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> ~~ Diospyros bracteata, Roxb. Catal. Pl. class. xu. Polyandr. monog. Dooab, E. Indies. Diospyros glutinosa, Roxb. Hort. Bengal. p. 40. India. Diospyros Sapota, Roxb. Hort. Bengal. p. 40. Mauritius [cult. 7] Diospyros Mabola, Roxb. Hort. Bengal. p. 40. Philippine Islands. Diospyros racemosa, Roxb. Hort. Bengal. p. 40. Tipperah, E. Indies. Diospyros ramiflora, Roxb. Hort. Bengal. p. 40. Tipperah, E. Indies. Ferriola buxifolia, Roxb. Hort. Bengal. p. 72. Coromandel. Diospyros sapotanigra, DC. Ess. Prop. Med. Pl. p. 200. Mexico. Royena lycioides, [Desf.] Cat. Hort. Paris. ex Poir. in Encyclop. Méth. Suppl. vol. 1v p. 435. Cape of Good Hope. Monodora microcarpa, Dunal Monogr. Anonac. p. 80. Australia. Diospyros caroliniana, Muhlenb. ex Rafin. Florula Ludovic. p. 139. Mississippi. Diospyros acapulcensis, Kunth in Humb. and Bonpl. Noy. Gen. m1. p. 254. Mexico. Diospyros psidioides, Kunth m Humb. and Bonpl. Nov. Gen. m1. p. 254. S. America. Diospyros conduplicata, Kunth in Humb. and Bonpl. Nov. Gen. mL. p. 254 $8. America. Celastrus crispus, Thunb. Fl. Cap. edit. 1. vol. ii. p. 115. Cape of Good Hope. Diospyros apeibacarpos, Raddi, Quar. nuoy. del Bras. p. 12. n. 10. Brazil. Diospyros rubiginosa, Roth, Nov. pl. sp. p. 385. E. Indies. Royena myrtifolia, Wendl. ex Steud. Nomencl. Bot. p. 705. Cape of Good Hope. Royena decidua, Burch. Tray. int. $8. Afric. vol. 1. p. 317. South Africa. Royena microphylla, Burch. Trav. int. S. Afric. vol. 1. p. 348. South Africa. Euclea ovata, Burch. Tray. int. 8. Afric. vol. L p. 387. South Africa. Diospyros chinensis, Blume, Cat. Hort. Buit. p. 110. China. Cavanillea Mabolo, Lamarck in Encyclop. Méth. tab. 454. Philippine Islands. Maba rufa, Labill. Sert. Austr. Caled. p. 33. t. 36. New Caledonia. Euclea myrtina, Burch. Trav. int. S. Afric. vol. 11 p. 588. South Africa. Maba Ebenus, Spreng. Syst. Veg. vol. m. p. 126. Molucca Islands. Diospyros vaccinioides, Lindl. in Hook. Exot. Fl. t. 139. China. Diospyros serrata, Hamilt. ex D. Don, Prodr. Fl. Nep. p. 143. Nepal. Diospyros cerasifolia, D. Don, Prodr. Fl. Nepal. p. 144. Nepal. Diospyros cauliflora, Blume, Bijdr. Fl. Ned. Ind. p. 668. Java. Diospyros frutescens, Blume, Bijdr. Fl. Ned. Ind. p. 668. Java. Diospyros maritima, Blume, Bijdr. Fl. Ned. Ind. p. 669. Java. Diospyros macrophylla, Blume, Bijdr. Fl. Ned. Ind. p. 670. Java. Leucoxylum buxifolium, Blume, Bijdr. Fl. Ned. Ind. p. 1169. Java. Diospyros exculpta, Hamilt. in Trans. Linn. Soc. Lond. vol. xv. p. 110. E. Indies. Diospyros insculpta, Hamilt. in Trans. Linn. Soc. Lond. vol. xv. p. 112. E. Indies. Diospyros Toposia, Hamilt. in Trans. Linn. Soc. Lond. vol. Xv. p. 115. Bengal. Noltia tricolor, Schum. and Thonn. Plant. Guin. p. 189. Guinea, Africa. Diospyros edulis, Lodd. ex Sweet. Hort. Brit. p. 270. E. Indies. 1828—32. Diospyros incisa, Hamilt. Hb. ex Wallich, list n. 4122 p. E. Indies. 1828—32. Diospyros glutinifera, Hb. Madr. ex Wallich, list n. 4128 8. Quilon, E. Indies. 1828—32. Diospyros oblonga, Wallich, list n. 4124. Penang, India. 70 Mr HIERN, ON EBENACE:. A.D. 1828—32. Diospyros (?) frondosa, Wallich, list n. 4125. Penang, India. 1828—32. Diospyros venosa, Wallich, list n. 4126. Penang, India. 1828—32. Diospyros lucida, Wallich, list n. 4127. Singapore, India. 1828—32. Diospyros oleifolia, Wallich, list n. 4128. Amherst, India. 1828—32. Diospyros (?) acuminata, Wallich, list n. 4129. Singapore, India. 1828—32. Diospyros Mabolo, Wallich, list n. 4131 A. 1828—32. Diospyros (?) pilosula, Wallich, list n. 4132. Sillet, India. 1828—32. Diospyros Roylii, Wallich, list n. 4134 India. 1828—32. Diospyros (?) chartacea, Wallich, list n. 4135. Burmah, India. 1828—32. Diospyros undulata, Wallich, list n. 4136. Amherst, India. 1828—82. Diospyros ebretioides, Wallich, list n. 4137. Amherst, India. 1828—382. Diospyros heterophylla, Wallich, list n. 4138, Ava. 1828—32. Diospyros amoena, Wallich, list n. 4139. Sillet. 1828—32. Diospyros densiflora, Wallich, list n. 4140. Moulmyne and Amherst. 1828—32. Diospyros grata, Wallich, list n. 4142. Nepal. 1828—32. Diospyros (?) foliolosa, Wallich, list n. 4143. §S. India. 1828—32. Diospyros multiflora, Wallich, list n. 4144, Sillet. 1828—32. Diospyros Wightiana, Wallich, list n. 4406. India. 1828—32. Diospyros dubia, Wallich, list n. 4407. India. 1828—32. Diospyros nigricans, Wallich, list n. 6351. Suillet. 1828—32. Guatteria (?) flavicans, Wallich, list n. 7295. Penang. 1832. Diospyros Schitze, Bunge, En. Chin. bor. n. 237. p. 42. N. China. 1834. Diospyros Persimon, Wikstr. Jahr. Schwed. 1830. pp. 92, 96. N. America. 1834. Diospyros punctata, Decaisne in N. Ann. Mus. Hist. Nat. vol. 1. p. 407. Timor. 1834. Diospyros malabarica, Kosteletsky, Med. Pharmac. Flora (11) p. 1099. India. 1835. Diospyros microcarpa, Spanoghe in Hook. Comp. Bot. Mag. vol. 1 p. 348. Timor. 1835. Diospyros dioica, Spanoghe in Hook. Comp. Bot. Mag. vol. L p. 348. Timor. 1835—36. Diospyros albens, Presl, Reliq. Haenk. m1. p. 62. Mexico. 1836. Diospyros angustifolia, Lodd. Cat. ex Loudon, Arb. et Frut. Brit. um p. 1197 (1838). N. America. 1836. Diospyros fertilis, Lodd. Cat. ex Loudon, Arb. et Frut. Brit. m p. 1197 (1838). N. America. 1836. Diospyros ciliata, Rafin. New Flora and Bot. N. Amer. part UL p. 25. Florida, N. America. 1837. Diospyros biflora, Blanco, Fl. Filipin. p. 303. Philippine Islands. 1837. Diospyros pilosanthera, Blanco, Fl. Filipin. p. 804. Philippine Islands. 1837. Sapota nigra, Blanco, Fl. Filipin. p. 409. Philippine Islands. 1837. Diospyros pterocalyx, Bojer, Hort. Maurit. p. 200. Mauritius. 1837. Diospyros Loureiriana, G. Don, Gen. Syst. Gard. and Bot. vol. tv. p. 39. E. Trop. Africa. 1837. Embryopteris gelatinifera, G. Don, Gen. Syst. Gard. and Bot. vol. 1v. p. 41. E. Indies, 1837. Embryopteris discolor, G. Don, Gen. Syst. Gard. and Bot. vol. tv. p. 41. Philippine Islands, Mr HIERN, ON EBENACE, 71 Embryopteris racemosa, G. Don, Gen. Syst. Gard. and Bot. vol. rv. p. 41. Sillet. Embryopteris Loureiriana, G. Don, Gen. Syst. Gard. and Bot. vol. 1v. p. 41. Cochin China. Embryopteris Kaki, G. Don, Gen. Syst. Gard. and Bot. vol. rv. p. 41. Japan, China and Cochin China. Diplonema elliptica, G. Don, Gen. Syst. Gard. and Bot. vol. iv. p. 42. Cape of Good Hope. Diplonema ambigua, G. Don, Gen. Syst. Gard. and Bot. vol. rv. p. 42. Cape of Good Hope. Maba (?) Ebenoxylon, G. Don, Gen. Syst. Gard. and Bot. vol. Iv. p. 43. Cochin China. Royena cordata, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Royena brachiata, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. A 7. Cape of Good Hope. Royena cuneifolia, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Royena rugosa, E. Meyer, Cat. Pl. Exsicc. Afr. Austr. Dreg, p. 7. Cape of Good Hope. Euclea rufescens, E. Meyer, Cat. Pl. Exsicc. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea macrophylla, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea lanceolata, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea polyandra, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. Pp: p- rf 7. Cape of Good Hope. Euclea tomentosa, E. Meyer, Cat. Pl. Exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea acutifolia, E. Meyer, Cat. Pl. exsicc. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea rigida, E. Meyer, Cat. Pl. exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Euclea pseudebenus, E. Meyer, Cat. Pl. exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Leucoxilon laurinum, E. Meyer, Cat. Pl. exsice. Afr. Austr. Dreg. p. 7. Cape of Good Hope. Myrsine Kellau, Schimper in Pl. Abyss. exsicc. sect. I. n. 159. Abyssinia. Diospyros mollis, Wall. ex Steud. Nomencl. Bot. edit. 11. part 1. p. 514. Tavoy, E. Indies. Diospyros Paralea, Steud. Nomencl. Bot. edit. u. part. 1. p. 514. 8S, America. Diospyros phyllomegas, Steud. Nomencl. Bot. edit. m. part 1. p. 514 Java. Patonia Walkerii, Wight, Ilustr. vol. 1. p. 19. Ceylon. Diospyros calycina, Audib. Cat. Hort. Tonn. ex Spach, Hist. Végét. mx. p. 405. N. America. Royena media, Hort. ex Steud. Nomencl. Bot. edit. mu. vol. m1. p. 475. Cape of Good Hope. Diospyros tetrandra, Spanoghe, Prodr. Fl. Timor. in Linnea xv. p. 336. Timor. Kellaua Schimperi, Alph. DC. in Ann, Se. Nat. Ser. 1. vol. xvi. p. 209. Abyssinia. Diospyros mespiliformis, Hochst. in Pl. Schimp. Abyss. exsice. sect. 11. nn. 655, 1243. Abyssinia. Euclea Kellau, Hochst. in Pl. Schimp. Abyss. exsice. sect. 11. n 1078. Abyssinia. Diospyros intermedia, Hort. ex Loudon Ene. Tr. and Shr. p. 627. N. America. Royena rufescens, E. Meyer, Pflanzengeogr. Doc. Drég. p. 154 in Flora. xxv1. ii. Cape of Good Hope. Royena opaca, E. Meyer, Pfanzengeogr. Doc. Drég. p. 217 in Flora. xxvr. ii. Cape of Good Hope. Mr HIERN, ON EBENACE. Royena falcata, E. Meyer, Pflanzengeogr. Doc. Drég. p. 217 in Flora. xxviii. Cape of Good Hope- Euclea ochrocarpa, E. Meyer, Pflanzengeogr. Doc. Drég. p. 184 in Flora. XXVI. i. Cape of Good Hope. Royena sericea, Bernh. in Flora. XXvil. ii. p. 824. Cape of Good Hope. Euclea Kraussiana, Bernh. in Flora. xxv. u. p. 824. Cape of Good Hope. Euclea ferruginea, Bernh. in Flora. XXviI. ul. p. 825. Cape of Good Hope. Royena ramulosa, E. Meyer ex Alph. DC. Prodr. vui. p. 212. n. 6. Cape of Good Hope. Euclea elliptica, Alph. DC. Prodr. vol. vur. p. 216. n. 1. Cape of Good Hope. Euclea Dregeana, Alph. DC. Prodr. vol. vu p. 216. n. 2. Cape of Good Hope. Euclea coriacea, Alph. DC. Prodr. vol. vu. p. 216. n. 4. Cape of Good Hope. Euclea natalensis, Alph. DC. Prodr. vu. p. 218. n. 10. Natal. Royena macrophylla, E. Meyer ex Alph. DC. Prodr. vol. vir. p. 218. n. 10. Natal. Diospyros (2) pilosa, Alph. DC. Prodr. vol. vim. p. 219. Cochinchina. Gunisanthus pilosulus, Alph. DC. Prodr. vol. vin. p. 220. Sillet. Rospidios vaccinioides, Alph. DC. Prodr. vol. vim. p. 220. China and Malacca. Macreightia caribeea, Alph. DC. Prodr. vol. vin. p. 221. n. 1. St Domingo. Macreightia albens, Alph. DC. Prodr. vol. vir. p. 221. n. 2. Mexico. Macreightia acapulcensis, Alph. DC. Prodr. vol. vitr. p. 221. n. 3. Mexico. Macreightia psidioides, Alph. DG. Prodr. vol. vim. p. 221. n. 4? S. America. Macreightia conduplicata, Alph. DC. Prodr. vol. vim. p. 221. n. 5. S. America. Macreightia inconstans, Alph. DC. Prodr. vol. vin. p. 221. n. 6. New Granada. Macreightia Pavonii, Alph. Prodr. vol. vit. p. 222. n. 7. America. Diospyros cayennensis, Alph. DC. Prodr. vol. vir. p. 224 n. 8. Cayenne, &e. Danzleria axillaris, Bert. ex Alph. DC. Prodr. vol. vin. p. 224 n. 8. Cayenne. Diospyros Pceppigiana, Alph. DC. Prodr. vol. vir. p. 224. n. 9. Brazil? Diospyros mauritiana, Alph. DC. Prodr. vol. vu. p. 226. n. 15. Mauritius, Diospyros macrocalyx, Alph. DC. Prodr. vol. viii. p. 226, n. 17. Mauritius (or Bourbon 2). Diospyros capensis, Alph. DC. vol. vir. p. 226. n. 19. Cape of Good Hope. Diospyros membranacea, Alph. DC. Prodr. vol. vit. p. 227. n. 20. Mauritius. Diospyros anonzefolia, Alph. DC. Prodr. vol. vii1. p. 227. n. 21. Mauritius (or Bourbon ?). Diospyros Neraudii, Alph. DC. Prodr. vol. vim. p. 227. n. 23. Mauritius. Diospyros philippinensis, Alph. DC. Prodr. vol. vir. p. 231. n. 43. Philippine Islands. Diospyros squamosa, Bojer ex Alph. DC. Prodr. vol. vit. p. 232. n. 49? Madagasear. Diospyros levis, Bojer ex Alph. DC. Prodr. vol. vim. p. 232. n. 50. Madagascar. Diospyros senegalensis, Perrottet ex Alph. DC. Prodr. vol. vi. p. 284 n. 59? Senegambia. Diospyros Berterii, Alph. DC. Prodr. vol. vit. p. 234 n. 61. New Granada. Diospyros citrifolia, Wallich ex Alph. DC. Prodr. vi. p. 235. n 65. Burmah, Diospyros sericea, Alph. DC. Prodr. vol. vir. p. 236. n. 67. Brazil. Diospyros hispida, Alph. DC. Prodr. vol. vit. p. 236. n. 682 Brazil. Diospyros Boutoniana, Alph. DC. Prodr. vol. viii. p. 236. n. 72. Mauritius (or Bourbon 2). A.D. 1844. 1844. 1844. 1844, 1844. 1844. 1844. 1844. 1844. 1844. 1844. 1845, 1845, 1845. 1845. 1846, 1846. 1847. 1847. 1847. 1847, 1847. 1848. 1848, 1848. 1848. 1849. 1849. 1849. 1850. 1850. 1850. 1850. 1850. 1850. 1850. 1850. 1850. 1850. 1850. 1850. Mr HIERN, ON EBENACEA, 73 Diospyros Blancoi, Alph. DC. Prodr. vol. vit. p. 237. n. 74. Philippine Islands. Diospyros Malacapai, Alph. DC. Prodr. vii. p. 237. n. 75. Philippine Islands. Diospyros Canomoi, Alph. DC. Prodr. vit. p. 237. n. 78. Philippine Islands. Diospyros (?) Cunalon, Alph. DC. Prodr. vol. vu. p. 237. n. 79. Philippine Islands. Diospyros feminina, Hamilt. ex Alph. DC. Prodr. vim. p. 238. n. 83. Nepal. Maba Cumingiana, Alph. DC. Prodr. vol. vin p. 241. n. 4 Philippine Islands. Maba madagascariensis, Alph. DC. Prodr. vol. vit. p. 241. n. 7. Madagascar. Maba guineensis, Alph. DC. Prodr. vol. vu. p. 241. n. 8. Guinea, Africa. Maba Smeathmanni, Alph. DC. Prodr. vol. yur p. 241. n. 9. Sierra Leone. Maba sandwicensis, Alph. DC. Prodr. vol. vir. p. 242. n. 16. Sandwich Islands. Cargilia maritima, Hassk. Cat. Pl. Hort. Bot. Bogor. m1. p. 159. Java. Vaccinium fragrans, Wall. ex Voigt Hort. Suburb. Caleutt. p. 345. n. 13. China. Diospyros grandifolia, Wall. ex Voigt Hort. Suburb. Calcutt. p. 345. n. 18. Mauritius. Diospyros nigra, Blane. Flora de Filipinas, edit. i. p. 211. Philippine Islands. Diospyros brachysepala, Alex. Braun in Leonhard and Bronn, Neues Jahrb. Mineral. p- 170. Germany. Diospyros japonica, Sieb. and Zuce. Fl. Jap. u. 12 in Abh. Bayer. Acad. Iv. 3. p. 136. n. 459. Japan. Diospyros truncata, Zoll. and Mor. in Mor. Syst. Verz. Jav. Pflanzen. p. 43. Java. Brachycheila pubescens, Harv. ex Zeyh. in Linnewa xx. p. 192. Cape of Good Hope. Euclea pubescens, Eckl. and Zeyh. in Linnea xx. p. 192. Cape of Good Hope. Euclea linearis, Zeyh. in Linnea xx. p. 192. Cape of Good Hope. Euclea desertorum, Eckl. and Zeyh. in Linnea xx. p. 192. Cape of Good Hope. Euclea humilis, Eckl. and Zeyh. in Linnea xx. p. 192. Cape of Good Hope. Diospyros Umlovok, Griffith, Itinerary Notes, p. 355. India. Diospyros pendula, Hasselt ex Hassk. Plant. Javan. p. 468. Java. Diospyros hexasperma, Hasselt ex Hassk, Plant. Javan. p. 468. Java. Diospyros ferruginea, Spltgbr. in Vriese Ned. Kruidk, Arch. p. 327. Guiana. Euclea angustifolia, Benth. in Hook. Niger Fl. p. 441. W. Tropical Africa. Maba vacciniefolia, Benth. in Hook. Niger Fl. p. 442. W. Tropical Africa. Diospyros texana, Scheele in Linnea xxi. p. 145. Texas, N. America. Diospyros Candolleana, Wight, Icon. tt. 1221—2. India. Diospyros capitulata, Wight, Icon. tt. 1224, 1588 bis. India. Diospyros ovalifolia, Wight, Icon. t. 1227. Madras. Maba neilgherrensis, Wight, Ie. Pl. Ind. Or. nn. 1228—9. Neilgherries, India. Plumeria flos-Saturni, Unger, Gen. et Sp. Pl Foss. p. 433. Croatia. Diospyros Wodani, Unger, Gen. et Sp. Pl. Foss. p. 435. Croatia. Diospyros Auricula, Unger, Gen. et Sp. Pl. Foss. p. 436. Croatia. Diospyros Myosotis, Unger, Gen. et Sp. Pl. Foss. p. 436. Croatia. Anona Lignitum, Unger, Gen. et Sp. Pl. Foss. p. 441. Europe. Celastrus europeus, Unger, Gen. et Sp. Pl. Foss. p. 459. Croatia. Tetrapteris Harpyiarum, Unger, Foss. Fl. Sotzka, p. 46. t. 29. f£ 9, 10. Europe. Getonia macroptera, Unger, Foss. Fl. Sotzka, p. 51. t. 33. ff 6—8.. Europe. VOL ile PARnret. 10 74 Mr HIERN, ON EBENACE. A.D. 1851. Diospyros amplexicaulis, Lindl. and Paxt. Fl. Gard. vol. 1. p. 11. n. 271. f. 139. Mauritius. 1851. Diospyros Scheuzeri, Al. Br, ex Unger, Pflanzenwelt, p. 233. Europe. 1851. Diospyros lancifolia, Al. Br. ex Unger, Pilanzenwelt, p. 233. Europe. 1851. Diospyros pamnonica, Ettingsh. Foss. Fl. Wien, p. 19. t. mr f 8. Austria. 1851. Diospyros heringiana, Ettingsh. Tert. Fl. Haring. p. 61. t. 21. f. 26. t. 22.£11. ‘Fyrol. 1851. Diospyros longifolia, Stizenberger, Verzeichniss, p. 83. Europe. 1852. Diospyros paniculata, Dalzell in Kew Journ. Bot. vol. Iv. p. 109. Bombay. 1852. Diospyros pruriens, Dalzell in Kew Journ. Bot. vol. tv. p. 110. Bombay. 1852. Diospyros Goindu, Dalzell in Kew Journ. Bot. vol. tv. p. 111. India. 1852. Holochilus micranthus, Dalzell in Kew Journ. Bot. vol. Iv. p. 291. Bombay. 1852. Diospyros eriantha, Champion in Kew Journ. Bot. vol. 1v. p. 302. Hong Kong. 1852. Diospyros Morrisiana, Hance ex Walpers Annal. vol. m1. p. 14. Hong Kong. 1854. Diospyros argenteus, Griffith, Notule, vol. Iv. p. 288. Malacca. 1854. Maba hermaphroditica, Zollinger, Syst. Verzeichniss Ind. Archip. p. 135. Java. 1854. Arbutus diospyrifolius, Massal. Lett. Scarab. p. 29. n. 203 in Ann. Se. Nat. Bologn. Italy. 1845—55. Diospyros laurifolia, Rich. Fl. Cub. in Ramon de la Sagra, Hist. de Cuba, vol. X1. p. 86. tab. 55 ex Walp. Ann. bot. Syst. vol. v. p. 480 (1858). 1851—5. Diospyros sumatrana, Mig. Plant. Jungh. vol. I. p. 203. Sumatra. 1851—5. Maba sumatrana, Miq. Plant. Jungh. vol. 1. p. 204 Sumatra. 1855, Diospyros aurea, Teijsm. and Binn. Pl. n. h. Bogor. in Neder]. Kruidk. arch. 11. p. 405. Java. 1855. Diospyros laxa, Teijsm. and Binn. Pl. nov. hort. Bogor. in Nederl. Kruidk. arch. 11. p- 406. Java. 5. Rhipidostigma Zollingen, Hassk. Retzia, 1. p. 104 Java. 55. Rhipidostigma Teijsmanni, Hassk. Retzia, I. p. 106. Java. 55. Getonia truncata, Goéppert, Tert. Fl. v. Schoussnitz, p. 37. t. 25. f 11. Silesia. 1856. Diospyros gaultheriwfolia, Mart. Fl. Brasil. Eben. p. 5. t. 2. f ) 5 2. £ 1. Brazil 1856. Diospyros brasiliensis, Mart. Fl. Brasil. Eben. p. 5. t. 2. f. 2. Brazil. 1856. Diospyros coccolobefolia, Mart. Fl. Brasil. Eben. p. 6. t. 1. f. 1. Brazil. led i. 1856. Diospyros artantheefolia, Mart. Fl. Brasil. Eben. p. 1856. Diospyros (?) myrmecocarpus, Mart. Fl. Brasil. Eben. p. 7. Brazil. 1856. Diospyros (?) xylopioides, Mart. Fl. Brasil. Eben. p. 8. Guiana, S. America. 1856. Macreightia obovata, Mart. Fl. Brasil. Eben. p. 9. t. 2. f. 3. Brazil. 1856. Diospyros timoriana, Miq. Fl. Ind. Bat. vol. 1. p, 1045. Timor. 1857. Maba javanica, Zollinger, Obs. Bot. Nov. p. 14 in Natuurk. tydschr. Neerl. Ind. vol. xIv. Java. 1857. Diospyros Kuhlii, Zollinger, Obs. Bot. Nov. p. 15 in Natuurk. tydschr. Neerl. Ind. vol. XIv. Java. 1857. Diospyros penduliflora, Zoll. Obs. Bot. Nov. p. 15 in Natuurk. tydschr. Neer]. Ind. vol. XIv. Java, 1857. Diospyros Hasseltii, Zollinger, Obs. Bot. Nov. p. 15 in Natuurk. tydschr. Neer], Ind. vol. XIV. Java, Mr HIERN, ON EBENACEZD. 75 Drebbelia subarborescens, Zoll. Obs. Bot. Nov. p. 16 in Natuurk. tydschr. Neerl. Ind. XIv. Java. Brachynema ramiflorum, Benth. in Trans. Linn. Soc. Lond. vol. xxu. (part ii.) p. 126. t. 22. IN: Brazil. Diospyros incerta, Massalongo, Synops. Fl. Foss. Senigall. p. 76. n. 197. Europe. Diospyros anceps, Heer, Fl. Tert. Helv. m1. p. 12. t. ci ff 15—18. Oeningen, &c., Europe. Macreightia germanica, Heer, Fl. Tert. Helv. vol. m1. p. 13. t. cr. ff. 1, 2. Oeningen, &e., Europe. Cassia phaseolites, Heer, Fl. Tert. Helv. vol. m1. tab. 138. f. 2 (solwm).. Europe. Diospyros laurina, Massalongo, Syllab. Pl. Foss. Tert. Venet. p. 77. Italy, Europe. Diospyros Weberii, Massal. Syllab. Pl. Foss. Tert. Venet. p. 77. Italy. Macreightia italica, Massalongo, Syllab. Pl. Foss. Tert. Venet. p. 77. Italy, Europe. Macreightia (?) umbellata, Massal. Syllab. Pl. Foss. Tert. Venet. p. 77. Italy. Diospyros pyrrhocarpa, Miq. Fl. Ind. Bat. Suppl. 1. p..583. W. Sumatra. Diospyros Diepenhorstii, Miq. Fl. Ind. Bat. Suppl. 1 p. 583. W. Sumatra. Diospyros Teysmanni, Miq. FI. Ind. Bat. Suppl. 1. p. 583. S. Sumatra. Diospyros (?) cystopus, Mig. Fl. Ind. Bat. Suppl. 1. p. 584. S. Sumatra. Maba (2) lamponga, Miq. Fl. Ind. Bat. Suppl. 1. p. 584 S. Sumatra. Diospyros crumenata, Thwaites, Enum. Ceylon PJ. p. 179. n. 5. Ceylon. Diospyros affinis, Thwaites, Enum. Ceylon Pl. p. 179. n. 6. Ceylon. Diospyros queesita, Thwaites, Enum. Ceylon Pl. p. 179. n. 7. Ceylon. Diospyros oocarpa, Thwaites, Enum. Ceylon Pl. p. 180. n. 9. Ceylon. Diospyros insignis, Thwaites, Enum. Ceylon Pl. p. 180. n. 10. Ceylon. Diospyros oppositifolia, Thwaites, Enum. Ceylon Pl. p. 181. n. 11. Ceylon. Diospyros Gardneri, Thwaites, Enum. Ceylon Pl. p. 181. n. 12. Ceylon. Diospyros Moon, Thwaites, Enum. Ceylon Pl. p. 182. n. 16. Ceylon. Diospyros acuta, Thwaites, Enum. Ceylon Pl. p. 182. n. 17. Ceylon. Diospyros attenuata, Thwaites, Enum. Ceylon Pl p. 182. n. 18. Ceylon. Maba angustifolia, Mig. ex Thwaites, Enum. Ceylon Pl. p. 183. Ceylon. Macreightia oblongifolia, Thwaites, Enum. Ceylon Pl. p. 183. Ceylon. Diospyros vetusta, Giebel, Flora Braunkohl. in Zeitschrift, vol. xvi. p. 57. Prussia. Maba nigrescens, Dalzell in Dalz. and Gibs. Bomb. Fl. p. 142. Bombay. Macreightia intricata, A. Gray in Proceed. Amer. Acad. vol. v. p. 163. Lower Cali- fornia. Ebenacites rugosus, Saporta, Exam. Anal. Fl. Tert. Prov. p. 31. S.E. France. Diospyros samoénsis, A. Gray in Proceed. Amer. Acad. vol. v. p. 326. Navigators’ Island. Maba foliosa, Rich. ex A. Gray in Proceed. Amer. Acad. vol. v. p. 326. Feejee Islands. Diospyros senensis, Klotzsch in Peters Mossamb. 1 p. 183. Mozambique. Diospyros squarrosa, Klotzsch in Peters Mossamb. 1. p. 184 Mozambique. Diospyros bicolor, Klotzsch in Peters Mossamb. 1. p. 184, Mozambique. Diospyros Waldemarii, Klotzsch in Prinz. Waldem. Preuss. p. 101. t. 55. India. 10—2 76 A.D. 1862. 1863. 1864. 1864. 1864. 1864. 1864. 1865. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 13866. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 1866. 1366. 1866. 1866. 1866. 1867. Mr HIERN, ON EBENACE. Diospyros rugosa, Saporta in Ann. Sc. Nat. ser. Iv. vol. Xv. p. 264.t.11.£3. S.E. France. Maba natalensis, Harvey, Thes. Capens. vol. u. 7. Natal. Maba inconstans, Griseb. Fl. Brit. W. Ind. p. 404. Tropical America. Diospyros Arnottiana, Miq. ex Thwaites, Enum. Ceylon Pl. p. 423. E. Indies. Macreightia ovalifolia, Thwaites, Enum. Ceylon Pl. p. 424 n, 2. Ceylon. Macreightia acuminata, Thwaites, Enum. Ceylon Pl. p. 424. n. 3. Ceylon. Cargillia pentamera, Woolls and F. Muell. in F. Muell. Fragm. rv. p. 82. Australia. Diospyros varians, Saporta in Ann. Se. Nat. ser. v. vol. m1. p. 111. t. iv. fi 14, t. vi. f. 4. S.E. France. Diospyros halesioides, Griseb. Cat. Pl. Cubens. p. 168. Cuba. Macreightia buxifolia, Griseb. Cat. Pl. Cubens. p. 169. E. Cuba. Cargillia mabacea, F. Muell. Fragm. v. p. 162. Australia. Maba quadridentata, F. Muell. Fragm. v. p. 162. Australia. Maba Cargillia, F. Muell. Fragm. v. p. 162. Australia. Maba pentamera, F. Muell. Fragm. v. p. 163. Australia. Cargillia megalocarpa, F. Muell. Fragm. v. p. 163. Australia. Maba megalocarpa, F. Muell. Fragm. v. p. 163. Australia. Maba interstans, F. Muell. Fragm. v. p. 163. Australia. Maba fasciculosa, F. Muell. Fragm. v. p. 163. Australia. Maba cupulosa, F. Muell. Fragm. v. p. 164. Australia, Maba sericocarpa, F. Muell. Fragm. v. p. 164. Australia. Maba Hillebrandii, Seem. Fl. Vit. p. 151. Sandwich Islands. Maba Andersoni, [Solander] ex Seem. Fl. Vit. p. 152. Tonga Islands. Euclea miocenica, Unger, Syll. Pl. Foss., pug. iii, in Denksehrift. xxv. p. 25. t. vill. f. 8. Croatia. Euclea Apollinis, Unger, Syll. Pl. Foss., pug. iii., in Denkschrift. xxv. p. 26. t. viii. f. 10. Croatia. Rhododendron Apollinis, Ettingsh. ex Ung. Syll. Pl. Foss., pug. iii., in Denkschrift. xxv. p- 26. Croatia. eDiospyros Zollikoferi, Unger, Syll. Pl. Foss. pug. iii, in Denkschrift. xxv. p, 27. t. ix. f. 6. Styria. Diospyros obliqua, Unger, Syll, Pl. Foss., pug. iii, in Denkschrift. xxv. p. 29. t. ix. f. 17. Croatia. Diospyros Royena, Unger, Syll. Pl. Foss. pug. iii, in Denkschrift. xxv. p. 29. t. ix. ff. 18, 19. Croatia. Diospyros Parthenon, Unger, Syll. Pl. Foss, pug. iii, in Denkschrift. xxv. p. 29. t. ix, f. 8. Europe. Diospyros Lignitum, Unger, Syll. Pl. Foss. pug. iii, in Denkschrift. xxv. p. 30. t. ix. f. 9. Europe. Diospyros lotoides, Unger, Syll. Pl. Foss. pug. iii, in Denkschrift. xxv. p. 30. t. x. ff. 1—12. Europe. Diospyros assimilis, Bedd, Rep. Ind. For. Madr. p. 20. “t. i” §S, Canara, India. A.D. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1867. 1868. 1868. 1869. 1869. 1869. 1869. 1870. 1870. 1871. 1871. 1871. 1871. 1871. 1871. 1871. 1871. 1871. Mr HIERN, ON EBENACEZ, v6 Diospyros mabacea, F. Muell. Austral, Veg. in Intercolonial Exhibition Essays, 1866—67, p- 35. East Australia, Diospyros megalocarpa, F. Muell. loc. cit. p. 35. North Australia. Diospyros fasciculosa, F. Muell. loc. cit. p. 35. East Australia. Diospyros cupulosa, F. Muell. loc. cit. p. 35. Queensland. Diospyros sericocarpa, F. Muell. loc. cit. p. 35. Queensland. Diospyros Cargillia, F. Muell. Joc. cit. p. 35. East Australia. Diospyros pentamera, Woolls and F. Muell. ex F. Muell. loc. cit. p. 35. East Australia. Diospyros humilis, F. Muell. loc. cit. p. 85. Queensland and North Australia. Diospyros geminata, F. Muell. Joc. cit. p. 35. Queensland. Euclea relicta, Unger, Foss. F]. Eub. in Denkschrift. xxvii. p. 68. t. xi. f. 39. Negropont. Royena greca, Unger, Foss. Fl. Eub. in Denkschrift. xxvu. p. 68. t. xi. ff 40—51. Negropont. Royena Amalthee, Unger, Foss. Fl. Eub, in Denkschrift. xxvi. p. 69. t. xiv. fi 1. Negropont. Royena Euboea, Unger, Foss. Fl. Eub. in Denkschrift. xxvu. p69) te xave is 2-4 Negropont. Royena Myosotis, Unger, Foss. FJ. Eub. in Denkschrift. xxvit. p. 69. t. xiv. ff 5—S. Negropont. Royena Pentelici, Unger, Foss. Fl. Eub. in Denkschrift. xxvi. p. 70. t. xiv. f. 9. Negropont. Diospyros Loveni, Heer, Fl. Foss. Arct. p. 118. t. vii. ff. 7, 8. t. xlvii. f. 8. N. Greenland. Diospyros oligandra, Bedd. Rep. Forests Madras, 1867—68, p. 25. Madras, India. Diospyros hebecarpa, A. Cunn. ex Benth. Fl. Austr. rv. p. 286. Australia. Maba hemicycloides, F. Muell. ex Benth. Fl Austr. Iv. p. 290. Australia. Maba laxiflora, Benth. Fl. Austr. Iv. p. 290. Australia. Diospyros speciosa, Wood, Rep. Forests Oudh, 1867—68, p. 33. Oudh, India. Diospyros costata, Carriere in Rev. Hortic. p. 134 China. Macreightia andamanica, Kurz, Rep. Veg. Andam. edit. m. p. 42. S. Andaman. Diospyros microphylla, Bedd. Ic. Pl. Ind. Or. p. 27. t. exxxii. §. India. Diospyros canarica, Bedd. Ic. Pl. Ind. Or. p. 27. t. exxxiv. S. Canara. Diospyros Thwaitesii, Bedd. Ic. Pl. Ind. Or. p. 27. t. exxxv. Ceylon. Diospyros nilagirica, Bedd. Ic. Pl. Ind. Or. p. 27. t. exxxvi. 8S. India. Diospyros rhodocalyx, Kurz in Journ. Asiat. Soc. Bengal. xb. Pt. ii. p. 71. Siam. Diospyros dasyphylla, Kurz in Journ. Asiat. Soc. Beng. vol. XL. Pt. ii. p. 71. E. Indies. Diospyros Brandisiana, Kurz in Journ. Asiat. Soc. Beng. XL. ii. p. 72. Burmah. Diospyros burmanica, Kurz in Journ. Asiat. Soc. Beng. XL. ii. p. 73. Pegu. Diospyros variegata, Kurz in Journ. Asiat. Soc. Beng. XL. i. p. 73. Pegu. 78 Mr HIERN, ON EBENACE. DESCRIPTION OF THE GENERA AND SPECIES, EXCLUSIVE OF FOSSILS. I. Royena, Linn. Gen. Plant. p. 114. n. 325 (1737). Flores seepius hermaphroditi et pentameri. Calyx plerumque accrescens, campanulatus vel urceolatus vel raro depresso-hemisphericus. Corolla urceolata vel campanulata ; lobis in preefloratione sinistrorse contortis. Stamina numero loborum corolle dupla raro plura, in verticillum unicum disposita. Ovarium hirsutum, 4-10-loculare ; ovula in loculis solitaria. Frutices rarius arbores africani; foliis alternis, plerumque coriaceis; pedunculis axillari- bus, sepius unifloris. Alph. DC. Prodr. vir1. p. 210 (1844); J. G. Agardh, Theor. Syst. Pl. tab. x. f. 13 (1858) ; Harv. MSS.; non Houston in Linn. Sp. Pl. p. 628 (1753) (= Loeselia). Pistachia (sp.) Pluknet. Almag. p. 298. t. 63. f. 4. t. 317. f£. 5 (1691, 1696). Vitis Idea (sp.) Plukn. Almag. p. 391. Phytogr. t. 321. f. 4 (1696). Staphylodendrum, Commelin. Hort. Amstelod. 1. p. 187. t. 96 (1697). Staphylodendron, Hermann, Paradisus Batavus, p. 232 cum tab. (1698). Arbutus (sp.) Linn. Hort. Cliff. p. 163 (1737). Buzxus (sp.) Linn. in Herb. Gronov. Vaccinium (sp.) Mill. Gard. Dict. edit. vi. (1771). Royenia, auct., non Houst. Flowers usually hermaphrodite and pentamerous, in one species tetramerous, and in R. ambigua 5-7-merous. Calyx 5-4-partite 5-fid or 5-toothed at apex, pubescent, usually accrescent in fruit. Corolla usually 5-fid, urceolate or campanulate, with obtuse reflexed lobes. Stamens 10, rarely 12—14, in one species 8; mmserted in one row at the base of the corolla, usually 2 opposite each of its lobes; filaments very short, glabrous; anthers lanceolate- linear, hairy or in &. sessilifolia glabrous, dehiscing longitudinally by lateral slits, rarely in subhermaphrodite Howers barren. Ovary pubescent, 4-10-celled; cells 1-ovuled; rarely abortive in male flowers. Styles 2-5 or -style 2-5-partite or -lobed. Fruit coriaceous, globose ovoid or oblong, sometimes 5-sided and splitting by valves. Seeds as in the family; albumen not ruminated. Shrubs or small trees or even large trees (see Burchell, Trav. 1. 390) mostly limited to South Africa, but two species (2. pallens and R. cistoides) reaching the tropics. Leaves alternate, simple, entire, shortly petiolate or subsessile or in one species quite sessile, according to Dr Harvey evergreen. Bracts 1—5. Flowers axillary, peduncled, solitary or in &. glabra 1-5 together or in R&R. parviflora in 3-5-flowered cymes. Named after Adrian van Royen, Professor of Botany in the University of Leiden, who died in 1779 at the age of 74 English name; African bladder-nuts. Alph. De Candolle describes 10 small glands at the base around the ovary; I do not, however, notice any such in any of the species of the genus. Mr HIERN, ON EBENACEA, ROYENA. KEY TO THE SPECIES. Flowers 5—7-, usually 5-, merous. Fruit not glandular or rarely so. Calyx divided half way down or deeper. Calyx 5-lobed only at the apex. Leaves cordate or sub-cordate or rarely rounded at base. Style 2-lobed. Leaves subsessile. Leaves smooth. Flowers hermaphrodite. Leaves scabrous. 6 Flowers with rudimentary ovary. | | Style 4—5-lobed. Leaves distinctly petiolate. | Leaves narrowed at base, not cordate. Peduncles short, not or scarcely longer than the flowers. Leaves subsessile. Anthers 10, hirsute. | Leaves sessile. Anthers 14, glabrous. | Flowers solitary. Calyx patent or reflexed in fruit. Leaves more than } in. long. | Flowers polygamous, 5—7-merous. Leaves under 4 in. long. | Calyx appressed to fruit. Flowers in 1—5-flowered cymes. | Leaves narrowly elliptical, }- 1 in. long. | Leaves obovate, 2—64 in. long. Flowers tetramerous or rarely pentamerous, Fruit glandular. 79 1. R. lucida. 2. R. cordata. 3. R. scabrida. 4. R. villosa. 5. R. hirsuta. 6. LR. sessilifolia. | Peduncles nearly as long as the leaves or much longer than the flowers. | Flowers hermaphrodite, 5-, rarely 4-, merous. 7. R. pallens. 8. R. ambigua. 9. R. nitens. 10. R. cistoides. 11. &. glabra. 12. R. parviflora. 13. R. glandulosa. 380 Mr HIERN, ON EBENACE. 1. RoyeENA Lucrpa, Linn. Sp. Pl. p. 397. (1753). R. foliis ellipticis vel ovatis, basi rotundatis vel cordatis, coriacets, nitidis, breviter petiolatis ; floribus hermaphroditis, pentameris; pedunculis unifloris; calyce campanulato, ampliato, utrinque hirsuto, apice breviter dentato, in fructu accrescente ; stylo bifido; ovario 4-loculari, 4-ovulato. Gertn. Fruct. et Sem. pl. ii. p. 80. t. 94. f 4 (1791). Jacq. Fragm. Bot. p. 3. t. 1 f 6 (1800—1809). Lam. Tabl. Encycel. ii. p. 492. t. 370. f. 1. (not good) (Anno. vim. 1800 2). Desf. in Annal. Mus. vi. p. 445. t. 62. f. 3 (An. x1m1.