Double HandedClub Certificate2025
BRAINSTORMNED 3338
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 Angles42.6°42.6°40.8°40.2°39.6°39.2°39.2°39.3°40.6°
Beat VMG2.813.784.454.835.015.105.135.155.02
52°4.315.706.577.027.217.317.357.407.33
60°4.555.966.807.227.427.527.567.637.61
75°4.706.126.957.377.627.787.878.008.07
90°4.576.006.897.387.687.908.088.348.49
110°4.155.686.807.427.788.018.188.418.71
120°4.035.546.677.377.798.138.438.809.05
135°3.585.016.197.067.597.998.359.109.96
150°2.974.265.426.407.157.638.008.749.60
Run VMG2.573.694.695.566.306.907.368.088.76
Gybe Angles145.3°145.3°149.8°152.6°157.9°164.8°170.5°177.0°175.9°
APH ToD:547.9CDL: 9.391APH ToT: 1.0951CertNo:3609
BOAT
ClassX-412DesignerN. JeppesenBuilderX-YachtsAge date01/1992Series date01/1990Offset filed0006_a.offData fileNED3338
HULL
Length Overall12.512 mMaximum Beam3.916 mDraft2.081 mDisplacement8,013 kgDLR8.0792IMS DivisionCruiser/RacerDynamic Allowance0.130%Age Allowance0.487%
PROPELLER
InstallationStrutTypeFeathering 2 bladesDiameter0.420m
CREW
Maximum weight170 kg(default)Minimum weight120 kg ✱ when appliedNon Manual PowerNoCrew Arm Extension
SAIL AREAS (m²)
MeasuredRatedMainsail40.2841.01Headsail Luffed57.1457.14Headsail FlyingSymmetric117.54117.54Asymmetric
STORM SAIL AREAS (m²)
Trysail11.86Storm Jib13.80Heavy Weather Jib37.26
SAIL LIMITS
Headsails5Spinnakers4
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.
Double HandedClub Certificate2025
BRAINSTORMNED 3338
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 VMG1281.8952.5808.3746.1719.1706.3701.9699.3717.3
52°834.4632.0547.8513.0499.3492.7490.1486.8491.3
60°791.4604.5529.1498.6484.9478.7476.0471.6472.9
75°765.6588.5518.3488.4472.7462.9457.3450.1446.4
90°788.0600.1522.7487.7468.6455.5445.5431.8424.1
110°867.8634.1529.4485.0462.4449.2440.0427.9413.5
120°892.2650.0539.5488.6462.0442.8427.1409.3397.9
135°1005.2718.6581.9510.3474.3450.8430.9395.8361.4
150°1212.7844.6664.1562.4503.8472.1449.8411.9374.9
Run VMG1400.4975.3766.9647.6571.4521.9488.8445.3410.9
Selected Courses
Windward / Leeward1339.0963.9787.6696.9645.3614.1595.4572.3564.1
All purpose1019.4746.9622.9562.7530.7511.9499.9483.3474.2
Single Number Scoring Options
CourseTime On
Distance
Time On
Time
Windward / Leeward670.90.8944
All purpose547.91.0951
 
Custom scoring options for Australia
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Custom scoring options for Austria
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Brazil
Scoring OptionTime On
Time
Upwind "bolina"0.7875
Downwind "popa"0.9508
70% up, 30% down "Ilhabela"0.8303
30% up, 70% down "Ilhabela-inverso"0.8951
Custom scoring options for Bulgaria
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Predominantly Upwind590.31.0165
Predominantly Downwind553.71.0835
Custom scoring options for Canada
Scoring OptionTODTOT
Triple Number AP Low684.90.8761
Triple Number AP Medium544.41.1021
Triple Number AP High496.71.2081
Triple Number Windward/Leeward Low913.20.6570
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Single Number Predominant Upwind611.10.9819
Single Number Predominant Reaching542.61.1059
Single Number Predominant Downwind577.81.0384
Predominant Upwind - Low804.20.7461
Predominant Upwind - Low/Med635.00.9449
Predominant Upwind - Medium574.61.0441
