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SPECIES:  Cynodon dactylon
Bermudagrass. Image by Doug Goldman, hosted by the USDA-NRCS PLANTS Database.


SPECIES: Cynodon dactylon
AUTHORSHIP AND CITATION: Carey, Jennifer H. 1995. Cynodon dactylon. In: Fire Effects Information System, [Online]. U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, Fire Sciences Laboratory (Producer). Available: []. Revisions: On 2 October 2018, the common name of this species was changed in FEIS from: Bermuda grass to: Bermudagrass. Images were also added.
ABBREVIATION: CYNDAC SYNONYMS: NO-ENTRY NRCS PLANT CODE: CYDA CYDAA CYDAD COMMON NAMES: Bermudagrass TAXONOMY: The currently accepted scientific name for Bermudagrass is Cynodon dactylon (L.) Pers. (Poaceae) [28,31,40,41]. Two varieties are recognized [77]: Cynodon dactylon var. dactylon (L.) Pers. Cynodon dactylon var. aridus Harlan & de Wet Numerous cultivars have been developed. Many of the studies cited in this report were conducted using Bermudagrass cultivars. The specific cultivar name is only mentioned here if it is compared to another cultivar. LIFE FORM: Graminoid FEDERAL LEGAL STATUS: No special status OTHER STATUS: NO-ENTRY


SPECIES: Cynodon dactylon
GENERAL DISTRIBUTION: Bermudagrass is native to Africa but occurs throughout the world in tropical to warm temperate climates between 45 degrees north and 45 degrees south latitude [58]. In the United States Bermudagrass is most common in the subtropical regions from southern California east to the Gulf Coast and southeastern states. It is adventive north to Washington, Idaho, Utah, Colorado, Iowa, Michigan, New York, Massachusetts, and New Hampshire [25,28,31,41]. Populations occurring in cool temperate climates may be winter hardy cultivars [5].
Distribution of Bermudagrass in the United States. Map courtesy of USDA, NRCS. 2018. The PLANTS Database. National Plant Data Team, Greensboro, NC. [2018, October 1] [77].
   FRES12  Longleaf-slash pine
   FRES13  Loblolly-shortleaf pine
   FRES14  Oak-pine
   FRES15  Oak-hickory
   FRES16  Oak-gum-cypress
   FRES17  Elm-ash-cottonwood
   FRES20  Douglas-fir
   FRES28  Western hardwoods
   FRES29  Sagebrush
   FRES30  Desert shrub
   FRES31  Shinnery
   FRES32  Texas savanna
   FRES33  Southwestern shrubsteppe
   FRES34  Chaparral-mountain shrub
   FRES35  Pinyon-juniper
   FRES36  Mountain grasslands
   FRES38  Plains grasslands
   FRES39  Prairie
   FRES40  Desert grasslands
   FRES41  Wet grasslands
   FRES42  Annual grasslands

     AL  AZ  AR  CA  CO  CT  DE  FL  GA  HI
     ID  IL  IN  IA  KS  KY  LA  MD  MA  MI
     MS  MO  NE  NV  NH  NJ  NM  NY  NC  OH
     OK  OR  PA  RI  SC  TN  TX  UT  VA  WA

    1  Northern Pacific Border
    3  Southern Pacific Border
    4  Sierra Mountains
    5  Columbia Plateau
    6  Upper Basin and Range
    7  Lower Basin and Range
   11  Southern Rocky Mountains
   12  Colorado Plateau
   13  Rocky Mountain Piedmont
   14  Great Plains


   Bermudagrass probably occurs on suitable sites within most
   SAF Cover Types that fall within its distribution.

   201  Blue oak woodland
   202  Coast live oak woodland
   203  Riparian woodland
   409  Tall forb
   422  Riparian
   717  Little bluestem-Indiangrass-Texas wintergrass
   718  Mesquite-grama
   719  Mesquite-liveoak-seacoast bluestem
   727  Mesquite-buffalograss
   728  Mesquite-granjeno-acacia
   729  Mesquite
   730  Sand shinnery oak
   731  Cross timbers-Oklahoma
   732  Cross timbers-Texas (little bluestem-post oak)
   801  Savanna
   804  Tall fescue
   807  Gulf Coast fresh marsh
   808  Sand pine scrub
   809  Mixed hardwood and pine
   810  Longleaf pine-turkey oak hills
   811  South Florida flatwoods
   812  North Florida flatwoods
   813  Cutthroat seeps
   815  Upland hardwood hammocks
   819  Freshwater marsh and ponds