—1805). Alph. DC. Prodr. vii. p. 211. n. 1 (1844). Lindl. in. Bot. Reg. xxxir. t. 40 (1846). Pappe, Silva Capensis, p. 20 (1854). Pistachia africana s. Staphylodendron Athiopicum MovoracioxadAnvomevodurroy singu- lari hirsuto folio nitente, Pluknet! Almag. p. 298, Phytogr. Tab. 63, f. 4, tab. 317. f. 5 (1696, 1691). STAPHYLODENDRUM africanwm semper virens, foliis splendentibus, Commelin. Hort. Amstelod. i. p. 187. t. 96 (bad). (1697.) STAPHYLODENDRON africanum folio ee lucido, Herm. Parad. Batay. p. 232 cum tab, (1698), Linn. Hb. Hort. Cliff! Royena foliis ovatis scabriusculis, Linn. Syst. Veg. p. 410 (1784). An evergreen shrub 5—12 feet high with numerous branches: Stem 6—12 inches thick. Bark black, rather smooth. Wood hard, tough, yellowish with brownish stripes when polished, well adapted for furniture, tools, screws, &c., but chiefly used for wagon work (Dr Pappe). Young parts covered with subferruginous pubescence. Leaves elliptical or somewhat ovate, usually pointed or apiculate at apex, obtuse or sub- acute, rounded or cordate very rarely somewhat narrowed at base, shortly petiolate, coriaceous, 4—2} inches long by }—1}in. wide, glabrescent and shining above, hirsute especially on the midrib and margin or glabrate beneath; midrib in slight relief on both sides; lateral veins not conspicuous ; petioles 2, — in. long, pubescent. Bracts small or foliaceous, sericeous, caducous. Peduneles axillary, solitary, 1-flowered, pubescent, patent or arching, }—1 in, long, on young branches, bearing 1—3 bracts. Flowers }—}in. long, hermaphrodite. Calyx urceolate, sericeous on both sides, 41—}in. long, 5-toothed at apex, much accres- cent in fruit; teeth short, acute. Corolla urceolate, 5-fid, with rounded lobes puberulous on both sides, Stamens (9—) 10, inserted at the base 2 opposite or corresponding to each lobe of the corolla, tin. long, equal; filaments very short, glabrous ; anthers lanceolate-linear, hispid on upper half, glabrous eiage Ovary conical, pubescent, 4-celled, 4-ovuled; surmounted by bifid style, glabrous above ; stigmas punctiform. Mr, HIERN, ON EBENACE. 81 Fruit ovoid or subglobose, }—1 in. in diameter, enclosed by inflated pubescent or subgla- brate calyx, 2—4-celled and -seeded, red and fleshy when ripe; flesh firm, whitish. Seeds glabrous, rather shining, yellowish; testa thin; albumen cartilaginous, hard, white; embryo half to two-thirds of the length of the albumen, somewhat curved inwards; cotyledons ovate, rather shorter than the radicle. In Cape Colony known by the name of Zwartbaste (blackwood). See Burchell, Travels, vol. 1. p. 317 (1822). Grows in forests, stony places, on the sides of mountains, &c. South Africa. From Cape Town eastwards to Natal. Reeves!; Ecklon! 698 (« R. hirsuta,’ on the eastern side of Devil’s Mountain) ; Drege A. (above the waterfall at Duivelsberg 1000—2000 ft. alt. May), B. (Bosch- rivier, in a wood, below 500 ft. alt. October), c! (Katrivier, in a wood on a hill, 1000— 2000 ft. alt. November); 7. Cooper! 1062 (Orange Free State); Miller !; Bowie!; Dr Pappe! (slope of Devil’s Mountain); Burchell! 5256 (Hartebeest-Vlakte and Kaatje’s Kraal), 5415 (in a forest close to Melkhout Kraal); Roxburgh!; Harvey!; Alexander!; Mac Owan! 309 Eastern districts; Zeyher! 3352. Natal. Dr Sutherland! (a low scrubby bush growing among stones). It is cultivated in St Helena, Gen. Walker! (stamens 5, abortive); and has long been introduced into Europe. 2. RoyENA corpDATA, E. Mey. Catal. Pl. Exs. Afr. Austr. Drég. p. 7 (1887). R. folirs ellipticts vel oblongis, basi cordatis, nitidis, coriaceis, apice obtusis vel subacutis, subsessilibus ; floribus pentameris, hermaphroditis ; pedunculis unifloris; calyce 5-partito, ac- erescente ; stylo bilobo; ovario 4-loculart. Alph. DC. Prodr. vir. 211. n. 2 (1844). R. opaca, E. Mey. Pflanz. Doc. Drég. p. 217 in Flora xxvu. ii. (1843), Alph. DC. Prodr. vil. p. 211. n. 3 (1844). R. swpra-cordata, Burch. MSS. in Hb. Afr. Austr. n. 4907 (1814). A shrub with numerous branches, and a brown-ferruginous pubescence on young parts, quickly glabrescent and nitescent. Leaves elliptical or oblong, cordate at base, usually obtuse-pointed mucronate or api- culate at apex, coriaceous, subsessile, often pubescent underneath, }—2in. long by 2—1 in. wide. Peduncles 1—#in. long, arching, bearing 2 alternate caducous ovate acute ciliate bracts similar to the leaves in shape fin. long. Flowers about double the length of the calyx, 4 in. long. Calyx 5-partite, villous on both sides, 4 in. long; nearly glabrate in fruit; lobes ovate- lanceolate, acute, hirsute and ciliate; calyx much accrescent in fruit with wide ovate cordate or auricled lobes, often nearly an inch long. Corolla 5-lobed, with a short cylindrical tube and reflexed rounded lobes; lobes oblong, 3 length of corolla, puberulous on both sides. Stamens 10, inserted at base of corolla, reaching to the mouth of the corolla, pilose. Style 2-lobed; ovary 4-celled, pilose, cells 1-ovuled. Fruit subglobose, half an inch or more in diameter. Vor, XU PART a: 11 82 Mr HIERN, ON EBENACES. Flowers in November and December. Grows by river-sides and among mountains. It reaches 4300 feet altitude. South Africa. Eastern district of the Cape of Good Hope, and Natal. Drige!; Zeyher! Uitenhage; Mac Owan! 429, 527, Mountains near Great Reynet; Mrs Hutton! Keiskamma, British Kaffraria; 7. Cooper! 35, 186, 306, British Kaffraria ; Gueinzius! Natal; Gerrard and M‘Ken! 12, 18, 99, 1608, Natal; Barber! 307, Queenstown district, a shrub, grows amongst other bushes, blossoms in spring and summer, flower pale cream colour; Burchell! 4166, 4186, 4907. 3. RoyENA SCABRIDA, Harv. MSS. R. foliis ovatis, basi cordatis, presertim subtus scabris, coriaceis, subsessilibus ; floribus pentameris, diecis ; pedunculis unifloris; calyce 5-partito; stylo im floribus masculis bifido, ovario abortivo. A shrub with “branches simple, 8—15 feet high,” pilose at the extremities with pale hairs. Leaves ovate, cordate at base, acute or obtuse at apex, scabrous especially beneath, subsessile, shining above, sericeous when young, ranging up to 2} in. long by 1$ in. wide; margins subrevolute. 3, Flowers nearly } in. long, white. Peduncles axillary, bracteate, much shorter than the leaves, 1-flowered. Bracts ovate, acuminate. Calyx 2 in. long, 5-partite, finely setose, erect; lobes ovate, acuminate, widened near base. Corolla appressedly pilose, campanulate-urceolate, divided {ths way down into 5 ovate- oblong acute lobes. Stamens 10, in one row, inserted at base of corolla, 4 in. long; filaments very short, hairy at apex; anthers hairy at and near apex, linear, acute. Ovary rudimentary, hairy; style bifid, hairy below, glabrous above. Tugela, Natal. Gerrard and M‘Ken! n. 1609. Grassy plains. Flowers in September and October. Near &. cordata, EK. Mey. 4. RoyENA vILLosa, Linn. Systema Nature, ed. xu. tom. 2. p. 302 (1767). R. foliis obovato-oblongis, basi cordatis, apice obtusis, villosis, petiolatis ; floribus penta- meris, hermaphroditis; pedunculis 1—3-floris ; calyce 5-partito; stylo 4—5-lobo ; ovario 8—10- loculart. Alph. DC. Prodr. vit. p. 218. n, 11 (1844). R. scabra, Burm. Prodr. p. 13 (1768). R. scandens, Burch. MSS. in Hb. Afr. Austr. nn. 3673, 3793 (1813). Pubescent trailing shrub with patent branches, 5 to 40 feet long. Leaves obovate-oblong with cordate base and rounded emarginate or shortly-pointed apex; pubescent especially beneath, glabrescent and dark green above, paler beneath, some- Mr HIERN, ON EBENACE. 83 times minutely pellucid-punctate, coriaceous, petiolate ; edges recurved; veins distinct, depressed above; 1 to 43 in. long by } to 2} in. wide. Petioles 1—% in. long, pubescent. Peduneles axillary, either 1-flowered about + in. long or 3-flowered longer and with pedicels about =, in. long, pubescent. Bracts leaf-like, but smaller and narrower than the leaves, caducous. Flowers densely pubescent. 1 5 Calyx with 5 ovate or lanceolate partitions, ovate and accrescent in fruit, closely pubes- cent on both sides. Corolla with 5 oblong lobes reaching 3rds down, tomentose outside except near base, glabrous inside. Stamens 10, anthers densely villous. Style 5-lobed; ovary 8- or 10-celled; stigmas punctiform. Fruit globose-pentagonal, tomentose or hispid, $—{$in. long, surrounded by the widely ovate enlarged lobes of the calyx which reach nearly as high, sometimes dehiscing by 5 valves from apex. Grows in woods. South Africa. Eastern districts and Natal. Drége!; T. Cooper! 1, British Kaftraria, (im flower and fruit.) “Stem 30—40 ft., trailing or twining among trees. Flowers yellow;’ J. Sanderson! 150, 613, 715, Natal (in flower); W. T. Gerrard! 30, Natal (in flower); Krauss! 226, 472, 482, Natal (in flower-bud); Gueinzius! Natal (an flower); Dr Stuart’ (in flower); P. Mac Owan/ 516, Grahamstown (in flower); Bowie! (in flower-bud); Burchell! 3673 (in flower), 3793 (in leaf), 4506 (in flower), 6054? (in leaf); Masson’ Ecklon and Zeyher! 464, Uitenhage; Gerrard and M:Ken, 613, 614, 2013, Natal; Alexander / 5. RoyEeNA HirsuTA, Linn. Sp. Pl. p. 397 (1753). R. foliis oblanceolatis, bast cuneatis, subsessilibus, hirsutis; floribus pentameris, herma- ? phroditis ; pedunculis brevibus, unifloris; calyce profunde 5-lobo ; corolld urceolatd ; stamini- bus 10; stylo plerumque bilobo et ovario 4-loculari. Lam. Tabl. Encycl. 1. p. 493. t. 370. f. 2 (anno vit—1800). Alph. DC. Prodr. vir. p. 212. n. 8 (1844), non Jacq. nec Eckl. nec Sieb. R. angustifolia, Willd. Spec. Plant. 1m. p. 633 (1799), Alph. DC. Ze. n. 5. 2? R. cuneata, Poir. in Encyclop. Méth. vi. p. 322 (1804), Alph. DC. Zc. p. 215. n. 18, non Spreng. R. microphylla, Burch. Trav. Int. S. Afr. 1. p. 348 (1822), Alph. DC. Le. p. 212. n. 9. R. rugosa, E. Mey. Cat. Pl. Exsice. Afr. Austr. Drég. p. 7 (1837), Alph. Zc. n. 7. Diospyros hirsuta, Desf. Ann. Mus. vi. p. 447. t. 62. f. 2 (1805), non L. D. pubescens, Pers. Synops. 11 p. 625 (1807), non Pursh. Arbutus foliis lanceolatis integerrimis hirsutis, Linn. Hort. Cliff. p. 163 (1737). A much-branched rigid shrub, 2—8 feet high, more or less downy-canescent or tomentose. Leaves oblanceolate, obtuse or acute at apex, cuneate at base, subsessile, crowded, coriaceous, hairy and rugose with raised veins or pitted beneath, {—1 in. long by 74—1 in. wide; margins flattish or recurved. 11—2 84 Mr HIERN, ON EBENACEA‘ Peduncles 1-flowered, arching, shorter than the flowers, ;;—} in. long, usually bibract- eate in the middle. Calyx deeply 5-lobed, hairy on both sides; lobes ovate, aecrescent, erect or reflexed in fruit. Corolla urceolate, 5-fid, grey-felted outside, puberulous inside ; lobes rounded or obtuse. Stamens usually 10; anthers hairy; filaments dilated. Styles 2—4, usually 2; stigmas more or less dilated, glabrous. Ovary villous, 4-, 6- 8-celled. Fruit globose, about } in. in diameter, more or less tomentose, often dehiscing by 2 or 3 valves. Fruiting calyx-lobes rounded, erect or reflexed. Grows among mountains and rocks and along banks of rivers and reaches 5000 feet in Natal. Flowers in August. Cape of Good Hope, Kalihari region and Natal. Dr Sutherland! Natal (in flower); Dr Zeyher / 3350, 3351, Uitenhage and Clanwilliam (in flower); Burchell! 7531 (in leaf), 7537 (in fruit), 7446 (in fruit), 1696 (in flower), 2502 (in fruit), 4898 (in fruit); Drége!; P. Mac Owan/ 269, Humansdorp (in flower); TZ. Cooper / 212, Queenstown (in flower); Verreaux!; Krauss, 1719; 8. Africa, Masson Auge and Olden- burg/; Barber! 311, Queenstown district, on stony hill-sides, flowers white, blossoms in spring. 6. ROYENA SESSILIFOLIA, sp. nov. R. foliis oblongo-obovatis, membranaceis, sessilibus, obtusis, basi angustatis ; floribus pen- tameris, diecis; pedunculis unifloris, brevibus; calyce 5-partito; corollé urceolatd ; stamini- bus in flore masculo 14, glabris; ovario abortivo. A shrub with erect stem; branches pubescent, spreading at about 70°. Leaves oblong-obovate, sessile, submembranous, pubescent beneath and on both sides when young, rounded or retuse at apex, narrowed to an obtuse base, 1{}—2in. long by }—}in. wide ; veins inconspicuous, depressed on the upper surface. Peduncles axillary, solitary, bearing one flower longer than itself, pubescent. Flowers fragrant, 4 in. long. Calyx pubescent outside, 5-partite, with lanceolate erect-patent 3-veined lobes } in. long. Corolla urceolate, 5-lobed at apex, glabrous inside, pubescent outside; lobes recurved, ob- tuse, ;1,in. long. Stamens 14, glabrous; anthers dehiscing from apex; filaments short; pollen globular, smooth, ;J%57 in. in diameter. Ovary rudimentary, rounded, pubescent. Pubescence whitish. Described from a living specimen cultivated in Hort. Kew! Approaches R. ambigua, Vent. by having more than 10 stamens, but differs from it by its shorter peduncles ; differs also from all other species of Royena by its absolutely sessile leaves. A specimen in the Leiden herbarium with sessile leaves, which however are coriaceous and usually pointed at the apex and have the veins in relief on both sides, may be the same species ; it was cultivated in 1785. Cfr. R. latifolia, Willd. Enum. PI. Hort. Berol. Suppl. p. 28 (1813, sine descriptione), Alph. DC, Prodr. vil. p. 215, Mr HIERN, ON EBENACE, 85 7. ROoYENA PALLENS, Thunb. Prodr. Fl. Capens., pars prior, p. 80 (1794). R. foliis anguste obovato-ellipticis, apice plerumque obtusis, basi in petiolum brevem angustatis, coriaceis ; floribus hermaphroditis, plerumque pentameris; pedunculis plerumque unifloris flore longioribus; calyce 5-partito ; stylo 83—5-fido; ovario 6—10-loculart. Alph. DC. Prodr. vimt. p. 213. n. 13 (1844) ; non Willd. Hb.! n. 8363. R. hirsuta, Jacq. Collect. v. p. 110. t. 13. f. 1 (1796), et Fragm. bot. t. 1. f. 2 (1800— 1809), non Linn. nec Eckl. nec Sieb. Diospyros lycioides, Desf. in Annal. Mus. vi. p. 448. t. 62. f. 1 (An. x11—1805). RR. pubescens, Willd. Hort. Berol. p. 457 (1809), Bot. Reg. t. 500 (1820), Alph. DC. 2. ¢. n. 12. Lt. lyciordes, (Desf.) Cat. Hort. Paris ex Poir. in Encycl. Méth. Suppl. rv. p, 435 (1816), Alph. DC. Le. p. 214 n. 17. R. decidua, Burch. Trav. Int. 8. Afr. 1. p. 317 (1822). R. cuneata, Speng. Syst. Vegetab. 11. p. 360 (1825), non Poir. &. brachiata, E. Mey. Cat. Pl. Exsice. Afr. Austr. Dr g. p. '7 (1837), Alph. DC. 2. c. p. 213. n. 10. R. cunetfolia, E. Mey. Cat. Pl. Exsice. Afr. Austr. Drég. p. 7 (1837), Atph. DC. dc. p. 214. n. 16. R. ramulosa, EK. Mey. ex Alph. DC. Prodr. vim. p. 212. n. 6 (1844). R. sericea, Bernh. in Flora 1844, p. 824. R. oleifolia, Desf. MSS. (1824) ex Gay MSS. in Herb. hispidula, Harv. MSS, by A shrub or small tree, ranging up to 15 feet in height. Bark reddish brown. Branches silky-pubescent or often glabrescent, pale or cinereous. Leaves more or less narrowly obovate-elliptical, obtuse or rarely acute at apex, narrowed at base into short petiole, silky especially beneath or glabrate, coriaceous, evergreen, }—2 in. long by 4—# in. wide; petioles ~,—+ in. long. Peduncles 4in. long, longer than the flowers, usually considerably so, 1- (rarely 2-) flowered, arching, bearing 2—3 narrow bracts about or above the middle. Flowers white or yellowish, hermaphrodite, pentamerous (or rarely tetramerous), }—? in. long. Calyx partite; lobes ovate or lanceolate, acute, hirsute, accrescent, in fruit spreading or 2 5 reflexed. Corolla deeply lobed, hairy outside; lobes lanceolate, acuminate. Stamens (9—) 10, about half the length of the flower; anthers hirsute. Style 3—5-fid, hairy; stigmas punctiform, glabrous. Ovary 6-, 8-, 10-celled, hairy. Fruit subglobose or ovoid, pubescent or rarely glabrate, }—1in. in diameter, sometimes bursting in a valvate manner. Albumen of seeds not ruminated. Reaches 5000 feet in Natal, 2500 at Graaf Reinet. Flowers in Sept., Oct. and Nov., Jan. Feb. Grows at margins of woods, &e. In South Africa on the banks of the Gariep it is called by the natives Zwartebast (black- wood). Cape of Good Hope, Kalihari region and Natal; also in West tropical Africa. Drege!; Peddie! Natal; Col. Bolton! Grahamstown; T. Williamson! Albany; Alexan= der!; Ecklon and Zeyher! 1127; Burchell! 745—1, 1750, 2371, 2930, 3301, 3325, 3396, 3472, 3789, 4184, 4501, 5490, 5529, 5632, 6490, 6813; Dr Zeyher! 3348, 3353, 3354; Burke! Great 86 Mr HIERN, ON EBENACEZ. Fish River, and Crocodile River; Harvey! 544; Mac Owan, 1646; W. 7. Gerrard! 129, 615, 1157, 1238, 1607, 1610, 1611, Natal; 7. Cooper! 272, Queenstown; 418, Beaufort; 1238, 1157, Natal; Hutton! Howison’s Port, Eastern Districts; J. Sanderson! 140, 318, 511, 527, 717, Natal; Dr Sutherland! Natal 3000—5000 feet alt.; Bowker! (103?) Albany; Wyley ! 103, Namaqualand; Bolus! 128, Graaf Reinet (flowers in Oct., fruits in Nov.); Krauss / 423, Natal, 1721, Knysna; Cape. Niven/ 51, large shrub 6 or 8 feet high, dry elevated plains near Goud river. Tropical Africa, Dr Kirk! Seshike (alt. 3000 feet); C. J. Meller! Manganja Hills (tree: always found by streams). A form with leaves acute at both ends and turning black in drying and with globose fruits thinly sprinkled with rigid hairs is R. hispidula, Harv. MSS. Burchell! Cat. no. 3789 at the Kowi Station, 26 Sept. 1813; and no. 4501 at the Lead mine, 29 January, 1814. Benguela. Distr. Huilla. Dr Welwitsch! no. 2533. A shrub 4—6 ft. high, rarely a small tree of 8 ft. Leaves broad. Flowers white, rather fleshy. Fruit puberulous. Fruiting calyx reflexed, not much increased. In woods and thickets between Lopollo and Monino. Do. Dr Welwitsch ! no. 2534, Leaves narrower. Do. Dr Welwitsch! no. 1255. A small shrub, a few inches high, much branched. Leaves densely sericeous, with some species of Aeidiwm growing on them, A sickly specimen probably belonging to R. pallens. 8. RoyENA AMBIGUA, Vent. Jard. Malm. n. 17 (1803). R. foliis obovato-ellipticis, obtusis, bast angustatis, coriaceis, breviter petiolatis; floribus 5—T7-meris,. diwcis; pedunculis unifloris, flore longioribus; calyce partito; corolld urceolaté; staminibus 10—14, sterilibus (2); stylo 5—7- (2) lobo. Alph. DC. Prodr. vir p. 214. n. 14 (1844). Diospyros ambigua, Vent. Malm. t. 17 (1808). h. polyandra 8. ambigua, Pers. Synops. 1. p. 486 (1805). Diplonema ambigua, G. Don, Gen. Syst. Gard. and Bot. rv. p. 42 (1837). Shrub with numerous erect-patent or ascending branches tomentose-pubescent (at least in wild specimen) throughout, about 3 feet high when in cultivation. Leaves obovate-elliptical, somewhat narrowed at base and rounded or apiculate at apex, dull green, sometimes minutely pellucid-punctate, coriaceous, shortly petiolate, 1 to 2 in. long by 4 to #in. wide. Petioles 4;—1 in. long. Peduncles (2) arching downwards, 1-flowered, bearing 2 (or 8) alternate linear bracts about their middle, three times the length of the petiole in flower, $ to $in. long in fruit. Flowers not hermaphrodite (?), drooping, orange-yellow, slightly scented. Calyx with 5 (or 62) lanceolate acute partitions. Corolla ureeolate, 5—7 ?-lobed: lobes rounded, shorter than the tube. Stamens in ? flower 10—14? shorter than the tube of the corolla, barren. Style 5—7 ?-lobed. Ovary with 5—7? external longitudinal furrows, 10-celled. Fruit globular, bright pale brown, pubescent, nearly } in. in diameter, sometimes dehiscing by 5 valves, (in one case) 3-seeded. Fruit-calyx accrescent, reflexed, with 5 oblong-lanceolate partitions 4 in. long. Seeds oblong, } in. long, pendulous, Mr HIERN, ON EBENACE. 87 South Africa. Burke / (in fruit); Ventenat; Ecklon and Zeyher! 1126, Magalisberg. Perhaps ought to be united to &. pallens, Thunb., of which Dr Harvey considered it to be a garden variety. 9. ROYENA NITENS, Harvey MSS. f. foliis anguste ellipticis, utringue plus minus angustatis, coriaceis, subsessilibus, dense sericeis, parvis ; pedunculis unifloris, semiuncialibus ; calyce fructifero profunde 5-lobo, paulum aucto ; fructu ellipsoideo, solitario. A closely branched shrub about 4 feet high with young shoots and underside of leaves densely covered for the most part with close sericeous persistent pale hairs. Branches terete, ascending, with dark rather shining cuticle. Leaves narrowly oval, crowded, narrowed more or less at both ends, coriaceous, dark and shining above, 1-veined, subsessile, }—2 in. long by ;4—} in. wide. — Flowers unknown. Fruit on the young branches, solitary, on arching pubescent peduncles nearly } in. long. Fruiting calyx deeply 5-lobed, spreading, rather more than 4 in. across, with lanceolate lobes which are about +in. long. Fruit ellipsoidal, puberulous with very short inconspicuous hairs, splitting into 5 (2) parts at the apex, }—1 in. long by 4—?in. thick, 1-celled., S. Africa. Natal. W. 7. Gerrard! n. 1158, February. 10. RoyYENA CISTOIDES, Welw. MSS. R. foliis anguste obovatis, apice obtusis et mucronulatis, ad basim obtusam angustatis, utrinque incano-sericeis, breviter petiolatis, margine reflexo; fructibus solitarvis ; pedunculis fructum fere equantibus ; calyce fructibus appresso. A low shrub, 1—1} ft. high, branched from the base. Wood very hard, strong. Branches terete, ultimately glabrate; shoots softly pubescent, erect; the fruiting branches arcuate- ascending. Leaves alternate, narrowly obovate, obtuse and mucronulate at apex, narrowed to an obtuse base, incano-sericeous on both sides, sub-coriaceous, }—1}in. long by }—in. wide, shortly petiolate ; margins reflexed ; subvenose beneath, Fruiting peduncles axillary, solitary, }—3 in. long, patent, hairy, 1-fruited. Fruiting calyx deeply 5-lobed, hairy on both sides, appressed to the fruit, }—in. long, articulated to the peduncle, with 10 little pits at the base on the concave surface of the articulation probably cor- responding to the 10 cells of the ovary; lobes elliptical, obtusely pointed. Fruit subglobose, puberulous, of shining golden colour, hard, 8—12-celled, }—3 in. in diameter, often bursting downwards from the apex, 3—5-seeded. Seeds }in. long. Albumen of seeds white, cartilaginous, not ruminated. Angola, W. Tropical Africa. Distr. Pungo Andongo, 3500 ft. altitude. Dr Welwitsch / no, 2532. In sandy thickets between Condo and Quisonde, near river Cuanza, Truit ripe in March. 88 Mr HIERN, ON EBENACE/, 11. RoyenA GLABRA, Linn. Sp. Pl. p. 397 (1753). R. foliis anguste ellipticis, utringue angustatis, nitescentibus, subcoriacets, subsessilibus, glabrescentibus ; floribus pentameris, subhermaphroditis ; pedunculis 1—5-floris ; calyce partito, paulum accrescente ; stylo bilobo ; ovario 4-locularc. Alph. DC. Prodr. vm. p. 214, n. 15 (1844). Vaccinium pensylvanicum, Miller, Gard. Dict. edit. vi. (1771). R. myrtifolia, Wendl. ex Steud. Nomencl. Bot. p. 705 (1821), Alph. DC. Le. p. 215. a R. hirsuta, Sieber! Fl. Cap. Exsicc. n. 94 (1824), non Linn. nee Jacq. nee Eckl. R. falcata, E. Mey.! Zwei pflanz. doc. Drég. p. 217 in Flora 1843, Alph. DC. Zc. p. 211. n. 4. Vitis Idea cthiopica, myrtinis folio, flosculis dependentibus, Plukn.! Almag. p. 391. Phytogr. t. 321. fig. 4 (1696). Vitis Idea xthiopica, buxi minoris folio, floribus albis, Commel. Hort. Amstelod. 1. p, 125. t. 65 (1697). Vitis Idea foliis angustissimis longis alternis, Linn. in Hb. Hort. Cliff! ? Buxus africana folio oblongiori non serrato, Linn. in Hb. Gronoy. ! A shrub with erect or ascending branches, 2—6 feet high. Stem 5—6 in. thick. Bark thin, grey, smooth. Wood light, porous, little used except for fuel (Dr Pappe). . Young parts pilose. Leaves narrowly elliptical, usually narrowed at both ends, crowded, subsessile, at length glabrous, shining above, thinly coriaceous, $—1 in. long by }—} in. wide. Peduncles about as long as the leaves, bearing 1—5 flowers, hairy ; equal to or longer than the pedicels, arching. Flowers subhermaphrodite, pentamerous. Bracts lanceolate. Calyx partite, usually but little accrescent ; lobes lanceolate or subulate, acute, hairy. Corolla exceeding the calyx, glabrous ; lobes reflexed. Stamens (9—) 10, not always fertile. Style bilobed, hairy below. Ovary nearly glabrous, 4-celled. Fruit oblong or globose, thinly glandular-pubescent, {—3 in. long, subtended by the usually reflexed calyx. South Africa. Cape of Good Hope. Southern and Western districts. Robertson /, Drege!, Sieber! 94, Wallich!, Mund !, Ecklon! 699, Pappe!, Thom!, Mac Gillivray / 610, Krauss /, Masson /, Roxburgh !, Niven! 48, Hb. Ammann /, Nelson /, Forster /, Thunberg !, Oldenburg/, W. Elliot /, Zeyher! 3349, Harvey! 572, Burchell! 2, 808, 5093, 5367, 5784, 6788, 7186, 7208, 7288, Stekmann! 12. ROYENA PARVIFLORA, sp. noy. h. foliis obovatis, basi cuneatis, apice rotundatis vel ad apicem emarginatum brevissime et abrupte angustatis, membranaceis vel junioribus subcoriaceis, petiolatis; floribus pentameris, hermaphroditis, cymosis; calyce depresso-hemispherico, 5-fido, lobis deltoideis ; stylo apice 5-lobo; ovario 10-loculari. Mr HIERN, ON EBENACEA, 89 A large scandent shrub with terete branches. Young parts and inflorescence softly shortly and appressedly pubescent. Leaves alternate, obovate, cuneate at base, rounded or very shortly and abruptly narrowed to an emarginate apex, membranous or the smaller ones subcoriaceous, green when dry, glabrous and with inconspicuous veins above, somewhat paler delicately veined and puberulous beneath, 2—6} in. long by 1—34in. wide; petiole 3-3 1n. long. Cymes axil- lary on the young shoots, }—% in. long, bearing 3—5 flowers ; common peduncle 4—t in. long; lateral pedicels }—} in. long, with a narrow bract at base about as long as themselves. Flowers hermaphrodite, small, creamy-white, articulated at base to pedicel; in bud depresso-conical, about +in. high and broad. Calyx depresso-hemispherical, short, 5-fid, with flat base, puberulous out- side; lobes deltoid. Corolla much contorted sinistrorsely as regarded from within, shortly pubescent outside except on imbricated sides of the lobes, glabrous inside, 5-lobed; lobes obtuse, rounded, {ths of the depth of the corolla. Stamens 10, hairy, equal, in one row, inserted at base of corolla. Ovary covered with very short hair, depresso-conical, 10-celled, cells l-ovuled ; style 5-lobed at apex, shortly ha‘ry. S. Africa, Zulu-land, Incansla. Gerrard and M*Ken! no. 2015. 13. RoYENA GLANDULOSA, Harvey MSS. Rh. foliis ovato-ellipticis, obiusis, basi rotundatis, subcoriaceis, subsessilibus ; floribus herma- phroditis, plerumque tetrameris; pedunculis unijfloris; calyce 4-partito; corolld wrceolaté ; staminibus 8; stylo apice 4-lobo; ovario 8-loculari; fructibus ellipsoideis, glanduloso-hispidis. A large shrub, “with pretty foliage and habit,” 8—10 feet high. Young shoots, peduncles and fruit glanduloso-hispid, subferrugious. Branches spreading. Leaves alternate, ovate-ellip- tical, obtusely pointed at apex, rounded at or near base, thinly coriaceous or firmly membranous, ciliate and pilosulous beneath, }—1in. long by }—+4in. wide; petioles about +4; in. long, hirsute. Flowers hermaphrodite, axillary on the young shoots, about } in. long, urceolate, articulated to the peduncle, tetramerous. Peduncles spreading, 2in. long, 1-flowered, solitary. Calyx pilose outside, pubescent inside, 4-partite; lobes }in. long, lanceolate, acute, rather patent. Corolla urceolate, glabrous but margin minutely ciliate, deeply 4-lobed ; lobes rounded, recurved above. Stamens 8, in one row, inserted at base of corolla, short, equal, 2 opposite each lobe of the corolla, pilose ; filaments short. Ovary hairy (except perhaps at middle), 8-celled ; style hairy, 4-lobed and glabrous at apex. Fruit ellipsoidal, scarcely } in. long by jin. thick, glandular- hispid. Fruiting calyx much enlarged, 4 in. long, loosely enclosing the fruit or reflexed, 4-partite; lobes ovate-oblong, foliaceous, reddish when dry, about 8-nerved inconspicuously. Rarely a flower is pentamerous. S. Africa, Port Natal, Tugela. Gerrard and M‘Ken! no. 1608. Prats II. Flowering and fruiting branches, natural size. a. Peduncle, magnified 5 dia- meters. b. Hair of peduncle, magnified 30 diameters. c. Flowering calyx, magnified 5 dia- meters. dd. Interior of corolla with stamens, laid open, magnified 5 diameters. e. Stamen, magnified 15 diameters. f. Pistil, magnified 5 diameters. g. Transverse section of ovary, magnified 5 diameters. Vou. XIT. Parr 1. 12 90 Mr HIERN, ON EBENACEZ. EXCLUDED AND NoMINAL SPECIES. Royena latifolia, Willd. Enum. pl. Berol. Suppl. p. 23 (1813). Name only. Cfr. R. sessi- hfolia. Royena media, Hort. ex Steud. Nomencl. bot. edit. ii. vol. ii. p. 475 (1841). Name only. Cape of Good Hope. Royena polyandra, Linn. fil. Suppl. p. 240 (1781) = Euclea polyandra, E. Mey. Royena (sp.) n. 15, Eckl. and Zeyh. ex Harv. and Sond. Fl. Cap. i. p. 71 (1859—60) = Aberia tristis, Sond. Royena 9140, Drég. ex Alph. DC. Prodr. vil. p. 216. n, 4 (1844)= Huclea coriacea Alph. DC. II. EUCLEA, Linn. Syst. Nat. edit. xi. p. 747 (1774), non Lour. Flores dicect, rarius polygami, 4—7-meri, racemosi vel paniculati. Calyx non accrescens- Corolla campanulata vel urceolata, lobis in preefloratione sinistrorse contortis. FLos Mascutus: Stamina 10—30, sepius geminata. Ovarium plerumque abortivum. FLos FEeMIneus: staminodia 0, rarius 2—4. Ovarium 4-loculare, varius 2- vel 6-loculare ; ovula in loculis solitaria, rarius bina im ovariis bilocularibus. Fructus parous, sepius 1- locularis et 1-spermus. Frutices vel rarius arbores Africani, foliis alternis vel oppositis vel rarius in tribus verti- cillatis, cymis axillaribus. a Alph. DC. Prodr. vit. p. 215. n. 1. (1844), Padus (sp.) Burm. Rar. Afric. pl. p. 238. t. 84, f, 1. (1788). Royena (sp.) Linn. fil. Suppl. p. 240 (1781). Celastrus (sp.) Thunb. Fl, Cap., pars post., p. 115 (1800). Diplonema G. Don, Dict. Gard. and Bot. tv. p, 42 (18387). Myrsine (sp.) Hochst. in Pl, Schimp. Abyss. exsice, sect. i n, 159 (1840). Rymia Endl. gen. pl. n, 4250 p, 743 (1835—40). Kellaua Alph, DC, in Ann, Se. Nat. ser. ii. vol. xvI. p. 96 (1841). Brachycheila Harv. in Linnea xx, p, 192 (1847). Mr HIERN, ON EBENACE. 91 Dicecious or occasionally polygamous. Calyx campanulate or small and flattish, 4—7-lobed, usually 4—5-fid; lobes lanceolate ovate or deltoid; not accrescent. Corolla campanulate or hemispherical, 4—7-lobed, 4—5-fid or -partite or -lobed only near the apex. é Stamens 10—30, usually 12—20, either free or in pairs or combined at base of filaments, in one or two rows, inserted at base of interior of tube of corolla or hypo- gynous or partly in both ways, sometimes inserted on an hypogynous ring; anthers hairy or glabrous, oblong or lanceolate, 2-celled, dehiscing laterally ; filaments short, usually slender and glabrous. Styles 1—2. Ovary usually abortive. Q Staminodes usually absent, sometimes 2—4, glabrous; anthers 0, Styles 2 (or 1, bifid), usually clabrous, rarely 3; stigmas emarginate or bifid at apex; ovary ovoid or globular, hairy or glabrous, usually 4-celled, rarely 2- or 6-celled; ovules 4, or rarely 6 when the ovary is 6-celled, pendulous. Fruit globular or rarely ovoid-conical, usually 1-celled and 1-seeded ; peri- carp fleshy. The fruit is edible and is called Guarry. Seed globular, usually marked outside by 3 longitudinal depressed lines. Albumen cartilaginous, usually with a normal intrusion of the testa at the micropyle, distinctly ruminated in a few species; embryo usually some- what curved with its concavity towards the centre of the seed, tending to be incumbent ; radicle superior, about as long as the foliaceous cotyledons. Flowers in axillary racemes or rarely in panicles or solitary. African shrubs or trees with alternate or opposite leaves, or rarely verticillate 3 together. Leaves quite entire except FE. ovata and LE. coriacea in which they are sometimes minutely or obscurely crenulate; usually coriaceous, often obovate, not acuminate except in £. ovata, evergreen. The name is derived from the Greek evxAeia, glory, in consequence of the beautiful evergreen foliage. 