}}={\rho _{\text{sample}} \over \rho _{\text{water}}}}
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The density of a wort is largely dependent on the sugar content of the wort. During alcohol
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1348:{\displaystyle n=P(P^{-1}(m)+1-{\frac {\rho _{\text{EtOH}}(A_{w})}{\rho _{\text{water}}}})}
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2220:{\displaystyle A_{v}=132.715({\text{OG}}-{\text{FG}})=({\text{OG}}-{\text{FG}})/0.00753\,}
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scale used by the wine industry. For example, OG 1.050 is roughly equivalent to 12
2648:= 20 °C. The equivalent relation giving SG at 20 °C for a given °P is:
2799:. Vol. 2. Translated by Kathleen Barton-Wright. London: Chapman & Hall.
2570:{\displaystyle ^{\circ }P\approx p_{t}/4=1000({\text{SG}}-1)/4=250\cdot SG-250.}
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1873:{\displaystyle A_{v}=A_{w}{{\text{SG}}_{\text{beer}} \over 0.79661}}
951:{\displaystyle A_{w}={(p-n) \over (2.0665-1.0665p/100)}=f_{pn}(p-n)}
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262:. Statements consisting only of original research should be removed.
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at various stages in the fermentation. The concept is used in the
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where the specific gravity is to be measured at a temperature of
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which depends on the OE. De Clerck tabulated
Ballings values for
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338:(abbreviated SG), or relative density compared to water, of the
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and wine-making industries. Specific gravity is measured by a
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and the apparent degree of fermentation (ADF) is based on AE
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2812:"Physical Equations Relating Extract and Relative Density"
1652:{\displaystyle {\left \over (2.0665-1.0665p/100)}-A_{w}=0}
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is the density of an aqueous ethanol solution of strength
1808:{\displaystyle f_{pm}=0.39661+0.001709p+0.000010788p^{2}}
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to other brewers to facilitate accurate replication.
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2695:{\displaystyle SG={\frac {260.4}{260.4-^{\circ }P}}}
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may be too technical for most readers to understand
101:. Unsourced material may be challenged and removed.
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2287:{\displaystyle {\text{RDF}}=100{(p-n) \over p}}
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2010:{\displaystyle p\approx 1000({\text{SG}}-1)/4}
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2461:{\displaystyle p_{t}:=1000({\text{SG}}-1)\,}
668:Original gravity (OG); original extract (OE)
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441:. Unsourced material may be challenged and
380:(°P), which is essentially the same as the
64:Learn how and when to remove these messages
1452:{\displaystyle \rho _{\text{EtOH}}(A_{w})}
1378:is a function that converts SG to °P (see
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701:Final gravity (FG); apparent extract (AE)
461:Learn how and when to remove this message
296:Learn how and when to remove this message
278:Learn how and when to remove this message
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200:, without removing the technical details.
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1061:but also on the multiplicative factor
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1754:De Clerk also tabulates values for
1744:{\displaystyle A_{w}=f_{pm}(p-m)\,}
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110:"Gravity" alcoholic beverage
45:This article has multiple issues.
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1932:{\displaystyle A_{v}=k(p-m)\,}
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403:Low vs. high gravity beers
2818:. Center for Open Science
2795:De Clerck, Jean (1958).
