Club Certificate2025
LillefixNOR 9280
Norges Seilforbund
Sognsveien 73
0854 OSLO
Rated boat velocities in knots
Wind Velocity 4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Beat Angles42.3°42.3°40.4°38.9°38.0°37.9°37.7°37.8°38.6°
Beat VMG2.853.874.625.085.295.405.465.505.45
52°4.405.836.747.227.457.557.617.667.64
60°4.666.096.947.387.617.737.807.877.89
75°4.836.257.067.507.767.958.088.218.31
90°4.686.127.007.487.798.038.248.548.71
110°4.215.766.907.537.918.208.418.728.87
120°4.075.606.757.457.888.248.569.099.47
135°3.605.046.237.117.658.058.429.2110.21
150°2.984.285.456.447.197.678.058.789.66
Run VMG2.583.714.725.596.336.947.418.128.79
Gybe Angles145.1°145.1°149.6°152.5°157.9°164.6°170.5°177.2°176.3°
APH ToD:534.2CDL: 9.965APH ToT: 1.1231CertNo:7062
BOAT
ClassX-412DesignerNielsJeppesenBuilderX-yachtsAge dateSeries date01/1990Offset fileIX412.OFFData file7062
HULL
Length Overall12.476 mMaximum Beam3.858 mDraft2.208 mDisplacement7,599 kgDLR8.0838IMS DivisionCruiser/RacerDynamic Allowance0.030%Age Allowance0.487%
PROPELLER
InstallationStrutTypeFolding 2 bladesDiameter0.440m
CREW
Maximum weight668 kg(default)Minimum weight538 kg ✱ when appliedNon Manual PowerNoCrew Arm Extension
SAIL AREAS (m²)
MeasuredRatedMainsail39.3839.98Headsail Luffed58.7958.79Headsail FlyingSymmetric120.52120.52Asymmetric
STORM SAIL AREAS (m²)
Trysail11.85Storm Jib14.31Heavy Weather Jib38.64
SAIL LIMITS
Headsails6Spinnakers5
STABILITY
Righting MomentN/AStability IndexN/A
The owner and any other person in charge is responsible that boat is complying with her certificate in accordance with RRS 78.1 and ORC Rule 304.
Club Certificate2025
LillefixNOR 9280
Norges Seilforbund
Sognsveien 73
0854 OSLO
Time Allowances in secs/NM
Wind Velocity 4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Beat VMG1264.3930.5779.7709.1680.8667.0659.3654.8660.8
52°818.4618.0533.8498.6483.5476.9473.2470.2471.1
60°771.8591.1518.5487.5472.9465.5461.5457.5456.2
75°745.5575.9509.7480.2463.7452.9445.7438.4433.2
90°769.9588.4514.5481.1462.1448.1437.1421.7413.5
110°855.7625.2521.8478.2455.4439.1428.2412.7405.9
120°883.7642.9533.0483.1456.8437.1420.4396.2380.3
135°999.8713.9577.5506.2470.5447.2427.4390.9352.5
150°1207.3840.7660.9559.2500.7469.1447.3410.0372.8
Run VMG1394.2970.8763.1644.2568.5519.0485.9443.1409.4
Selected Courses
Windward / Leeward1327.7950.7771.4676.7624.6593.0572.6549.0535.1
All purpose1006.5735.6611.2549.5517.2497.6484.4466.6454.4
Single Number Scoring Options
CourseTime On
Distance
Time On
Time
Windward / Leeward650.70.9221
All purpose534.21.1231
 
Custom scoring options for Australia
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Custom scoring options for Austria
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Brazil
Scoring OptionTime On
Time
Upwind "bolina"0.8266
Downwind "popa"0.9557
70% up, 30% down "Ilhabela"0.8615
30% up, 70% down "Ilhabela-inverso"0.9129
Custom scoring options for Bulgaria
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Predominantly Upwind571.61.0497
Predominantly Downwind545.81.0994
Custom scoring options for Canada
Scoring OptionTODTOT
Triple Number AP Low673.40.8910
Triple Number AP Medium531.01.1300
Triple Number AP High481.01.2474
Triple Number Windward/Leeward Low898.40.6679
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Single Number Predominant Upwind592.21.0132
Single Number Predominant Reaching532.21.1274