12—2 92 Mr HIERN, ON EBENACEZ. EUCLEA. Key TO THE SPECIES. |Ovary hairy. Stamens 15—30. Corolla 4—7-lobed only at apex. Leaves elliptical or obovate, flat or nearly so, not or very rarely cordate at base. Stamens 20—30. ¢ racemes }—1}in. long. 1. £. polyandra. Stamens 18. d¢racemes short. 2. E. tomentosa. Leaves ovate, subcordate, wavy. Stamens 16—17. 3. £. coriacea. Leaves linear or lanceolate, flat, not cordate at base. ‘Flowers pentamerous or hexamerous. Leaves not falcate. | Leaves oblong-lanceolate, about tin. wide, apiculate. 4 LE. acutifolia. Leaves linear or linear-lanceolate, about ;;—14 in. wide. _ | Lower leaves obtuse, not apiculate. Flowers nearly glabrous. 5. . lancea. Leayes apiculate. Flowers hairy. 6. #. pseudebenus. Flowers tetramerous or rarely pentamerous. Leaves faleate. 7. £. linearis. Corolla 4—5-fid or -partite. Fruiting calyx-tube receiving the base of the fruit. $ Flowers racemose, 3—9 together. Leaves quite entire, obtuse or subacute. 8. £. lanceolata. Leaves minutely crenulate or acutely apiculate. 9. £. ovata. é Flowers panicled or many together. Leaves glabrous, subglaucous, opposite. 10. £. divinorum. Leaves pubescent or not glaucous, alternate. ll. E. multiflora. Fruiting calyx-tube consolidated and articulated to thickened pedicel. |F ruits many together. Albumen not ruminated. 12. E. fructuosa. |Fruits 83—4 together. Albumen ruminated. 13. E. natalensis. Ovary usually glabrous or chiefly so. Stamens 10—18 usually about 12. Leaves flat or nearly so. Ovary quite glabrous or rarely pubescent all over. Racemes dense. Leaves usually opposite or verticillate 3 together. Ovary 2—6-celled. Leaves obovate. Ovary 2-celled. Staminodes 0. 14. EF. bilocularis. Leaves obvate-oblong. Ovary 2-, 4-, 6-celled. Staminodes 0. 15. E. macrophylla. Leaves oblanceolate-oblong. Ovary 4-, 6-celled. Staminodes 0-4 16. L. daphnoides. | Male racemes lax. Leaves subopposite or alternate. Ovary 4-celled. Ovary glabrous. Staminodes 0. Abyssinian. 17. £. Kellau. Ovary pubescent or rarely glabrous. Staminodes 2—4. 8. African. 18. E. racemosa. Leaves wavy or very small. Ovary hairy at base, glabrous above. 19. £. wndulata. 1, EvciEs potyanpra, E. Mey, Cat. Pl. exsice. Afr. Austr. Drég. p. 7 (1837). E. foliis ellipticis, alternis vel suboppositis, obtusis, basi subangustatis rotundatis vel raris- sime cordatis, breviter petiolatis, coriaceis, planis ; cymis racemosis; floribus 5—7-meris, diecis, corolla apice lobaté; staminibus 20—30, in floribus femineis 0; ovario hirsuto. Royena polyandra, Linn. fil. Suppl. p. 240 (1781), non Willd. Hb. n. 8366; Diplonema elliptica, G. Don, Gen. Syst. Gard. and Bot. Iv. p. 42 (1837); Rymia polyandra, Endl. Cat. hort. Acad. Vindob. 1. p. 123, n, 4583 (1843) ; E. elliptica, Alph. DC. Prodr. yin. p. 216, n, 1 (1844) ; Mr HIERN, ON EBENACEA, 93 E. Dregeana, Alph. DC. Le. n. 2; E. ferruginea, Bernh. in Flora xxvu. ii. p. 825 (1844) ; E. pubescens, Eckl. et Zeyh. in Linnea xx. p. 192 (1847); Brachycheila pubescens, Harv. ex Eckl. et Zeyh. lc. A shrub 3—7 feet high, pubescent often ferruginous but sometimes glabrescent at least in the male plant, diccious. Branches terete or subterete, alternate or subopposite. Leaves more or less elliptical, alternate or subopposite, more or less obtuse at apex, some- what narrowed, rounded, or even in rare cases cordate at base, coriaceous, quite entire, flat, shortly petiolate, 1—3in. long by }—I}in. wide; petioles ;{—}in. long. 6. Cymes racemose, axillary, pubescent, 3—9-flowered, usually drooping, }—1} in. long; pedicels ;;—} in. long, the lower ones the longer; bracts lanceolate, caducous. Flowers }in. long, urceolate, 5—7-merous, pubescent. Calyx ;4— in. long, glabrous inside; lobes lanceolate or deltoid. Corolla urcevlate, lobed only near apex. Stamens 20—30, more or less united at base in pairs or otherwise, hairy. Ovary more or less abortive, with two slender styles. @. Cymes usually 3- rarely 4—5-fiowered, axillary, =;—}in. long, pubescent or tomen- tose, usually drooping ; pedicels short; bracts caducous. Flowers jin. long, ellipsoidal, 5—7- merous. Calyx shorter than the corolla, 5—7-fid; lobes ovate or deltoid. Corolla shortly lobed at apex. Staminodes 0. Ovary ovoid-conical, hairy, 4-celled, 4-ovuled, in. long, sur- mounted by 2 short styles glabrous above which just appear at the mouth of the corolla. Stigmas emarginate. Fruit usually solitary, occasionally 2—3 together, tomentose, usually ferruginous, globular }—+in. in diameter, 1-celled, 1-seeded. Seed globular; albumen some- what ruminated. The shrub is called Kersse-bosch by the natives im South Africa. Frequent in S. and SW. districts of Cape Colony up to 2000 ft. alt. Masson! ; Niven! 47, 53; R. C. Alexander !; Burchell! 4807 2, 48737, 4998 ?, 6941; Ecklon! 727; Krauss; Zeyher / 3362, 3363, 3364; Drege! 2. EvucLEA TOMENTOSA, E. Mey. Cat. Pl. exsicc. Afr. Austr. Drég. p. 7 (1837). E, foliis alternis, ellipticis, basi cuneatis, apice obtusiusculis vel obtuse angustatis, tomen- tosis, planis, coriaceis, breviter petiolatis ; cymis breviter racemosis, 1—8-floris ; floribus 5—7- meris, diecis; corollé apice lobatd ; staminibus 18, in floribus femineis 0; ovario tomentoso, 4-loculari. Alph. DC. Prodr. vii. p. 216. n. 3 (1844). ‘ &. Kraussiana Bernh. in Flora xxviii. p. 824 (1844). A shrub about 4 feet high or more with dark brown bark and branches cinereo- tomentose at the extremities. Leaves alternate, elliptical, in most cases obtusely pointed or mucronate at apex and wedge-shaped or obtusely narrowed at base, coriaceous, tomen- tose, shortly petiolate, 1}—12in. long by 4—}in. wide; petioles ;;—,4 im. long. | g. Cymes racemose, axillary, few—S-flowered, much shorter than the leaves, pedicels rather longer than the flowers, crowded. Stamens 18, free or somewhat connate at the base. 94 Mr HIERN, ON EBENACEZ. @. Cymes tomentose, densely racemose, axillary, 1—several-flowered, pendulous, shorter than the leaves; pedicels ,in. long. Flowers }in. long, 5—T7-merous, when solitary with numerous imbricated caducous bracts on the short peduncle. Corolla } in. long, shortly lobed, villous outside, glabrous inside, urceolate or campanulate, nearly 8 times the length of the calyx. Staminodes 0. Ovary tomentose, 4-celled, 4-ovuled. Styles 2, nearly glabrous. Fruit solitary, on peduncle ;4—}in. long, pubescent erect or erect-patent. Immature fruit ovoid, somewhat conical at apex, incano-tomentose, 4-celled, 3—} in. long by }—,§,in. thick. Fruiting calyx 5—7-fid, very tomentose, shallow ; lobes deltoid. Called Kersboschjes also Faahdls-bosch by the natives in South Africa. Occurs in Western districts of Cape Colony up to 2000ft. alt. Masson/; Drege! ; Krauss; (2) Burchell! 987. Namaqualand, Whitehead! 3. Euciea cortAcea, Alph. DC. Prodr. vu. p. 216. n. 4 (1844). E, foliis ovatis, alternis, plerumque acutis, apiculatis, bast latis et subcordatis, subgla- brescentibus, breviter petiolatis, undatis; cymis densis, 6 1—8-floris, 9 3—T-floris; floribus 5—6-meris, diecis; corolld apice lobatd; staminibus 16—17; fructibus globosts, subglabratis. Euclea n. 9140, E. Mey. Zwei Pflanz. Doc. Drég. in Flora xxvi. i. p, 48(1849). Royena n. 9140, Drég. ex Alph. DC. Le. (Hb. DC}). A dense shrub with strong dark-cinereous branches. Young parts and inflorescence slightly pubescent. Leaves alternate, ovate, more or less acute, apiculate, wide and subcordate at base ; coriaceous, pubescent, nearly glabrescent, without evident veins above, veined and duller beneath, 1—2in. long by }—1}in. wide; margins wavy, sometimes obscurely crenulate; petioles ranging up to iin. long. Bracts ovate, small, caducous. g Flowers ,in. long, axillary, 1—3 together, crowded ; pedicels shorter than the flowers. Calyx 5—6-fid ; lobes ovate, acute. Corolla urceolate, 4 times the length of the calyx, 5—6- lobed at the apex. Stamens 16—17, sometimes in pairs; anthers linear lanceolate, silky at the back. Ovary rudimentary. 9. Flowers 3—7 together; peduncles very short; pedicels ranging up to jin. long. Calyx (in fruit) 5—6-fid, nearly flat, stellate, }in. in diameter; lobes ovate or lanceolate, acute. Fruit globose, }—2in. in diameter, subglabrate or minutely puberulous, 1-celled, 1-seeded ; seeds subglobose, about }in. in diameter, marked outside with depressed curved lines; testa intruded into the hard ruminated albumen. East-midland districts of Cape Colony, S. Africa. Tafelberg, Drége/, in moist and rocky places, 6000—7000 ft. alt. (in g flower, December); side of Mount Oudeberg near Graaff Reinet, 4500 ft. alt., November, Bolus/ n. 638 (in fruit). 4. EucLEA ACUTIFOLIA, E. Mey. Cat. Pl. Exsicc. Afr. Austr. Drég. p. 7 (1837), E. foliis alternis, oblongo-lanceolatis, apiculatis, coriaceis, glabris, basi cuneatis, subsessili- bus ; cymis femineis dense racemosis ; calyce 6-lobato ; corollé apice lobatd ; ovario dense piloso; fructibus dense racemosis, globosis, glabrescentibus, Mr HIERN, ON EBENACEA, 95 Alph. DC. Prodr. vit. p. 217. n. 5 (1844). Shrub with glabrous leaves and branches. Leaves oblong-lanceolate, apiculate, thickly coriaceous, alternate, cuneate at base, subsessile, erect, subglaucescent, 14—2} in. in length by about 4 in. in width. @. Fruit densely racemose on cymes }in. long; pedicels very short, 3—7; flowers 3 in. long, cylindrico-urceolate, pubescent, pentamerous. Calyx short. Corolla lobed at apex. Ovary densely pilose; styles 2, erect, glabrous; stigmas dilated. Fruit globose, glabrescent, finely netted, dark, 3—4 in. in diameter. Fruit-calyx very small, with 6 or more lobes; seeds unequally divided by three depressed lines ; albumen slightly ruminated. South Western districts, Cape of Good Hope. I have seen this plant only in fruit. The flower is unknown. Between Vierentwintig-rivier and Pikenierskloof on the plain, under 500 feet, January, Drége!; Ecklon and Zeyher! 5. EvcLea LANcEA, Thunb.! Prodr. Pl. Capens., pars posterior, p. 85 (1800). EL. folvis alternis, lineari-lanceolatis vel oblanceolatis, inequalibus, inferioribus apice rotun- datis superioribus acutis, subsessilibus, glabris; cymis 3-floris; floribus subhermaphroditis (2), 5—6-meris ; corolla apice lobatd, subglabra; staminibus 15; ovario hirsuto. Alph. DC. Prodr. vir. p. 219. n. 16 (1844). A glabrous shrub, erect, 3 ft. or more high. Branches alternate, terete, erect-patent Leaves alternate, subsessile, linear-lanceolate or -oblanceolate, unequal, the lower ones rounded, the upper acute at the apex, attenuate at base, coriaceous, 1—2 in. long by about } in. wide, entire, inconspicuously reticulato-venose. Flowers axillary, “in 3-flowered cymes,” very nearly glabrous, urceolate, 4 in. long by 4 in. wide (imperfectly hermaphrodite ?). Calyx short, ;, in. high by + in. wide, obscurely 5—6-lobed, coriaceous. Corolla 5—6-lobed at apex, shortly ciliate, imbricated in estivation. Stamens 15, alternately (?) in pairs and single; the pairs consisting of 2 equal or unequal anthers placed back and front on a common filament or combined by their filaments, alternating with the corolla-lobes. Anthers pointed, with short patent pale sete on upper half, dehiscing laterally; filaments dark glabrous slender, shorter than the majority of the anthers, mostly inserted at the base of the corolla. Ovary hairy, #; in. wide and long, ovoid-conical, rudimentary or 4-celled? and -ovuled? Styles 2, glabrous, erect; stigmas punctiform, emarginate at apex, Cape of Good Hope. Thunberg/ Near EZ. pseudebenus, E. Meyer, but differs by its obtuse lower leaves and nearly glabrous corolla; it may possibly include 2. pseudebenus as a form of the same species. 6. EUCLEA PSEUDEBENUS, E. Mey. Cat. Pl. Exsicc. Afr. Austr. Drég. p. 7 (1837). E. foliis alternis, linearibus vel lineari-lanceolatis, apiculatis, glabris, breviter petiolatis ; cymis masculis racemosis 8—7-floris, femineis parvis 1—38-floris; floribus diacis, plerumque 5-meris; corolld pubescente, apice lobatéd; staminibus 16—22, in flore femineo 0; ovario pubescente, 4-loculari. 96 Mr HIERN, ON EBENACEZ, Alph. DC. Prodr. vim. p. 217. n. 7 (1844). E. rigida, E. Mey. lc. Alph. DC. l. c. n. 6. E. angustifolia, Benth. Niger Fl. p. 441 (1849). Leaves and branches glabrous or pu- bescent. Leaves linear or linear-lanceolate, apiculate, erect or patent, alternate, coriaceous, very shortly petiolate, crowded, 1—2} in. in length by ~;—} in. in width; petioles 3,—+# in. in length. 6. Cymes racemose, hairy, bearing 3—7 flowers, erect or erect-patent, }—2 m. in length; pedicels slender, ,—1in. in length; flowers ~;—+ in. ia length, puberulous or incano-pubescent, usually pentamerous, rarely hexamerous; calyx with deltoid lobes reaching half way down; corolla lobed at apex; stamens 16—22, with a few bristles on the lanceo- late anthers or glabrous; filaments more or less combined at the base, inserted around base of rudimentary ovary. 9. Flowers solitary or two or three together, or in small cymes, } in. in length, pen- tamerous ; peduncles 3;—+} in. in length, not drooping. Stamens 0; styles 2; ovary 4-celled, pubescent; fruit 1-celled, I-seeded, glabrescent, globular, 1 in. in diameter; albumen not or scarcely ruminated ; fruit edible, fleshy, sweet and slightly astringent; seeds marked by three depressed lines. There are three forms of this species according as the plant is glabrous with linear leaves, pubescent with linear leaves, or glabrous with linear-lanceolate glaucescent leaves. The two latter forms belong to £. angustifolia, Benth. and #. rigida, E. Mey. respectively. It is known by the names of Orange river ebony, black ebony, zwartebbenhout, and sneezewood. It is a large shrub, 6—8 ft. high or a tree, the heart-wood of which is extremely hard and black. It occurs in the western districts of South Africa, up to an elevation of 4000 ft., and reaches the tropics. Drege!/; Niven! n. 46. Namaqualand, Dr Atherstone! n. 2; Wyley! S.W. Tropical Africa, lat. 23°, Chapman and Baines!; Curror/; Angola, Distr. Mossamedes, shrub, 5—8 feet high, flowers white, dicecious, fruit the size of a pea, edible, glaucous-bluish (as in Juniperus communis), called by the natives (as also Huclea lanceolata) Embolo, quite frequent in thickets and woods in company with Tamaria and Cordia near the rivers Bero and Maiombo, Dr Welwitsch / nos. 2543, 2544. Note, This species may prove identical with E. lancea, Thunb. 7. Evcira Linearts, Zeyher in Linnea xx. p. 192 (1847, sine descriptione). E, foliis alternis suboppositis vel oppositis, linearibus, acutis, falcatis, nwmerosis, sessilibus, glabris ; cymis racemosis, 3—T7-floris; floribus diccis, tetrameris; corolld breviter 4-fidd ; staminibus 16, in flore femineo 0; ovario hirsuto. Plant quite glabrous and subglaucous, dicecious, 2}—8ft. high, Branches numerous, at about 35° with stem. Leaves alternate opposite or subopposite, linear, acute, usually some- what faleate, sessile, numerous, 1 to 24 in. in length by ,'; in. in width. Cymes racemose, bearing 3—7 flowers, g } to 4 in, in length (excluding flower), usually drooping; 9 } to } in. in length, pedicels not exceeding #5 in. in length, less on the ? plant, opposite or alternate, falling short of or equalling the bracts; bracts at base of pedicels, caducous. Mr HIERN, ON EBENACE, 97 é. Flower 4 in. in length. Calyx small, flattish, 4-fid. with wide lobes, Corolla barrel- shaped, 4-lobed, many times higher than calyx; lobes about } depth of corolla, not reflexed, semi-circular and imbricated in flower. Stamens 16, the few inner ones smaller, glabrous or nearly so, ;';—y5 in. long; anthers oblong, 2-celled, dehiscing laterally, thick; filaments very short, thinner than anthers, inserted with corolla, Ovary rudimentary, slightly hairy, terminated by 1 or 2 styles. ?. Bracts linear, rather longer than pedicel; flower about +; in. in length. Calyx campanulate, j;in. in height, 4lobed; lobes not quite half the depth of the calyx, with intervening sinuses in the form of ares of circles. Corolla openly campanulate, shortly 4-fid, with spreading not reflexed oval or ovate lobes, j; in. in length; stamens 0. Ovary ellipsoidal, hairy, terminated by 2 thick glabrous styles, #, in. in height; styles 34, in. long, erect, con- tiguous, dilated, and emarginate at apex; ovary 4-celled, 4-ovuled, two of the septa being very slender, namely, those opposite the styles. Rarely a flower is pentamerous. Western districts of Cape Colony, South Africa, Zeyher!’ 1125, Windhoek, Olifant River ; Burke! Great Fish River. 8. EUCLEA LANCEOLATA, E. Mey. Cat. Pl. Exsicc, Afr. Austr. Dreg. p. 7 (1837). E, foliis alternis vel oppositis, lanceolatis ovatis vel anguste ellipticis, apice obtusis vel subacutis, plerumque undulatis et basi in petiolum brevem angustatis, integerrimis ; cymis race- mosis, 3—9-floris ; floribus dicecis, 4- raro 5- meris; calyce campanulato ; corolld 4—5-fidd ; staminibus 16—17, in flore femineo 0; ovario hirsutissimo. Alph. DC. Prodr. vir. p. 218. n. 12 (1844). E. ochrocarpa, E. Mey. Zwei Pflanz. Doc. Dreg. p. 184; in Flora, 1843; Alph. DC. Zc. p. 217. n, 9. E. desertorum, Eckl. and Zeyh. in Linnea xx. p. 192 (1847). Pubescent glabrous or glaucescent shrub or tree, ranging up to 20—25 feet high and trunk up to 10—15 inches thick ; dicecious; branches terete, at 30’—45°; young shoots angular. Leaves lanceolate ovate or narrowly elliptical, alternate or opposite, coriaceous, obtuse or subacute at apex (very rarely acute), more or less undulating at the entire margins, often narrowed at base into the short petiole, 1—8 in. long by ~,—1,) in. wide; petioles jj;—} in. long. Inflorescence racemose, often with leaf-like bracts. 6. Racemes }—1 in. long, 3—9-flowered; occasionally two racemes proceed from the 1 Flowers usually tetramerous, occasionally pentamerous, same axil; pedicels };—¥}, in. long. 35—3; in. long. Calyx widely campanulate, small; lobes deltoid, about half the length of the calyx. Corolla campanulate; lobes ovate or oval, about half the length of the corolla, somewhat pubescent outside. Stamens 16—17 (very rarely 8—10), mostly inserted in pairs at base of corolla, shorter than the corolla; anthers hispid or nearly glabrous, as long as the slender filaments. Ovary rudimentary, hirsute ; styles 2, glabrous. Q. Racemes } in. long, pubescent, 3—5-flowered; pedicels 35 in. long. Flowers tetra- merous or pentamerous. Calyx and corolla as in the male plant. Staminodes 0. Ovary sub- globose, very hirsute, 4-celled ; styles 2, glabrous, as high as the corolla. Fruit globular, } in. Wei, JUG Jee bh 18 98 Mr HIERN, ON EBENACE. in diameter, pubescent or glabrescent, 1-celled, 1-seeded. Testa intruded some distance into the albumen. A very variable species, and in some cases difficult to separate from JL. ovata, Burch., to which possibly it ought to be united; it is also nearly related to H. divinorum. It is called Omgwali by the Kaffirs, according to Dr Pappe. South Africa; Cape Colony, Namaqualand, Natal and Trans-Vaal; common. Masson /; Burchell! 4880, 4938, 5648; Drege!; Ecklon! 1123. Uitenhage, Harvey! 575, 690; Zeyher / 3355, 8357, 33592; Bothasberg, in stony places at 2000 ft. alt. Mac Owan/ 902; Bruintjies Hoghte, 4000 ft. alt. Mac Owan! 1740; Albany, T. Wilhkamson/; Caffraria, Bowker ! 324; Namaqualand, Drége/; Natal, Gerrard! 33, 528, 1155, 1156, 1605; Macalisberg, Trans-Vaal, Burke ! Dr Welwitsch has collected the following forms from Benguela: a. Leaves glabrous and shining, young ones lepidote; branches spreading. Benguela, Distr. Bumbo, 15° South Latitude, 2000 ft. altitude ; shrub, 8 ft. high, in thickets ; in male flower at end of October; Dr Welwitsch / n. 2548. Distr. Mossamedes; much branched shrub, 5—8 ft. high, branches occasionally 3 or 4 together; ¢ flowers of pale rose-colour; frequent in rocky places near the river Meriombo in company with Tamari« articulata and Ximenia americana, from Pedra de Rei almost to Bumbo; Dr Welwitsch! n. 2547. Distr. Huilla; shrub 4—6 ft. high, with rather rigid and tortuous ramification; 2 flowers fallen; ovary hairy; at margins of woods between Mumpulla and Nene, at end of October; Dr Welwitsch! n. 2549. Distr. Benguela; shrub 4—6 ft. high, with virgate usually opposite branches; in maritime thickets near the city of Benguela; fruit in middle of June; Dr Telwitsch ! n. 2545. Distr. Mossamedes; shrub, occasionally arborescent, 7—12 ft. high, ever- green; frequent in sandy and rocky thickets very near the river Bero; July; native name Nboto or Emboto; fruit edible, berries red; Dr Welwitsch! n. 2546. 8. Leaves and shoots pubescent. Branches ascending. Benguela, Distr. Huilla; small shrub 1—1} ft. high; flowers white; in somewhat stony thickets near Mumpulla, not unfrequent; male flower in October; Dr Welwitsch! n. 2550; Cfr. £. ovata, Burch. Distr. Huilla; small shrub 1—1} ft. high, subeespitose; frequent in steep pastures on right bank of river Lopollo in company with small myrtaceous plants; flowers in November, fruits in February; Dr Welwitsch! nn. 2551, 2552. The following two specimens may also belong to this variable species: Distr. Huilla; a small shrub, 6—8 in. high, from a woody base; fruit dark purple, edible, iin. in diameter; in moist sandy thickets on the right bank of the river Lopollo in com- pany with small species of Eugenia and Celastrus; fruit in January; Dr Welwitsch! n. 2553. Arborescent shrub; in thickets on the sides of large rocks; Pedras de Guinga, Angola. Distr. Pungo Andongo; March, in young fruit; Dr Welwitsch/ n. 1247. 9. Evctea ovata, Burch. Trav. Int. S. Afr. 1. p. 387 (1822). E. foliis ellipticis vel acutd ovatis, oppositis vel alternis, plerumque apiculatis, breviter petiolatis, rigidis, margine planis et minute crenulatis vel undulatis et integerrimis; cymis racemosis, 3—7-floris ; floribus sub-diwcis vel polygamis ; calyce depresso-campanulato ; corolla Mr HIERN, ON EBENACES, 99 4—5-fidd ; staminibus 16 vel 20, in floribus sub-hermaphroditis circiter 12; ovario hirsuto ; Ffructibus globosis, primiim pubescentibus, dem&m glabris. Alph. DC. Prodr. vi. p. 218. n. 13 (1844). Celastrus crispus, Thunb. Fl. Cap. edit. ii. vol. m1. p. 115 (1820). Cfr. Sond. in Harv. et Sond. Fl. Cap. L p. 461 (1859—60). E. rufescens, EH. Mey. Cat. Pl. Exsice. Afr. Austr. Drég. p. 7 (1837). Royena rufescens, EK. Mey. Cat. Pl. Dreg. p. 154 (Flora, 1843). A densely leafy shrub pubescent or sometimes glabrescent, 3—7 ft. high; branches terete, at 50°—60°. Leaves opposite or alternate, elliptical or narrowly ovate, usually apiculate, acute or obtuse, coriaceous, shortly petiolate, minutely crenulate or quite entire, flat or undulated, 1—2 in. long by {—1 in. wide; petioles ,—5 m. long. Racemes 3—7-flowered, at length drooping, 3,—2 in. long; pedicels j,—, in. long; flowers tetramerous or occasionally penta- merous, sub-dicecious or polygamous, pubescent, #—} in. long. Calyx shortly campanulate ; lobes deltoid. Corolla campanulate, 4—5-fid. Stamens 16 or 20, in subhermaphrodite flowers about 12, hairy; filaments slender, glabrous. Styles 2, glabrous; ovary hirsute, globose or ovoid, (2—) 4-celled. Fruit globose, black, of the size of a pea, at first pubescent but at length glabrous. The flavour of the fruit is pleasant with a little astringency and perfectly wholesome. The variety with undulated leaves 20 stamens and less deeply divided corolla (Z. ru- fescens) much resembles EL. coriacea, Alph. DC. Occurs in midland districts of Cape Colony: and northwards into the Kalihari region . of South Africa. Burchell! 1706, 2487—2, 2487—7, 2542, 2920, 3058—1, 3058—2, 3102; Drege! 10. EUCLEA DIVINORUM, sp. nov. E. foliis ellipticis, oppositis, a medio utrinque angustatis, obtusis, breviter petiolatis, supra glaucescentibus, undulatis; cymis masculis conferto-~racemosis vel-paniculatis; floribus 4—5-merrs, diecis; corolld profunde 4—5-lobd ; staminibus 16. Shrub, ‘nearly glabrous and somewhat glaucous, with opposite or subopposite leaves and branches; branches terete, making 30°—40° with the stem. Leaves elliptical, narrowed more or less from the middle towards both ends especially towards the base into the short petiole, obtuse, coriaceous, glaucescent above, reddish and somewhat farinaceous beneath; margins undulated; veins inconspicuous; 14—2}in. long by }—$in. wide; petioles about jin. long. Bracts small, shorter than the pedicels, caducous. Male flowers in crowded racemes or panicles, about 10 or more together, globular in bud, hemispherical when expanded, tetramerous or pentamerous. Cymes not exceeding 5% in. long by 4in. broad, usually erect; pedicels about 75 in. long, spreading. Calyx jin. high, 4.—5-fid; lobes small or depresso-deltoid. Corolla deeply 4—5-lobed, with a few whitish appressed hairs outside especially along the middle of the lobes, nearly glabrous inside; lobes rounded. Stamens 16, not in pairs, ,in. long, as high as the expanded corolla; anthers oblong, hairy; filaments shorter, glabrous. Ovary rudimentary, consisting of an ovoid bunch of hairs. 138—2 100 Mr HIERN, ON EBENACE. Female plant unknown. Called by the natives in Batoka country Matlakula, Mosakola, where it is the medicine of the diviners being rubbed in the hands. South Tropical Africa, Victoria Falls, Dr Kirk/; Delagoa Bay, Forbes! 11. EUcCLEA MULTIFLORA, sp. nov. Puate III. E. foliis ellipticis vel oblongis, apice rotundatis vel obtusis, basi subangustatis, alternis vel raro suboppositis; cymis presertim masculis paniculatis, multifloris; floribus polygamus, 5- raro 6-meris; calyce campanulato; corolla profunde lobaté; staminibus numero corolle loborum quadruplis, in flore femineo 0; ovario hirsuto. Pubescent subglabrous or even subglaucous shrub, usually subferrugimous, polygamous but usually dicecious, sometimes hermaphrodite, 2—10 ft. high. Branches usually angular near the extremities. Leaves elliptical or oblong, usually rounded or obtuse at apex and somewhat narrowed at base into the petiole, coriaceous, alternate or rarely sub-opposite, often dark and shining on the upper surface; veins usually not conspicuous; margins undulated or plane; 1—4in. in length by ;3—1in. in width; petiole ,8—1in. in length. Flowers especially the male ones paniculate, sometimes as many as 30in one panicle, variable in size, tetramerous or pentamerous or rarely hexamerous. Calyx campanulate, hairy, with ovate or deltoid lobes extending about half way down the calyx. Corolla about twice as long as the calyx, dark, deeply lobed; lobes oval usually with a hairy keel outside. Stamens 4 times as numerous as the lobes of the corolla in the male or hermaphrodite plants, none in the female plant, subglabrous or somewhat hairy, in pairs inserted at base of corolla or around base of ovary, outer ones longer, enclosed in corolla; filaments glabrous. Ovary in male flowers often abor- tive, in female or hermaphrodite flowers globular, hairy, 4-celled, cells 1-ovuled; styles 2, glabrous or nearly so, included within the corolla. A variable and widely distributed plant. Flowers in August and fruits from September to October. Fruit at first pubescent, in most cases ferruginously so, subsequently black and glabrous, globular, 3,in. in diameter, 1-celled, 1-seeded. Embryo curved and tending to be incumbent. Cape of Good Hope, Natal and Angola. Wallich!; ? Bergius!; Zeyher! 767, 778, 3361; Grahamstown, Mac Owan/; Burchell, 3510, 3572, 3980 (seeds consumed by a species of Apion), 4835; Albany, Miss Bowker / ; Eastern districts, Hutton!; British Kaffraria, Cooper/ 44; Clanwilliam, Zeyher/; Natal, Gueinzius !, Cooper / 1253, Gerrard! 92, 699. Benguela, Distr. Huilla, Dr Welwitsch! n. 2557, arborescent shrub 5—8, sometimes 10 ft. high, forming a dense dark green head, young fruit 1- rarely 2-seeded, hirsute-tomen- tose, in thickets, Matus de Monino, February. Do. Dr Welwitsch! n. 2555, bush 4—8 ft. high, in high thickets near Tau, in bud, May. Do. Dr Welwitsch! n. 1258, handsome shrub 5—8 feet high, in thickets at the skirts of woods near Lopollo, leaves frequently attacked by a fungus (Spheria). Angola, distr. Pungo Andongo, Dr Welwitsch! n. 1257, bush 5—7 feet high with erect trunk 2—2}in. in diameter and spreading branches towards the top, branches and fruit tomentose, at first whitish, soon becoming rufous, leaves dark-green with a high polish, in stony woods at Barrancos de Catele, young fruit in December. Mr HIERN, ON EBENACE. 101 Plate Ill. Fig. 1. Flowering branch, natural size. a. Flower unexpanded, magnified 5 diameters. b. Flower expanded, magnified 5 diameters. c. Interior of corolla with stamens, laid open, magnified 6 diameters. d. Pistil, magnified 5 diameters. Fig. 2. Flowering branch of another form of the same species, natural size. e. Flower unexpanded, magnified 5 diameters. jf. Flower expanded, magnified 5 diameters. 12. EUCLEA FRUCTUOSA, sp. nov. E. folixs obovato-oblongis, basi in petiolum cuneatis, alternis vel suboppositis ; cymis femineis racemosis vel paniculatis, 3—20-floris ; calyce 4—5-lobo, tubo in fructu farcto, lobis deltoideis parvis ; coroll4 4—5-fida (2), interdum ad fructus apicem marcescente; fructibus numerosis, fulvo-pubescentibus. Varying in size from a small to an arborescent shrub with softly pubescent fulvous and terete branches spreading at 35’—40° with the stem. Leaves obovate-oblong, cuneate at base into the petiole, coriaceous, quickly glabrescent and nitescent, alternate or subopposite ; margins reflexed, net-veins numerous and delicate; 1—4}in. in length by 3—1} in. in width; petioles 3,—2in. in length, pubescent. 2 Q Racemes or panicles j—1in. in length, bearing from 3—20 flowers and nearly as many fruits, pubescent; pedicels short not exceeding ;;in. in length, dilated upwards in fruit to articulation with calyx. Fruit pale or darker, fulvo-pubescent, about 4in. in diameter, 1-celled, 1 seeded; embryo somewhat curved and tending to be incumbent; albumen not ruminated. Calyx 5-lobed; lobes deltoid, acute, small; tube consolidated in fruit and bearing fruit at its apex. Corolla 4—5-fid (?) ; sometimes marcescent ; lobes ovate. Known only in fruit. Grows in dry places, &c. East Tropical Africa. Zambesia, Luame river mouth, 8 Feb. 1861, Dr Kirk!; between ,Tette and the sea coast, 16 March, 1860, Dr Kirk!; Zanguebar, Dar Salam, October to December, 1868, Dr Kirk! 13. EvcLEA NATALENSIS, Alph. DC. Prodr. vit. p. 218, n. 10 (1844). E. foliis alternis, angusté ellipticis, basi cuneatis, petiolatis, undulatis, glabrescentibus ; cymis femineis racemosis, 8—10-floris ; calyce fructifero 4—5-fido, tubo farcto, lobis deltoideis ; fructibus subglabris. E. macrophylla, E. Mey. d, non a, b, Zwei Pflanz. Doc. Drég. p. 184 in Flora 1843. Royena macrophylla, E. Meyer, d! in Hb. DC. (Prodr. 