Predominant Upwind - Med/High546.11.0988
Predominant Upwind - High535.11.1213
Predominant Downwind Low833.30.7200
Predominant Downwind Low/Med623.30.9626
Predominant Downwind Medium535.51.1205
Predominant Downwind Med/High482.21.2442
Predominant Downwind High452.51.3261
Predominant Reaching Low731.50.8202
Predominant Reaching Low/Med571.31.0502
Predominant Reaching Medium510.01.1766
Predominant Reaching Med/High474.91.2635
Predominant Reaching High455.21.3182
SA/DSPL Upwind24.16SA/DSPL Downwind39.14
Scoring OptionTODTOT
5-Band Windward/Leeward Low1014.10.5917
5-Band Windward/Leeward Low/Med760.50.7889
5-Band Windward/Leeward Medium656.30.9142
5-Band Windward/Leeward Med/High598.81.0020
5-Band Windward/Leeward High574.91.0437
Windward/Leeward 60-40 Low1011.00.5935
Windward/Leeward 60-40 Low/Med767.30.7819
Windward/Leeward 60-40 Medium670.50.8948
Windward/Leeward 60-40 Med/High619.80.9681
Windward/Leeward 60-40 High600.50.9992
5-Band AP Low756.20.7935
5-Band AP Low/Med606.30.9896
5-Band AP Medium538.01.1153
5-Band AP Med/High501.71.1960
5-Band AP High484.61.2382
5-Band SF Bay Tour - Low876.90.6842
5-Band SF Bay Tour - Low/Med727.70.8245
5-Band SF Bay Tour - Medium635.30.9444
5-Band SF Bay Tour - Med/High586.51.0231
5-Band SF Bay Tour - High565.61.0608
Chicago-Mac Upwind0.9562
Chicago-Mac All Purpose0.9825
Chicago-Mac Downwind1.0027
Bayview-Mac Cove Island0.7343
Bayview-Mac Shore0.7406
Harvest Moon Regatta478.11.2551
Custom scoring options for Cyprus
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Denmark
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Spain
Scoring OptionTime On
Distance
Coastal/Long Distance594.7
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Estonia
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Custom scoring options for Finland
Scoring OptionTime On
Distance
Time On
Time
FinRating (Coastal/Long Distance)594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for France
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Custom scoring options for Germany
Scoring OptionTime On
Time
Coastal/Long Distance1.0089
Triple Number Coastal/Long Distance Low0.8761
Triple Number Coastal/Long Distance Medium1.1022
Triple Number Coastal/Long Distance High1.2081
Triple Number Windward/Leeward Low0.6852
Triple Number Windward/Leeward Medium0.9002
Triple Number Windward/Leeward High1.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
For information
General Purpose Handicap (GPH)610.6
Custom scoring options for Greece
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Hungary
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Custom scoring options for Israel
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Japan
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Korea
Scoring OptionTime On
Distance
Triple Number Coastal/Long Distance Low684.9
Triple Number Coastal/Long Distance Medium544.4
Triple Number Coastal/Long Distance High496.6
Triple Number Windward/Leeward Low875.7
Triple Number Windward/Leeward Medium666.5
Triple Number Windward/Leeward High591.4
Predominantly Upwind590.3
Predominantly Downwind553.7
Predominantly Reaching531.9
Custom scoring options for Lithuania
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Netherlands
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8761