Single Number Predominant Downwind569.41.0537
Predominant Upwind - Low789.30.7602
Predominant Upwind - Low/Med617.90.9710
Predominant Upwind - Medium555.61.0800
Predominant Upwind - Med/High525.11.1426
Predominant Upwind - High512.21.1714
Predominant Downwind Low824.90.7273
Predominant Downwind Low/Med615.40.9749
Predominant Downwind Medium527.41.1377
Predominant Downwind Med/High473.61.2668
Predominant Downwind High442.91.3548
Predominant Reaching Low720.70.8325
Predominant Reaching Low/Med561.71.0682
Predominant Reaching Medium500.21.1996
Predominant Reaching Med/High464.11.2928
Predominant Reaching High442.91.3546
SA/DSPL Upwind25.23SA/DSPL Downwind41.09
Scoring OptionTODTOT
5-Band Windward/Leeward Low1000.50.5997
5-Band Windward/Leeward Low/Med742.70.8078
5-Band Windward/Leeward Medium635.80.9436
5-Band Windward/Leeward Med/High576.51.0408
5-Band Windward/Leeward High550.71.0896
Windward/Leeward 60-40 Low995.60.6026
Windward/Leeward 60-40 Low/Med746.70.8036
Windward/Leeward 60-40 Medium646.60.9279
Windward/Leeward 60-40 Med/High593.51.0109
Windward/Leeward 60-40 High571.91.0492
5-Band AP Low745.40.8049
5-Band AP Low/Med593.91.0103
5-Band AP Medium524.31.1443
5-Band AP Med/High486.41.2337
5-Band AP High467.61.2833
5-Band SF Bay Tour - Low862.90.6954
5-Band SF Bay Tour - Low/Med709.80.8453
5-Band SF Bay Tour - Medium614.70.9761
5-Band SF Bay Tour - Med/High563.81.0641
5-Band SF Bay Tour - High540.81.1095
Chicago-Mac Upwind0.9860
Chicago-Mac All Purpose1.0048
Chicago-Mac Downwind1.0168
Bayview-Mac Cove Island0.7474
Bayview-Mac Shore0.7540
Harvest Moon Regatta470.21.2760
Custom scoring options for Cyprus
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Denmark
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Spain
Scoring OptionTime On
Distance
Coastal/Long Distance579.6
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Estonia
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Custom scoring options for Finland
Scoring OptionTime On
Distance
Time On
Time
FinRating579.61.0352
FinRating, kevyt tuuli984.11.0161
FinRating, kova tuuli470.81.0621
Custom scoring options for France
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Custom scoring options for Germany
Scoring OptionTime On
Time
Coastal/Long Distance1.0352
Triple Number Coastal/Long Distance Low0.8910
Triple Number Coastal/Long Distance Medium1.1301
Triple Number Coastal/Long Distance High1.2474
Triple Number Windward/Leeward Low0.6968
Triple Number Windward/Leeward Medium0.9284
Triple Number Windward/Leeward High1.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
For information
General Purpose Handicap (GPH)596.6
Custom scoring options for Greece
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Hungary
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Custom scoring options for Israel
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Japan
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Korea
Scoring OptionTime On
Distance
Triple Number Coastal/Long Distance Low673.4
Triple Number Coastal/Long Distance Medium530.9
Triple Number Coastal/Long Distance High481.0
Triple Number Windward/Leeward Low861.1
Triple Number Windward/Leeward Medium646.3
Triple Number Windward/Leeward High568.9
Predominantly Upwind571.6
Predominantly Downwind545.8
Predominantly Reaching523.1
Custom scoring options for Lithuania
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Netherlands