1. c.) Young parts pubescent. Leaves alternate, erect, narrowly elliptical and cuneate at base into petiole, coriaceous, glabrescent, 2—4in. in length by }—4in. in width; margins undu- lated; petioles ;3—+in. in length. 9 Racemes solitary about }in. in length, bearing 8—10 flowers and about 4 fruits. Pedi- cels very short, dilated upwards in fruit to articulation with calyx. Calyx 4—5-fid, glabrescent ; lobes deltoid, acute; tube consolidated in fruit, with small spreading limb, and bearing fruit at its apex. Berry dark, sub-glabrous, globular }—4in. in diameter, 1-celled, 1-seeded; seed globose, black, marked outside with 3 longitudinal lines, albumen somewhat ruminated. Port Natal. Drege!; Peddie! 102 Mr HIERN, ON EBENACE. 14. EUcLEA BILOCULARIS, sp. nov. E. foliis alternis oppositis et in tribus verticillatis, obovatis, apice rotundatis, basi cuneatis, breviter petiolatis ; cymis femineis racemosis, sub-9-floris ; pedicellis brevissimis; floribus tetra- meris; calyce 4-dentato ; corolla breviter 4-lobd ; staminodiis 0; ovario biloculari, glabro, loculis biovulatis. Glabrous. Branches at 50°, sometimes whorled 3 together. Leaves obovate, cuneate at base, rounded at apex, somewhat undulating, coriaceous, alternate opposite and in whorls of 3: veins inconspicuous, in relief on both sides, dark green above, ruddier beneath; 2—3 in. long by 4—12 in. wide, shortly petiolate; petioles about 5 in. long. Racemes of 9 flowers (in bud) short, about 4in. long, bearing about 9 very short pedicels. Flower-buds ;4, in. long, tetramerous, with short cup-shaped 4-toothed calyx and corolla shortly 4-lobed. Staminodes 0. Ovary 2-celled, with 2 ovules in each cell, glabrous. East tropical Africa, Zanzibar, Dr Kirk! A male plant from Madagascar collected by Bojer! may belong to this species; it has 16—18 stamens. 15. E. MACROPHYLLA, E. Mey, Cat. Pl. Exsicc. Afr. Austr. Drég. p. 7 (1837). E. foliis in tribus verticillatis vel oppositis, obovato-oblongis, breviter petiolatis, subcoriaceis ; cymis femineis 8—15-floris, floribus tetrameris, calyce 4-fido, corolla 4-fidd, staminodiis 0, ovario 4- vel raro 6-loculari, glabro. Alph. DC. Prodr. viii. p. 218. n. 11 (1844). Glabrous. Stem nodose; branches at 60°, often verticillate three together, straight. Leaves obovate-oblong, rounded at apex, cuneate at base, shortly petiolate, sub-coriaceous, opposite or subverticillate three together; margins reflexed, plane or wavy; veins delicate ; 2—4 in. in length by }—1} in. in width; petiole 3; to } in. in length. Q. Flowers in cymes which measure }—1 in. in length and bear 8—135 flowers; pedicels + to Lin. in length; flowers tetramerous. Calyx 4-fid, shortly cup-shaped, with deltoid-pointed lobes. Corolla campanulate, 4-fid; lobes obtuse or mucronate. Stamens 0 or represented by a few hairs at circumference of disk. Ovary glabrous, 4- or rarely 6-celled, with one ovule in each cell; at the upper part the ovary is frequently 2-celled, according to Dr Atherstone, in consequence of two of the dissepiments being false; styles 2. South Eastern districts, Cape of Good Hope. Enon in woods under 500 feet high, March, Uitenhage, Drege! in @ flower; Grahamstown, Dr Atherstone! 461. 16. EUCLEA DAPHNOIDES, sp. nov. E. foliis alternis oppositis vel in tribus verticillatis, oblanceolato-oblongis, subsessilibus ; cymis racemosis; floris 4—5-meris ; calyce 4—5-fido ; corolla profunde 4—5-lobd ; staminibus circiter 12, uniserialibus, in flore femineo 0 vel + effetis; stylis 2—3; ovario glabro, 4- vel 6-loculart. Mr HIERN, ON EBENACE. 103 Glabrous shrub, 2—4 feet high or more, or even a low tree. Stem shining and turning pale yellowish; branches at 40°—50° with stem, alternate opposite or subverticillate. Leaves alternate opposite or 3 in a whorl, varnished on surface, crowded, oblanceolate-oblong, thickly coriaceous, flat or wavy, subsessile with thick articulation, 1}—3 in. in length by } to } in. in width. Cymes racemose, much shorter than the leaves; pedicels 3, in. long; bracts small and slender. $- Flowers tetramerous, nearly glabrous, small. Calyx 4-fid. Corolla deeply 4-lobed. Stamens about 12, in one row. Ovary rudimentary. . Flowers 11—21 in cyme, 7; in. in length by +; in. in width, ovoid, glabrous. Calyx zy In. in height by 7; in. in width, 4—5-fid. Corolla openly campanulate, with nearly erect lobes, 7; in. in height by 5 in. in width, 4—5-lobed. Staminodes 0 or 4, inserted at base of interior of corolla or around base of ovary, glabrous, without anthers. Styles 2—3, 4. in. in length, somewhat concave as seen from inside; stigmas bilobed at apex, projecting beyond the corolla; ovary glabrous, ovoid, j; in. in length and width, 4—6-celled ; ovules pendulous, solitary in the cells. Fruit globular, } in. in diameter, dark, glabrous, 1-seeded, 1-celled; seed marked outside by 3 depressed longitudinal lines; fruiting calyx small; albumen not ruminated but testa introverted at apex; embryo slightly curved. Nearly related to #. racemosa L. from which it differs by its oblanceolate-oblong and longer leaves and its longer and more numerously flowered racemes, by its ovary being sometimes 6-celled, and by its 12 stamens being in one row in the only g specimen examined. South Africa. South-western districts of Cape Colony and Natal. In a walk by the Baaken’s river under Fort Frederick, Algoa Bay, 14 Dec. 1813, Burchell! 4356, in 9 flower; on the rocky side of the mountain, also on the western bank of Wagenbooms river on the north side of Lange Kloof, 11 March 1814, Burchell! 4909, in 9 flower; Cape of Good Hope, Ecklon and Zeyher! ; Natal, W. T. Gerrard! 1506, 1606, in g flower-bud. 17. Evuciea KELLavu, Hochst. in pl. Schimp. Abyss. exsice, sect. i. n. 1078 (1842). E. foliis suboppositis, obovatis vel oblanceolatis, apice rotundatis, bast cuneatis, breviter petiolatis; cymis racemosis; floribus diacis, 4—5-meris ; corolla 4—5-fidd; staminibus 12, in flore femineo 0; stylis 2; ovario glabro, 4-loculart. Hochst. Nov. Gen. pl. Afr. in Flora 1843, p. 83; Alph. DC. Prodr. vim. p. 672 (1844) ; Rich. Fl. Abyss. ii. p. 24. t. 66 (1847). Myrsine Kellau, Hochst. in pl. Schimp. Abyss. exsicc. sect. i, n, 159 (1840). Kellaua Schimperi, Alph. DC. in Ann, Se. nat. ser. i, vol. Xvi. p. 209 (1842), Prodr. Vill. p. 290 (1844). Shrub or small tree, glabrous, dicecious ; branches at 88’—45° with stem, subopposite, straight. Leaves obovate or oblanceolate, shortly petiolate, sub-coriaceous, rounded at apex, wedge-shaped at base, subopposite, shining and of a rich brown colour on upper face, paler beneath; veins delicate, flat or wavy at margins, spreading; 1—2in. in length by }—1in. in width; petiole ;i—} in. in length, Flowers racemose with lanceolate bracts at base of pedicels, tetramerous or pentamerous. 104 Mr HIERN, ON EBENACE. ¢ Racemes 3—1in. in length, bearing 9—13 flowers, spreading, dark; pedicels slender —1in. in length, the lower ones the longer, alternate or opposite, patent; flowers }—1in. in length. Calyx short, 4-fid, with apiculate deltoid erect lobes. Corolla campanulata, 4-lobed; lobes 1—1 depth of corolla, erect, oval. Stamens 12, free, 8 in one row and 4 interior and inserted lower at base of interior of corolla, included; anthers erect, with a few hairs at top or glabrous, 2-celled, dehiscing laterally from apex. Ovary rudimentary ; styles 1—2. g@ Racemes 2—}in. in length, bearing usually 11 flowers, dark; glabrous or glandular ; pedicels ;,—, in. in length, patent, sub-opposite; flower ;4,in. in length, campanulate. Calyx ;\; in. in height with 4 or 5 deeply divided erect deltoid acuminate lobes, persistent. Corolla campanulate, twice the height of the calyx, with 4—5 lobes divided more than half the depth of the corolla. Stamens 0. Ovary 3,in. in height by jin. in thickness, conical, glabrous, 4-celled, cells 1-ovuled; styles 2. Fruit globose, }in. in diameter, glabrous, 1-celled, 1-seeded; seed filling the cavity of the fruit, marked externally by 3 longitudinal lines; albumen horny, not ruminated but with introversion of testa at apex; embryo slightly curved, tending to be incumbent. Abyssinian name of the fruit; Aédlaw. Abyssinia. Schimper! i. n. 159, among hills, valleys and low places near Adoa. In fruit, 1 June 1837. u. 1078. On mount Sina, near Adoa. In ¢ flower, 13 November 1838. ll. 1527, 1.1919. Near Axus. In ¢ flower. 913. Agrima, 6000 ft. alt.; Legua, 5000 ft. alt. 1852. Fs 35 80. = 5500—6500 ft. alt. October 1862. - Quartin-Dillon and Petit./ Scholoda. 18. EvucieA RAcEMosA, Linn. Syst. veg. edit. xm. p. 747. Cur. Murray (1774). E. foliis alternis vel oppositis, obovatis vel oblongo-obovatis, apice rotundatis, basi cuneatis, breviter petiolatis; cymis racemosis; floribus diwcis, 4- raro 5—6-meris; corolla profunde lobata ; staminibus 12—18, in flore femineo 2—4 effetis; stylis 2; ovario toto pubescent vel glabro, 4-loculari. Jacq. Fragm. t. 1, f. 5, t. 63, f. 8 (1800—9); Alph. DC. Prodr. vim. p. 219. n. 15 (1844). Padus foliis subrotundis, fructu racemoso, Burm. Afr. p. 238, t. 84, f. 1 (1739). Glabrous shrub 24 to 6 feet high, or small tree 18 feet high. Branches making 30° with stem, purplish. Leaves obovate or oblong-obovate, coriaceous, alternate, subopposite, or opposite, marked with obscure transverse veins, green on the upper surface, pale beneath, margins somewhat reflexed, wavy or nearly flat, subsessile or shortly petiolate, rounded at apex, cuneate at base, 4—2}in. in length by }—1}in. in width; petioles j4—s%, in. in length. Bracts narrow, at base of pedicels, solitary, linear-lanceolate, Mr HIERN, ON EBENACEA, 105 & Racemes }—1}in. in length, shorter than the leaves from the axils of which they spring, drooping; pedicels ;—}in. in length, 5—13 in cyme, articulated at apex; flowers 5—t in. in joncite campanulate, 4- or rarely 5—G6-merous, glabrous. Calyx short, lobes del- toid, about half length of calyx. Corolla campanulate, open, deeply lobed, much raised above the calyx; lobes oval, obtuse or acutish, spreading or erect but not reflexed, of a dirty white colour. Stamens 12—18, in two rows, inserted at base of interior of corolla or on an hypogynous ring; the inner ones smaller and often connate at the base with outer ones; anthers lanceolate, thick, 2-celled, with a few hairs or glabrous, included or exserted, erect, dehiscing laterally and widely from apex, ;t4— 4 in. in length; pollen white; filaments slender, 3o—sy In. in length, often united in pairs at base, glabrous. Ovary rudimentary, hairy or glabrous; styles 2, distinct, erect, terete, white. ¢g. Racemose cymes }—1}in. in length, usually shorter than the leaves but sometimes longer, drooping in fruit; pedicels about #, im. length, 9—13 in cyme. Flowers ovoid, rather smaller than the ¢ apes tetramerous or rarely pentamerous. Calyx hemispherical; lobes ovate, acute, about half depth of calyx. Corolla ovoid, deeply lobed; lobes not reflexed. Staminodes 2—4, glabrous. Ovary hairy or glabrous, 4-celled, cells 1-ovuled; styles 2; fruit globular, glabrescent or glabrous, black, 1-celled, 1-seeded, }in. in diameter. Bark grey, smooth. Wood hard, heavy, employed by wheelrights and turners; answers very well for wooden screws, but is chiefly used as fuel. {Dr Pappe, Silva Capensis, p. 21 (1854)}. The variety Burchellii with glabrous ovary may be a distinct species. It is a tree 18 feet high with erect trunk and ascending branches and oblong-obovate leaves; bark entire, turning white; ovary globose; styles 2, short; staminodes 2—4, inserted on corolla or around base of ovary. Cape of Good Hope, southern and western districts. Drege!; Talbot! ; Reeves! ; Wright! ; Boivin! ; Bowie! ; Alexander-Prior!; Oldenburg! ; Nelson!; Hove!; Fenhen ! 3356; Harvey ! 574; Burchell! 397, 807, 3219 (var. Burchellii), 3806, 8295; Hondeklip Bay, Clanwilliam, Rev. H. Whitehead! 19. EvcLea UunpULATA, Thunb. Nova Genera Plantarum (v.) p. 86 (1784). E. foliis alternis vel oppositis, obovatis (vel in var. oblanceolatis), apice obtusis, basi cuneatis, breviter petiolatis, wndatis (in var. parvis et subplanis) ; cymis racemosis, 3—S8-floris; floribus dioecis, tetrameris; staminibus 10—15, plerumque geminatis, in flore femineo 0; stylis 2; ovario basi subpubescente, 2- vel 4-loculari, 4-ovulato. . Alph. DC. Prodr. vit. p. 219. n. 14 (1844). E. myrtina, Burch. Trav. Int. 8. Afr. 11 p. 588 (1824), Alph. DC. Zc. p. 217. n. 8. E. humilis, Eckl. et Zeyh. in Linnea, xx. p. 192 (1847). Glabrous dense shrub, extremities and flowers glandular but not hairy, 4 to 9 feet in feeht or a moderate sized tree, dicecious. Branches alternate or opposite, at 40° to 60° with stem, numerous. Leaves obovate or oblanceolate, coriaceous, shortly petiolate, wavy or in var. myrtina nearly flat, opposite or alternate, veins inconspicuous, cuneate at base, rounded or nearly so at apex, evergreen, }—14 in. in length by }—3 in. in width .(or 1 in, in width in variety myrtina); petioles 33—y5 in. in length. Bracts sometimes large Vou. XII. Parr I. 14 1 10 106 Mr HIERN, ON EBENACE. and leaf-like, caducous; flowers racemose, with divisions of corolla reaching to level of apices of calycine lobes. é. Racemes 2—4 in. in length, shorter than leaves, lax, bearing 5—7 flowers; pedicels 1—1 in. in length, slender; flowers 4; in. in length. Calyx openly cup-shaped, 4-fid, short; lobes deltoid, acute. Corolla hemispherical, 4-lobed, somewhat glandular outside; lobes oval, more than half the depth of the corolla. Stamens 10—15, mostly in pairs, inserted at base of interior of tube of corolla; anthers oblong, mucronate, with a few bristles near apex, dehiscing widely from apex; filaments slender. @. Racemes }—4 in. in length, nearly erect in flower, drooping in fruit, bearing 3—8 flowers; pedicels under ;; in. in length; narrow bracteoles sometimes present on middle of pedicels. Flowers 3;in. in length. Calyx short, campanulate, 4-lobed, lobes deltoid, extending less than half-way down the calyx. Corolla 4-partite, erect or spreading, in bud cylindrical, somewhat glandular outside along middle of lobes; lobes oval. Stamens 0. Ovary and 2 styles together rather longer than corolla; styles as long as ovary, at first erect and conti- guous, glabrous, bifid at apex, deciduous; ovary somewhat hairy at base, hairs white, glabrous above, 2—4-celled, 4-ovuled; ovules oblong. Berry globular, =;—1 in. in diameter, purple or red, glabrous, edible, 1—2-celled, ultimately only 1-celled, 1-seeded. Bark whitish grey rough, wood brown hard close-grained and fit for joiners’ fancy-work, veneering, &e. (Dr Pappe, Silva Capensis, p. 21 [1854)). Var. 8. myrtina. Leaves +;—1 in. wide, oblanceolate, nearly plane; fruit black; about 4 ft. high. Known only in fruit, but probably a form of this species. (#. myrtina, Burch.) The fruit is sweet, with some astringency; called, as well as other species of the genus, guarribosches, and the fruit guarri, by the Hottentots in South Africa. Cape of Good Hope, Kalahari region and Trans-Vaal. Drege!; Reeves!; Dr Pappe!; Burke! (Trans-Vaal); Masson/; Alexander-Prior!; Dr Thom! 243, 386; Cooper! 408; Mac Owan!; Zeyher! 3358; Ecklon and Zeyher! 1124 (K. humilis, Eckl. et Zeyh.); Burchell! 1792 (2162, 2573, E. myrtina, Burch., Kalahari Region), 2943, 3168, 7198. EXCLUDED SPECIES OF EUCLEA. Euclea herbacea, Lour. Fl. Cochinch. p. 629 (1790). Cfr. Euphorbiacez. Luclea pilosa, Lour. loc. cit.= Diospyros pilosa, Alph. DC. Ill. MABA, J. R. et G. Forster, Characteres Generum Plantarum, p. 121. t. 61 (1776). Flores diect, rarissime monaci vel polygami, plerumque trimeri rarius 4—6-meri. Calya campanulatus vel oblongus, non plicatus, lobatus vel truncatus ; corolla campanulata vel tubulosa, lobis in prefloratione sinistrorse contortis. Flos masculus; stamina 3—, plerumque glabra rarius pilosa vel pubescentia. Ovarium abortiwum. Flos femineus ; staminodia 0—2 , plerumque pauca; ovarium 3- vel 6-loculare, 6-ovulatum ; fructus plerumque mediocris, baccatus. Arbores vel frutices, foliis alternis integerrimis, inflorescentia axillari vel rarius laterali. Alph. DC. Prodr. vu. p. 240. n. vit. (1844). Mr HIERN, ON EBENACE. 107 Pisonia (sp.) Rottb. in Nye Saml. Kong. Danske Skrift. vol. m. p. 536. t. 4 f. 2 (1788) Ehretia (sp.), Willd. Phytogr. I. p. 4 t. 2. f. 2 (1794). Ferreola, Roxburgh, Pl. Coromandel, I. p. 35, t. 45 (1795). Ferriola, Roxburgh, Hort. Bengal, p. 72 (1814), Fl. Ind. edit. 1832. vol. m1. p. 790. Macreightia, Alph. DC. Prodr. vim. p. 220. n. v. (1844). Holochilus, Dalzell in Kew Journal of Botany, Iv. p. 290 (1852). Rhipidostigma, Hasskarl, Retzia, 1. p. 103 (1855). Flowers dicecious or rarely polygamous or very rarely moncecious, usually trimerous, occa- sionally 4—6-merous. Calyx usually 3-fid, sometimes 4—6-fid or -partite or shortly lobed, rarely truncate and entire; more or less campanulate at least in flower, sometimes accrescent but less so than in many species of Diospyros, not plicate. Corolla usually 3-lobed, exceeding the calyx, campanulate or tubular; lobes contorted sinistrorsely as regarded from within. Stamens in 6 flower 3—20 usually about 9 and glabrous except in § Trichanthera, distinct or some or all united by their filaments in pairs or otherwise; anthers oblong or lanceo- late-linear, dehiscing longitudinally by lateral slits; filaments inserted at base of corolla or hypogynous; staminodes in @ flower 0—2, usually fewer than in ¢g flower, glabrous or hairy. Ovary in 6 rudimentary, hairy or glabrous; in 2 3- or 6-celled, hairy or glabrous ; style 3-lobed or styles 3; ovules 6, solitary in the cells or 2 together in 3-celled ovaries; rarely an ovary is 3-celled with 3 imperfect septa between the pairs of ovules not reaching the central axis of the ovary. Fruit usually globose or ovoid, glabrous or hairy, 1—6-celled and -seeded, usually not exceeding 1 in. long, baccate or dry; seeds as in the Order, in a few species with ruminated albumen. Fruiting calyx spreading or cupuliform. Trees or shrubs usually with hard wood, widely distributed in most countries where the Order is represented but absent from the Cape of Good Hope, though occurring in Natal and other parts of Africa. Leaves always alternate simple and quite entire, smaller for the most part than in Dio- spyros, but reaching 10} in. in length in M. punctata. Flowers solitary or in short cymes either axillary or very rarely lateral on the older branches. The name is adopted from that locally used in the Friendly Islands for plants of this genus, Maba is also given by the natives in the vicinity of the Congo river, West tropical Africa, to the fruit of the oil-palm (Eleis guineensis). Maza may be divided into the following sections, a key to which is subjoined. Anthers glabrous or in a few species slightly hairy. Flowers trimerous or occasionally tetramerous or rarely in J. lancea pentamerous. Calyx-lobes not much imbricated. Ovary densely hairy (except in M. obovata, R. Br.) Staminodes 0. Ovary 3-celled. § 1. FERREOLA. Staminodes 3—6. Ovary 6-celled. § 2. MACREIGHTIA. Ovary glabrous (pubescent or nearly glabrous in M. Seychellarwm). Flowers sessile or subsessile. Ovary 3- or 6-celled. § 3. HoLocuiLus. Flowers crowded in short branched or fascicled cymes (@ flowers solitary in M. lamponga). Ovary 6-celled. § 4, RHIprIDosTieMa. Calyx-lobes rounded and much imbricated so as to make the ca- lyx appear subtruncate. § 5. BARBERIA, Anthers pilose. Flowers 8—6-merous. Ovary hairy, 6-celled. § 6. TRICHANTHERA. 14—2 108 Mr HIERN, ON EBENACE4‘. § 1. FERREOLA. Fruit reddish, brown, or dark-coloured. Fruiting calyx very small, usually not cupuliform, flat or reflexed. ¢ flowers tubular. Fruit subglabrous, shining. Fruiting calyx trifid. | Fuliginous-hispid. Leaves about 1 in. long. 1. M. diffusa. | Glabrescent. Leaves 1$—44 in. long. Fruit globose, } in. in diameter. Calyx glabrate. 2. M. Mualala. Fruit ellipsoidal-oblique, }in. long. Calyx somewhat hairy. 3. JL hemicycloides. Fruit tomentose. Fruiting calyx tripartite. | Stamens 4—5. 4. ML. acuminata. | Stamens 12—16. Leaves oblong. Stamens about 12. 5. M. oblongifolia. Leaves oblong-ovate. Stamens 13—16. 6. M. ovalifolia. Fruiting calyx accrescent or not very small, usually cupuliform (sometimes small in M. buxifolia). g flowers subsessile in short cymes. @ flowers sessile or subsessile. Fruit hairy. ¢ flowers with tubular corolla. | Stamens 3—6 (-7). Flowers trimerous. Leaves cordate at base, subsessile. 7. MW. foliosa. Leaves not cordate at base, shortly petiolate. 8. MW. rufa. Stamens 9 (in trimerous flowers, 4 in a tetramerous one). Leaves glabrous, elliptical. 9. M. laurina. Leaves hairy, lanceolate-oblong. 10. M. nigrescens. Fruit subglabrate. Corolla campanulate. Leaves without conspicuous net-veins. Calyx hairy. Leaves usually more than 1} in. long. Branches nigro-verrucose. Stamens 15—17. 11. M.sandwicensis. Branches smooth. Stamens 6—12. & flowers 1—3 together. 12. MW. bu«ifolia. é flowers several together. | Leaves lanceolate, paler beneath. 13. M. lancea. Leaves obovate, of same colour on both sides. Q flowers solitary. 14. M. obovata. 2 flowers 3 together. 15. M. geminata. Calyx glabrous, at least in fruit. Leaves about 1 in.long. 16. JL humilis. Leaves highly reticular. Bracts not much imbricated. Leaves elliptic-oblong, about 2 in. long. 17. I. reticulata. Leaves elliptical, 3—4 in. long. 18. M. compacta. Bracts much imbricated. 19, M. Hillebrandii. $ flowers pedicelled in manifest cymes. ¢ corolla tubular. @ flowers stalked. Stamens 3—6. Leaves oval, obtuse, glabrescent. 20. M. elliptica. Stamens 9. Leaves ovate-oblong, acuminate at apex, hairy. 21. Jf sumatrana. | Fruit covered with white efflorescence. 22. M. Vieillardi. Cfr. [23. JL major. 24. IM, Andersoni:] Mr HIERN, ON EBENACEA. 109 § 2. MAcrEIGHTIA. Glaucescent. (N. America and West Indian Islands.) Leaves rotund or ovate,. spinulose-apiculate. 25. M. Grisebachit. Leaves obovate, not. spinulose-apiculate. Albumen deeply ruminated. Net-veins conspicuous. 26. M. caribea. Albumen not ruminated. Veins few. 27. M. intricata. Dull or hairy. (S. America, Mexico, and West tropical Africa.) Albumen not ruminated. Stamens glabrous. Flowers campanulate. Fruiting calyx somewhat cupuliform, not very small. (S. America and Mexico.) Cymes usually 3-flowered, ,4—+ in. long. | Leaves whitish beneath. 28. I. albens. Iiiieaves not whitish beneath. Leaves oval or obovate. 29. M. inconstans. Leaves obovate-lanceolate, membranous. 30. IL. acapulcensis. Leaves lanceolate-oblong, coriaceous. 31. MW. salicifolia. | é. Cymes many flowered, } in. long, @ flowers solitary. 32. M. Pavoni. Albumen ruminated. Stamens somewhat hairy. § flowers tubular; fruiting calyx very small, flat. (Africa.) 33. M. Manni. § 3. HoLocuiLus. Flowers campanulate or with short tube. (Africa.) Ovary 3-celled; cells 2-ovuled. Calyx shortly 3-lobed, pubescent. Ovary more or less hairy. Leaves narrowly elliptical, obtuse. 34. M. Seychellarwm. Ovary: quite glabrous. Leaves lanceolate, acute. 39. M. lanceolata. . Calyx truncate entire, glabrous. 36. M. natalensis. Ovary 6-celled; cells 1-ovuled. Leaves lanceolate-oblong ; flowers several together. Branches dark. 37. I. abyssinica. Leaves oval. flowers 3 together. Branches argenteo-cine- reous. 38. IL. quiloénsis. Flowers tubular. (India.) 39, ML. micrantha. 110 Mr HIERN, ON EBENACEZ. § 4. RuaIpPmDosTIcMA. Dicecious. Stamens 8—18. Glabrous. Leaves not cordate at base. é. Cymes rather lax. Q. Flowers solitary. Corolla-lobes acuminate. 40. M. lamponga. 2. Flowers cymose. Corolla-lobes not acuminate. Leaves submembranous. Stamens glabrous. 41, M, merguensis. Leaves coriaceous. Filaments often minutely cili- ated, Albumen not ruminated. 42. M, fasciculosa. Leaves coriaceous. Albumen ruminated. 43. M. ruminata. &é. Cymes dense. 44, M. confertiflora. Pubescent. Leaves cordate or subcordate at base, 45, M. punctata. “Hermaphrodite. Stamens 4—5.” Cymes about }in. long. Leaves oblong, 46. M. Teijsmannt. Cymes very short. Leaves oblong-lanceolate. 47. M. hermaphroditica. Cfr. [48. I. javanica.] § 5. BARBERIA. Ovary glabrous. Staminodes about 8. 9 Cymes 3—5-flowered. 49. M. Maingayt. Staminodes about 16. Flowers subsolitary. 50. M. Motleyz. Ovary shortly pubescent. Glabrous. Leaves 2—4in. long, more or less narrowed at base; petioles }—2in. long. 51. M. myrmecocalyz. Leaves }—2in. long, rounded at base, with shortly tomentose mid- rib; petioles tin. long. 52. WM. Beccari. § 6. TRICHANTHERA. Polygamous. Ovary ovoid-conical. Leaves appressedly flavo-sericeous beneath, not cordate. 53. MW. sericea. Ovary globose at base, narrowly conical above. Leaves supra-cordate 54. J. cordata. Diecious. Ovary subglobose. Leaves not flayo-sericeous beneath. Flowers 3- (rarely 5-) merous. | Shoots with spreading hairs. 55, M. myrmecocerpa. | Shoots with appressed hairs. 56. M, myristicoides. Flowers 5—6-merous, Flowers arising from the old wood. 57. ML. cauliflora. Flowers axillary from the young branches, Staminodes 11—13, somewhat pilose 58. M. Hilatrei. Staminodes 25—380, nearly glabrous, 59. M. Mellinoni. Mr HIERN, ON EBENACEA!. 111 1. MABA DIFFUSA, sp. nov. ML. ramulis patentibus, fuligineo-pubescentibus ; foliis ovatis vel ovalibus, uncialibus, apice obtuse angustatis, basi subrotundis, glabris, nitentibus, subcoriaceis, breviter petiolatis ; fructibus ellipsoideis, appresse subsericeis, trilocularibus, 1—2-spermis, brevissime pedunculatis ; calyce fructifero minimo, trifido, non appresso. Stem and branches terete, dark or cinereous; branches fuliginous-pubescent, patent, slender. Leaves ovate or oval, nearly rounded at base, obtusely narrowed at apex, glabrous and shining on both sides, subcoriaceous, margins thickened, somewhat wavy ; midrib depressed on upper side; lateral veins patent very numerous and delicately raised on both sides; of a rich brown colour on both sides when dry; $ to lin. in length by 4 to 2in. in width; petioles j,in. in length, pubescent. Known only in fruit. Fruit shortly pedunculate, near ends of branches, solitary; fruiting peduncle J, or horizontal, small, somewhat pubescent, about ;5in. in length, roundedly 3-fid. Fruit jyin. in length. Fruiting calyx loosely concave somewhat appressedly silky, of rich brown colour, straight, ellipsoidal, 4 to 4in. in height by 4+ to $in. or more in thickness, 3-celled, 1—2-seeded; seeds black, fin. in length; albumen not ruminated; embryo nearly flat. N.W. Madagascar, Pervillé! 2. Masa MuALAta, Welw. MSS. DM. glabra, foliis ellipticis, apice seepius obtuse acuminatis, bast leviter angustatis vel sub-rotundis, tenuiter coriaceis, persistentibus, nitentibus, reticulatis, breviter petiolatis ; fructibus solitariis vel binis, subsessilibus, globosis, glabris; calyce fructifero trifido, minimo, patente, glabrato. A fine glabrous tree, 15—85 feet high in the interior of the country, or near the sea-coast scarcely more than a bush 3—5 feet high; very rarely flowering. Trunk strict; branches terete, leafy. Wood very hard, valuable, black in the centre but not always so. Leaves alternate, elliptical, in most cases obtusely acuminate, slightly narrowed at base or nearly rounded, thinly coriaceous, evergreen, deep green, highly polished, 1}—4}im. long by #—1}in. wide, delicately reticulated; midrib depressed above; margins slightly undu- lated; petioles ;4—in. long. Flowers unknown, @ axillary, in very short 1—3-flowered cymes. Fruit solitary or two together, subsessile, globose, shining, glabrous, black-purplish, slightly nerved, about }in. in diameter, 1-seeded; seed globose, nearly fin. in diameter; albumen white, cartilaginous, not ruminated; fruiting calyx 3-fid, spreading, {—}in. in dia- meter, glabrate; lobes ovate, subacute. West tropical Africa, Distr. Golungo Alto, in dense woods, fruits in March, Dr Welwitsch ! 2539, 2540, 2541; Do. Distr. Loanda, very rare, in thickets, Dr Welwitsch! 2542; native name Mualdla. 3. MABA HEMICYcLOIDES, F. Muell. ex. Benth. Fl. Austr. iv. p. 290. n. 3 (1869). M. glabrescens, foliis ellipticis vel oblongis, utrinque plus minus angustatis, apice obtusis, subcoriaceis, breviter petiolatis; fructibus solitariis, brevissime pedunculatis, subglabris, oblique ellipsoideis ; calyce fructifero minimo, patente, trifido, leviter pubescente. 112 Mr HIERN, ON EBENACE. A small tree; branchlets slender, somewhat hirsute with dark hairs at extremities, quickly glabrescent, dark cinereous or brown, terete. Leaves elliptical or oblong, narrowed more or less at both ends, usually with an obtuse apex, thinly coriaceous, glabrous; margins with small undulations, just reflexed; midrib depressed above; lateral veins delicate, nume- rous, raised on both sides, at 60° to 70°; 2} to 4}in. in length by 1—l4in. in width; petioles }1in. in length. Known only in fruit; fruiting peduncle ,,—j;in. in length, - not thick, pubescent ; fruit solitary, near ends of branches, glabrous or nearly so, pale brown, oblique, ellipsoidal, about }in. in height by 3—35 in. in thickness, tipped somewhat laterally with remains of style; fruiting calyx small, horizontal, 3-fid, —3im. in diameter, covered with scattered appressed short pale hairs; lobes deltoid. Australia, Queensland, Rockingham Bay, Dallachy! 4, MABA ACUMINATA, M. foliis ellipticis valde acuminatis, basi rotundatis vel parum angustatis, submembra- naceis, breviter petiolatis ; corolle tube quam calyce duplo longiore ; staminibus 4—5; fructibus globosis, tomentosis et sparse pilosis; calyce fructifero tripartite, minimo. Macreightia acuminata, Thw. Enum. Ceyl. Pl p. 424 n. 3 (1864). A moderate sized tree with terete erect-patent branches. Young parts pale brown sericeo- pubescent, afterwards becoming dark and glabrous. Leaves elliptical, long-acuminate, rounded or nearly so at base, in the dry state pale greenish glabrous and shining on upper side with scarcely raised veins, pale brown sericeous or subpubescent on under-side with raised clear lateral veins anastomosing near margin and sericeous prominent midnb, submembranous, shortly petiolate, 2—5in. in length by } to 1}im. in width; petioles j;—}in. in length, pubescent. Bracts imbricated, sericeous. g. Tube of the corolla twice as long as the calyx, Jin. in length; stamens 4—5; ovary pilose. 9. Fruit globular, pale brown, appressedly sub-tomentose-pubescent, 3—in. in diameter ; fruiting calyx not auricled. Ceylon, Thwaites! C.P. 3718. 