In the southeastern United States, Bermudagrass occurs in pastures and
fields and in the understory of open woods, forests, orchards, and pine
(Pinus spp.) plantations.  In Georgia it occurs in a 15-year fallow
field with blackberry (Rubus spp.), American plum (Prunus americana),
sassafras (Sassafras albidum), smooth sumac (Rhus glabra), and numerous
herbaceous plants [48].  In South Carolina it occurs in an 8-year fallow
field dominated by broomsedge bluestem (Andropogon virginicus) and
paintbrush bluestem (A. ternarius) [30].  Bermudagrass occurs with
slender woodoats (Chasmanthium laxum var. sessiliflorum) and big
bluestem (Andropogon gerardi var. gerardi) in the herbaceous layer of a
pine-oak (Quercus spp.) forest in eastern Texas [88].

In the southwestern United States, Bermudagrass occurs in riparian
areas and in grasslands adjacent to streams and marshes.  It is a
frequently encountered understory grass in velvet mesquite (Prosopis
velutina) bosques [11].  A mixed honey mesquite (Prosopis glandulosa
var. glandulosa)-saltcedar (Tamarix ramosissima)-Bermudagrass
association has replaced some native associations in the Rio Grande
floodplain in Big Bend National Park, Texas [8].  On Santa Rosa Island,
California, Bermudagrass is a common understory plant in a riparian
woodland composed of black cottonwood (Populus trichocarpa), arroyo
willow (Salix lasiolepis), and goosefoot (Chenopodium spp.) [16].  In
the Sacramento River valley, California, Bermudagrass occurs in a
gravel bar thicket community dominated by sandbar willow (Salix exigua)
where the willow canopy is not dense [18].


SPECIES: Cynodon dactylon
IMPORTANCE TO LIVESTOCK AND WILDLIFE: Bermudagrass is eaten by livestock [74]. No information was found discussing beneficial or detrimental effects of Bermudagrass on wildlife. PALATABILITY: Bermudagrass is highly preferred by cattle [23]. NUTRITIONAL VALUE: Bermudagrass has good forage value for cattle [74], and is acceptable for sheep [59]. Bermudagrass, sampled in September in Oklahoma, contained 8.1 to 10.2 percent crude protein and was 41.6 to 44.4 percent digestible [9]. Nutritional contents (% dry matter) of Bermudagrass stems and leaves sampled from the Edwards Plateau region of Texas are as follows [44]: phosphorus crude protein digestible organic matter May 24 0.22% 12% 58% June 28 0.21% 12% 56% Crude protein was measured for four Bermudagrass cultivars in New Mexico. Coastal Bermudagrass had the highest crude protein values: 5.5, 7.5, and 7.4 percent for July, August, and October, respectively. [52]. COVER VALUE: NO-ENTRY VALUE FOR REHABILITATION OF DISTURBED SITES: Bermudagrass is commonly used to revegetate lignite surface mine spoils in the southeastern and Gulf Coast states [37,69,81]. It provides good initial erosion control as well as high forage quality. However, its use is limited because optimal establishment requires planting by sprig rather than by seed. Seeding rates and sprig spacings are described [8]. Bermudagrass planted on uranium mine spoils should not be used for forage because of potentially high plant selenium concentrations [43]. Bermudagrass declines on mine spoils if not fertilized on a yearly basis. It may be a good species for initial erosion control followed by later replacement with lower maintenance plants [69,81]. Harris and Zuberer [37] found that Bermudagrass production increased when grown with subterranean clover (Trifolium subterranean) inoculated with rhizobia (Rhizobium spp.). The clover grows during the winter months and increases soil nitrogen which the Bermudagrass then utilizes during the summer growing season [37]. Bermudagrass increases streambank substrate stability during floods; it grows weel in sand and resists scouring [55]. In Arizona riparian areas, Bermudagrass enhanced postflood development of aquatic macrophyte communities [19]. OTHER USES AND VALUES: Bermudagrass is used as a turf grass for lawns, athletic fields, and golf courses [64,75,81]. OTHER MANAGEMENT CONSIDERATIONS: Bermudagrass is a widely planted turf, hay, and pasture grass in the southern United States. Many cultivars have been developed for increased drought resistance, cold hardiness, disease resistance, and forage production. Quisenberry [64] reviewed the research conducted in the southeastern United States relevant to the resistance of Bermuda grass cultivars to insects and mites. Bermudagrass requires regular fertilizing to maintain high yields and turf quality. Bermudagrass pastures can be safely and adequately fertilized with municipal sewage sludge [51]. Bermudagrass is considered a weed in corn, alfalfa, citrus, grape, cotton, sugarcane, and other crops, as well as in landscaping and nonBermudagrass lawns [33]. Bermudagrass is a troublesome weed in native plant restoration projects [1,35]. In Everglades National Park in Florida, Bermudagrass primarily colonizes disturbed sites and is not considered a threat to native vegetation [86]. No information was found concerning the ability of Bermudagrass to invade and outcompete undisturbed native vegetation. Bermudagrass is difficult to eradicate without herbicides. Numerous herbicides have been tested on Bermudagrass and its various cultivars. Herbicide application rates and effectiveness are described [4,22,46,47,66]. Soil solarization is only partially effective at killing Bermudagrass [3]. The phytotoxins of several fungi which utilize Bermudagrass have been isolated. Investigations of their possible use as a control are ongoing [72]. Bermudagrass is widely used in timber pastures. Timber pastures are usually fertilized annually. Bermudagrass is tolerant of competition from a periodically thinned pine overstory. However, pine litter restricts Bermudagrass reproduction by stolons. Use of prescribed burning to enhance Bermudagrass by removing litter without damaging young pines is described [17]. Shade reduces Bermudagrass forage yields, but as long as pines are small and spaced so that direct sunlight reaches grass during a portion of each day, yields are satisfactory [14]. Bermudagrass is suspected of having allelopathic qualities [54,84]. It inhibited the growth of newly planted peach (Prunus persica) [84]. Bermudagrass produces cyanogenic compounds [59].