Triple Number Coastal/Long Distance Medium1.1022
Triple Number Coastal/Long Distance High1.2081
Triple Number Windward/Leeward Low0.6852
Triple Number Windward/Leeward Medium0.9002
Triple Number Windward/Leeward High1.0145
Coastal/Long Distance1.0089
Predominantly Upwind1.0165
Predominantly Downwind1.0835
For information
General Purpose Handicap (GPH)610.6
Custom scoring options for Norway
Scoring OptionTime On
Time
Distanseseilas Singeltall  (APH)1.0951
Distanseseilas Trippeltall svak vind0.8761
Distanseseilas Trippeltall mellomvind1.1022
Distanseseilas Trippeltall sterk vind1.2081
Pølsebane Trippeltall svak vind0.6852
Pølsebane Trippeltall mellomvind0.9002
Pølsebane Trippeltall sterk vind1.0145
Motvind Singeltall1.0165
Medvind Singeltall1.0835
Custom scoring options for Portugal
Scoring OptionTime On
Distance
Time On
Time
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Coastal/Long Distance594.71.0089
Custom scoring options for Russia
Scoring OptionTime On
Distance
Time On
Time
Predominantly Upwind590.31.0165
Predominantly Downwind553.71.0835
Custom scoring options for South Africa
Scoring OptionTime On
Distance
Time On
Time
Cape Doctor Inshore566.61.0590
Cape Doctor Coastal476.91.2581
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Custom scoring options for Slovenia
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8761
Triple Number Coastal/Long Distance Medium1.1022
Triple Number Coastal/Long Distance High1.2081
Triple Number Windward/Leeward Low0.6852
Triple Number Windward/Leeward Medium0.9002
Triple Number Windward/Leeward High1.0145
Custom scoring options for Switzerland
Scoring OptionTime On
Time
Triple Number Coastal/Long Distance Low0.8761
Triple Number Coastal/Long Distance Medium1.1022
Triple Number Coastal/Long Distance High1.2081
Triple Number Windward/Leeward Low0.6852
Triple Number Windward/Leeward Medium0.9002
Triple Number Windward/Leeward High1.0145
Custom scoring options for Sweden
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Predominantly Upwind590.31.0165
Predominantly Downwind553.71.0835
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for Ukraine
Scoring OptionTime On
Distance
Time On
Time
Coastal/Long Distance594.71.0089
Triple Number Coastal/Long Distance Low684.90.8761
Triple Number Coastal/Long Distance Medium544.41.1022
Triple Number Coastal/Long Distance High496.61.2081
Triple Number Windward/Leeward Low875.70.6852
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Performance Curve4 kt 6 kt 8 kt 10 kt 12 kt 14 kt 16 kt 20 kt 24 kt
Coastal/Long Distance1425.1965.0742.8630.8565.9530.8504.2462.2452.8
Custom scoring options for United States of America
Scoring OptionTODTOT
Triple Number AP Low684.90.8761
Triple Number AP Medium544.41.1021
Triple Number AP High496.71.2081
Triple Number Windward/Leeward Low913.20.6570
Triple Number Windward/Leeward Medium666.50.9002
Triple Number Windward/Leeward High591.41.0145
Single Number Predominant Upwind611.10.9819
Single Number Predominant Reaching542.61.1059
Single Number Predominant Downwind577.81.0384
Predominant Upwind - Low804.20.7461
Predominant Upwind - Low/Med635.00.9449
Predominant Upwind - Medium574.61.0441
Predominant Upwind - Med/High546.11.0988
Predominant Upwind - High535.11.1213
Predominant Downwind Low833.30.7200
Predominant Downwind Low/Med623.30.9626