5. MABA OBLONGIFOLIA. M. foliis oblongis, acuminatis, subcoriaceis, basi rotundatis, subtus secus nervos cum petiolo brevi_ sub-ferrugineo-hispidis, denique glabris; floribus masculis solitariis crebris subsessilibus, calyce breviter lobato, staminibus 12 glabris; floribus femineis solitariis breviter pedunculatis, calyce tripartito, hispido, non accrescente, staminodiis 0, fructibus subglobosis tomentosis. Macreightia oblongifolia, Thw. Enum. Ceyl. Pl. p. 183. n. 1. (1860), p. 423 (1864); non Marcreightia oblongifolia, Kurz, A small tree; young parts very hispid, subferruginous; branches terete, quickly turning dark and glabrous, spreading at about 40°, Leaves oblong, acuminate, subcoriaceous ; upper side brown (often of a rich deep colour) shining and glabrous when dry, midrib and lateral veins depressed; under-side palish brown, subpubescent, lateral anastomosing veins and especially midrib raised prominent and pubescent; 3 to 7$in. in length by 1} to 3in. in width; petioles ;4—?in. in length, glabrescent, at first hispid. Mr HIERN, ON EBENACEA, 113 g. Flowers subsessile solitary crowded on short axillary densely pubescent branches ; buds oblong, subferruginous, sericeous-pubescent, about Jin. in length. Calyx 4 in. long, 3-lobed at apex. Stamens 12, glabrous, in several rows, unequal, partly hypogynous and partly at base of interior of tube of corolla. Ovary minute, hairy. @. Flowers solitary, ferruginous, shortly pedunculate, hispid; bracts imbricated, large, hispid; peduncle ;;in. in length, hispid. Flowers #8, in. in length (not expanded in specimens), ovoid-oblong. Calyx }in. in length, with 3 deep diverging ovate acute lobes. Corolla 3-fid, glabrous inside. Stamens 0. Ovary covered with light ferruginous vertical hairs, 3-celled or, according to Dr Thwaites, 6-celled. Style divided at apex into 3 glabrous stigmas. Fruit subglobose, ferruginous-tomentose, lin. in diameter, fruiting calyx not accrescent nor auricled; 2- or 3-seeded; seeds black, glabrous, about din. in length by }in. in thick- ness, bounded by 2 plane contiguous sides and a curved surface, a horizontal section being a sector of a circle; a reddish raised line runs down middle part of outer surface of the seed; albumen not ruminated; radicle cylindrical, half as long again as the oblong cotyledons. Ceylon, Thwaites! C.P. 3396. 6. MABA OVALIFOLIA. M. foliis oblongo-ovatis, parum acuminatis, obtusiusculis, basi sepius rotundatis, subcoriaceis, glabrescentibus, breviter petiolatis ; floribus masculis solitariis, crebris, calyce inequaliter tri- dentato, (corolla 4-fidd@), staminibus 13—16, glabris, ovarii rudimento hirsuto. Macreightia ovalifolia, Thw. Enum. Ceyl. pl. p. 424. n. 2 (1864). Tree of moderate size; young parts pubescent, soon glabrescent and cinereous; branches terete, erect-patent. Leaves oblong-ovate, shortly acuminate, subcoriaceous, usually rounded at base, brown on both sides when dry, darker above, glabrescent, flat, nrargins just recurved, patent, shortly petiolate, midrib and lateral anastomosing veins raised beneath depressed above, 2 to 3hin. in length by 1 to 1?in. in width; petioles +in. in length, stout. Bracts imbricated, large, caducous. é. Flowers solitary, crowded on young short branchlets, ferruginous sericeous, ;in. in length before expansion, oblong. Calyx jin. in length, tubular, with 3 short acute teeth chiefly on one side, a deeper division being opposite. Corolla often bent sideways (closed in specimens), somewhat constricted about the middle, 4-fid, dark and glabrous inside. Stamens 13—16 (14 in one case examined), unequal, glabrous; ovary rudimentary, repre- sented by a bunch of hairs. Ceylon, Thwaites! C.P. 3717. 7. Mapa FoLiosa, Rich. ex Asa Gray in Proceedings of the American Academy of Arts and Sciences, Vol. v. p. 326 (1862). IM, foliis ovalibus vel ovatis, basi cordatis, coriaceis, confertis, subsessilibus ; floribus masculis 3—5-nis, brevissime cymosis, calyce campanulato-oblongo, breviter trifido, corolla breviter trifidd, staminibus 3, glabris; floribus femineis subsessilibus, cymis 1—3-floris, fructibus ferrugineo- tomentosis. Vou. XII. Parr I. 15 114 Mr HIERN, ON EBENACE. Young parts rufous or fuliginous, hirsute ; branches glabrescent, cinereous ; leaves crowded, subsessile, oval or ovate, cordate at base, midrib depressed above, veins indistinct, coriaceous, rufous-hirsute when young, glabrescent except on margins and midrib beneath, 1—2}in. long by #—lin. wide; petioles shorter than the emargination at the base of the leaves. g. Flowers on very short nodose pubescent 8—5-flowered cymes; flowers (in bud) ovoid-oblong, rufous-hirsute. Calyx Jin. long, campanulate-oblong, shortly 3-fid, smooth inside ; lobes deltoid; corolla shortly 3-fid, hirsute outside, glabrous inside; stamens 3, hypogynous, glabrous; filaments distinct; anthers linear, dehiscing laterally by longitudinal slits; ovary pubescent, small, rudimentary. ¢. Fruiting peduncles 1—3-flowered ; calyx 3-lobed ; fruit ferruginous-tomentose. Feejee Islands, Wilkes!; New Caledonia, Pancher/ 301; Muthuata and Ovolau, alt. 2000 feet, Feejee Islands, Asa Gray, l. c. 8. Mapa pura, Labill. Sert. Austr. Caled. p. 33. t. 36 (1824). M. foliis ovalibus vel oblongis, apice lanceolatis vel breviter et obtuse acuminatis vel rotundatis, basi angustatis vel rotundatis, junioribus utrinque rufo-sericeis, sepius glabrescentibus, coriaceis, breviter petiolatis ; inflorescentid et fructibus rufo-serieets ; floribus masculis 3—5-nis, brevissime cymosis, axillaribus, trimeris, corolla tubulosd, staminibus 3—6, glabris; floribus femineis solitariis, subsessilibus, staminodiis 0, ovario dense sericeo 3-loculari, Fructibus sub- globosis vel ellipsoideis, calyce fructifero cupuliformt. Alph. DC. Prodr. vu. p. 241. n. 10 (1844). M. sericocarpa, F. Muell. Fragm. v. p. 164 (1866), Benth. Fl. Austral. Iv. p. 289, n, 2 (1869). Mf. cupulosa, F. Muell. Fragm. v. p. 164 (1866), vi. p. 253 (1868). Diospyros sericocarpa, F. Muell. Austr. Veg. in Intercol. Exh. Ess., 1866—67, p. 35 (1867). D. cupulosa, F. Muell. 1. c. M. revoluta, Vieill. MSS. in Hb. N, Caled. n. 2876. A shrub or tree 20 feet high; branches terete, slender, spreading at about 45°—40, rufous-sericeous when young, leafy, Leaves oval or oblong, lanceolate or shortly and obtusely acuminate or rounded at apex, narrowed or rounded at base, coriaceous, appressedly rufous- sericeous when young, usually glabrescent, 1—4}in. long by 3—22in. wide; midrib depressed on the upper surface, margins recurved (sides revolute in MZ. revoluta, Vieill.); petioles +,—} in. long. é. Inflorescence rufous-sericeous, axillary on young branches; cymes 3—5-flowered ; common peduncle in, long; pedicels very short; flowers ovoid-oblong, }—}in.long. Calyx tubular, shortly 3-lobed, }—}in. long, crass, tomentose on both sides. Corolla tubular, shortly 3-lobed, sericeous outside, glabrous inside; lobes ovate. Stamens 3—6 (-7), glabrous, hypogynous; filaments slender. Ovary rudimentary, pilose. 9. Flowers solitary, subsessile, about }in. long, ferruginous-hairy; bracts imbricated, caducous. Calyx campanulate, shortly 3-fid. Corolla tubular, 3-lobed at apex, with rounded imbricated lobes. Staminodes 0, Ovary 8-celled, densely sericeous; style 3-lobed at apex. Mr HIERN, ON EBENACEAS. 115 Fruit ellipsoidal or subglobose, }—1 in. high, more or less sericeous, 3-celled, 1—3—+4-seeded ; fruiting calyx accrescent, cupuliform, trifid, reaching half way up fruit or higher, pubescent. Seeds oblong; albumen cartilaginous, not ruminated; embryo nearly straight. Australia, Queensland, Rockingham Bay, Dallachy! ; New Caledonia, Deplanche! 312, 446; Labillardiere!; Pancher !; Caldwell! ; Vieillard / 891, 892, 894, 895, 896, 2872 (2), 2876, 2880. 9. Mapa Laurina, R. Br. Prodr. Fl. Nov. Holl. p. 527. n. 1 (1810). M. foliis ellipticis vel oblongis, apice rotundatis, glabris, nitentibus, tenuiter coriaceis, petiolatis; jfloribus trimeris, subsessilibus, calyce late campanulato, crasso, corolla tubulosd, staminibus 9, glabris; in floribus femineis staminodiis 0, ovario 3-loculari, subglabro, dense service. Alph. DC. Prodr. vu. p. 241. n. 3 (1844), Benth. Fl. Austral, iv. p. 289 n. 1 (1869). A small tree with smooth dark bark and quite glabrous shoots; buds and inflorescence rufous-hairy. Leaves elliptical or oblong, rounded at apex, thinly coriaceous, glabrous, shining especially above, 3—5in. long by 1$—2}in. wide, margins incrassato-recurved, veins slender, raised on both sides; petioles 4in. long. 6. Flowers few together subsessile (ex Benth. Jc.) solitary or sometimes 2 together very shortly peduncled (ex R. Br. MSS.), trimerous; calyx fin. long, globose-campanulate, coriaceous, rather crass, with numerous soft subappressed cinereous-ferruginous hairs outside, glabrous inside; corolla yellowish white, tube cylindrical, twice the length of the calyx, hairy outside above the calyx, lobes rounded, one third the length of the corolla; stamens 9, glabrous, hypogynous, alternately in pairs and single, equal, pollen white; ovary subglobose, hairy, rudimentary (?); style and stigma wanting. @. Flowers solitary, subsessile, trimerous, rufous-tomentose, scarcely $in. long by }in. thick; calyx lin. long, semi-ellipsoidal, crass, appressedly hairy inside, shortly 3-fid, lobes obtuse; corolla urceolate-oblong, glabrous inside, lobes short spreading obtuse; staminodes 0; style 3-lobed at apex, stigma dilated; ovary subglobose, densely sericeous, rufous, 3-celled, cells 2-ovuled. Cumberland Islands, Australia, R. Brown!, Oct. 17, 1802. 10. MABA NIGRESCENS, Dalz. in Dalz. et Gibs. Bomb. Fl. p. 142 (1861). M. foliis lanceolato-oblongis, sub-coriaceis, undulatis, ciliatis, breviter petiolatis, nervis in- conspicuis ; floribus 1—5-nis, 3—4-meris, ferrugineo-pubescentibus, subsessilibus, staminibus 9 (vel in fl. 4-meris, 4—6) glabris; in floribus femineis staminodiis 0, ovario pubescente, 3- loculari, fructibus ellipsoideis, sericeis, calyce cupuliformu. A tree from 15 to 35 feet high with dense ferruginous pubescence on the shoots petioles and flowers; older branches dark-cinerous ; branches at about 50°, rigid. Leaves lanceolate-oblong, narrowed at least at apex, sometimes nearly rounded at base, coriaceous, 1—3lin. long (including petiole ;j,—1in. long) by }—I}in. wide, midrib depressed above, hairy beneath, margins ciliate, wavy. Flowers subsessile. g. Flowers 1—5 together in very short cymes, }in. long, trimerous or tetramerous ; 15—2 116 Mr HIERN, ON EBENACE. calyx lin. long, 3—4-lobed, lobes j,in. deep, deltoid acute; corolla campanulate-oblong, lin. long, 3—4+fid, lobes spreading; stamens 9, 6 in 3 pairs and 3 distinct, or all in one row, or in tetramerous flowers 4—6, glabrous, hypogynous, anthers jin. long, linear, acute, filaments slender; ovary rudimentary, hairy. 9. Flowers 1—2 together, trimerous, ?in. long; calyx din. long, funnel-shaped, shortly 3-fid, lobes obtuse; corolla 3-fid, lobes somewhat spreading, rounded at apex; staminodes 0; ovary hairy, 3-celled, cells 2-ovuled. Fruit rufous, sericeous, ellipsoidal, obtuse, } in. long in the specimens, often with the remains of the corolla at the apex which has been pushed forward during the growth of the fruit; fruiting calyx }in. wide by +in. high, somewhat accrescent and cup-shaped. Flowers in July, February; fruits in May. India, Canara, Goa, Dalzell!; Moollis, Dr Ritchie! n. 85. Pretty common in the Ghaut jungles, native name “ Ruktroora.” The leaves turn black in drying, and appear quite veinless. Allied to MM. guineensis ex Dalz. and Gibs.l.c. I have not seen an authentically named specimen. 11. Mapa sanpwicensis, Alph. DC. Prodr. vi. p. 242. n. 16 (1844). M. ramis nigricantibus verrucosis, foliis ellipticis, obtuse acuminatis, basi angustatis, nervis inconspicuis ; floribus subsessilibus plerumque trimeris, corollé. campanulatd, staminibus 15—17, glabris ; fructibus solitariis ellipsoideis vel subglobosis, glabratis, calyce paulum aucto brevt. I. elliptica, Seem. Fl. Vit. p. 152 (1866), non Forst. A tree or shrub, glabrous except the young parts and inflorescence which are pubescent ; branches dark-cinereous, rough, verrucose; leaves elliptical, subacute or rounded at apex, coriaceous, glabrous, petiolate, 1—2}in, in length by 4 to 1}in. in. width; petioles j5—3% in. in length. g . Flowers subsessile; calyx 3-fid with deltoid acute lobes, hairy ; corolla similar; stamens 15—17, glabrous, anthers of same length as filaments. 2. Fruit ellipsoidal or subglobose but somewhat oblique, solitary, downy or subglabrous, 3—in. in height, reddish; fruiting calyx, cup-shaped, not or scarcely accrescent, usually with rounded lobes, very rarely 4-lobed, somewhat hairy. Fruit-peduncle patent, } in. in length or shorter. Flora Hawaiiensis, no. 124, H. Mann and W. T. Brigham! 1867; in woods, Sandwich Islands, Capt. Wilkes! U.S. South Pacific Expl. Exp.; Gaudichaud!; Oahu and Numan, Dr Hillebrand! 273, Remy! 473; Hawaii, Dr Hillebrand ! 274, Remy! 470 (2%; Fiji Islands, Dr Seemann! 295. 12. Masa Buxiroiia, Pers. Synops. Plant. ii. p. 606 n. 2 (1807). M. foliis ellipticis vel obovatis vel lanceolatis, apice obtusis, bast angustatis, coriaceis vel submembranaceis, glabris, breviter petiolatis ; floribus 1—38-nis subsessilibus trimeris pubescen- tibus, cymis brevissimis, calyce corolldque breviter trifidis, staminibus 6—12 glabris ; in floribus Femineis staminodiis 0, ovario hirsuto, 3-loculart; fructibus globosis vel ellipsoideis, glabratis, monospermis ; albumine non ruminato. Mr HIERN, ON EBENACE. Lie Wight, Ic. pl. Ind. Or. vol. iii, pt. i, p. 4. t. 763 (1843), Alph. DC, Prodr. vim. p. 240. n, 2 (1844), Thw. En. Ceyl. pl. p. 183 (1860). HIGHULHAENDA, Herm. Mus. Zeyl. p. 21 (1717). Pisonia (?) buxifolia, Rottb. in Nye Saml, Kong. Danske Skrift. vol. m. p. 536. t. 4 f. 2 (1783). Ehretia ferrea, Willd. Phytogr. 1. p. 4 t. 2. f. 2 (17 Ferreola buxifolia, Roxb. Coromand. vol. 1. p. 35. t. (1795), Juss. in Aun. Mus. v. p. 418 (1804), Corréa de Serra in Ann. Mus. vii. p. 399. t. 65. f. 2 (1806). Maba littorea, R. Br. Prodr. Fl. Austral. p. 527. n. 5 (1810) [Mr Bentham unites this with JZ geminata, R. Br.]. Ferriola buxifolia, Roxb. Hort. Bengal. p. 72 (1814). Ferreola guineensis, Schum. Plant. Guin. p. 448 (1827), in Kong. Danske Vid. Selsk. iv. p. 222 (1829). Maba Cumingiana, Alph. DC. Prodr, vin. p. 241, n. 4 (1844). M. madagascariensis, Alph. DC. le. n. 7. MW. guineensis, Alph. DC. Le. n. 8. M. Smeathmanni, Alph. DC. Le. n. 9. (2) M. vacciniefolia, Benth. in Hook. Niger Fl. p. 442 (1849). MW. neilgherrensis, Wight, Ic. pl. Ind. Or. (iv.) nn, 1228—9 (1850), Ilust. Ind. Bot. 1. p. 147. t. 148 bis E. (1850). MW. Ebenus, Wight. Le. tt. 1228—9 (1850), non Spreng. M. angustifolia, Miq. ex Thw. En. Ceyl. pl. p. 183 (1860). A shrub or tree; young parts pubescent, glabrescent; branches terete, spreading at 35°—60°. Leaves elliptical obovate or lanceolate, obtuse at apex, more or less narrowed at base, coriaceous or submembranous, }—5 in. long by }—2 in. wide, margins usually thickened 94). 45 t. or reflexed and often undulated, veins inconspicuous, petioles ;,—} in. long, sometimes hairy. Flowers subsessile, trimerous, pubescent, about 1in. long, 1—3 together, in very short axillary cymes, on the young branches. Calyx |}; in. long, campanulate, with short deltoid lobes. Corolla campanulate-oblong, shortly 3-fid, lobes elliptical. Stamens 6—12 in male flower, 0 in female, hypogynous, glabrous; ovary rudimentary and hairy in male flower, 3-celled in female flower, style 3-lobed at apex. Fruit globose or ellipsoidal, glabrate, !—3 in. thick ; fruiting calyx cupuliform, shorter than the fruit; seeds solitary; albumen white, car- tilaginous, not ruminated. Dr Thwaites, who has seen growing in Ceylon many forms of this polymorphic and widely distributed species, gives the following varieties: Var. 8. microphylla, foliis parvulis. Var. y. Ebenus, foliis majoribus membranaceis parum acuminatis vel retusis sape subor- biculatis. : Var. 8 angustifolia, foliis lanceolatis vel lineari-lanceolatis, obtusis. Dr Thwaites Ic. adds: “I have devoted much time to the examination of the several very different-looking varieties of this plant, expecting to discover some sufficiently important 118 Mr HIERN, ON EBENACEZA!. constant characters to enable me to separate them specifically, but I find them so com- pletely connected together by intermediate forms that I have no hesitation in considering them all as representing only one very variable species; variable it may truly be called, since the leaves in var. &. are sometimes not a quarter of an inch in length, whilst im var. 6. they reach. to five inches in length.” East Indies, Wallich! list 4145, 7461, 7535; Dr Wight! 1729, 1730, 1731; Koenig !; Perrottet!; Dr Abel!; Malacca, Chr. Smith! 99, Dr Maingay! 979; Helfer and Griffith! 3641; Ceylon, Walker! 263, Dr Thwaites! 477, 1916, 1917, 3395; Philippine Islands, Cuming! 1694; Sooloo I., Wilkes / New Caledonia, Pancher! 249, Vieillard! 2864, 2873, 2877 (2). Australia, North Coast Bay, R. Brown! Madagascar, Gerard! 28, Bernier! 112, Pervillé! 700. Tropical Africa, Congo, Chr. Smith !, Dr Welwitsch ! 2527; Sierra Leone, Smeathmann /; I. St Thomé, Don/(?); Guinea, Leprieur ! In Ceylon it is called Kaloo-habaraleya-gass, in Godaveri forests Nella maddi, and in Madagascar Cacason matntt. The following specimens seem to me to belong to this widely-spread and variable species; namely, a plant in fruit from the Isle of Pines, Loyalty Islands, Oceania, collected by Sir E. Home (1853, Hb. Mus. Brit.) and Milne, n. 12 (1853, in Hb. Kew.); and a plant with subsessile g¢ flowers and fruit from the Fiji Islands collected by J. Storck, n. 898 in 1860, which Dr B. Seemann in Fl. Vit. p. 152 (1866) refers to JZ elliptica, Forst. var. glabrescens. A specimen stated to have been brought from the Straits of Magellan (but probably by mistake) in Herb. Commerson in fruit seems also to belong to this species. According to Dr Roxburgh, this species among the mountains of the Coromandel coast of India grows to a small tree, but in the low countries it is only a shrub; it flowers during the hot season; the berries when ripe are there universally eaten and are very well tasted ; the wood is dark-coloured, remarkably hard and durable, and when its size will allow it is employed for such uses as require the most durable and heavy wood. 13. MABA LANCEA, sp. nov. M. foliis lanceolato-oblongis, apice acut? acuminatis, basi angustatis, subglabris, subtus pallidis, supra nervis inconspicwis, petiolatis ; floribus masculis subsessilibus, dense cymosis, tri- meris rarius pentameris, staminibus 5—6 (2), antheris basi pubescentibus, ovario 0. Young parts and inflorescence puberulous; branches straight, terete, dark, spreading at about 50°. Leaves lanceolate-oblong, alternate, firmly submembranous, opaque, acutely acu- minate at apex, somewhat narrowed at base, nearly glabrous except the veins beneath, dark green on upper side, pale beneath, with veins inconspicuous on upper side; 3—4 in. long by 1 in. or rather more wide; petioles ;';—} in. long. $. Flowers small, several together, crowded on very short ferruginous-hairy axillary cymes, ferruginous hairy (closed in the specimen); bracts rounded; calyx openly campanu- late, yin. long, deeply 3-fid; with ovate acute lobes pubescent on both sides; corolla (closed) ty in. long, ovoid-conical, covered outside with pale ferruginous shining hairs, 3 ?-lobed, glabrous Mr HIERN, ON EBENACE. 119 inside; stamens 5—6 (?), hypogynous, erect, anthers subsessile, hairy towards the base, subu- late; ovary 0. Occasionally a calyx is pentamerous. Africa, Sierra Leone, Smeathman ! 14. Masa opovaTa, R. Br. Prodr. Fl. Austr. p. 527. n. 2 (1810). M. foliis obovatis, apice rotundatis vel retusis, basi cuneatis, breviter petiolatis, nervis in- conspicuis ; floribus masculis 83—7-nis, trimeris vel rarius tetrameris, brevissime cymosis, cam- panulatis, staminibus 6—12, sepius 9, ovarvi rudimento villoso; floribus femineis solitariis sub- sessilibus trimeris, staminodiis 0, ovario glabro triloculart. Alph. DC. Prodr. vit. p. 241. n. 5 (1844); Ettingsh. Blatt-skel. dikot. p. 90. t. 29. f. 6. t. 32. figs. 1, 2 (1861). Young parts appressedly pubescent; branches terete, smooth. Leaves obovate, usually retuse or rounded at apex, cuneate at base, thinly coriaceous, about 14 in. long by 1 in. wide, veins inconspicuous, margins undulated, scarcely recurved, of same colour on both sides; petioles ;5 in. long. 8. Flowers campanulate, {—1 in. long, 3—7 together, in very short axillary cymes crowded on the young shoots; calyx 3-fid or unequally 4-fid, somewhat pubescent outside, glabrous inside, lobes ovate; corolla whitish, exceeding the calyx, 3—4-fid, lobes obtuse, somewhat patent appressedly subsericeous outside; stamens 6—12, usually 9 and alternately in pairs, glabrous; pollen white; ovary rudimentary, hairy. @. Flowers solitary, axillary, subsessile, like g but rather thicker; trimerous; stami- nodes 0; ovary glabrous, 3-celled, subglobose, cells 2-ovuled; style shorter than the ovary, stout, deeply 3-fid, glabrous; stigmas emarginate at apex, glabrous. Australia, Carpentaria Islands, R. Brown /, flowers in November. Mr Bentham unites this species with Jf humilis, R. Br. The glabrous ovary in the 2 is exceptional in this section of the genus, but the rudiment of the ¢ ovary is hairy; possibly the two sexes belong to different species, but the foliage is quite alike in both. 15. Mapa ceminata, R. Br. Prodr. p. 527. n. 4 (1810). I, foliis obovatis, apice subretusis vel obtusis, bast cuneatis, coriaceis, glabris, petiolatis ; Fructibus 1—3-nis, subsessilibus, subglabris, ellipsoideis; calyce fructifero breviter cupuliforms, trilobo, subglabro ; floribus masculis 5—7-nis, subsessilibus, trimeris, campanulatis, calyce pube- rulo, staminibus 9, glabris. Alph. DC. Prodr. vit. 242. n. 13 (1844); Benth. Fl. Austr. Iv. p. 291. n. 8 (1869), excl. syn. Diospyros geminata, F. Muell. Austral. Veg. in Intercolonial Exhibition Essays, 1866— 67, p. 35 (1867). A tree, glabrous except the flowers and fruit, with a diffuse irregular head; branches terete, cinereous, smooth, spreading at 45°. Leaves obovate, coriaceous, subretuse or obtuse at apex, cuneate at base, 1} to 3 in. in length, by ? to 2 in. in width; petioles }—} in. in length. 120 Mr HIERN, ON EBENACE. g. Flowers in subsessile clusters, about 5 to 7 together, } in. in length, oblong; calyx dark, with scattered short hairs, 3-lobed at apex, 3; in. in length, lobes depresso-deltoid ; corolla pale, sericeous, 3-fid. stamens 9, free, equal, glabrous, mostly hypogynous; ovary rudi- mentary. . Flowers 3 together, subsessile; fruit 1 to 3 together, ellipsoidal, subglabrous, 2 in. in length by } in. in width, subsessile, 1—2-celled, terminated by remains of style, 1-seeded, straight, rarely 3-celled and 3-seeded ; fruiting calyx, j;—+ in. high, cup-shaped, with 3 broad and shallow lobes; seed 14 in. in length by 3, im. in thickness with a depressed longitudinal line; albumen not ruminated; radicle more than double the length of the cotyledons. A slender tree attaining 50—60 feet in height and 9 to 12 inches in diameter, with dark scaly bark, found growing in the scrubs; wood soft and tough; fruit eaten by the natives (Thozet). E. Australia, from Moreton Bay to Rockingham’s Bay; Queensland, Dallachy!; Rodd’s Bay, N. E. Australia, A. Cunningham / 306; Moretown Island, Dr Mueller /; Brisbane River, Fraser!, Mueller! ; Queensland Woods, London Exhibition, 1862, no. 50, Hill!; Keppel Bay, Shoalwater Bay, Thirsty Sound, Broad Sound, &. Brown! 16. Masa HumILis, R. Br. Prodr. p. 527. n. 3 (1810). M. foliis obovatis, parvis, apice rotundatis vel subretusis, basi cuneatis, coriaceis, glabris, subsessilibus; floribus masculis 3-nis, brevissime cymosis, trifidis, campanulatis, calyce sub- glabro, staminibus 8—9, glabris, ovarii rudimento hirsuto; jfloribus femineis solitariis, sub- sessilibus, trilobis; fructibus glabris, apice hirtellis, ellipsoideis, calyce fructifero cupuliformi, glabro. Alph. DC. Prodr. vill p. 242. n. 12 (1844); Benth. Fl. Austr. Iv. p. 291. n. 9 (1869); non Ettingsh. Blatt-skel. dikot. p. 90. t. 36. £ 8 (1861). Diospyros humilis, F. Muell. Austral. Veg. in Intercolonial Exhibition Essays, 1866—67, p. 35 (1867). An erect bush glabrous or puberulous except the flowers, 2—5 feet or sometimes 20 feet high, much branched ; branches terete, subcinereous. Leaves obovate, rounded or retuse at apex, narrowed at base, coriaceous, $—1} in. in length by }—?in. in width; petioles 3,—; in. in length; veins not conspicuous ; young leaves with a few depressions on the lower surface which disappear from the older leaves. &é. Cymes 3-flowered, jt, in. in length; flowers not much exceeding +; in. in length; calyx in the dry state of a chestnut brown colour, 3-fid, subglabrous; corolla not much exceeding calyx, 3-fid, lobes straight, light-hairy outside; stamens 8 or 9, some in pairs, hypogynous, glabrous; ovary rudimentary, hairy. ?. Flowers solitary, subsessile, } in. in length, oval; calyx pubescent, subferruginous, with 3 shallow rounded lobes, turbinate; corolla not much exceeding calyx, hairy outside. Fruit solitary, 3-celled with cells 2-seeded, or 1—2 cells often abortive and seed solitary, glabrous except at apex, 3; in. in length ellipsoidal; fruiting calyx between a half and a third of Mr HIERN, ON EBENACE. 121 the length of fruit, cupshaped, at first with a short cylindrical base, glabrous; radicle longer than the cotyledons. Australia, from Arnhem Land to the islands in the Gulf of Carpentaria and to the tropic in East Australia. Rockhampton, Dallachy and O’Shanesy!; Point Pear, Mueller /; Dawson River, Mueller’; Burnett River, Mueller’; Gilbert River, Mueller /; Cliffs on the entrance of the Victoria River, Mueller /; Sweers Island, Henne’; Broad Sound near upper head, in thickets not far from the shore, R. Brown! 17. Masa ReticuLata, R. Br. Prodr. p. 528. n. 6 (1810). M. foliis obovatis vel ovalibus, apice emarginatis vel rotundatis, supra valde reticulatis, coriaceis, glabris, breviter petiolatis; floribus masculis 3—4-meris, 3—5-nis, brevissime cymo- sis, campanulatis, calyce subglabro, staminibus 7—W4, glabris; floribus femineis solitariis, subsessilibus, corollé 3—4-fidd, staminodiis 0, ovario sericeo, 3-loculari; fructibus glabratis, subglobosis, calyce fructifero leviter aucto, intus breviter tomentoso, extus glabro. Alph. DC. Prodr. vi. p. 241. n. 6 (1844); Benth. Flora Austr. Iv. p. 291. n. 7 (1869). M. interstans, F. Muell. Fragm. bot. v. p. 163 (1866). A shrub of 8 ft. or a tree from 20 to 30 feet in height, erect, glabrous or very quickly gla- brescent, much branched; branches terete, spreading at about 45°—50°, bark cinereous, thinly rimose. Leaves oval or obovate, emarginate or rounded at apex, suddenly narrowed or rounded at base, margins often recurved, highly reticulated above, midrib depressed 2 in. in length. above, coriaceous, 1} to 4in. in length by $ to 24 in. im width; petioles #4 6. Cymes 3—5-flowered, hairy, ;4—+ in. in length, crowded on young branches; pedicels very short, with oval ciliate caducous bract at base; flower tin. in length, usually tri- merous, occasionally tetramerous ; calyx campanulate, dark, #5 in. in height, with 3 or rarely 4 roundly deltoid lobes reaching about halfway down calyx, subglabrous; corolla 3—4-fid, argenteo-sericeous outside, narrowly urceolate; stamens 7—14, hypogynous, equal, glabrous, when numerous many in pairs, about } in. in length; anthers about im. in length, nar- row ; ovary rudimentary, hairy. Q. Flowers solitary, subsessile, thick, about 1 in. in length; calyx 3-fid, nearly hemi- spherical, nearly glabrous outside ; corolla urceolate, 3- or unequally 4-fid, silky; staminodes 0; style scarcely any; stigma 3-lobed; ovary globular-pointed, silky, pale, 3-celled, cells 2-ovuled; fruit globular or depresso-globular, } in. thick, glabrate and shining; fruiting calyx 3-celled, 3-seeded, somewhat accrescent, finally recurved or spreading, covered inside with dense furlike hair, glabrate outside, 1 in. across. Australia, Cape York, Voyage of Rattlesnake, October 1848, John Macgillivray! 439 ; Mr Daniel! March 1868 ; Rockingham Bay, Ferd. Mueller! Dallachy; Prince of Wales and Cumberland Islands, R. Brown! Nov. 2, 1802, in male flower. 18. Masa compacta, R. Br. Prodr. p. 528. n. 7 (1810). U. foliis ovalibus, apice emarginatis vel rotundatis, coriaceis, glabris, reticulatis, breviter petiolatis; fructibus solitariis, subsessilibus, subglobosis, glabratis, nitentibus, 3-locularibus, 3-spermis ; calyce fructifero patente vel reflexo, intus tomentoso, extus glabro. Vout. XII. Part L 16 122 Mr HIERN, ON EBENACEZ. Alph. DC. Prod. vit. p. 242. n. 11 (1844); Benth. Fl. Austral. rv. p. 290. n. 6 (1869). Known only in fruit; shrub 4—5 feet high, erect, branched ; shoots terete, bark dark cinereous; glabrous except the inside of the spreading or recurved calyx. Leaves oval, suddenly narrowed or rounded at base, emarginate or rounded at base, coriaceous, highly reti- culated, 2—4 in. long (including dark petiole 4—4in. long) by 1{—2}in. wide; midrib depressed above. Fruit subsessile, solitary, depresso-globose, yellow, about 4 in. thick, glabrate and shining, 3-celled, 3-seeded; fruiting calyx }im. across, spreading or recurved, densely covered on reflexed surface with short furlike tementum, glabrous outside. Differs from Maba reticulata by wider leaves and more spreading or reflexed not cupuli-form fruiting calyx. Australia, North Coast Island, Feb. 18, 21, 1803, R. Brown! 19. Mapa HILtesranpu, Seem. FI. Vit. p. 151 (1866). M. foliis oblongis vel ovato-oblongis, apice obtusis, basi rotundatis vel cordatis, glabris, tenuiter coriaceis, supra crebre reticulatis, breviter petiolatis; floribus solitariis sessilibus bast bracteatis, masculis 3-meris, femineis 3—4-meris; staminibus 9, glabris; fructibus oblongis subglabratis, calyce fructifero glabro, lobis deltoideis. Glabrous except the inflorescence; branches dark cinereous. Leaves oblong or ovate- oblong, rounded or cordate at the base, usually obtuse at the apex, thinly coriaceous, 2—6 in. long by 1—31 in. wide; veins except midrib in relief on both sides, remarkably prominent on the upper side, reticulated; petioles =;—,3, in. long. Flowers solitary sessile with several imbricated ciliate bracts at base. &é. Flowers pubescent, trimerous; stamens 9, 6 in 3 pairs alternating with the corol- la-lobes and 3 distinct opposite the corolla-lobes, all glabrous ; ovary rudimentary, hairy. Q@. Fruit oblong, 3 in. long by 4—2 in. thick, subglabrate, somewhat oblique; fruiting calyx fin. long by 2—} in. wide at apex, 3—4-fid, glabrous; lobes deltoid acute, some- what spreading. Sandwich Islands, Mountains, Oahu, Dr Hillebrand!, Remy! 472. 