SPECIES: Cynodon dactylon
GENERAL BOTANICAL CHARACTERISTICS: Bermudagrass is an introduced, perennial, mat-forming, warm season grass. It is both rhizomatous and stoloniferous. Erect or ascending culms grow 0.3 to 1.3 feet (0.1-0.4 m) tall. The panicle has two to seven digitate branches [40]. Rhizomes are hard, scaly, and 0.06 to 0.13 inch (1.5-3.3 mm) in diameter. Stolons are flattened and several feet long, rooting at nodes [33]. Main root length per plant of four cultivars ranged from 2.3 to 37.1 feet (0.7-11.3 m) while total root length ranged from 43 to 4,854 feet (13.0-1,480 m). Root hairs contributed 64 to 95 percent of the total root length [32]. RAUNKIAER LIFE FORM: Geophyte Hemicryptophyte REGENERATION PROCESSES: Although Bermudagrass reproduces by seeds, it spreads most rapidly by stolons and rhizomes. Both stolon and rhizome branching intensities were reduced in response to lower light and lower nutrient levels. Average stolon and rhizome internode and total lengths are reported under differing light and nutrient conditions [21]. In a study of six Bermudagrass variants present in southern Africa, vegetative reproduction was greater by rhizomes than by stolons [26]. Seeds, eaten by animals, are widely dispersed. Bermudagrass seeds present in domestic sheep dung germinated in "large numbers" [39]. Fernald [25] stated that seeds are rarely perfect. Seed viability of the six variants from southern Africa ranged from 0 to 3.5 percent [26]. Germination of viable seeds is low unless scarification occurs. Seeds treated with sulfuric acid for 10 minutes had 68 percent germination after 4 days, but untreated seeds had only 4.5 percent germination after 10 days [12]. Prolonged exposure to acidic conditions decreases seed germination. Bermudagrass seeds did not germinate in sulfuric acid solutions of pH 3 or less. In pH 4 and 5 solutions, germination was about 5 and 10 percent, respectively, after 12 days [68]. Bermudagrass seeds were present in intact soil/litter samples collected for a germination study from an upland site in Arizona; seeds may have come from an aerial seeding of an adjacent property. Equal amounts of Bermudagrass seeds germinated in the control sample and in the scarified soil surface treatment. No seeds germinated from soil samples which had the litter manually removed or burned [29]. SITE CHARACTERISTICS: Although adaptable to most soil types, Bermudagrass grows best on fertile, sandy to silty soils or alluvium [75,81]. Bermudagrass occurs in regions that receive more than 16 inches (410 mm) of rainfall a year. In areas with less rainfall, it requires a surface source of water or irrigation [75]. Bermudagrass is classified as a facultative to facultative upland species [67]. In the southwestern United States, Bermudagrass occurs in irrigated areas and along streambanks [40,85]. Bermudagrass can expand a short distance into the upland by transferring water via stolons. In a laboratory study, Bermudagrass plants in separate moist and dry-soil compartments transferred water from one compartment to the other [79]. In Organ Pipe National Monument, Arizona, Bermudagrass occurs in damp areas but shows no tendency to spread [6]. Bermudagrass has deep roots and is capable of extending roots during drought stress. Ten cultivars distributed at least some roots 47 to 59 