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37:
30:
21:
2864:
2863:
2859:
2858:
2857:
2855:
2854:
2853:
2834:
2833:
2832:
2831:
2821:
2819:
2808:
2804:
2793:
2784:
2779:
2761:
2754:
2747:
2740:
2733:
2726:
2723:
2705:
2680:
2676:
2672:
2667:
2656:
2653:
2652:
2621:
2616:
2598:
2595:
2593:
2590:
2589:
2538:
2524:
2507:
2501:
2497:
2485:
2482:
2480:
2477:
2476:
2443:
2428:
2424:
2422:
2419:
2418:
2411:
2409:Brewer's points
2375:
2371:
2362:
2354:
2350:
2348:
2320:
2318:
2307:
2305:
2302:
2301:
2262:
2260:
2249:
2247:
2244:
2243:
2237:
2208:
2200:
2192:
2178:
2170:
2155:
2151:
2149:
2146:
2145:
2138:
2118:
2114:
2112:
2109:
2108:
2083:
2078:
2077:
2075:
2067:
2059:
2047:
2043:
2031:
2027:
2025:
2022:
2021:
1999:
1985:
1971:
1968:
1967:
1947:
1944:
1943:
1901:
1897:
1895:
1892:
1891:
1859:
1854:
1853:
1851:
1845:
1841:
1832:
1828:
1826:
1823:
1822:
1799:
1795:
1765:
1761:
1759:
1756:
1755:
1716:
1712:
1703:
1699:
1697:
1694:
1693:
1673:
1669:
1667:
1664:
1663:
1637:
1633:
1616:
1600:
1582:
1578:
1568:
1564:
1555:
1551:
1550:
1548:
1521:
1517:
1516:
1512:
1502:
1498:
1496:
1494:
1491:
1490:
1470:
1466:
1464:
1461:
1460:
1440:
1436:
1427:
1423:
1421:
1418:
1417:
1393:
1389:
1387:
1384:
1383:
1363:
1360:
1359:
1334:
1330:
1320:
1316:
1307:
1303:
1302:
1300:
1273:
1269:
1255:
1252:
1251:
1232:
1229:
1228:
1207:
1203:
1168:
1152:
1147:
1135:
1131:
1129:
1126:
1125:
1102:
1098:
1096:
1093:
1092:
1072:
1068:
1066:
1063:
1062:
1034:
1031:
1030:
1005:
989:
984:
972:
968:
966:
963:
962:
924:
920:
903:
887:
870:
868:
859:
855:
853:
850:
849:
839:
837:Alcohol content
818:
815:
814:
788:
783:
782:
776:
771:
770:
768:
762:
758:
750:
747:
746:
735:
710:
707:
706:
703:
682:
679:
678:
670:
636:
631:
630:
619:
615:
609:
605:
603:
594:
589:
588:
579:
574:
573:
571:
568:
567:
534:
530:
524:
520:
518:
509:
504:
503:
501:
498:
497:
485:
480:
467:
456:
450:
447:
432:
416:
405:
385:
302:
291:
290:
289:
284:
273:
267:
264:
249:
237:
233:
222:
211:
205:
202:
194:help improve it
191:
182:
178:
167:
156:
150:
147:
104:
102:
92:
80:
39:
35:
28:
23:
22:
15:
12:
11:
5:
2862:
2852:
2851:
2846:
2830:
2829:
2802:
2781:
2780:
2778:
2775:
2774:
2773:
2767:
2766:
2752:
2738:
2722:
2719:
2703:
2702:
2688:
2683:
2679:
2675:
2671:
2666:
2663:
2660:
2642:
2641:
2627:
2624:
2620:
2615:
2612:
2609:
2606:
2601:
2597:
2578:
2577:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2541:
2537:
2534:
2531:
2523:
2520:
2517:
2514:
2510:
2504:
2500:
2496:
2493:
2488:
2484:
2470:
2469:
2456:
2453:
2450:
2442:
2439:
2436:
2431:
2427:
2410:
2407:
2403:
2402:
2388:
2385:
2382:
2374:
2369:
2361:
2353:
2347:
2344:
2339:
2335:
2332:
2329:
2326:
2323:
2317:
2314:
2295:
2294:
2281:
2277:
2274:
2271:
2268:
2265:
2259:
2256:
2236:
2233:
2228:
2227:
2215:
2211:
2207:
2199:
2191:
2188:
2185:
2177:
2169:
2166:
2163:
2158:
2154:
2124:
2121:
2117:
2105:
2104:
2091:
2074:
2066:
2058:
2053:
2050:
2046:
2042:
2039:
2034:
2030:
2006:
2002:
1998:
1995:
1992:
1984:
1981:
1978:
1975:
1951:
1940:
1939:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1904:
1900:
1881:
1880:
1867:
1848:
1844:
1840:
1835:
1831:
1802:
1798:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1771:
1768:
1764:
1752:
1751:
1739:
1736:
1733:
1730:
1727:
1722:
1719:
1715:
1711:
1706:
1702:
1676:
1672:
1660:
1659:
1648:
1645:
1640:
1636:
1632:
1626:
1623:
1619:
1615:
1612:
1609:
1606:
1603:
1598:
1593:
1581:
1576:
1571:
1567:
1563:
1554:
1547:
1544:
1541:
1538:
1535:
1532:
1527:
1524:
1520:
1515:
1511:
1508:
1505:
1501:
1473:
1469:
1448:
1443:
1439:
1435:
1426:
1399:
1396:
1392:
1367:
1356:
1355:
1344:
1333:
1328:
1323:
1319:
1315:
1306:
1299:
1296:
1293:
1290:
1287:
1284:
1279:
1276:
1272:
1268:
1265:
1262:
1259:
1236:
1224:
1223:
1210:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1178:
1175:
1171:
1167:
1164:
1161:
1158:
1155:
1151:
1146:
1141:
1138:
1134:
1108:
1105:
1101:
1078:
1075:
1071:
1050:
1047:
1044:
1041:
1038:
1015:
1012:
1008:
1004:
1001:
998:
995:
992:
988:
983:
978:
975:
971:
959:
958:
947:
944:
941:
938:
935:
930:
927:
923:
919:
913:
910:
906:
902:
899:
896:
893:
890:
885:
882:
879:
876:
873:
867:
862:
858:
838:
835:
822:
807:
806:
761:
757:
754:
734:
731:
714:
702:
699:
686:
669:
666:
662:
661:
650:
647:
644:
629:
618:
608:
602:
587:
553:
552:
533:
523:
517:
484:
481:
479:
476:
469:
468:
419:
417:
410:
404:
401:
330:of fermenting
304:
303:
286:
285:
268:September 2013
240:
238:
231:
224:
223:
185:
183:
176:
169:
168:
83:
81:
74:
69:
43:
42:
40:
33:
26:
9:
6:
4:
3:
2:
2861:
2850:
2847:
2845:
2842:
2841:
2839:
2817:
2816:OSF Preprints
2813:
2806:
2798:
2791:
2789:
2787:
2782:
2772:
2769:
2768:
2764:
2763:Liquor portal
2758:
2753:
2750:
2744:
2739:
2736:
2730:
2725:
2718:
2716:
2712:
2708:
2686:
2681:
2677:
2673:
2669:
2664:
2661:
2658:
2651:
2650:
2649:
2647:
2625:
2622:
2618:
2613:
2610:
2607:
2604:
2599:
2596:
2588:
2587:
2586:
2582:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2539:
2532:
2529:
2518:
2515:
2512:
2508:
2502:
2498:
2494:
2491:
2486:
2483:
2475:
2474:
2473:
2451:
2448:
2437:
2434:
2429:
2425:
2417:
2416:
2415:
2406:
2383:
2380:
2359:
2345:
2342:
2337:
2330:
2327:
2324:
2315:
2312:
2300:
2299:
2298:
2279:
2272:
2269:
2266:
2257:
2254:
2242:
2241:
2240:
2232:
2213:
2209:
2197:
2186:
2175:
2164:
2161:
2156:
2152:
2144:
2143:
2142:
2122:
2119:
2115:
2089:
2064:
2051:
2048:
2044:
2040:
2037:
2032:
2028:
2020:
2019:
2018:
2004:
2000:
1993:
1990:
1979:
1976:
1973:
1965:
1949:
1922:
1919:
1916:
1910:
1907:
1902:
1898:
1890:
1889:
1888:
1884:
1865:
1846:
1842:
1838:
1833:
1829:
1821:
1820:
1819:
1816:
1800:
1796:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1769:
1766:
1762:
1734:
1731:
1728:
1720:
1717:
1713:
1709:
1704:
1700:
1692:
1691:
1690:
1674:
1670:
1646:
1643:
1638:
1634:
1630:
1621:
1617:
1613:
1610:
1607:
1604:
1596:
1591:
1579:
1569:
1565:
1552:
1545:
1542:
1539:
1533:
1525:
1522:
1518:
1513:
1509:
1506:
1503:
1499:
1489:
1488:
1487:
1471:
1467:
1441:
1437:
1424:
1415:
1397:
1394:
1390:
1381:
1365:
1331:
1321:
1317:
1304:
1297:
1294:
1291:
1285:
1277:
1274:
1270:
1263:
1260:
1257:
1250:
1249:
1248:
1234:
1208:
1204:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1173:
1169:
1165:
1162:
1159:
1156:
1149:
1144:
1139:
1136:
1132:
1124:
1123:
1122:
1106:
1103:
1099:
1076:
1073:
1069:
1045:
1042:
1039:
1010:
1006:
1002:
999:
996:
993:
986:
981:
976:
973:
969:
942:
939:
936:
928:
925:
921:
917:
908:
904:
900:
897:
894:
891:
880:
877:
874:
865:
860:
856:
848:
847:
846:
845:
834:
820:
812:
759:
755:
752:
745:
744:
743:
741:
730:
728:
712:
698:
684:
677:. The symbol
676:
665:
645:
642:
616:
606:
600:
585:
566:
565:
564:
562:
558:
531:
521:
515:
496:
495:
494:
491:
475:
465:
462:
454:
444:
440:
436:
430:
429:
425:
420:This section
418:
414:
409:
408:
400:
396:
389:
383:
379:
374:
372:
367:
365:
361:
357:
356:refractometer
353:
349:
345:
341:
337:
333:
329:
325:
319:
315:
310:
300:
297:
282:
279:
271:
261:
257:
253:
247:
246:
241:This article
239:
230:
229:
220:
217:
209:
199:
195:
189:
186:This article
184:
175:
174:
165:
162:
154:
151:December 2008
143:
140:
136:
133:
129:
126:
122:
119:
115:
112: –
111:
107:
106:Find sources:
100:
96:
90:
89:
84:This article
82:
78:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
18:Final gravity
2820:. Retrieved
2815:
2810:Buhl, Josh.
2805:
2796:
2709:
2704:
2645:
2643:
2583:
2579:
2471:
2412:
2404:
2296:
2238:
2229:
2106:
1941:
1885:
1882:
1817:
1753:
1661:
1357:
1225:
960:
840:
808:
736:
704:
671:
663:
557:U-tube meter
554:
486:
472:
457:
448:
433:Please help
421:
390:
375:
371:fermentation
368:
323:
322:
292:
274:
265:
242:
212:
206:October 2009
203:
187:
157:
148:
138:
131:
124:
117:
105:
93:Please help
88:verification
85:
61:
54:
48:
47:Please help
44:
2771:Plato scale
2749:Wine portal
2735:Beer portal
2235:Attenuation
1793:0.000010788
727:Plato scale
675:Plato scale
378:Plato scale
314:thermometer
2838:Categories
2822:12 October
2777:References
1201:0.00001561
395:Stammwürze
360:pycnometer
352:hydrometer
252:improve it
121:newspapers
50:improve it
2844:Equations
2682:∘
2678:−
2614:−
2600:∘
2562:−
2553:⋅
2530:−
2495:≈
2487:∘
2449:−
2381:−
2360:−
2343:≈
2328:−
2270:−
2198:−
2176:−
2065:−
1991:−
1977:≈
1920:−
1732:−
1631:−
1608:−
1580:ρ
1553:ρ
1546:−
1523:−
1507:−
1425:ρ
1395:−
1332:ρ
1305:ρ
1298:−
1275:−
1192:0.0024688
1183:≈
1160:−
1043:−
997:−
940:−
895:−
878:−
643:−
617:ρ
607:ρ
601:−
532:ρ
522:ρ
422:does not
326:, in the
256:verifying
56:talk page
2721:See also
2711:Software
1784:0.001709
809:See the
638:apparent
596:apparent
451:May 2023
2849:Brewing
2715:BeerXML
2214:0.00753
2165:132.715
2090:0.79661
1866:0.79661
1778:0.39661
1186:0.48394
844:Balling
443:removed
428:sources
348:brewing
328:context
324:Gravity
250:Please
192:Please
135:scholar
2139:
1942:where
1611:1.0665
1605:2.0665
1382:) and
1358:where
1163:1.0665
1157:2.0665
1000:1.0665
994:2.0665
961:where
898:1.0665
892:2.0665
526:sample
386:
137:
130:
123:
116:
108:
2674:260.4
2670:260.4
2619:260.4
2611:260.4
1584:water
1414:Plato
1412:(see
1380:Plato
1336:water
811:Plato
778:recon
764:recon
740:Plato
621:water
536:water
142:JSTOR
128:books
2824:2023
2707:1).