20. Mapa ELLiprica, J. R. et G. Forst. Char. Gen. Pl. p. 122. t. 61 (1776). M. foliis ellipticis vel oblongo-lanceolatis, apice obtusis, basi cuneatis, subcoriaceis, glabres- centibus, breviter petiolatis ; cymis axillaribus, 3—8-floris, pubescentibus; floribus trimeris, campanulato-tubulosis ; staminibus 3 vel 6; ovario 3-loculari, pubescente; fructibus ellipsoideis, pedunculatis, pubescentibus. J. R. et G. Forst. Beschreib. Gatt. Pflanz. edit. Kerner, p. 127. t. xv. f. 61 (1779); Poiret in Lam. Encyel. Méth. Suppl. m1. p. 566. t. 803 (1813); Labill. Sert. Austro-Caled. p. 32. t. 35 (1824); Alph. DC. Prodr. vir. p. 240. n. 1 (1844); Ettingsh. Blatt-skel. Dikot. p. 90. t. 40. f, 2 (1861); non Seem. Fl. Vit. p. 152 (1866). Ebenus vulgaris, Rumph. Amb. Vol. m1. p. 1. t. 1 (1750). ? Ebenoxylum verum, Lour. Fl. Cochinch. p. 613 (1790). Mr HIERN, ON EBENACEA, 123 Maba Ebenus, Spreng. Syst. Veg. 1. p. 126. n. 8 (1825), Alph. DC. lc. p. 242. n. 17, Hassk. Retz. 1. p. 107 (1855), non Wight. ? Maba? ebenoxylon, G. Don, Dict. Gard. and Bot. Iv. p. 43. n. 10 (1837). Diospyros hexasperma, Hasselt ex Hassk. Pl. Javan. p. 468. n. 353 (1848). A shrub of 6 ft. or more or a moderate-sized tree or sometimes a lofty tree; branches slender, cinereous, terete, rather rough; shoots hairy ; glabrescent; leaves elliptical or oblong- lanceolate, obtuse at apex, cuneate at base, glabrescent, subcoriaceous, 14—44 in. long by $—12 in. wide; petioles ;j—1 in. long. é.Cymes longer than the petioles, 1—1in. long exclusive of the flowers, pubescent, 0 3—8-flowered, crowded on the young branches; common peduncle ;4—1 in. long; bracts linear, small, caducous; flowers trimerous, } in. long, campanulate-tubular, pubescent; calyx cam- panulate, $in. long, lobes deltoid-acute ; corolla tubular, 3-fid, yellowish white, lobes acute, ;4, in. long, rather patent; stamens 3 or 6, ae ee glabrous, distinct ; ovary rudimentary, hirsute. Q@. Cymes }—} in. long; flowers as in g; staminodes 0; ovary hairy, ovoid, 3- (or according to Labillardiére 4- or by abortion 2-) celled; cells 2-ovuled; style short; stigma 3 (—4)-lobéd; fruit fleshy, pedunculate, crowded, greenish, ellipsoidal, scarcely lin. long by } 1m. thick, pubescent or nearly glabrous, 2—3-celled; seeds triquetrous; albumen car- tilaginous; ‘plumule indistinct; fruiting calyx not accrescent, somewhat spreading, 3-fid, 4—1} in. across; lobes deltoid. Friendly Islands, Forster !, Capt. Cook !, A. Matthews ! 144; Navigator’s Islands, Wilkes / var. foliis acuminatis; Amboina, Rumf, Teijsmann!, Hasskarl ; Java, Hasselt; Cochinchina (%), Loureiro; New Caledonia, Labillardiére!, Vieilard/ 893; “Amsterdam Insula Oceani pa- cifici” (= Tonga Tabboo, Friendly Islands), J. R. and G. Forster /. Called Maba, by the natives in the Friendly Islands, and Avhar&pat in Java. The plant called Anzime in Navigator’s Islands (see Rev. Thomas Powell in Seemann’s Journal of Botany, Vol. vi. p. 278, 1868) may belong to this species; it is eaten by children, and flowers in June or J uly and in January or February. Difficult when young to distinguish from J. rufa, and approaching also M. buwifolia. 21. Masa suMATRANA, Miq. Pl. Junghuhn. i. p. 204 (1851—55), Fl. Ned. Ind. vol. 11. p. 1051, tab. xxxvi. B (1856). M. foliis ovato- vel ovali-oblongis, acuminatis, basi rotundatis, costatis, subtus secus costas hirtellis; cymis masculis axillaribus, multifloris ; calyce trilobo ; corolld ovoideo-tubulosd ; sta- minibus 9, glabris ; ovarw rudimento pubescente. A subferruginous, pubescent tree, about 30 feet in height. Branches terete. Leaves ovate- or oval-oblong, acuminate, rounded at base; margins flat, dark green, and with scat- tered appressed long hairs on upper face; velutinous and subferruginous, especially on veins beneath ; lateral veins numerous (about 8), plain beneath ; petiolate ; subcoriaceous ; 24—4 in. in length by 2—1} in. in width; petioles =;—+ in. in Vent $. Cymes pedunculate, Hanchawered, 7—1 in. in length; flower (in bud) }in. in length, oblong, subferruginous, tomentose ; calyx in. in length, 3-lobed at apex; corolla ovoid-tubular, with a slight constriction near middle, 3-fid; lobes cordate, sub-acute; sta- mens 9, 6 in 3 pairs, 3 distinct, glabrous; anthers as long as filaments; ovary rudimentary, hairy. 16—2 124 Mr HIERN, ON EBENACE. Sumatra, Dr Fr. Junghuhn! 719; in woods near Tobing, ex Miq. in Pl. Jungh. 1. p. 204; Java, De Vriese! Marcreightia andamanica, Kurz in Rep. Veg. Andam. I. edit. i. p. 42 (1870), WM. oblongi- folia, Kurz 1. c. edit. i: p. XI. (1867), is said by Mr Kurz in Journ. Asiat. Soc. Beng. vol. Xt, pt. ii. p. 74 (1871) to belong to Maba sumatrana, Migq.; it is a dull dark green shrub, with oblong submembranous leaves 7—8 inches long by 2}—3 in. wide, subcordate at base, and robust petioles + in. long; it was collected in South Andaman by Mr Kurz! in which island he states that it is common. 22, MABA VIEILLARDI, sp. nov. M. foliis obovato-ellipticis, apice rotundis vel retusis, basi cuneatis, coriaceis, glabris, undatis, breviter petiolatis ; floribus masculis brevissime cymosis, monstrosis in speciminibus ; floribus femineis solitariis breviter pedunculatis ; fructibus glabratis, albido-pulverulentis, sub- globosis, calyce trifido. A tree of about 13 feet high; glabrous or on quite young parts slightly pubescent; branches numerous, terete, smooth; leaves oval or somewhat obovate, coriaceous, alternate, rounded or somewhat emarginate at apex, more or less narrowed at base, shining, of same metallic lustre when dry and without conspicuous veins on each side, coriaceous, 1—2 in, long by {—1 in. wide; petioles 1,—4 in. long, dark and rather stout; wavy (in the dry state) and with revolute margins. é. Cymes axillary on young branches, about ;4; in. long, recurved, puberulous ; flowers about 4 in. long, monstrous in the specimen (Deplanche, 449) by the stamens being petaloid, puberulous; calyx and corolla campanulate, about } in. long, deeply 3-fid; ovary 0. 2. Fruit solitary, on peduncles about jin. long, puberulous or glabrate, subglobose, glabrous, covered with white efflorescence, nearly 4 in. in diameter, 3-celled, 5—6-seeded ; seeds about } in. Jong; albumen scarcely ruminated, but with slight sinuous intrusion of the rather thick testa; fruiting calyx, puberulous outside, glabrous inside, not accrescent, appressed to base of r it, 3-fid, 1 in. across. New Caledonia, Vieillard! n. 897; Deplanche! 448 (in fruit), 449, Kanala; Pancher !, Tron Mountains of Kanala, 1862. The following two species are very imperfectly known: 23. Mapa ANDERSONI, Soland. MSS. in Herb. Mus. Brit., Seem. Fl. Vit. p. 152 (1866). M. arborea, ramis cinereis glabris; foliis ellipticis, apice obtusis, basi subrotundis, petio- latis ; floribus pubescentibus, subsessilibus, masculis glomeratis ; fructibus solitariis. A tree with cinereous branches, glabrous except the inflorescence, apparently dicecious, Leaves alternate, elliptical, obtuse at apex, rounded or nearly so at base, of uniform colour, with minute net-veins, 43—5} in. long by 21—3} in. wide; petioles about 4 in. long. 3(%). Flowers subsessile, clustered several together on the young branches. ?. Fruit solitary, subsessile, with wide articulation at base to the very short peduncle. Tonga Islands, Capt. Cook!, third voyage. Possibly identical with M. major, Forst. The foliage is somewhat like that of Wl. com- pacta, R. Br. . Mr HIERN, ON EBENACE. 125 24. Masa magor, G, Forst. Pl. Escul. Insul. Ocean. Austr. p. 54, n. 21 (1786). M. arborea, fructibus edulibus bipollicaribus, ceterum M. elliptice similibus, 2—3-spermis ; seminibus triquetris. Cook, Voyage to the Pacific Ocean in 1776—80, edit. ii. p. 393 (1785); Alph. DC. Prodr. yin, p. 242. n. 15 (1844). A tree known only from its fruit, which is 2 in. long, “roundly oval,” like that of W. elliptica Forst., but three times the size, tough, egg-shaped, and containing 2 or 3 trique- trous seeds in cells. The taste is insipid, but nevertheless is used by the natives of the Friendly Islands for food, and is frequently planted near their houses; they call it Maba or Mabba. Tongatabu, Namoka, E-uwa, Hapa-i, and other of the Friendly Islands, G. Forster, Capt. Cook. 25. MABA GRISEBACHII. M. glaucescens, foliis rotundato- vel ovali-ovatis, apice spinuloso-apiculatis, coriaceis, basi rotundis vel subcordatis, brevissime petiolatis, reticulatis ; floribus femineis solitariis, axillaribus, brevissime pedunculatis, trimeris ; corolle lobis ovatis, acutis; staminodiis 6, glabris, uniseriali- bus ; ovario ovoideo-conico, hirsuto, apice glabro, 6-loculari, 6-ovulato. Macreightia buaxifolia, Grisebach, Catal. Plant. Cubens. p. 169 (1866). Pale glaucescent shining stiff (shrub ?), with terete branches spreading at about 50°—60°, glabrous except the flowers. Leaves alternate, crowded, rotund, oval, or ovate, spinulose- apiculate, coriaceous, rounded or subcordate at base, shortly petiolate, average size 4t in. long (including petiole and apiculus) by ;3; in. wide; petioles 4 in. long by 1, in. wide, dilatato- concave ; veins reticulated, in relief on both sides, more conspicuous on under-side. 2. Flowers solitary, crowded, in axils of upper leaves, shortly pedunculate, 2 in. long, trimerous ; peduncle equalling or slightly exceeding the petiole, hairy; calyx 1 in. long, thickly coriaceous, covered outside with close short pale hairs and inside with denser hair except near base; lobes + in. long, broadly ovate, suddenly acuminate at apex, with sides revolute and sub-auricular at base, somewhat concave within to make room for the ovary. Corolla 3% in. long, hairy like the calyx outside except near base, glabrous inside; lobes 2; in. long, ovate, acute, spreading; tube triangularly prismatic. Staminodes 6, jin. long, glabrous, neatly equal, uniseriate, inserted near base of corolla. Ovary + in. long (including style), ovoid- conical, continuous with the 3-lobed style, covered except at apex with short dense pale hair, 6-celled, cells 1-ovuled. E. Cuba, near St Antonio, Wright! No. 2938. 26. MABA CARIBEA. M. glaucescens, foliis obovatis, apice rotundatis vel emarginatis, basi angustatis, coriacets, glabris, reticulatis, breviter petiolatis ; floribus masculis brevissime cymosis, pubescentibus, tri- meris, staminibus 8; floribus femineis solitarivs, sessilibus vel breviter pedunculatis, trimeris, staminodiis 3—6, ovario dense hirsuto, 6?-loculari, 6-ovulato; fructibus subglobosis, glabris, nitentibus ; albumine ruminato, 126 Mr HIERN, ON EBENACEZ. Macreightia caribea, Alph. DC. Prodr. vint. p. 221. n. 1 (1844), non Griseb. Veg. Karab. Ins. Guadal. p. 91. n. 846 (1857, = Casasia calophylla Rich.). Tree, glaucescent, glabrous except very young parts and flowers, which are pale fulvous and softly pubescent ; branches making 60° with stem. Leaves obovate, rounded or emarginate at apex, somewhat narrowed at base, coriaceous, midrib depressed above, glabrous, plane but margins reflexed ; net-veins very closely and clearly reticulated, raised on both sides; 14—3 in. in length by 2—14 in. in width, rather paler beneath ; petioles 75 in. in length. g. Cymes very short, usually 3-flowered, pubescent, pale fulvous; flowers narrowly oval; calyx tubular, with 3 shortly deltoid lobes at apex; corolla 3-fid; glabrous and dark inside ; stamens 8, unequal; ovary rudimentary, hairy. Q. Flowers solitary, sessile or on peduncles }—? in. in length, pubescent, 5—} in. in height ; bracts small, pubescent; calyx coriaceous, thick, with wide undulating diverging and auricled lobes; openly campanulate, deeply 3-fid, 2 in. in width, hairy on both sides; corolla 3-fid, 2 in. long, lobes acute, glabrous inside, hairy outside; staminodes 3—6, equal, inserted near base of corolla; ovary densely hairy, 6 ?-celled, 6-ovuled. According to Grisebach (Fl. Br. W. Ind. p. 404) the ovary is 3-celled, with 3 other incomplete dissepiments separating the gemi- nate ovules. Fruit squarely subglobose, glabrous and shining, orange-coloured, about 1 in. in diameter ; fruiting calyx nearly as wide, but not accrescent, horizontal; lobes with replicative sinuses ; albumen deeply ruminated. Cuba, C. Wright / 1331, near village called Monte Verde, E. Cuba; fugel, 662; Haiti, C. Ehrenberg! ; Nectowa!/; Antilles!; “America meridionalis,” Richard / in Hb. Vahl. - 7. MABA INTRICATA., M. glaucescens, intricato-ramosa, foliis obovatis, apice rotundatis, basi cuneatis, coriaceis, brevissime vpetiolatis; fructibus globosis, glabratis, uncialibus, breve pedunculatis, 6-spermis, albumine non ruminato, calyce fructifero patente, trilobo. Macreightia intricata, A. Gray in Proceed. Amer. Acad. v. p. 163 (Jan. 1862). Pale glaucescent (shrub ?), with intricate branches spreading at 60°—80°; young parts weakly and appressedly pubescent. Leaves obovate, cuneate at base, rounded at apex, few- veined, appressedly and inconspicuously pubescent on midrib and beneath, about 1 in. long by } in. wide; coriaceous; petioles very short. Fruiting peduncles arching-reflexed, }—+ in. long, tough, glabrous, solitary; fruiting calyx flat, } in. in diameter, covered with very short inconspicuous and weak pale hairs, with 3 rounded lobes, tin. long, reflexed at tip; fruit of bright orange colour, glabrate, globular, about 1 in. in diameter, 6-seeded; albumen not ruminated. Lower California, Cape St Lucas, &e., Xantus/ 68, Aug. 1859—Jan. 1860. 28, MABA ALBENS. M. foliis obovato-oblongis, utrinque angustatis, confertis, molliter puberulis, subtus albenti- bus, subcoriaceis, breviter petiolatis ; floribus masculis 3-nis, brevissime cymosis, 3—4—5-merts ; staminibus 12—11, glabris ; ovarti rudimento pubescente. Mr HIERN, ON EBENACEA. 127 Diospyros alhens, Presl, Reliq. Haenk. 1. p. 62 (1835-6). Macreightia albens, Alph. DC. Prodr. vu. p. 221. n. 2 (1844); Ettingsh. Blatt-skel. Dikot. p. 89. t. 38. f. 11 (1861). A shrub or tree with pallid or cmereous bark and dull leaves; branches terete, glabrescent ; young parts pubescent; leaves obovate-oblong or lanceolate, more or less narrowed at both ends, crowded, softly puberulous, dull green above, paler beneath and with minute scales, subcoriaceous, midrib slightly depressed beneath, veins slender; 1}—3 in. long by 3—12 in, wide; petiole 4;—+ im. long. g. Flowers arranged on short (;—4 in. long) pubescent 3-flowered cymes, which grow on the youngest shoots; $ in. long by 3 in. wide; calyx campanulate or ovoid, } in. long by Lin. wide, unequally 3-fid (occasionally 4—5-fid vit lanceolate lobes), pubescent on both sides; lobes usually ovate; corolla shortly 3—4-lobed, urceolate-oblong, pubescent outside, glabrous inside, lobes oblique, imbricated sinistrorsely ; stamens 12—11 (6 filaments, 2 together ex Pres] lc.) all or some inserted at the base of the corolla, glabrous; ovary rudimentary, pubescent. Flowers in June. Mexico, Acapulco, Presl, Haenke! 47; Soledad, Dr Wawra! 168. 29. MABA INCONSTANS, Grisebach, Fl. Brit. W. Ind. p. 404 (1864). M. foliis oblongo-obovatis vel oblongis, apice obtusis, basi angustatis, subglabris vel sub- tomentosis, tenuiter reticulatis, subcoriaceis, interdum minute pellucido-punctatis, breviter peti- olatis ; floribus masculis brevter cymosis, 3—4-meris ; staminibus 6—12, sepius 9, inequalibus, glabris ; floribus fenineis subsolitartis, 3—(4-)meris ; staminodiis 3—4; ovario hirsuto, 6-locu- lari; fructibus solitarws, 6-locularibus, depresso-globosis, subglabratis ; seminibus oblongis ; albu- mine non ruminato. Macreightia inconstans, Alph. DC. Prodr. vim. p. 221. n. 6 (1844). Diospyros inconstans, Jacq. Amer. p. 276, t. 174. f. 67 ees Macreightia conduplicata, Alph. DC. Prodr. vim. p. 221. n. 5 (1844). Diospyros conduplicata, Kunth in Humb. et Bonpl. Nov. Con ii. p. 254 (1818). Diospyros Berterw, Alph. DC. Prodr. vit. p. 234. n. 61 (1844). Diospyros obtusifolia, Bert. in Alph. DC. 1. c., non Humb. es Bonpl. Macreightia obovata, Mart. in Fl. Bras. vit. Eben. ee 9. t. 2. f. 3 (1856). Macreightia psidioides, Alph. DC. Prodr. vim. p. 221. n. 4 (1844). Diospyros psidioides, Kunth in Humb. et Bonpl. Nov. Gen. iii. p. 254 (1818). A moderate-sized dicecious (moncecious, according to Jacquin) tree or shrub, with young parts and inflorescence fulvo- or ferruginous-pubescent, more or less glabrescent. Leaves alter- nate, oblong-obovate or oblong, subglabrous or subtomentose-pubescent, reticulated, subcoria- ceous, somewhat narrowed at base, and more or less pointed or obtuse at apex; sometimes minutely pellucid-punctate; margins just recurved, 1?—6 in. long, 2—22 in. wide; midrib depressed above ; petioles 4—} in. long; cymes short, drooping, 3-flowered or 3-several-flowered in male plants, ;;—i—+ in. long; bracts small, caducous, acute, ovate or lanceolate or obovate- oblong. g . Flowers ~;—}in. long, 3—4- (usually 3-) -merous, campanulate-oblong ; calyx ;4,—1 in. long, campanulate, pubescent; lobes ovate, somewhat spreading, about equalling the tube or 128 Mr HIERN, ON EBENACE. exceeding it; corolla glabrous (villous, according to Jacquin) within, pubescent outside, conical at apex in bud; lobes ovate-lanceolate, about equalling the tube; stamens 6—12, usually 9 (8—10, according to Jacquin), unequal, either distinct or in pairs or 3 together, inserted at base of corolla or partly hypogynous, glabrous; ovary abortive ; receptacle hairy. Q@. Cymes soon becoming 1-flowered by lapse of the lateral flowers, $;—1 in. long. Calyx openly campanulate, 3- (4-)fid, with reunded lobes, about 4} im. across, puberulous out- side, tomentose inside; corolla widened below, the lobes extending only + way down, densely ferruginous-pubescent outside; staminodes glabrous, (in one flower) 4, 2 being distinct and 2 combined by their filaments; in another flower 3, alternating with the corolla-lobes; ovary 6-celled, 6-ovuled, ;1;in. high, covered outside with short appressed drab hairs; style simple, columnar, 4 in. high, trifid at apex, hairy at base ; stigmas punctiform ; fruit solitary, 6-celled, yellowish, with black bitter pulp, depresso-globose, subsessile or shortly stalked, }—? in. thick, subglabrate and shining; fruiting calyx about 3 in. in diameter, reflexed or nearly flat, with 3 (4) rounded or bifid lobes, tube thickened within; seeds oblong; albumen not ruminated. The following varieties may be noticed: a. obovata. A tree or shrub with obovate-oblong leaves. 8. granatensis. A shrub with oblong leaves. Occasionally the leaves are conduplicate in the dry state). ay Flowers in February, July, and September. S. America, St Vincent, Guilding!; Martinique, Plée! 762; New Granada, Carthagena, Jacquin, Triana! 2613; Sabanilla, Karsten! ; S* Martha, Purdie! Goudot! No. 1; Guayaquil, Bonpland! 3846; Brazil, Pohl! 1980, Sello! 1211, 1689, 2301; Rio Janeiro, Gaudichaud ! ; Minas Geraes, Weddell! ; Dr Regnell! iu. 1516. 30. MABA ACAPULCENSIS. M. foliis obovato-lanceolatis, apice acutis, basi cuneatis, utrinque hirtellis, subtus subcanes- centibus, reticulatis, submembranaceis, petiolatis ; fructibus solitariis, subsessilibus, subglobosis, uncialibus ; calyce fructifero patente, profunde 3-fido; albumine non ruminato. Macreightia acapulcencis, Alph. DC. Prodr. vir. p. 221. n. 3 (1844), excl. Syn. Diospyros salicifolia. Diospyros acapulcensis, Kunth in Humb. et Bonpl. Nov. Gen. iii. p. 254 (1818). Terminal buds oblong, sericeous-tomentose ; the axillary ones smaller, pubescent; shoots glabrous, dark-cinereous, smooth; leaves obovate-lanceolate, acute, cuneate at the base, hir- tellous on both sides, especially beneath where they are subcanescent, reticulato-venose, mem- branous, 24 in. long or more, by about $ in. wide, petiolate; fruit solitary, subsessile, sub- globose, 1 in. in diameter; fruiting calyx flat, nearly 1 in. across, deeply 3-fid; lobes widely ovate, felted inside, puberulous outside ; albumen not ruminated; cotyledons oblong, rather obtuse, double the length of the radicle. Mexico, Acapulco, Bonpland ! Mr HIERN, ON EBENACEZ. 129 31. MABA SALICIFOLIA. M. ramis teretibus cinereis, junioribus pubescentibus ; foliis lanceolato-oblongis, utrinque angustatis, apice obtusis, coriaceis, supra glabrescentibus, subtus pubescentibus, breviter petio- latis, nervis inconspicuis; fructibus solitartis, globosis, glabris, breviter pedunculatis ; calyce fructifero trifido, utrinque puberulo, appresso. Diospyros salicifolia, Humb. et Bonpl. ex Willd. Sp. Pl. rv. p. 1112. n. 18 (1805); Hb. Willd. n. 19250. Young leaves and shoots pubescent; branches cinereous, terete. Leaves lanceolate-oblong, narrowed at both ends, obtuse at apex, glabrescent above, coriaceous, with inconspicuous veins, about 24 in. long by 3 in. wide; petioles } im. long, puberulous. Fruit solitary, globose, glabrous, shining, of a pale orange colour, about 1} im. in diameter; peduncles } in. long, stout, puberulous; calyx 3-fid, appressed to base of fruit, puberulous on both sides, about 1 in. across, lobes semi-elliptical, with obscure parallel veins. Equatorial America, Humboldt and Bonpland ! Alph. De Candolle unites this species with M. acapulcensis, but the foliage is sufficiently different. 32. Masa PAVONII. M. foliis ovalibus, apice acutis, basi obtusis, supra subglabris, subtus velutinis, subcoria- ceis, breviter petiolatis; floribus masculis cymosis, brevissime pedicellatis, 3-meris, pubescenti- bus; floribus femineis solitaris, breviter pedunculatis. Macreightia Pavonit, Alph. DC. Prodr. vit. p. 222. n. 7 (1844). Branches puberulous. Leaves oval, acute at apex, rather glabrous above, velutinous and paler beneath and on the petioles, 5—6 in. long by 2#in. wide; midrib puberulous above, thinly subcoriaceous; petioles } in. long. 3. Flowers 2 in. long, several together on axillary fulvo-tomentose peduncles which are about } in. long; pedicels scarcely 4; 1m. long. Calyx } in. long, ovoid, hairy on both sides; lobes ovate, acute. Corolla fulyvo-sericeous outside except at base, glabrous outside, twice the length of the calyx. . Q@. Flowers solitary, 4; in. long; peduncles } in. long; calyx deeply 3-fid; lobes oval, submucronate. Local name Orlaca. Peru (?) or Mexico (?) ex Alph. DC., Pavon/ 33. Mapa MANNII, sp. noy. M. glabrescens, foliis ovalibus, apice obtusis, basi rotundatis vel parum angustatis, subco- riaceis, breviter petiolatis; floribus masculis 3-nis, brevissime cymosis, trimeris, staminibus 6—9, leviter hirsutis, basi corolle insertis ; ovarw rudimento hirsuto; fructibus solitariis, subsessilibus, subglobosis, glabratis, 5—6-locularibus ; calyce parvo, patente, leviter puberulo ; seminibus 5—6, albumine ruminato. A small tree, growing by rivers; glabrescent, dark when dried; branches terete, erect- patent. Leaves oval, browner beneath, subcoriaceous, spreading, midrib and lateral veins Vou. XII. Parr I. 17 130 Mr HIERN, ON EBENACE. clear, raised beneath, depressed above, 3 to 5 in. in length by 14 to 2} in. in width; petioles 1,—1 in. in length; flowers subsessile, &. Cymes very short, very slightly pubescent, dark, 3-flowered, thick. Flower trimerous, 42 in. in length, slightly hairy, white when living, dark when dry. Calyx ciliate and slightly hairy, j; in. in height, with 3 rounded lobes about 3; in. in length, campanulate, not appressed to corolla, not accrescent; corolla tubular, glabrous, 3-lobed near apex; lobes } of the depth of the corolla, rounded. Stamens 8 (6—9), linear, acute, somewhat hairy, inserted at base of corolla, in 2 series (6 in outer series). Ovary rudimentary, hairy. Q. Fruit glabrous, of bright orange colour when ripe (glabrescent), sub-globose, ob- scurely 5—G6-sided, (5-) 6-celled, 5—6-seeded, nearly 1 in. in diameter. Fruiting calyx hori- zontal, small, 3; in. across, faintly puberulous; albumen ruminated. Flowers in April, near the Equator, West Africa, Niger Expedition, Barter! 1220; Bagroo River, Mann! 839; Quorra, Vogel! 34. MABA SEYCHELLARUM, sp. nov. M. fruticosa, foliis anguste ellipticis, apice obtusis, glabris, subcoriaceis, distichis, subsessi- libus ; floribus femineis solitarirs, subsessilibus, pubescentibus, trimeris, calyce breviter 3-lobo, staminodiis 3—6, glabris, basi corolle insertis, ovario ovoideo, 3-loculari, loculis biovulatis, stylo apice 3-lobo; fructibus ellipsoideis, glabris ; calyce fructifero cupuliformi, appresso ; seninibus solitartis, albumine non ruminato. Shrub 10—12 ft. high; ‘branches dark-cinereous, terete, at 35°, with short patent hairs at extremities, glabrescent; terminal bud with light brown pubescence. Leaves narrowly elliptical, obtuse or notched at apex, slightly narrowed at base, subsessile, distichous, somewhat convex from above in dried state; midrib depressed above, other veins inconspicuous; sub- coriaceous, glabrous, 1 to 2 in. in length by } to }in. in width; internodes } to } in. in length. @. Flowers solitary, subsessile, with light brown pubescence, } in. long. Calyx campanu- late, } in. long, with 3 shallow depresso-deltoid apiculate lobes, pilose outside, glabrous within. Corolla 3-lobed, divided moge than half-way down, hairy outside except near base, glabrous inside; lobes obtuse, imbricated. Staminodés 3 or 6, glabrous, inserted at base of corolla-tube. Ovary ovoid, glabrous except near apex or pubescent all over, 3-celled, cells 2-ovuled. Style erect, 3-lobed at apex, hairy except at apex. Fruit glabrous, ovoid, pallid, rather more than 8 in. long by rather more than } in. thick, 1 (?)-seeded. Albumen not ruminated, white; fruit- ing calyx 3-cornered, shortly cup-shaped, about } in. high by } in. wide. Seychelles L, Pervillé/ 36, mountains near the cascade; Mahé, 13 Febr. 1840. A specimen with similar foliage but rather more slender branches and peduncles (spines?) 4—1 in. long without flowers may belong here. Seychelles, Boivin! Mahé. Fruit subsessile, solitary, axillary, ellipsoidal, }—} in. long by 4—} in. thick, of a pale colour, shining, glabrous except at the apex where the remains of the hairy style project; fruiting calyx pubescent or glabrescent, cup-shaped, appressed to base of fruit, 3-lobed usually with short depresso-deltoid lobes, ;—} in. high; fruit 1-celled, 1-seeded; seed rather more than } in. long (in one case) by } in. thick; albumen not ruminated, bony. Seychelles, Dr Percival Wright / 1867, 30 May—238 Nov., n. 122. -<* Mr HIERN, ON EBENACEA, 131 35. MABA LANCEOLATA. M. foliis lanceolatis vel ovato-lanceolatis, utrinque acutis, glabris, coriaceis, breviter petio- latis ; floribus masculis 1—3-nis, sessilibus, basi bracteatis, bracteis imbricatis, calyce hirsuto, breviter 3-lobo, corolld 3—4-lobd, staminibus 24—32, glabris, basi corolle insertis ; floribus femineis brevissime pedunculatis, ovario glabro, globoso, 3-loculari, loculis biovulatis ; fructibus ovoideis, glabris, nitidis. Diospyros lanceolata, Poir. Encycl. Méth. v. p. 434 (1804); Alph. DC. Prodr. vir. p. 236. n. 69 (1844); non Wall. A tree; with glabrous, lanceolate or ovate-lanceolate leaves, acute at both ends, especially at the apex, coriaceous, in the dry state brown on both sides, 14—2# in. long by 3—1,, in. wide; petioles spreading, }—; in. long; veins confluent at the margin, shining above; margins recurved. 3. Flowers 1—3 together, sessile, ovoid, acute in bud, 1} in. long; surrounded at the base by 7 imbricated rounded ciliolated unequal coriaceous bracts, glabrous except at margin, the inferior ones very short. Calyx nearly }in. long, densely hirsute, ferruginous, shortly 3-lobed, 3-cornered, campanulate in flower (spreading in fruit much less hairy and } in. across). Corolla glabrous but with broad hairy patches outside lobes, }—;, in. long, deeply 3—4-lobed; lobes oblong, emarginate, spreading and recurved. Stamens 24—32, glabrous, inserted at base of corolla, the outer ones shorter; filaments short. Ovary wanting. @. Peduncles very short, recurved; calyx urceolate, shortly 3-lobed, not accrescent ; corolla narrowed at the throat, deeply 3—4-(?) lobed; staminodes ...; ovary quite glabrous, shining, spherical, 3-celled ; cells 2-ovuled with imperfect septa in middle; style 3-lobed, erect; fruit ovoid, glabrous, shining. Madagascar, Commerson / The leaf described and figured by Eétingshausen in Blatt-skel. Dikot. p. 89. t. 37. fig. 12 (1861) is decidedly larger than in the specimens that I have seen of this species; it pro- bably belonged to a different species. 36. MABA NATALENSIS, Harv. Thes. Cap. 1. p. 7. t. 110 (1863). M. fruticosa, ramis gracilibus patentibus ; foliis ovalibus, obtusis, glabris, supra nitentibus, subtus pallidioribus, breviter petiolatis; floribus feminers solitariis, brevissime pedunculatis ; calyce cupuliformi, glabro, integro; corolla trilobd, extus sericed; staminodiis 6—9, glabris, uniserialibus ; ovario conico, glabro, 3-loculari, loculis bi-ovulatis. A quickly glabrescent shrub; branches pale, slender, spreading at 60°—65°; shoots flexuous, puberulous. Leaves oval, obtuse or mucranate at apex, submembranous, flat, 4—1 in. in length by }—,4 in. in width, veins delicate and inconspicuous, shining and dark green above, paler beneath; petioles ;,—,, in. in length. ?. Flower solitary, axillary, very shortly pedunculate, 3 in. in length; peduncle ;;—,, in. in length. Calyx 7, in. in length, truncate, entire, dark green, glabrous, semi-ellipsoidal, Corolla 2; in. in length, argenteous-sericeous outside, 3-lobed ; lobes } in. in depth, diverging, ob- long, acute. Staminodes 6—9, free, uniseriate, 75 in. in length, glabrous. Ovary conical, drab, 17—2 132 Mr HIERN, ON EBENACE. glabrous, terminated by a style 3-lobed at the apex, 3-celled, cells bi-ovuled; style as long as ovary, glabrous. Fruit ellipsoidal, glabrous, of pale chestnut colour, 4 in. in height by fin. in width; style persistent; fruiting calyx not increased in height, hemispherical, like an acorn-cup, 1-seeded. S. Africa, Natal, W. Z. Gerrard / 110; D’Urban, Macken! 675. 37. MABA ABYSSINICA, sp. nov. M. fruticosa, foliis lanceolato-oblongis, plerumque apice obtusis et basi rotundatis, glabris, subcoriaceis, planis, breviter petiolatis, nervis inconspicuis ; floribus masculis subsessilibus, aggregatis, 3—4-meris, calyce laxo, lobis rotundatis ciliatis, corolld glabré, staminibus circiter 14, glabris ; floribus femineis 3—5-nis, aggregatis, brevissime pedicellatis, 3—5- sepius 3-me- ris, calyce campanulato, non accrescente, corolla glabra, aperte campanulatad ; staminodiis 3—4, glabris; ovario ovoideo, glabro, 6-loculara, loculis uni-ovulatis, stylo apice 3-lobo ; fructibus glabris, subglobosis ; seminibus solitartis, albwmine non ruminato. A large shrub, glabrous except the inflorescence; shoots dark, terete. Leaves lanceo- late-oblong, obtuse and often somewhat acuminate at apex, more or less narrowed at base, subcoriaceous, flat, of the same dull colour on each face, somewhat shining above, patent or erect-patent, 2—5 in. long by }—13 in. wide; petioles {—} in. long; midrib slightly depressed above, veins inconspicuous. Bracts pubescent, small, caducous; flowers subsessile, clustered, axillary ; flowers mostly trimerous, sometimes 4—\5-merous. g. Flowers in. long; calyx 3; in. long, lax, usually 3-fid, lobes rounded, minutely ciliate; corolla widely campanulate, glabrous, 3—4-fid, lobes rounded; stamens about 14, glabrous, appearing at the mouth of the open corolla, mostly in pairs, inserted at the base of the corolla, filaments short; ovary rudimentary, glabrous. ¢. Flowers 8—5 together, on very short puberulous pedicels, }—} in. long; calyx } in. long, or after the fall of the corolla about }in. long, glabrous or margins of the lobes minutely ciliate, deeply 83—5- usually 3-lobed, campanulate, lobes wide or cordate at base, 4 in. long; corolla glabrous, openly campanulate, deeply 3—5-lobed, not always isomerous with the calyx, lobes spreading or recurved, oval-ovate, obtuse; staminodes 3—4, glabrous, alternate with the lobes of the corolla; ovary glabrous, ovoid, conical at apex, 6-celled, cells l-ovuled; style thick, 3-lobed at apex; fruiting calyx lax, not accrescent ; fruit glabrous, sub- globose, shining, of dark coppery colour, 7;—{ in. long by }—} in. thick, 1—few-seeded, bearing remains of style at apex; albumen of fads cartilaginous, not ruminated. Abyssinia, on the sides of the valley of Bellagass near Loegga, 5000 feet altitude, Schimper / 1854, n. 1080 ; Sila 5000—6000 feet altitude, Schimper! 