Predominant Downwind Medium535.51.1205
Predominant Downwind Med/High482.21.2442
Predominant Downwind High452.51.3261
Predominant Reaching Low731.50.8202
Predominant Reaching Low/Med571.31.0502
Predominant Reaching Medium510.01.1766
Predominant Reaching Med/High474.91.2635
Predominant Reaching High455.21.3182
SA/DSPL Upwind24.16SA/DSPL Downwind39.14
Scoring OptionTODTOT
5-Band Windward/Leeward Low1014.10.5917
5-Band Windward/Leeward Low/Med760.50.7889
5-Band Windward/Leeward Medium656.30.9142
5-Band Windward/Leeward Med/High598.81.0020
5-Band Windward/Leeward High574.91.0437
Windward/Leeward 60-40 Low1011.00.5935
Windward/Leeward 60-40 Low/Med767.30.7819
Windward/Leeward 60-40 Medium670.50.8948
Windward/Leeward 60-40 Med/High619.80.9681
Windward/Leeward 60-40 High600.50.9992
5-Band AP Low756.20.7935
5-Band AP Low/Med606.30.9896
5-Band AP Medium538.01.1153
5-Band AP Med/High501.71.1960
5-Band AP High484.61.2382
5-Band SF Bay Tour - Low876.90.6842
5-Band SF Bay Tour - Low/Med727.70.8245
5-Band SF Bay Tour - Medium635.30.9444
5-Band SF Bay Tour - Med/High586.51.0231
5-Band SF Bay Tour - High565.61.0608
Chicago-Mac Upwind0.9562
Chicago-Mac All Purpose0.9825
Chicago-Mac Downwind1.0027
Bayview-Mac Cove Island0.7343
Bayview-Mac Shore0.7406
Harvest Moon Regatta478.11.2551
Double HandedClub Certificate2025
BRAINSTORMNED 3338
Data in meters/kilograms (Metric)
HULL AND APPENDAGES (Lightship Trim)
ClassX-412Measurement13/05/1992HINPlan reviewHull constructionCoredCarbon rudderNoLight stanchionsNoTrim tabNo
LOA12.512Max. Beam3.916Draft2.081Displacement8,013Wetted area30.01IMS L10.158LSM09.972Acc. length12.207Sink (kg/mm) 22.10
VCGD-0.062VCGM0.060RM Rated (kg·m) 174.8Limit of positive stability(°) N/AStability IndexN/ADefault crew weight170Declared crew weightN/A
PROPELLER
Propeller TypeFeathering 2 blades
InstallationStrutTwin screwNoHydro generatorNo
PRD0.420PBWPIPA0.0029
EDL2.670ST10.042ST2
ST3ST40.093ST50.317
RIG
Forestay tensionAftInner stayAdjustableMast materialAluminumSingle headsail furlerNoMainsail furlerNoArticulated bowspritNoNon-circular riggingNoFiber riggingNoAdjustable mast footNoRunning backstays1CheckstaysSpreaders3Rotating mastNo
P14.280IG16.425ISP16.345MDT10.110MDL10.215MDT20.100MDL20.180TL1.600MW0.180GO0.235
E4.742J4.895BAS1.733FSD0.022SFJ0.000SPL4.988WPLTPSBD0.190MWT0.00MCG0.000
Flying Headsail Foretriangles IdISPTPS
FLOTATION AND STABILITY
Calculation methodRM entered directlyRM entered177.2Flotation Date13/05/1992
BALLAST
TANKS
INVENTORY
INVENTORY
Fixed Ballast Total0
Total deductible
Total deductible
Double HandedClub Certificate2025
BRAINSTORMNED 3338
MAINSAIL
IdMHBMUWMTWMHWMQWAreaMeas.DateComment010.160.981.753.074.0040.28
HEADSAIL
IdHHBHUWHTWHHWHQWHLPHLUBtnFlyingFTAreaMeas.DateComment016.8016.51NoNo 57.14
SYMMETRIC SPINNAKER
IdSLUSLESLSHWSFLAreaMeas.DateCommentORC16.1216.1216.128.808.55117.54* Copied from legacy *
CRUISER/RACER REQUIREMENTS*
Accommodation length12.2 mTable area0.55 m²Headroom1.82 mMinimum number of berths5Personal gear stowage0.20 m³Fresh water capacity155 lMinimum number of seats in cockpit5Food stowage0.30 m³Fuel capacity94 lCabin soleStovesSinksGalley gear stowageToiletWash basinNavigation areaHanging locker
*This list is a summary of the requirements. See IMS Appendix 1 for details.