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8910
Triple Number Coastal/Long Distance Medium1.1301
Triple Number Coastal/Long Distance High1.2474
Triple Number Windward/Leeward Low0.6968
Triple Number Windward/Leeward Medium0.9284
Triple Number Windward/Leeward High1.0547
Coastal/Long Distance1.0352
Predominantly Upwind1.0497
Predominantly Downwind1.0994
For information
General Purpose Handicap (GPH)596.6
Custom scoring options for Norway
Scoring OptionTime On
Time
Distanseseilas Singeltall  (APH)1.1231
Distanseseilas Trippeltall svak vind0.8910
Distanseseilas Trippeltall mellomvind1.1301
Distanseseilas Trippeltall sterk vind1.2474
Pølsebane Trippeltall svak vind0.6968
Pølsebane Trippeltall mellomvind0.9284
Pølsebane Trippeltall sterk vind1.0547
Motvind Singeltall1.0497
Medvind Singeltall1.0994
Custom scoring options for Portugal
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Coastal/Long Distance579.61.0352
Custom scoring options for Russia
Scoring OptionTime On
Distance
Time On
Time
Predominantly Upwind571.61.0497
Predominantly Downwind545.81.0994
Custom scoring options for South Africa
Scoring OptionTime On
Distance
Time On
Time
Cape Doctor Inshore539.31.1126
Cape Doctor Coastal458.11.3099
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Custom scoring options for Slovenia
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8910
Triple Number Coastal/Long Distance Medium1.1301
Triple Number Coastal/Long Distance High1.2474
Triple Number Windward/Leeward Low0.6968
Triple Number Windward/Leeward Medium0.9284
Triple Number Windward/Leeward High1.0547
Custom scoring options for Switzerland
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8910
Triple Number Coastal/Long Distance Medium1.1301
Triple Number Coastal/Long Distance High1.2474
Triple Number Windward/Leeward Low0.6968
Triple Number Windward/Leeward Medium0.9284
Triple Number Windward/Leeward High1.0547
Custom scoring options for Sweden
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Predominantly Upwind571.61.0497
Predominantly Downwind545.81.0994
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for Ukraine
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance579.61.0352
Triple Number Coastal/Long Distance Low673.40.8910
Triple Number Coastal/Long Distance Medium530.91.1301
Triple Number Coastal/Long Distance High481.01.2474
Triple Number Windward/Leeward Low861.10.6968
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1415.5952.7728.5614.4550.3516.0489.5448.6436.1
Custom scoring options for United States of America
Scoring OptionTODTOT
Triple Number AP Low673.40.8910
Triple Number AP Medium531.01.1300
Triple Number AP High481.01.2474
Triple Number Windward/Leeward Low898.40.6679
Triple Number Windward/Leeward Medium646.30.9284
Triple Number Windward/Leeward High568.91.0547
Single Number Predominant Upwind592.21.0132
Single Number Predominant Reaching532.21.1274
Single Number Predominant Downwind569.41.0537
Predominant Upwind - Low789.30.7602
Predominant Upwind - Low/Med617.90.9710
Predominant Upwind - Medium555.61.0800
Predominant Upwind - Med/High525.11.1426
Predominant Upwind - High512.21.1714
Predominant Downwind Low824.90.7273
Predominant Downwind Low/Med615.40.9749
Predominant Downwind Medium527.41.1377
Predominant Downwind Med/High473.61.2668
Predominant Downwind High442.91.3548
Predominant Reaching Low720.70.8325
Predominant Reaching Low/Med561.71.0682