1854, n. 1334; Keren, bank of river, Beccari! n. 55, May, 1870. 38. MABA QUILOENSIS, sp. nov. M. glabrescens, foliis ellipticis, apice obtusis, basi subcordatis, submembranaceis, breviter petiolatis, nervis lateralibus inconspicuts ; floribus femineis subsessilibus, sub-3-nis, calyce trilobo, ovario glabro, ovoideo-conico, 6-loculari, loculis unti-ovulatis, stylo apice trilobo, glabro, caly- cem superante. Mr HIERN, ON EBENACE. 155° Glabrous shrub; branches terete, at about 60°, argenteous-cinereous, except when very young and then they become blackish in the dried state. Leaves alternate, elliptical, dark, without conspicuous lateral veins, submembranous, obtuse or rounded at apex, usually sub- cordate at base, 1 to 2 in. in length by } to 1 in. in width; petioles scarcely 3, in. in length. Q. Flowers subsessile, about 3 together, dark when dry; calyx 3, in. in height, with 3 ovate diverging lobes extending % down the calyx; corolla fallen from specimens; ovary glabrous, ovoid-conical, 6-celled, cells 1-ovuled; style 3-lobed at apex, glabrous, higher than the calyx. East Tropical Africa, Quiloa, Dr Kirk/ Fl. Zangueb. n. 110, 10 January, 1867. 39. MABA MICRANTHA. M. foliis ellipticis vel oblongis, basi attenuatis, apice obtuse acuminatis, glabris, coriaceis, petiolatis ; floribus femineis solitariis, sessilibus, awillaribus ; calyce tubuloso, integro, truncato, tn squamis paucis bifariis imbricatis insidente ; corolla tubulosa, trifidd, quam calyce triplo lon- gore, lobis ovatis obtusis patentibus ; staminodis 6, bast corolle insertis ; ovario hemispherico, glabro, 6-loculart ; loculis uni-ovulatis ; stylis 3, erectis. Holochilus micranthus, Dalz. in Kew Jour. Bot. Iv. p. 291 (1852), Dalz. et Gibs. Bomb. Fl. p. 142 (1861). A middle-sized tree. Leaves elliptical or oblong, attenuate at base, obtusely acuminate at apex, coriaceous, glabrous, 4—5 in. long by 2 in. wide; petioles $ in. long. Flowers dic- cious; ¢ flowers unknown. 2. Flowers } in. long, white, solitary, sessile, axillary; calyx tubular, entire, truncate, placed among a few bifarious scales; corolla tubular, 3 times the length of the calyx, 3-lobed nearly to the middle; lobes ovate, obtuse, spreading. Staminodes 6, inserted at the base of the corolla-tube, distinct, filaments double of the length of the barren anthers; styles 3, erect, rather thick, obtuse at the apex; ovary hemispherical, glabrous, 6-celled; ovules solitary. Fruit cylindric-oblong, supported at the base by the accrescent truncate calyx, dry, hard, 1 in. long, 6-celled; seeds solitary. In the Syhadra hills, on the Southern Ghauts, Bombay. Flowers in February and March, Dalzell. 40. Mapa LAMPONGA, Mig. Fl. Ind. Bat. Suppl. 1 n. 1179. p. 584 (1860). M. foliis obovato-oblongis vel ellipticis, apice rotundatis retusis, basi angustatis, glabres- centibus, coriaceis, breviter petiolatis, venis inconspicuis; floribus masculis (monstrosis in spe- ciminibus?) axillaribus supra-axillaribus et lateralibus, paniculatis fasciculatis wmbellatis vel interdum solitarvis ; floribus femineis solitariis, subsessilibus vel breviter pedunculatis, axilla- ribus, calyce tridentato, corolld subcampanulata, lobis elliptico-oblongis, acuminatis, ovario ovoideo, glabro, stylo brevissimo, stigmatibus 3, patentibus. , Buds velutinous. Leaves minutely and appressedly downy when young, quickly glabres- cent, obovate-oblong or elliptical, acute or sub-cuneate at the base, in most cases widely rounded and retuse at the apex, coriaceous, rather shining, griseo-pallid when dry, nearly 134 Mr HIERN, ON EBENACEA, veinless, but with slender net-veins when old, 3}—2 in. long; petioles short, sub-trigonous. Flowers dicecious ? g. Flowers (all monstrous?) axillary supra-axillary and lateral, sometimes in short panicles, at other times fascicled or umbelled, occasionally solitary, pedicelled; corolla repre- sented by 3 ovate scales united at the base, alternating with the calyx-teeth, pubescent ; moreover there are placed inside numerous narrower scales (monstrous stamens?) in several series, free or united in pairs, more or less hairy at the back, plainly imbricated. @. Flowers solitary, subsessile or shortly pedunculate, axillary; calyx coriaceous, cupu- liform-globose, tridentate; teeth triangular acute, appressedly downy outside; tube of the corolla short, subeampanulate, glabrous as high as the calyx; lobes elliptic-oblong, acuminate, densely hirsute along the middle of the back. Ovary ovoid, glabrous; style very short, thick, stigmas 3, spreading, canaliculate in front. South Sumatra in prov. Lampong; on sea coast. Teysmann. 41. MABA MERGUENSIS, sp. nov. M. foliis ovalibus oblongis vel ovato-oblongis, apice acwminatis, basi subrotundis vel parum angustatis, glabris, submembranaceis vel tenuiter coriaceis, petiolatis; floribus masculis 8-nis, paniculatis, parvis, acxillaribus, 3—4-meris, pedicellis brevissimis ; calyce aperte campanulato, ‘minute puberulo, ciliato; corolld subglabrd; staminibus 14—16, glabris; ovarit rudimento subtus glabro; floribus femineis dense cymosis, 3—9-nis, plerwmque trimeris ; staminodiis 3 vel 6, glabris, basi corolle insertis; ovario glabro, 6-loculari, loculis uni-ovulatis ; fructibus globosis, glabris; seminibus oblongis, albumine non ruméinato. Cfr. Diospyros frutescens, var., Blume, Bijdr. fl. ned. Ind. p. 668 (1825); 8. Tallak, Alph. DC. Prodr. vu. p. 230. n. 38 (1844). A small tree, glabrous, with brown or dark-ashy branches, spreading at about 50°. Leaves oblong or ovate-oblong, glabrous or sometimes with puberulous midrib beneath, sub-mem- branous, dark above, paler and brownish beneath, rounded or slightly narrowed at base, acuminate at apex, not black-punctate, nearly flat, veins delicately raised on both sides, or sometimes slightly depressed on the upper surface, 2}—6} in. in length by 1—3 in. in width; petioles 1—4 in. in length. g. Cymes panicled, bearing numerous flowers, pubescent with short lightish brown hairs, 4 to 1 in. in length (excluding the flowers); pedicels very short; flowers small; calyx +; in. in height, openly campanulate, with slight short pubescence outside, and 3 or 4 widely del- toid lobes about half the depth of the calyx, ciliate; corolla nearly glabrous, about 1; in. long, 8—4-lobed, lobes short; stamens 14—16, mostly or all in pairs, inserted at base of interior of corolla or hypogynous, glabrous, the interior ones the smaller, anthers about equalling the longer filaments; ovary rudimentary. ?. Cymes dense, short, many-flowered, with thick pedicels, pubescent in flower, gla- brescent in fruit; bracteoles ovate, pubescent, caducous, jin. in length; flower usually trimerous, occasionally tetramerous, } in. in length; calyx pubescent } in. in length, gla- brescent, spreading, lobes #5 in. long, diverging, ovate, sides reflexed; corolla pubescent, lobes ;; in. in length, oval, somewhat spreading; staminodes 3 or 6, inserted at base of the Mr HIERN, ON EBENACE. 135 tube of the corolla, } in. in length, linear, glabrous; styles 3, distant, glabrous, 3; in. in length ; ovary semi-ellipsoidal, glabrous except at base where there is a band of hairs, 6-celled, cells l-ovuled. Fruit glabrous, smooth and shining, globular, about 4—? in. in diameter, some- times 4-celled when young; albumen of seeds not ruminated. Flowers in January, fruits in February. Mergui Archipelago, Griffith! (in fruit), Helfer! 3618; Sumatra, Korthals/; Borneo, O. Beccari! n. 1670; ?Java, Blume!, Kurl and Hasselt / 42. Mapa FAscIcuLosa, F. Muell. Fragm. v. p. 163 (1866). M. foliis ovato-lanceolatis vel oblongis, apice angustatis vel acuminatis, obtusis, basi angus- tatis, glabris, coriaceis, petiolatis; floribus masculis numerosis, dichotome cymosis, 3—4-meris, staminibus 8—18, antheris glabris, filamentis sepe minute ciliatis ; floribus femineis 3—x -nis, 3—4-meris, staninodiis 0—4, ovario glabro, 6-leculari, loculis uni-ovulates ; fructibus subglo- bosis, glabris. Benth. Fl. Austral. rv. p. 290. n. 5 (1869). Diospyros fasciculosa, F. Muell. Austral. Veg. in Intercol. Exh. Ess. 1866—67, p. 35 (1867). M. laxiflora, Benth. l.c. n. 4. A tall shrub or lofty tree, glabrous with terete branches spreading at about 45°. Leaves ovate-lanceolate oval or oblong, narrowed or acuminate at apex, obtuse, more or less narrowed at base, minutely black-punctate beneath, 2—4} in. long by 3—1} in. wide; petioles 1—2 in. long; midrib and veins more or less raised on both surfaces; margins somewhat recurved. 6. Flowers numerous, 3—4-merous, {—,% in. long, in fascicled axillary cymes 3—4 in. long exclusive of the flowers; pedicels slender, subglabrous; bracteoles small, ovate, slightly pubescent, caducous. Calyx ;,;—% in. long, with short lobes, somewhat pubescent outside, glabrous inside. Corolla campanulate, 8—4-fid, glabrous or obsoletely pubescent, lobes rounded; stamens 8—18, anthers glabrous, filaments often minutely ciliate; ovary rudimentary, gla- brous. 2. Flowers 3 or many together, clustered, 3—4-merous; cymes axillary short; stami- nodes 0—4, glabrous; ovary shortly conical, glabrous, 6-celled with cells 1-ovuled, or 3-celled with 2 ovules in each cell separated by an incomplete dissepiment; style very short, 3-lobed at apex. Fruit subglobose or shortly ellipsoidal, shining, of pale colour, 4;—3 in. long; fruiting calyx 3—4-fid with spreading and reflexed deltoid lobes, tube cupuliform; seeds 1—4, albu- men not ruminated. Called in New Caledonia Médeso. Australia, Rockingham, Dallachy!; Queensland Woods, Hill! 100; Brisbane River, F. Mueller !; Rockhampton, O’Shanesy!, Thozet. New Caledonia, Deplanche! 48, 206; Vieillard! 899. 43. MABA RUMINATA, sp. NOY. M. foliis anguste ellipticis, utrinque angustatis, glabris, coriaceis, petiolatis; floribus femineis w-nis, trimeris; fructibus subglobosis, glabris; calyce fructifero 3-fido, tubo hema- spherico, lobis late ovatis, patentibus; albumine seminum ruminato. 136 Mr HIERN, ON EBENACE. Young parts and inflorescence puberulous ; branches somewhat cinereous. Leaves narrowly elliptical, narrowed at both ends, glabrous, alternate, coriaceous, 3—5 in. long by 4—13 in. wide; petioles {—} in. long; margins recurved; midrib slightly dilatato-depressed above ; veins not conspicuous above, of same colour as lamina beneath. g. Cymes axillary, many-flowered, }—} in. long, spreading; pedicels 7;—% in. long, puberulous; fruiting calyx 8-fid, nearly } in. across, glabrous or very nearly so, tube hemi- spherical; lobes widely ovate, convex from above, spreading; fruit about } in. long, sub- globose, glabrous, pale and shining; seeds with ruminated albumen. New Caledonia, Deplanche/ 311. 44. MABA CONFERTIFLORA, sp. nov. M. foliis ovali-oblongis, apice obtuse vel emarginate acuminatis, glabris, coriaceis ; floribus masculis aggregatis, brevissime cymosis, subsessilibus, trimeris, corolla urceolato-tubulosd ; sta- minibus 12, geminatis, glabris; floribus femineis axillaribus, subsessilibus, aggregatis, trimeris ; staminodiis 2—8, stylo apice 3-lobo, ovario glabro, 6 ?-loculard. A small tree; shoots subglabrous with scattered short appressed hairs, cinereous. Leaves crowded, oval-oblong, obtusely or emarginately acuminate, coriaceous, glabrous except the midrib beneath, shining, with depressed veins on the upper surface, somewhat paler beneath ; lateral veins inconspicuous and weak; leaves 14—3} in. long (including subglabrescent petiole + in. long) by $—1} in. wide. Bracts shortly ovate. g. Flowers many, crowded, on short slightly hairy cymes, subsessile, rufous; calyx small, trifid, with scattered short hairs, spreading; corolla urceolate-tubular, shortly trifid, with 3 hairy lines down the middle lines of the lobes; stamens 12, united by their fila- ments in 6 pairs, the inner ones being the shorter, glabrous; anthers dehiscing widely from apex downwards; ovary rudimentary, glabrous. 9. Flowers subsessile, crowded in axils of leaves, several abortive; calyx sub-glabrescent, coriaceous, spreading, 3-lobed; corolla 3-fid; staminodes 2—38; style 3-lobed at apex; ovary glabrous, 6 ?-celled, with a few hairs at base. . Labuan, Lobb/, Motley! 205. 45. MABA PUNCTATA, sp. nov. M. foliis oblongis, apice breviter acuminatis vel apiculatis, basi subcordatis, coriacets vel submembranaceis, minute pellucido-punctatis, supra glabris nitidis, subtus secus nervos puberulis, breviter petiolatis ; floribus masculis dense cymosis, pubescentibus, trimeris, staminibus 9, glabris ; frribus femineis 3—® -nis, breviter cymosis, supra-axillaribus, trimeris ; fructibus sub-globosis, glabris, 6-locularibus. Diospyros punctata, Korthals, MSS. in Hb. Lugd. Batav. Ebenac. n. 15. A small tree; young parts, inflorescence, &e. softly ferruginous-pubescent ; shoots terete, pubescent. Leaves oblong, alternate, coriaceous or submembranous, minutely pellucid-pune- tate, subcordate at base, suddenly and sharply acuminate apiculate or mucronate at apex, glabrous and shining above with depressed midrib and lateral veins; midrib and about 9 Mr HIERN, ON EBENACEA, sy or 10 lateral veins on each side, puberulous, distinct and in relief beneath; lower surface slightly and appressedly puberulous, with evanescent reddish pulverulence ; 33—10} in. long by 14—3} in. wide; petioles ¢in. long, thick, terete, pubescent; usually some depressed glands are visible on the lower surface of the leaves especially at the base. 6. Inflorescence axillary, dense, many-flowered, short, ;3—2in. long (exclusive of the flowers); bracts acute, numerous, hairy ; pedicels varying in length up to } in.; flowers + in. long, white; calyx $in. long, campanulate, shortly 3-fid, shortly pubescent outside, alnies inside, lobes deltoid; corolla tubular, hypocrateriform, #,im. wide, shortly 3-lobed, sericeous outside, glabrous inside ; lobes acute, spreading, #5 in. long; stamens glabrous, 9, hypogynous, equal, 6 united by thes filaments in 3 pairs of which ihe inner ones are the shorter, and 3 distinct; anthers longer than the filaments, linear, acute ; ey 0; receptacle glabrous. @. Inflorescence supra-axillary; cymes 3—many-flowered, 1—1 in. long CaOuENs of the flowers); bracts small acute caducous; fruiting pedicels thickened upwards, ;,—1 in. long; pericarp rather thick; fruit ovoid nt long by 2 in. thick, terminated by style 35 in. lone 3-lobed at apex, glabrous (in dry state), subverrucose, shining, 6-celled, Gabededs fruiting calyx appressed to young fruit and sericeous, quite patent and nearly glabrate in older fruit, not accrescent, }in. across, 3-fid with ovate or deltoid lobes. Borneo, Foot of Gunong Pautie on Serpentine Rocks. Mr Motley! n. 766; Korthals /, Beccari! n. 1423. Pratt TV. A fruiting branch, natwral size. a. A piece of a male branch in flower, natural size. b, A piece of a female branch, natural size. c. 3. DIOSPYROS OPPOSITIFOLIA, Thw. En. Ceyl. Pl. p. 181. n. 11. (1860). D. foliis oppositis, obtusis, breviter acuminatis, basi rotundatis, coriaceis, glabratis, breviter petiolatis ; floribus masculis anguste tubulosis, subsessilibus, tetrameris, calyce 4-fido, campa- nulato, corolla breviter 4-sidd, staminibus circiter 8, incequalibus. Bedd: Te. Pl Ind. Or. (Pt vin.) p. 27. t. 131 (871). A moderate-sized tree; branches glabrous terete. Leaves oval, firmly coriaceous, glabrous (or the younger ones slightly pubescent), rounded at base, obtuse or shortly acuminate at apex, opposite or subopposite, 2—Gin. long by 14$—8in. wide; net-veins inconspicuous, nearly transverse and feebly depressed on the lower surface; petioles j4—jin. long, tumid- erass, dark, glabrous. 3 Flowers sessile or subsessile, few together, 2in. long. Calyx ;,in. long, not quite glabrous, 4-lobed nearly to the middle, with acute lobes. Corolla slender, hispid, }in. long, lobes about 4 the depth of the corolla. Stamens about 8, very unequal; (the filaments and connectives are figured as having short hairs). The timber of this tree resembles that of D. quesita, Thw., Calamander. Ceylon, Thwaites!; C.P. 3011; Hinidoon Corle, up to elevation of 1000 feet ; local name ‘“ Kaloomidereya-gass.” 158 Mr HIERN, ON EBENACE. 4. Drosprros Tupru, Buch. Journey vol. I. p. 183 (1807). D. foliis alternis et suboppositis, ellipticis ovatis vel subrotundis, apice obtusis rarius acuminatis, basi rotundatis vel rarius angustatis, coriaces, subtus tomentosis, petiolatis; pe- dunculis florum masculorum longitudine petioli, apice 3—4-floris, calyce campanulato apice 4—6-lobo, staminibus 12—18; floribus femineis solitariis brevissime pedunculatis, staminodits 0—6, fructibus subglobosis vel ovoideis glabris, calyce profunde 4—6-fido, lobis ovatis margine extus reflexis; albumine ruminato. Hamilt. (olim Buch.) in Trans. Linn. Soe. xv. p. 111 (1827). Diospyros exculpta, Hamilt. lc. p. 110, D. exsculpta, Alph. D.C. Prodr. vit p. 228. n. 3 (1844), Bedd. Fl Sylv. t. 66 (1870) 2 fr. Diospyros insculpta, Hamilt. lc. p. 112, Alph. DC. Lc. n. 6. Diospyros tomentosa, Roxb. Hort. Beng. p. 40 (1814), Fl. Ind. edit. 1832 vol. 2. p. 532, Roxb. draw. n. 1728 in Hb. Kew, R. Wight Ic. tt. 182, 183 (1840), non Poi. 2D. speciosa, Wood, Rep. For. Oudh 1867—68, p. 33 (name only, 1869). Called Tupru (Carnatic), Kend (Hindoo), Kendu (Bengal) according to Hamilton, and Kallindoo (Sanscrit), Kyou and Ywmala (Bengal), according to Roxburgh; Tunki in the Cuddapah district, and Zumboornee in the Bombay presidency, according to Beddome. A tree either of small moderate or large size up to 60—80 ft. high; dicecious or poly- gamous; the heart-wood is black in some trees and of a hard and heavy substance called at Munghur Batii and at Saseram Abnus. The latter word is said to be of Persian origin and a source from which our word Ebony is derived. Trunk grey-black, bark very closely cracked both transversely and longitudinally. Branches cinereous, alternate or opposite, ramified as in the oak ; young shoots ferruginous-pubescent. Leaves opposite subopposite and alternate, elliptical ovate or subrotund, bright green, more or less coriaceous, usually almost glabrous on the upper side and tomentose beneath, sometimes glabrous on both sides; obtuse or rounded at the base; emarginate rounded or obtusely narrowed or sometimes apiculate at apex; 3—I4in. long by 14—7}in. wide; petioles }—fin. long; lateral veins usually prominent beneath; deciduous. & Flowers 3 or 4, on recurved thickened tomentose peduncles equalling or rather longer than the petioles, 4—5-merous, white, }— ,;in. long; bracts small; pedicels very short; calyx tomentose, campanulate; corolla much longer than the calyx, with short lobes, hairy outside; stamens 12—18, inserted on the receptacle, glabrous (7); ovary rudimentary, hairy. Q Flowers solitary, subsessile or shortly stalked, 4—6-merous; peduncles about ;\in. long; bracts 3—4, scale-like, caducous; calyx campanulate, 4—6-fid; corolla shortly 4—6- lobed; staminodes 0O—6; ovary 4 (-6%) -celled, somewhat hairy; styles 2 (—8). ~ Fruit egg- shaped or globose, glabreseent, about lin. long by fin. thick, usually 4-celled and 3-seeded ; seeds din. long by fin. wide and fin. thick; fruiting calyx surrounding the base of the Mr HIERN, ON EBENACEA. 159 fruit or spreading, pubescent on both sides, 3—{in. across, not or scarcely accrescent ; testa shining, marked with reticulated depressions; albumen cartilaginous, ruminated, grey. “The fruit when ripe is sweet and not very bad tasted;” according to Hamilton the cotyledons in J. insculpta are conduplicate. This valuable tree sheds all its leaves in the cold season, and they appear again in the beginning of the hot weather (Beddome); not uncommon in the Cuddapah, Salem and Kurnool forests in Madras. Difficult to distinguish from D. melanoxylon, Roxb., to which species it ought perhaps to be united. N.W. India. Hb. Royle! ; Brundekund, Edgeworth ! 6004; Hb. Stocks! ; plains of Behar, &e., Dr Hooker! 440, 441; Magadi (used for small beams and posts), Hejuru, S.W. Mysore (a large tree; timber good), Buch. Ham. Journey vol. 1. p. 183. vol. 11. p. 125, W. Himalaya, Dr Stewart ! ~ 5. DIOSPYROS MELANOXYLON, Roxb. Coromand, p. 36, t. 46 (1795). D. foliis oppositis suboppositis vel alternis, ovalibus vel eblongis, apice rotundatis vel leviter angustatis, basi cuneatis vel rarius rotundatis, pubescentibus tenwiter coriaceis, petiolatis ; pedun- culis florum masculorum longitudine petioli plurifloris, calyce campanulato, tomentoso, breviter 4—5-lobo, staminibus 12—16, rarius 8, glabris vel antheris leviter hirsutis ; floribus Semineis solitariis brevissime pedunculatis 5—4-meris, staminodiis 8 vel 10, ovario 4—(8-) loculari, fruc- tibus globosis vel ovoideis, pubescentibus vel glabratis, plerumque 4-spermis ; albumine ruminato. Alph. DC. Prodr. virt. p. 224. n. 7 (1844), non Blum. nec Hassk. D. Wightiana, Wall.! List n. 4406 (1828—32), Alph. DC. Prodr. vit. p. 223. n.2, Beddom. Fl. Sylv. Madras, t. 67 (1870). D. Roylii, Wall. List n. 4134 (1828—382), D. Roylei, Alph. DC. l.c. p. 239, n. 89. D. dubia, Wall. n. 4407, Alph. DC. lc. p. 223. n. 4, non Goepp. Cfr. D. rubiginosa, Roth, Nov. Pl. Sp. p. 885 (1821), et D. montana, Heyne ex Roth /.c., non Roxb, Tunki Tumi and Tumbi in Tamil and Telugu, ex Beddome, Fl. Sylv. Madr. t. 67 (1870) ; Tumida of the Telingas, ex Roxb. l.c.; Twmballi of the Tamuls, ex Roxb. Fl. Ind. edit. 1832, vol. 11. p. 531; Zindoo of the Hindoos ex Roxb. l.¢.; (Zendoo, Beddome); Coromandel ebony tree; Thomboorah Marum in Hb. Wight; Zoomrie, Dr Ritchie (Belgaum). A large tree with a trunk 8—10 feet in circumference, sometimes only a small shrub ; dicecious; young shoots very downy, pale-ferruginous. Leaves opposite subopposite and alternate, pubescent especially beneath, thinly coriaceous, oval or oblong, cuneate or rarely rounded at base, rounded or somewhat narrowed at apex, 2—6 in. long by 1—2! wide; petioles }—} in. long; veins less conspicuous than in D. Tupru; deciduous. é. Flowers }—} in. long, in panicled tomentose-ferruginous drooping cymes }—} in, long, longer than the petioles, several or many together, with small bracts at base and apex of short pedicels. Calyx shortly 4—6-lobed, campanulate, tomentose on both sides. Corolla 4—6-lobed at apex, glabrous inside, densely silky outside, 1}—3 times the length of the calyx. Stamens 12—16 (rarely 8 only), in pairs when 16, glabrous or with lines of short hairs back 160 Mr HIERN, ON EBENACEZ. and front on the anthers which are longer than the filaments; ovary wanting or rudimen- tary and hairy. @. Flowers rather larger than the male, solitary, subsessile, pentamerous or tetramerous ; calyx hairy on both sides, 5-winged (Gn D. Wightiana, Wall.!) by the patent projection of the margins of the lobes; staminodes 8 or 10; styles 2 bifid somewhat hairy; ovary 4! (—%) -celled, densely hairy; cells l-ovuled. Fruit globular or ovoid, somewhat hairy or glabrescent, usually 4-celled and 4-seeded, about 1 in. long; albumen of seed somewhat ruminated ; according to Roxburgh 2—8 seeds ripen; fruiting calyx nearly flat about 3 in. across Neilgherries and Serramallee Hills, India, R. Wight! (D. dubia); Adjeeghur, and Bisrum- gunge ghaut, Royle (D. Roylii); Belgaum, Dr Ritchie! 1108; Calicut!, Hb. Wight! 1723, Subbulpore, 1727, 1721, 1725; Hb. Grifith!, 3630, 3626 (1); Bababoodun Hills, Mysore, Mr Law!; common in dry forests in Madras, according to Major Beddome. The ebony tree of Malabar and Coromandel. Mysore, a small shrub, common, Dr Brandis/, May 1868. It is only the centre of large trees that is black and valuable, and tbe quantity found varies with the age of the tree. The outer portion of the wood is white and soft, and either decays soon or is destroyed by insects which leave the black part untouched. The ripe fruit is eaten by the natives in the Circars, but is astringent and not very palatable. The bark of the tree possesses tonic and astringent properties, and in decoction proves useful in atonic diarrhoea, dyspepsia and diseases of debility. [See E. J. Waring, Pharmacopeia of India, p. 132 (1868).] Ctr. D, decandra, Lour. 6. Diospyros DECANDRA, Loureiro, Fl. Cochinch. p. 227 (1790). D. foliis ovato-lanceolatis vel ellipticis, apice obtuse acuminatis, basi plus minus angustatis, alternis, tenuiter coriaceis, leviter pubescentibus, petiolatis ; floribus femineis sub- 3-nis, cymosis, 4—5-meris; corollé urceolaté; staminodiis 10, glabris; ovario 6—8-loculari; Jructibus subglo- bosis edulibus. Alph. DC. Prodr. vim. p. 238. n. 85 (1844), non Boj. A large tree with rather patent branches, producing excellent heavy timber, white but marked with many black veins and sometimes with black heart-wood. Leaves thinly coriaceous, slightly pubescent, especially on the midrib, which is somewhat depressed on the upper surface, of nearly the same brown colour (in the dry state) on both sides, alternate, elliptical or ovate-lanceolate, shortly and obtusely acuminate at apex, more or less narrowed at base, 2—3 in. long (besides petiole ~,—2 in. long) by ;4—1} in. wide; venation as in D. melanoxylon. &. Inflorescence rufous-hairy, more or less glabrescent; peduncles axillary, ranging up to 4 in. long or rather more, bearing 3 or more flowers on act pedicels. Flowers whitish. Flower-bud depresso-ovato, 4 in. long by 3% in. thick; calyx deeply 4—5-fid, enclosing the young corolla, with valvate (?) deltoid lobes whose sides are somewhat revolute; -corolla shortly lobed, glabrous inside, tube urceolate, lobes obtuse, reflexed in full flower; staminodes glabrous, 10 according to Loureiro, short, inserted at the base of the corolla. Ovary 6- or 8-celled and -ovuled; ovules pendulous. Style short, lobed at apex. Fruit compresso-rotund or subglobose, subglabrate at least in part, 6—8-celled in the cases examined, about 1 in, in Mr HIERN, ON EBENACE. 161 diameter or perhaps larger, yellow, edible, pulpy, sweet but astringently so, 6—8-seeded, strongly scented, not very pleasant to the taste. Fruiting calyx spreading, nearly as wide as the fruit when young, Seeds bony, “compresso-ovate.” The fruit according to Loureiro is brought to market for sale. N. Cochinchina. Lowreiro/ A.D. 1774 (seen in Hb. Mus. Brit.). Local name Cay Thi. Very possibly D. melanoxylon, Roxb. ought to be united with this species; but the leaves in the latter are all alternate, so far as the specimen seen by me shews. 7. DIOSPYROS SYLVATICA, Roxb. Coromand. p. 37. t. 47 (1795). D. foliis alternis, ovalibus, sepe acuminatis, basi angustatis, vix coriaceis, glabris vel sub- glabris, breviter petiolates ; floribus masculis cymosis, -nis, globosis, parvis, swpius 4-meris interdum 3- vel 5-meris ; staminibus 13—22, glabris; floribus femineis solitariis pedunculatis, sepe in ramulis junioribus racemose dispositis, 4—3-meris, globosis ; stanvinodiis 4, glabris ; ovarvo 8- vel 6-loculari; fructibus globosis ; albumine ruminato. Alph. DC. Prodr. vii. p. 231. n. 41 excl. var. 8 velutina (1844). Thw. Enum. Ceyl. Pl. p. 178. n. 3 (1860). Bedd. Ic. Pl. Ind. Or. (Part vit.) p. 25. t. 121 (1871). D. orixensis, Klein ex Willd. Sp. Pl. rv. p. 110 (1805), Alph. DC. lc. p. 230. n. 35, non Wight. Native names: Tella-gada of the Telingas; Nella-gada (Hb. Roxb.); Soodoo-Kadoombat- reya-gass in Ceylon. A pretty large tree; foliage turning black when dry; branches spreading at 60°—75°, glabrous or the young shoots pubescent. Leaves alternate, oval, pointed or acuminate, thin, usually somewhat narrowed at base; nearly or quite glabrous, 2—6 in. long by }—3 in. wide; petioles 1—*, in. long, often puberulous; midrib and veins depressed on upper side, but not conspicuous; lateral veins not very close. g. Cymes axillary, several- or many-flowered, }—,% in. long (excluding the flowers), more or less shortly-pubescent; ultimate pedicels short; flowers small, 7,—+ in. long, white and fragrant when growing, 8—5-merous, usually 4-merous; calyx very short 2,—,% in. high by 75 in. wide, 3—5-fid, pubescent or ciliate, glabrous inside; corolla obconic-subglobular, lobed at apex, nearly glabrous; stamens 13—22, mostly in pairs and inserted at base of corolla, glabrous (or rarely with a few short hairs); anthers about the length of the filaments, dehiscing laterally from apex; ovary rudimentary, somewhat hairy at apex or glabrous, 2. Flowers solitary, on peduncles }—} in. long, larger than the g¢ , 3—4 usually 4- merous; staminodes 4, glabrous, alternating with the corolla-lobes; ovary 6- or 8-celled, glabrous or hairy at apex; cells l-ovuled; styles 3 or 4; fruit globose, glabrous or with a few appressed hairs around apex, } in. or more in diameter; fruiting calyx spreading, accres- cent, # in. in diameter. Seeds 2—8; albumen somewhat ruminated. Wood very hard, used for fancy work. Vou, XII. Parr I. 21 162 Mr HIERN, ON EBENACE. India, Circars, Roxburgh !; Concan, Law!; Bombay, Law /, 3000 ft. alt. Ceylon, Thwaites ! C. P. 2729, damp forests up to 4000 ft. alt. 8. Dziosprros Kurzu, sp. nov. D. foliis alternis, ovato-ovalibus, mox glabratis, apice acuminatis, basi cuneatis, breviter petiolatis, nitentibus, nervis lateralibus crebris tenuibus; floribus femineis sub-3-nis, breviter cymosis, tetramers, urceolatis; staminodiis 4, glabris ; ovario 4-loculari, 4-ovulato, stylis 2, basi connatis. Young branches pubescent with short appressed silky fulyous or brown hairs; branches at about 40°. Leaves ovate-oval, quickly glabrescent, alternate, dark, very dark and shining above with crowded delicate lateral veins which are also in relief beneath where the leaf is slightly paler, acuminate at apex, more or less narrowed at base, thinly coriaceous; 2131in. long by 1—12in. wide; petioles i1—1in. long; midrib depressed above. @ Cymes axillary, }—}in. long (excluding flowers), about 3-flowered, with very short pedicels, pubescent, and with small caducous bracts at base of calyx. Flowers jin. long. Calyx jin. long, puberulous outside, glabrous inside, shortly 4-fid, bigger in young fruit. Corolla 4-lobed at apex, with rounded lobes pubescent on both sides, urceolate ; staminodes 4, glabrous, alternating with corolla-lobes; ovary glabrous except apex, 4celled; cells l-ovuled. Styles 2, straight, erect, slender, hairy, long, connate at base. South Andaman, S. Kurz! 9, DIOSPYROS EHRETIOIDES, Wall. List, n. 4137 (1828—32). D. foliis alternis, ellipticis, vix coriaceis, discoloribus ; floribus masculis c-nis, cymosis, subglobosis ; staminibus 2229, glabris, ovarit rudimento hirsuto; floribus femineis solitariis, breviter pedunculatis ; fructibus globosis, glabratis ; albwmine ruminato. Alph. D.C. Prodr. vit p. 231. n. 42 (1844). D. mollis, Wall. ex Steud. Nomencel. bot. edit. ii, 1 p. 514 (1840). Young shoots and inflorescence ferruginous-pubescent ; branches spreading, alternate, terete. Leaves elliptical, rounded or somewhat narrowed at base, rounded obtusely pointed or apiculate at apex, alternate, thinly coriaceous or submembranous, glabrous except the veins, ferruginous or reddish-brown beneath, greener or slaty-brown above, 3—9in. long by 215 in. wide; petioles }—}in. long. g Cymes compound, trichotomous, 4 times the length of the petioles, patent, abundant on the young shoots; bracts hooked at the apex; flowers {in. wide, pubescent, globose, reflexed on very short pedicels. Calyx 4-fid, pubescent, with obtuse lobes. Corolla cam- panulate, twice the length of the calyx, with ciliated much contorted lobes. Stamens 2229, glabrous, subequal, crowded on the receptacle, mostly distinct; filaments short. Ovary rudimentary, represented by a few hairs. 