inches (120-150 cm) deep during a drought stress laboratory test. The bulk of the root mass was within the top 24 inches (60 cm) [38]. Bermudagrass is susceptible to cold temperatures, especially those occurring in the early winter. Anderson and others [2] studied the freeze tolerance of six cultivars grown in containers and held overnight at freezing temperatures. The temperature resulting in fifty percent mortality ranged from 15 degrees Fahrenheit (-9.6 deg C) to 18 degrees Fahrenheit (-7.7 deg C) for the six cultivars [2]. A winter hardy cultivar survived three winters in Morgantown, West Virginia, even though temperatures reached as low as -8 degrees Fahrenheit (-22 deg C) [53]. Bermudagrass is generally tolerant of low soil pH and high salt concentration. Six strains collected from southern Africa survived at soil pH of 2.7 [26]. Vogel [81] reported Bermudagrass growing in soil with pH as low as 3.2. Bermudagrass dry matter yields were unaffected by one growing season of irrigation with brackish water, but were reduced in the second season [61]. Although tolerant of salty soils [75,81], Bermudagrass does not appear to occur in saltwater wetlands. It occurs only in the freshwater vegetation type in the Louisiana coastal region [15]. Although common in the lower Sacramento River valley, Bermudagrass does not occur in the tidal streambank community [87]. In California Bermudagrass occurs below 2,950 feet (900 m) elevation [40]. In Colorado it occurs from 4,200 to 5,300 feet (1,280-1,620 m) elevation [20]. In Utah it occurs along waterways below 465 feet (1,525 m) [85]. SUCCESSIONAL STATUS: Bermudagrass is an early successional species. Shade reduces Bermuda grass vigor, and complete canopy closure may eliminate Bermudagrass [14]. It inhabits open locations subject to frequent disturbances such as grazing, flooding, and fire [21]. After a major flood in March on the Hassayampa River in Arizona, Bermudagrass cover increased to near preflood levels by September [73]. In a study of unreclaimed lignite mines, Bermudagrass was most frequent on recently abandoned sites. It was not present on sites 20 years old or older [70]. On the Rio Grande Valley National Wildlife Refuge in southern Texas, Rooseveltweed (Baccharis neglecta), buffel grass (Pennisetum ciliare), and Bermudagrass were the dominant species after 5 years of old-field succession. The two grasses may have inhibited the establishment of other species by successfully competing for moisture and light [83]. In central Utah, Bermudagrass was present in young saltcedar communities (age 2 to 3.1 years) but was absent from older communities, possibly because saltcedar lowers the water table [10]. Horton [42] observed that spaces between individual saltcedar are usually dominated by Bermudagrass or salt grass (Distichlis spicata) if the water table is 5 feet (1.5 m) deep or less. SEASONAL DEVELOPMENT: Bermudagrass begins growth late in the spring, continues to grow during the hot summer months, and becomes dormant when the weather cools in the fall [76]. Near Bakersfield, California, Bermudagrass emerged when soil temperatures at a depth of 2 inches (5 cm) reached 63 degrees Fahrenheit (17 deg C) [49]. In Morgantown, West Virginia, growth did not begin until mid- to late May [53]. Bermudagrass flowers from July to October [25].