Predominant Reaching Medium500.21.1996
Predominant Reaching Med/High464.11.2928
Predominant Reaching High442.91.3546
SA/DSPL Upwind25.23SA/DSPL Downwind41.09
Scoring OptionTODTOT
5-Band Windward/Leeward Low1000.50.5997
5-Band Windward/Leeward Low/Med742.70.8078
5-Band Windward/Leeward Medium635.80.9436
5-Band Windward/Leeward Med/High576.51.0408
5-Band Windward/Leeward High550.71.0896
Windward/Leeward 60-40 Low995.60.6026
Windward/Leeward 60-40 Low/Med746.70.8036
Windward/Leeward 60-40 Medium646.60.9279
Windward/Leeward 60-40 Med/High593.51.0109
Windward/Leeward 60-40 High571.91.0492
5-Band AP Low745.40.8049
5-Band AP Low/Med593.91.0103
5-Band AP Medium524.31.1443
5-Band AP Med/High486.41.2337
5-Band AP High467.61.2833
5-Band SF Bay Tour - Low862.90.6954
5-Band SF Bay Tour - Low/Med709.80.8453
5-Band SF Bay Tour - Medium614.70.9761
5-Band SF Bay Tour - Med/High563.81.0641
5-Band SF Bay Tour - High540.81.1095
Chicago-Mac Upwind0.9860
Chicago-Mac All Purpose1.0048
Chicago-Mac Downwind1.0168
Bayview-Mac Cove Island0.7474
Bayview-Mac Shore0.7540
Harvest Moon Regatta470.21.2760
Club Certificate2025
LillefixNOR 9280
Norges Seilforbund
Sognsveien 73
0854 OSLO
Data in meters/kilograms (Metric)
HULL AND APPENDAGES (Lightship Trim)
ClassX-412MeasurementHINPlan reviewHull constructionCoredCarbon rudderNoLight stanchionsNoTrim tabNo
LOA12.476Max. Beam3.858Draft2.208Displacement7,599Wetted area29.50IMS L10.309LSM09.795Acc. length12.476Sink (kg/mm) 21.41
VCGD-0.053VCGM0.004RM Rated (kg·m) 172.1Limit of positive stability(°) N/AStability IndexN/ADefault crew weight668Declared crew weightN/A
PROPELLER
Propeller TypeFolding 2 blades
InstallationStrutTwin screwNoHydro generatorNo
PRD0.440PBWPIPA0.0033
EDL2.700ST10.050ST20.170
ST30.170ST40.100ST50.300
RIG
Forestay tensionAftInner stayNone FittedMast materialSingle headsail furlerNoMainsail furlerArticulated bowspritNoNon-circular riggingNoFiber riggingNoAdjustable mast footNoRunning backstays1CheckstaysSpreaders3Rotating mastNo
P14.400IG16.500ISP16.500MDT10.120MDL10.210MDT20.100MDL20.180TL1.520MW0.180GO0.300
E4.700J4.880BAS1.830FSD0.030SFJ0.000SPL4.978WPLTPS0.000BD0.000MWT204.00MCG5.500
Flying Headsail Foretriangles IdISPTPS
FLOTATION AND STABILITY
Calculation methodVCG entered directlyVCG entered-0.053Flotation Date01/01/2015
BALLAST
TANKS
INVENTORY
INVENTORY
Fixed Ballast Total0
Total deductible
Total deductible
Club Certificate2025
LillefixNOR 9280
Norges Seilforbund
Sognsveien 73
0854 OSLO
MAINSAIL
IdMHBMUWMTWMHWMQWAreaMeas.DateComment394-2017122112530.180.951.692.943.8639.3821/12/2017
HEADSAIL
IdHHBHUWHTWHHWHQWHLPHLUBtnFlyingFTAreaMeas.DateComment394-2021050514090.100.841.653.395.297.2916.72NoNo 58.7905/05/2021398-2013042211000.070.671.282.483.674.7816.70NoNo 41.0122/04/2013400-2007042017170.000.000.000.000.004.1415.41NoNo 32.4720/04/2007
SYMMETRIC SPINNAKER
IdSLUSLESLSHWSFLAreaMeas.DateComment376-20041124093016.1416.1416.140.000.00120.5224/11/2004398-20130422111116.3016.3016.308.808.60118.9922/04/2013
CRUISER/RACER REQUIREMENTS*
Accommodation length12.5 mTable area0.55 m²Headroom1.83 mMinimum number of berths5Personal gear stowage0.20 m³Fresh water capacity162 lMinimum number of seats in cockpit5Food stowage0.30 m³Fuel capacity100 lCabin soleStovesSinksGalley gear stowageToiletWash basinNavigation areaHanging locker
*This list is a summary of the requirements. See IMS Appendix 1 for details.