2565:250.
2519:1000
2438:1000
2085:beer
1980:1000
1861:beer
1557:EtOH
1429:EtOH
1309:EtOH
790:beer
581:true
511:true
490:ASBC
426:any
424:cite
388:°P.
382:Brix
344:must
340:wort
318:beer
114:news
2550:250
2346:100
2316:100
2309:ADF
2258:100
2251:RDF
2041:250
1622:100
1247:as
1174:100
1011:100
909:100
611:air
437:by
362:or
342:or
254:by
196:to
97:by
2840::
2814:.
2785:^
2526:SG
2445:SG
2435::=
2377:OG
2364:FG
2356:OG
2202:FG
2194:OG
2180:FG
2172:OG
2080:SG
2069:FG
2061:OG
1987:SG
1856:SG
1815:.
785:SG
773:SG
633:SG
591:SG
576:SG
506:SG
358:,
354:,
312:A
59:.
2826:.
2687:P
2665:=
2662:G
2659:S
2646:T
2626:G
2623:S
2608:=
2605:P
2559:G
2556:S
2547:=
2544:4
2540:/
2536:)
2533:1
2522:(
2516:=
2513:4
2509:/
2503:t
2499:p
2492:P
2468:,
2455:)
2452:1
2441:(
2430:t
2426:p
2387:)
2384:1
2373:(
2368:)
2352:(
2338:p
2334:)
2331:m
2325:p
2322:(
2313:=
2280:p
2276:)
2273:n
2267:p
2264:(
2255:=
2210:/
2206:)
2190:(
2187:=
2184:)
2168:(
2162:=
2157:v
2153:A
2123:m
2120:p
2116:f
2073:)
2057:(
2052:m
2049:p
2045:f
2038:=
2033:v
2029:A
2005:4
2001:/
1997:)
1994:1
1983:(
1974:p
1950:k
1926:)
1923:m
1917:p
1914:(
1911:k
1908:=
1903:v
1899:A
1847:w
1843:A
1839:=
1834:v
1830:A
1801:2
1797:p
1790:+
1787:p
1781:+
1775:=
1770:m
1767:p
1763:f
1738:)
1735:m
1729:p
1726:(
1721:m
1718:p
1714:f
1710:=
1705:w
1701:A
1675:w
1671:A
1647:0
1644:=
1639:w
1635:A
1625:)
1618:/
1614:p
1602:(
1597:]
1592:)
1575:)
1570:w
1566:A
1562:(
1543:1
1540:+
1537:)
1534:m
1531:(
1526:1
1519:P
1514:(
1510:P
1504:p
1500:[
1472:w
1468:A
1447:)
1442:w
1438:A
1434:(
1398:1
1391:P
1366:P
1343:)
1327:)
1322:w
1318:A
1314:(
1295:1
1292:+
1289:)
1286:m
1283:(
1278:1
1271:P
1267:(
1264:P
1261:=
1258:n
1235:m
1209:2
1205:p
1198:+
1195:p
1189:+
1177:)
1170:/
1166:p
1154:(
1150:1
1145:=
1140:n
1137:p
1133:f
1107:n
1104:p
1100:f
1077:n
1074:p
1070:f
1049:)
1046:n
1040:p
1037:(
1014:)
1007:/
1003:p
991:(
987:1
982:=
977:n
974:p
970:f
946:)
943:n
937:p
934:(
929:n
926:p
922:f
918:=
912:)
905:/
901:p
889:(
884:)
881:n
875:p
872:(
866:=
861:w
857:A
821:n
760:P
756:=
753:n
713:m
685:p
649:)
646:1
628:(
586:=
516:=
464:)
458:(
453:)
449:(
445:.
431:.
299:)
293:(
281:)
275:(
270:)
266:(
248:.
219:)
213:(
208:)
204:(
190:.
164:)
158:(
153:)
149:(
139:·
132:·
125:·
118:·
91:.
66:)
62:(
20:)
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