SPECIES: Cynodon dactylon
FIRE ECOLOGY OR ADAPTATIONS: In its native Africa, Bermudagrass occurs in grassland communities that regularly experience fire [5]. In North America, Bermudagrass has established in plant communities that experience fire such as grasslands and pine and oak forests. Grassland fires tend to burn quickly, consuming aboveground fuels but usually not heating the soil enough to damage rootstocks [82]. The ability of Bermudagrass to reproduce from rhizomes probably enables it to survive most fires [80]. FIRE REGIMES: Find fire regime information for the plant communities in which this species may occur by entering the species name in the FEIS home page under "Find Fire Regimes". POSTFIRE REGENERATION STRATEGY: Rhizomatous herb, rhizome in soil Secondary colonizer - on-site seed


SPECIES: Cynodon dactylon
IMMEDIATE FIRE EFFECT ON PLANT: Fire top-kills Bermudagrass but rhizomes probably remain undamaged except during severe fire that burns organic soil [80]. Cultivars that are strongly stoloniferous may be more damaged by fire than those that are predominantly rhizomatous [62]. Soil- or litter-stored Bermuda grass seed did not germinate after litter was removed by fire; seeds may have been destroyed by fire [29]. PLANT RESPONSE TO FIRE: Bermudagrass productivity and cover have both increased and decreased after early spring fires. Bermudagrass response depends on postfire moisture conditions and nutrient levels [34,56,57,63]. Four treatments (fertilized, burned in early April, burned and fertilized, and control) were applied to an Oklahoma grassland of prairie threeawn (Aristida oligantha), Bermudagrass, little bluestem (Schizachyrium scoparium), and paintbrush bluestem. Bermudagrass cover increased significantly (P<0.05) over control levels with fertilization and the burning-fertilizing combination, but increased only slightly with burning alone. The burning-fertilizing treatment resulted in slightly higher cover than the fertilized treatment [63]. A March 5 fire on a Georgia old field resulted in a decrease in Bermuda grass yield. The control produced 14.7 grams per square meter and the burned area produced 0.24 gram per square meter in the summer after the fire. Bermudagrass was a minor species on the site [60]. Postfire moisture conditions were not reported. No change was detected in Bermudagrass cover after a dormant season fire in a mid-grass community in Serengeti National Park, Tanzania, Africa [5]. Spring burning stimulates seed production of Bermudagrass. In Georgia Bermudagrass burned on March 29 produced 46 pounds of seeds per acre compared to 16 pounds per acre on the unburned control. The following year the site was burned on April 15, and Bermudagrass produced 29 pounds per acre on the burn compared to 3 pounds per acre on the control [13]. FIRE MANAGEMENT CONSIDERATIONS: Early spring prescribed burning is regularly used in Bermudagrass pastures to remove old stubble and manure, reduce insects and disease, control woody sprouts, and reduce weeds [34,36,45,56,57,62]. Prescribed burning of Bermudagrass may control leaf spot and stem blight caused by Helminthosporium spiciferum and H. rostratum [36]. Hamilton [34] recommended burning only in years with sufficient soil moisture to promote rapid postfire growth. However, in areas with high rainfall (30 to 40 inches [760-1,020 mm]) or where irrigation is available, burning can be done on an annual basis [34]. Pinkerton and Rice [62] reported that some cultivars can be burned as often as needed with either headfires or backfires. Burning should take place while plants are still dormant. The recommended time is 1 week before the average date of the last killing frost [34,56]. Average dry matter yield of Bermudagrass pasture burned March 1 in Georgia was 832 pounds per acre (1,000 kg/ha) higher than yields of unburned controls. Digestibility and crude protein content of the Bermudagrass were not affected by burning. Because of an increase in absorbed solar radiation, burning increased the soil temperatures at 1-, 2-, and 4-inch (2.5, 5, and 10 cm) depths for 2 to 3 weeks after burning. Dry matter yields were positively related to soil temperatures, but the relationship was subject to the modifying influences of rainfall, air temperature, and soil fertility [56]. Morris [57] reported that burning Bermudagrass pastures had differing effects on forage yields depending on fertilization levels. Yields increased by 1,017 pounds per acre (1,140 kg/ha) after annual spring burning followed by a high level of fertilization on a Georgia site, but yields remained unchanged with medium and low levels of fertilization. Burning reduced weeds regardless of fertilization level. Burning on April 1 provided better weed control than burning on January 1 or March 1 [57]. The effects of fire on Bermudagrass yield vary among cultivars. Pinkerton and Rice [62] investigated the effects of annual March backfires and headfires on the yields of six Bermudagrass cultivars. After 3 years of annual burning, yields of `Coastal,' `Common,' `Brazos,' and `Tifton 44' were unaffected by either backfire or headfire; `Tifton 78' was reduced by backfire only; and `Grazer' was reduced by both backfire and headfire. Fire-related yield reductions occurred during only the first two of the five yearly harvests. Yield reductions appeared to be related to how stoloniferous the cultivar was, with the more stoloniferous cultivars showing greater reductions, particularly when burned with backfires which traveled more slowly than headfires [62]. Spring burning prior to urea application may decrease the amount of gaseous ammonia lost when urea reacts with organic matter. In a Georgia study Bermudagrass yields from fields burned and fertilized with urea did not differ significantly from fields fertilized with ammonium nitrate fertilizer [45].


SPECIES: Cynodon dactylon
REFERENCES: 1. Anderson, Bertin W.; Disano, John; Brooks, Donald L.; Ohmart, Robert D. 1984. Mortality and growth of cottonwood on dredge-spoil. In: Warner, Richard E.; Hendrix, Kathleen M., eds. California riparian systems: Ecology, conservation, and productive management: Proceedings of the conference; 1981 September 17-19; Davis, CA. Berkeley, CA: University of California Press: 438-444. [5851] 2. Anderson, Jeffrey A.; Taliaferro, Charles M.; Martin, Dennis L. 1993. Evaluating freeze tolerance of bermudagrass in a controlled environment. HortScience. 28(9): 955. [25086] 3. Bainbridge, David A. 1990. Soil solarization for restorationists. Restoration & Management Notes. 8(2): 96-98. [14160] 4. Bedmar, F. 1992. Evaluation on postemergence grass herbicides against Cynodon dactylon in sunflower. Annals of Applied Biology. 120(Supp): 58-59. [19627] 5. Belesky, D. P.; Perry, H. D.; Windham, W. R.; [and others]. 1991. Productivity and quality of bermudagrass in a cool temperate environment. Agronomy Journal. 83(5): 810-813. [25225] 6. Bennett, Peter S.; Kunzmann, Michael R. 1989. A history of the Quitobaquito Resource Management Area, Organ Pipe Cactus National Monument, Arizona. Tech. Rep. No. 26. San Francisco, CA: U.S. Department of the Interior, National Park Service, Western Region. 77 p. [12097] 7. Bernard, Stephen R.; Brown, Kenneth F. 1977. Distribution of mammals, reptiles, and amphibians by BLM physiographic regions and A.W. Kuchler's associations for the eleven western states. Tech. Note 301. Denver, CO: U.S. Department of the Interior, Bureau of Land Management. 169 p. [434] 8. Boeer, William J.; Schmidly, David J. 1977. Terrestrial mammals of the riparian corridor in Big Bend National Park. In: Johnson, R. Roy; Jones, Dale A., tech. coords. Importance, preservation and management of riparian habitat: a symposium: Proceedings; 1977 July 9; Tucson, AZ. General Technical Report RM-43. 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Washington, DC: U.S. Department of the Interior, Fish and Wildlife Service. 786 p. [806] 21. Dong, Ming; de Kroon, Hans. 1994. Plasticity in morphology and biomass allocation in Cynodon dactylon, a grass species forming stolons and rhizomes. Oikos. 70(1): 99-106. [25226] 22. Edwards, M. B.; Dougherty, P. M. 1988. Controlling bermudagrass to establish loblolly pines. In: Environmental legislation and its effects on weed science: Proceedings, 41st annual meeting of the Southern Weed Science Society; 1988 January 18-20; Tulsa, OK. 41: 202-209. [16248] 23. Everitt, J. H.; Gonzalez, C. L.; Scott, G.; Dahl, B. E. 1981. Seasonal food preferences of cattle on native range in the south Texas plains. Journal of Range Management. 34(5): 384-388. [12981] 24. Eyre, F. H., ed. 1980. Forest cover types of the United States and Canada. Washington, DC: Society of American Foresters. 148 p. [905] 25. Fernald, Merritt Lyndon. 1950. Gray's manual of botany. [Corrections supplied by R. C. Rollins]. 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