9 Flowers solitary; peduncles }—,in. long, on the young shoots. Fruiting calyx with recurved lobes, somewhat pubescent or nearly glabrate, about #in. broad (when expanded) ; fruit glabrous, globular, 1}in. in diameter; albumen ruminated. Tavoy and Moolmyne, Wallich!; Pegu, M*Zelland ! its i ) : Ta Oo, Pouasmetitn local name Siti, Ds Brandis / Ce Mr HIERN, ON EBENACEZE. 165 10. DrospyROS ROTUNDIFLORA, sp. nov. D. foliis ovali-oblongis, alternis, apice acuminatis, bast rotundutis, tenwiter coriaceis, supra nitentibus glabris, subtus subglabris, breviter petiolatis; jfloribus masculis paniculatis, sub- globosis vel ovoideis, 4—3-meris, pubescentibus ; staminibus 14—16, biserialibus, glabris, ovario rudimentario hirsuto. Young parts and inflorescence subtomentose ; branches cinereous, terete. Leaves oval- oblong, alternate, acuminate at apex, rounded at base, thinly coriaceous, of rich brown colour when dry, shining and glabrous above with veins inconspicuous, nearly glabrous beneath, 3—7in. long by 1}—2}in. wide; petioles shortly pubescent, 1in. long; lateral veins about 10 on each side the midrib. g. Cymes axillary and lateral, many-flowered, less than lin. long; ultimate pedicels very short or obsolete; bracts ovate, sometimes larger than the flowers; flowers subglobose or ovoid, $—}in. in diameter, pubescent; calyx subhemispherical or widely campanulate, 4- occasionally 3-fid; lobes deltoid; corolla shortly 4—3-fid, lobes rounded; stamens 14 —16, biseriate, subequal, glabrous; ovary rudimentary, hairy. Borneo, O. Beccart! n. 3567. Near D. ehretioides, Wall. 11. Diospyros uirsuta, Linn. fil. Suppl. p. 440 (1781). D. foliis alternis, ellipticis oblongis vel ovatis, tenwiter coriaceis, breviter petiolutis, nervis lateralibus sceepius inconspicuis; floribus masculis dense cymosis, oblongis, pubescentibus, 4—5- meris; staminibus 5—16, subglabris; jfloribus femineis 1—6-nis; stanvinodiis 5—10; ovario 4—10-loculari, loculis 1-ovulatis; stylis 2—5, brevibus; fructibus globosis vel ellipsoideis, tomentosis vel glabratis, calyce fructifero, stellato- vel depresso-cupuliformi ; seminibus oblongis, albumine ruminato. Alph. DC. Prodr. vim. p. 223. n. 5 (1844), Thw. Enum. Ceyl. Pl. p. 181. n. 15 (1860), Bedd. Icon. Pl. Ind. Orient. (Part vu.) p. 28. t. 137 (1871), non Desf. A tree of moderate size, dicmcious or occasionally moncecious; produces an ebony. Buds inflorescence and in some cases the young branches and underside of leaves pubes- cent. Leaves alternate, more or less elliptical oblong or ovate, thinly coriaceous, obtusely or acutely acuminate at apex, narrowed or rounded at base, glabrous and shining above, sometimes pubescent beneath, 2—12in. long by 1—4in. wide; petioles 1—2in. long; mid- rib depressed on upper side; lateral veins usually inconspicuous beneath. Flowers subsessile, axillary, 4—5-merous; bracts rounded, caducous. According to Dr Thwaites female flowers are occasionally intermixed in the male cymes and in that case are much smaller than when occurring alone. $. Flowers 1—}in. long, oblong, in dense cymes. Calyx hairy on both sides, 4—5- fid; lobes acute. Corolla tubular, at least double the length of the calyx, 4—5-fid, glabrous inside. Stamens 5—16, glabrous or mainly so, when numerous often united by the filaments in pairs. Ovary rudimentary, hairy. 21—2 164 Mr HIERN, ON EBENACEA. ?. Flowers 4in. long, thicker than in the ¢, 1—6 together. Margins of calyx-lobes wavy and reflexed. Corolla-lobes reflexed, rounded, mucronate. Staminodes 5—10; barren anthers, glabrous or with setose tips, filaments glabrous or hairy. Ovary ovoid, covered with ferruginous or rufous hairs, 4—10-celled. Styles 2—5, short. Fruit globose or ellip- soidal, pale-glabrate or rarely tomentose and ferruginous or rufous, }—1}in. long, 1—10- seeded. Fruiting calyx stellate-flat or shallow-cupulifurm, }—1in. in diameter, 4—5-fid; lobes with reflexed margins. Seeds oblong, usually compressed, transversely scored outside; al- bumen ruminated. The following forms seem to me difficult to separate from D. hirsuta, L. f., but the combination of them all into a single species makes it a very variable and widely spread one. D. lucida, Wall. List, n. 4127 (1828—32), non Hort., Alph. DC. Prodr. van p. 283. n. 52 (1844). = (%) D. nilagirica, Bedd. Icon. pl. Ind. Or. (Pt. vii.) p. 27. t. 136 (1871). D. Candolleana, Wight, Icon. tt. 1221, 1222 (1850), non Thw. D. Moonii, Thw. Enum. Ceyl. Pl. p. 182. n. 16 (1860), Bedd. Zc. p. 28. t. 138 (1871). Perhaps a distinct species. D. canarica, Bedd. Le. p. 27. t. 134. = D. oligandra, Bedd. Rep. Forests Madras 1867 —68 p. 25 (1868) name only. D. Thwaitesti, Bedd. lc. p. 27. t. 135. =D. Candolleana, Thw. lc. p, 181. n. 14, non Wight. The following key serves to contrast the typical characters of these forms, but inter- mediate states exist. Ovary (6-) 8—10-celled; filaments of stamens often hairy. Ovary usually 10-celled. Stamens 5. Leaves elliptical, narrowed at base. hirsuta proper. Leaves oblong, wide near base. Moon. Ovary 8-celled. Stamens 16. nilagirica. Ovary 4-celled; filaments of stamens 10—12, glabrous. Staminodes 4—5. Staminodes quite glabrous. Fruit pale, glabrate. Candolleana. Staminodes setose at tip. Fruit rufous-hairy, at length glabrate. Thwaitesii. Staminodes 8—10, glabrous. canarica. Hirsuta proper. Ceylon, Thwaites/ 382. Moonii. Ceylon, Thwaites! 2833; Moon!; Walker!; (?) Tennaserim and Andaman, Hb. Helfer! 3632. Nilagirica. Sispara ghat, Nilgiris, India, Major Beddome ; (lucida) Singapore, Wallich ! 4127; Malacca, Maingay! 970, 973; Griffith! 3637. Candolleana. Courtallum and Quilon, Wight! 1715, Canara, Mangalor, Wight! 1728; Hohenacker! 591 (Native name Karmarn); Concan, Dr Gibson! 128; Goa, Dalzell/; Moollis, Dr Ritchie! 96, 3 (tree 24 ft. high); Phoondu Ghaut, Dr Ritchie! 96/2 (tree 36 ft.); Ram Ghaut, Dr Ritchie’ 96 (Native name Kalevin). Thwaitesti. Ceylon, Local name Homedereya-gass, Thwaites! 3394. Mr HIERN, ON EBENACE. 165 Canarica. S. Canara, Major Beddome! “yields an ebony,” native name Kara mara. A specimen from Malacca (Maingay! 969), with less dense cymes and 14 stamens with somewhat pilose anthers, may belong to this species. I cannot discover any authentic and satisfactory specimen of D. hirsuta, Linn. fil.; there is not a specimen so named by the younger Linnzus in the elder Linneus’ herbarium, and one in Sir J. E. Smith’s herbarium sent from Ceylon by Burmann and labelled “ Diospyros hirsuta, H.L. fil.” is not a Diospyros nor even a member of the family. 12. Drospyros MESPILIFORMIS, Hochst. in Pl. Schimp. Abyss. Exsice. sect. ii. mn. 655, 1243 (1842). D. foliis ellipticis vel oblongis, alternis, tenuiter coriaceis, glabrescentibus vel leviter pu- bescentibus, breviter petiolutis, nervis inconspicuis; floribus masculis axillaribus, 5—4-meris, Serrugineo-tomentosis, breviter cymosis, urceolato-oblongis, staminibus 10—16, subglabris; flori- bus femineis 1—3-nis, axillaribus, 5—4- rarius 3-meris, staminodiis 6—8, uniserialibus, glabris, ovario ovoideo vel conico, sericeo, 4- vel 8-loculari, loculis 1-ovulatis; fructibus subglobosis, glabratis, edulibus, calyce fructifero margine undulato; albumine ruminato. Alph. DC. Prodr. vii. p. 672 (1844). D. senegalensis, Perrott. ex Alph. DC. l.c. p. 234. n. 59. D. bicolor, Klotzsch in Peters Mossamb. I. p. 184 (1862). A shrub or tree from 6 to 40 feet high or more. Wood much thought of by the natives, white, compact, and useful for many purposes, or black in the centre like ebony. Branches terete, brown-cinereous, glabrescent, more or less patent; the young shoots and inflorescence ferruginous-tomentose. Leaves alternate, oblong or elliptical, somewhat narrowed or rounded at either end, thinly coriaceous (the younger ones very softly membranous) glabrescent and shining, or with scattered appressed pubescence beneath, often rubescent, especially on the midrib beneath, 2—6 in. long by 3—2}in. wide; petiole 1—4 in. long; midrib depressed above, lateral and net-veins delicate; margins just recurved, ‘Flowers white, dicecious. é. Inflorescence axillary, cymose, bearing few to many flowers, {—3 in. long exclusive of the flowers. Flowers ferruginous-tomentose, about 4 in. long, pentamerous, occasionally tetramerous; bracteoles lanceolate. Calyx about jin. long, 5- occasionally 4-fid, campanu- late or campanulate-oblong, hairy on both sides; lobes ovate or lanceolate, Corolla in general shortly 5-fid, urceolate-oblong, twice the length of the calyx or more, sericeous out- side, glabrous inside; lobes spreading, pointed. Stamens 10—16, often in pairs, nearly glabrous but with a narrow band of light-coloured hairs on the back of the anthers, in- serted at the base of the corolla; filaments short; connective produced at apex; pollen widely ellipsoidal, smooth. Ovary rudimentary, hairy, or 0. ?. Flowers pentamerous or tetramerous or rarely trimerous, solitary or in very short 1—3-flowered axillary cymes; peduncles }—# in. long; bracts narrow, caducous, Calyx hairy on both sides, campanulate, deeply lobed; lobes ovate acuminate with undulated margins, 166 Mr HIERN, ON EBENACE. Corolla pubescent outside, glabrous inside, exceeding the calyx, shortly lobed, lobes pointed. Staminodes 6—8, in one row, inserted at base of the corolla, glabrous. Ovary ovoid or conical, sericeous, terminated by 2 short hirsute bilobed styles, 4- or 8-celled and -ovuled. Fruit glabrate subglobose, 3—1 in. in diameter, edible, often slightly wrinkled, 4—5-seeded. Fruiting calyx somewhat or but little increased, with undulated margins, appressed to fruit or spreading. Seeds shining, }—3 in. long; albumen cartilaginous, somewhat ruminated ; embryo straight; cotyledons linear-lanceolate; radicle shorter than the cotyledons. Difficult to distinguish by technical characters from the Indian species Diospyros hirsuta, Linn. fil, of which it may be taken as the African representative; its forms also are subject to considerable variation. Tropical Africa. Abyssinia; native name Ajé or Ajejeh, near Docheladscheranne, Schimper / sect. ii. nn. 655, 1243, in & flower, June; Petit in Hb. Franq. iv. Coll. n. 434, in fruit; Schimper (1862)! n. 155, September, 4400 ft. alt.; Nubia, Fayohel, Kotschy! n. 470, in fruit; Dr Martin St Ange! in young fruit; Tinné Expedition, nn. 170, 394, fruit in flavour like that of Theobroma Cacao; Benischangul, Cienkowski, n. 966; Gallabat, Matamma, Schweinfurth! n. 973, 974; Mo- zambique, between Tette and the sea coast, Dr Kirk! in fruit ; near Lupata, Dr Kirk! in fruit, January; native names, Sechuana dialect, Makudima; Tette dialect, Kasinjamtolmera ; 50 miles above Tette, Kawrabassa; Sena, Dr Peters!; Niger, Barter! 1208, 1334; Senegambia, Le- prieur !, Whitfield !, Lelierre!, Perrottet!, Dr Daniell! (“ Monkey Guava”); Livingstone Expe- dition, (“ Mocheka”) Dr Kirk!; Angola, Distr. Golungo Alto, Dr Welwitsch! 2529, frequent, “ Musolveira ;’ Benguela, in woods from Serra da Xella towards Mumpulla, Dr Welwitsch! 2530; Congo, in rocky and sandy woods near Ambriz, Dr Welwitsch! 2528; Cape Coast, Brass ! 13. Diospyros BURMANICA, Kurz in Journ. Asiat. Soc. Bengal, Vol. xu. Pt. u. p. 73. n. 96 (1871). D. foliis alternis, ovalibus, apice obtusis vel brevissime acuminatis, basi rotundis vel sub- cuneatis, breviter petiolatis, tenuiter cortaceis, junioribus supra tomento tenui fugact adspersis subtus appresse tomentosis; floribus masculis 4—6-meris, breviter cymosis, tomentosis, wreeolato- oblongis, staminibus sepius 8, rarius 14—16, glabris; floribus femineis solitartis, fructibus globosis, glabris, nitentibus, vulgo 4-spernuis, albumine seminum pulcherrime ruminato, calyce Sructifero margine undulato. A small tree with young parts appressedly fulvo-pubescent. Branches cinereous. Leaves alternate, oval, obtuse or slightly acuminate at apex, rounded or wedge-shaped at base, thinly coriaceous, covered when young especially beneath with appressed pubescence, at length more or less glabrate, 1—6 in. long by 1—2in wide; veins not prominent; petioles 3—t in. long. &é- Flowers cream-coloured, urceolate-oblong, 4—6- usually 5-merous, 4—}in. long, fulyo-tomentose, in short 3—5-flowered axillary eymes, on young branches. Calyx 4—}in. long, hemispherical, tomentose on both sides, shortly 4—6-fid; lobes deltoid or cordate-ovate. Corolla shortly 4—5-lobed, glabrous inside; lobes rounded, reflexed. Stamens glabrous, Mr HIERN, ON EBENACE. : 167 usually 8, occasionally 14—16, hypogynous or inserted at base of corolla, about Lin. long ; anthers linear, acute, longer than the filaments. Ovary rudimentary, fulvo-pubescent. ?. Flowers solitary, pentamerous. Fruit globose, shining, 1—l1} in. in diameter, commonly 4-seeded. Seeds about jin. long, reticularly wrinkled outside, shining; albumen grey, beau- tifully ruminated. Fruiting calyx about in. in diameter, at base of fruit, tomentose; margins undulated. Burma, in sandy and hilly woods, 3rd Kioudweng, 13 May, 1837, Griffith! 3638 [see Journal of Travels, p. 104 (1847)]; Pegu, Zeebenthlah, October, 1861, Dr Brandis! 952 (vernacular name ZYayben) ; M*Lelland!; Kurz! 3010. Authentic specimens of this species seen during the printing of this paper prove that it embraces the specimens of Griffith and M*Lelland, which I had previously named D. octandra and so printed on pages 33 and 41. 14. DI0spyROS LATERALIS, sp. nov. D. ramis teretibus cinereis glabris, gemmis et inflorescentid ferrugineo-pubescentibus ; foliis alternis, ovalibus, apice acuminatis, bast angustatis, tenwiter coriaceis vel submembra- naceis, glabris, petiolatis, nervis tenuibus manifestis ; floribus masculis tubulosis breviter cymosis ; cymis lateralibus secus ramos et aillaribus, 3—9-floris; calyce oblongo apice 5—6-lobo, corolla tubulosd plerumque 5-fidd, staminibus 10—14, inaequalibus, fere glabris, antheris apiee minute setulosis, filamentis glabris ; ovario hirsuto rudimentario. Buds and inflorescence ferruginous-pubescent ; branches terete, glabrate, cinereous. Leaves oval, alternate, narrowly acuminate, cuneate at base, glabrous, submembranous or thinly coriaceous, 2—4tin. long by 1—2}in. wide; lateral veins slender, manifest, about 6 on each side the midrib, margins slightly recurved; petioles 3—2in. long, glabrous. 3. Cymes lateral on the older branches and axillary on the younger ones, 1—tin. long, 3—9-flowered ; flowers tubular; calyx oblong, ;%; in. long, 5—6-lobed at apex, glabrous inside, lobes somewhat spreading, often unequal; corolla tubular, exceeding the calyx, usually 5-fid, lobes obtuse; stamens 10—14, very nearly glabrous, unequal, anthers minutely se- tulose at apex, filaments glabrous; ovary rudimentary, hairy. Borneo, O. Beccari! n. 1600. 15. Di1ospyROS VERRUCOSA, sp. noy. D. fructicosa, foliis ovato-oblongis, alternis, apice angustatis, obtusis, mucronulatis, basi rotundatis, tenuiter coriaceis, supra subglabris, subtus appresse pubescentibus ; fructibus soli- tariis, pedunculatis, subtetragono-ellipsoideis vel -globosis, verrucosis, pubescentibus, 4-locularibus, 4-spermis ; pedunculis robustis, patentibus ; calyce fructifero parvo, patente, 4-lobo, pubescente ; seminibus oblongis, sulcatis, albumine ruminato. A shrub; branches numerous, at length glabrescent, terete; young shoots densely and 168 Mr HIERN, ON EBENACE. shortly pubescent, subferruginous. Leaves ovate-oblong, alternate, thinly coriaceous, some- what narrowed and mucronulate at apex, rounded at base, nearly glabrous above except that the depressed midrib is sometimes puberulous, paler with soft appressed pubescence beneath and rufous-pubescent on the raised midrib and lateral veins; 1}—3}im. long by $—Il#in. wide, including petiole ~,—1 in. long, rufous, densely puberulous; lateral ves about 6 on each side the midrib. @. Flowers solitary, on distinct densely puberulous rather slender peduncles, axillary ; bracts small, rufous-hairy, caducous, near base of peduncle ; fruiting peduncles stout, thickened upwards with wide articulation at apex, nearly }in. across, puberulous or subglabrate, 3—3 in. long, patent; fruiting calyx subtomentose-pubescent on both sides, spreading, 4-fid, }in. across; with depresso-deltoid lobes slightly recurved at apex. Fruit oblong or globose, pulpy, roundedly 4-sided, verrucose, at length smoother with pale ferruginous short pubes- cence between the raised warty prominences, obtusely umbonate at apex, 1—1}in. long by 5—lin. across from one side to the opposite side, 4-celled; cells 1-seeded; seeds 3—3in, long, enclosed in a thin chartaceous envelope, transverse section a quadrant of a circle of radius 3,in., shewing several intrusions of the testa into the albumen corresponding with depressed lines on the exterior of the seed; embryo nearly straight, nearly the length of the seed; radicle superior, much shorter than the compressed 1—3-nerved cotyledons. In one case the calyx is triangular and flat. The pulp of the fruit is eaten. E. Africa, Prov. Zanguebar, Dr Kirk /; Zambesia, Rovuma R., 20 miles above the mouth. Dr Kirk!, August 1862. 16. Diospyros KORTHALSIANA. D. glabra, foliis alternis ellipticis, apice obtuse acuminatis, bast cuneatis, coriaceis, costis tnconspicuis ; fructibus solitariis, axillaribus, pedunculatis, glabris, apice cum stylorum reliquiis appresse ferrugineo-hirsutis, ellipsoideis, 8-locularibus, 8-spermis ; seminibus oblongis, albumine ruminato. Diospyros macrocarpa, Korthals MSS. in Hb. Lugd. Batav. Ebenacee No. 2, non mihi. Glabrous; branches (in dry state) drab. Leaves elliptical, cuneate at base, obtusely acuminate at apex, alternate, coriaceous, 2—4in. long (besides petiole }—in. long) by j{—1l} in. wide, palish brown (in dry state) on both sides, shining above, midrib depressed above; veins inconspicuous. Fruit solitary, on axillary peduncles which are Zin. long, suberect, thickened upwards and thicker than the extension of the young branches from which they grow; fruit ellip- soidal, glabrous except at apex, dark and shining, about Iin. across, by scarcely 2in. (?) long, S-celled, 8-seeded, tipped by appressedly ferruginous-hairy remains of style. Pericarp in the dry state j;in. long; seeds fin. long by }in. wide and Jin. thick in the dry state, pendulous from inner side; albumen somewhat ruminated. Fruiting calyx nearly glabrate outside, appressedly hairy and smooth inside, very erass, Mr HIERN, ON EBENACE, 169 shallowly cup-shaped, lin. across, 4-cornered and shortly 4-lobed; about }in. high; tube with elevated rim; lobes much thinner, reflexed; verrucose-rugose outside, Borneo, Korthals / 17. Dtospyros AFFINIS, Thw. Enum. Ceyl. Pl. p. 179. n. 6 (1860). D, foliis alternis, ovalibus vel oblongis, apice obtusissimis, basi angustatis vel subrotundis, tenuiter coriaceis, glabris, petiolatis ; floribus masculis 3—T-nis, cymosis, pubescentibus, 4-meris, calyce apice lobato, corollé urceolato-oblongd ; staminibus 6—16 glabris vel leviter hirsutis ; Jloribus femineis solitariis, pedunculatis; staminodiis 6—8, glabris, ovario 6 (?) -loculari ; calyce inter lobos marsupto-dilatato, lobis acuminatis; albumine ruminato. Bedd. Ic. Pl. Ind. Or. (Part viz.) p. 26 t. 127 (1871). A moderate-sized tree; buds ferruginous hairy; branches quickly glabrescent. Leaves oval or oblong, alternate, quite obtusely narrowed at apex, narrowed at base, thinly coria- ceous, glabrous or puberulous below, 14—4$in. long by 3—Itin. wide; petioles 1—2in, long; midrib canaliculate above, net-veins numerous, raised on both sides. g. Flowers tin. long, cymose, 83—7 together; cymes }—?in. long excluding the flowers ; ultimate pedicels short, not exceeding jin. long. Calyx semi-ellipsoidal, with short hairs on both sides, 4-toothed at apex, in. long. Corolla shortly salver-shaped, tawny-hairy outside; tube inflated below, constricted at top; lobes 4, spreading, oval, somewhat pointed at apex, about 1} the length of the tube. Stamens 6—16, usually about 9 and some or all in pairs, short, usually hypogynous and unequal; filaments glabrous, shorter than the anthers which are glabrous or somewhat hairy. Ovary rudimentary, represented by a bunch of hairs. ?. Flowers solitary, 4in. long and wide; peduncles axillary, }—,%,in. long, equalling or rather shorter than the flower, puberulous. Calyx in. long, hairy inside, subglabrate outside, 4-fid, plicate; lobes acuminate with very wide sinuses; somewhat enlarged in fruit. Corolla shortly 4-fid with 6—8 glabrous staminodes at base inside. Ovary 6-celled (4-celled ?, conical), hairy; styles 2, bifid at apex. Fruit globular, apiculate, usually 4-seeded, 1 in. long, finally glabrous; seeds }in. long or more by 4in. wide, with ruminated albumen, Ceylon, Thwaites! C.P. 2924, 18. DIospyROS CRUMENATA, Thw. Enum. Ceyl. Pl. p. 179. n. 5 (1860). D. foliis alternis, ovalibus vel oblongis, obtuse et breviter acuminatis, coriaceis, glabris, petiolatis; floribus masculis breviter cymosis, 3—5-nis, tetrameris, pubescentibus, staminibus circiter 12, glabris; floribus femineis solitariis, breviter pedunculatis, tetrameris, calyce inter lobos marsupio-dilatato, staminodiis 8, glabris; ovario 8-loculari, hirsuto; fructibus subglobosis glabrescentibus, albumine ruminato. Bedd. Ic. Pl. Ind. Or. (Part vit.) p. 26. t. 126 (1871). A large tree; branches glabrous. Leaves oval or oblong, alternate, obtusely shortly Vou. XII. Parr I. 22 170 Mr HIERN, ON EBENACE. and abruptly acuminate at apex, rounded or narrowed at base, coriaceous, glabrous, with midrib channelled above and net-veins numerous and raised on both sides, 2—5in. long by 1—2in. wide; petioles 4—1in. long, canaliculate above. g. Cymes 3—5-flowered, near together, about }in. long, hairy. Calyx obscurely 4-dentate, glabrous inside, 1—}in. long. Corolla 3,in. long, shortly 4lobed ; lobes recurved. Stamens about 12, glabrous, hypogynous. ©. Flowers solitary, rather more than }in. long; peduncles hairy, fin. long, thickened upwards. Calyx }in. high by %in. wide, shallowly 4-lobed, plainly plicate, coriaceous, puberulous outside, hairy inside; lobes obtuse or apiculate and rounded; between the lobes marsupio-dilated. Corolla slightly exceeding the calyx ferruginous-tomentose, shortly 4-lobed ; lobes with undulated margins and tomentose on both sides. Staminodes 8, in one row, glabrous, inserted at base of interior of corolla-tube. Ovary 8-celled, hairy. Fruit subglobose, 6—S8-seeded, 1}—2in. in diameter, at length glabrous, resting at base on tetragonal thickened spreading calyx, 1}in. in diameter; seeds black, shining, 1in. long, }in. wide, with ruminated albumen. Ceylon, 2000—4000 ft. alt. Thwaites! C.P. 2438. 19. DrospyRos FRUTESCENS, Blume, Bijdr. Fl. Ned. Ind. p. 668 excel. var. (1825). D. foliis alternis, ellipticis, apice acuminatis, basi angustatis, firmiter submembranaceis, glabris, nitentibus; floribus femineis azxillaribus vel lateralibus, tetrameris, c-nis; calycis lobis margine revolutis; corolld suburceolatd, 4-fidd ; staminodws 8, wqualibus, uniserialibus ; ovario hispido ; fructibus globosis, subglabris, succulentis. Alph. DC. Prodr. vit. p. 230. n. 38 excl. var. (1844), non Hassk. Plant. Javan. rar. p. 467 (1848). Young shoots puberulous; branches dark, terete, smooth. Leaves alternate, elliptical, firmly submembranous, somewhat narrowed at base, acuminate at apex, glabrous and shining, 24—5in. long by 14—21in. wide, petioles 4—}in. long; veins inconspicuous above; midrib depressed above; margins neatly recurved. ?. Cymes axillary or lateral, fasciculate, many-flowered, 3—j}in. long (excluding the flowers), shortly pubescent, ferruginous (or fuliginous); common peduncle obsolete; pedicels 4—1in. long; bracts ovate acuminate, near base of pedicels. Flowers about iin. long; calyx about din. long, puberulous outside, deeply 4-lobed; lobes with sides reflexed from a longitudinal internal edge; corolla glabrous except 4 longitudinal puberulous lines outside, 4-fid, suburceolate; tube jin. long and thick, lobes gin. long spreading, ovate, subciliate and pointed at apex by inflexion of sides near apex, contorted in estivation; staminodes 8, equal, inserted in one row near base of corolla, 2in. long, appearing at mouth of corolla- tube, filaments longer than the barren anthers, hairy near top; ovary globose below, ovoid above, terminated by 2 hairy styles, ferrugineo- or (nigro-) hispid, 4- (10-) celled, 4- (10-) ovuled; stigmas emarginate. Fruit globose, subglabrate, 1}in. in diameter, succulent; (seed searcely lin. long by in. wide, transversely suleate outside; albumen ruminated). Fruiting ‘ Mr HIERN, ON EBENACE, Wt calyx $in. across, spreading, puberulous outside, with raised 4-sided thickened stellate border inside; lobes wide, short, undulated. Java, Blume/, Horsfield drawings n, 128 (part) in Hb. Kew. 20. DiI0spyROS DENSIFLORA, Wall. List, n. 4140 (1828—82). D. foliis alternis, anguste ovalibus, utrinque obtusis, interdum subacutis, glabris, coriaceis, supra nitidis venis inconspicuis, petiolatis; floribus cymosis, tetrameris, pubescentibus, calyce profunde lobato, lobis margine reflexis, corolla tubulosd, staminibus 15—16, antheris glabris, filamentis brevissimis hirsutis; fructibus globosis, ferrugineo-pilosis, calyce fructifero plicato, seminibus oblongis, transverse notatis, albumine ruminato (2). Alph. DC. Prod. vu. p. 233. n. 56. (1844). Branches glabrous terete. Leaves alternate, coriaceous, oval-oblong, glabrous, obtuse or obtusely acuminate at apex, slightly narrowed at base, 4—S8in. long by 14—32in. wide; petioles about tin. long, glabrous, strong, wrinkled; midrib depressed and lateral veins slightly raised on upper face. 6. Cymes panicled, about lin. long, hairy, many-flowered with hairy bracts and short pedicels; flowers about }in. long, tetramerous; calyx }in. long, hairy on both sides, 4- partite, lobes ovate with reflexed sides; corolla cylindrical, hairy outside, glabrous inside, 4 times the length of the calyx; stamens 15—16, anthers glabrous, on very short hairy filaments; ovary rudimentary, hairy. ?. Cymes 2in. long, puberulous, about 12-flowered, trichotomous ; pedicels longer than the peduncle; bracts lanceolate, at base of pedicels; flowers lin. long. Fruit globose, 3—4 in. long, ferruginous-pilose; seeds oblong, transversely scored, albumen ruminated (?). Fruiting calyx 4-partite, ? the length of the fruit, puberulous, lobes much widened at base with auricled imbricated bases forming 4 dependent processes, plicate; pedicels 1—}in. long. Moolmyne and Amherst, Wallich/; Martaban, Burma, below 500 feet alt., a small tree, Dr Brandis! 21. Diospyros oocaRpA, Thw. Enum. Ceyl. Pl. p. 180. n. 9 (1860). D. foliis alternis, glabris, ovatis vel ovalibus, apice obtuse acuminatis, basi rotandatis vel parum angustatis, tenwiter coriaceis, nervis inconspicuis; floribus masculis, 3—7-nis, bre- vissime cymosis, 3—4-meris, calyce subintegro, vel dentato, corolle prefloratione irregulart, staminibus 9—12, glabris; jfloribus femineis 1—3-nis, subsessilibus; ovario 6—8-locularz ; fructibus subglobosis vel oblongis, puberulis vel glabratis, rugoso-areolatis, albumine non rumi- nato; calyce fructifero vix aucto. D. Arnottiana, Mig. (in Pl. Ind. Or. Hohenacker, n. 562!) ex Thw. lc. p. 423 (1864). Ceylon name Kaloo-kadoombaireya-gass. A moderate-sized tree; young shoots pubescent or puberulous, quickly glabrescent. Leaves alternate, glabrous, ovate or oval, obtusely acuminate at apex, more or less rounded 22—2 172 Mr HIERN, ON EBENACEZ. towards base, inconspicuously veined with midrib canaliculate above, thinly coriaceous 2—44 in. long by 1—2in. wide; petioles $—3 in. long. 8. Flowers 3—7 together, arranged in dense axillary fulvous-silky cymes equalling or falling short of the petioles, with very short pedicels and rounded concave bracts. Calyx Zin. long, tubular, subentire or 3—4 dentate (or even 3—4-fid), glabrous inside. Corolla #,in. long, 3—4-fid, with obtuse lobes, one of which is completely enclosed by the others in bud, the other lobes imbricating sometimes dextrorsely and at other times sinistrorsely. Stamens 9—12, alternately in pairs and single, glabrous, inserted at the base of corolla- tube or some hypogynous; the outer ones of the pairs the longer; anthers shorter than the longer filaments (at least in bud); ovary rudimentary, hairy. . Flowers 1—8 together, scarcely longer than the male; ovary 6—8-celled. Fruit egg-shaped when ripe, cylindrical when young, scattered with short appressed ferruginous hairs, glabrescent, I4in. long by 2in. thick when ripe, rugoso- or sub-verrucose, resting at base on scarcely increased calyx, solitary, 2-or more-celled, “usually 6-seeded.” Seeds with albumen not ruminated. Ceylon, Thwaites/ C.P. 1914; Concan, Dalzell! ; Bababoodun Hills, Mysore, Mr Law / 22, Drtospyros TRUNCATA, Zoll. et Mor. in Moritzi, Systemat. Verzeichn. Javan. Pfinzn. p. 43. n. 1156 (1846). D. foliis alternis, ovali-oblongis, apice obtuse acuminatis, bast cuneatis, glabris, tenuiter coriaceis, breviter petiolatis, marginibus revolutis; floribus masculis 2—8-nis in alis sub- sessilibus vel brevissime cymosis, glabris; calyce tubuloso subintegro, corolle lobis acuminatis, staminibus 11—14, glabris; floribus femineis 1—2-nis, brevissime pedunculatis, tetrameris ; calycis lobis latissimis retusis reflexis; corolle lobis acutis, patentibus; staminodiis 8—10; stylis 4; fructibus 8-locularibus, glabris. D. laxa, Teijsmann et Binnendijk, Pl n. h. Bogor, in Kruidk. Arch. m1. p. 406 (1855). D. melanoxylon, Blume! Bijdr. FJ. Ned. Ind. p. 669 (1825), non Roxb. A tree with terminal buds slightly hairy; branches glabrous, terete, lax, widely spreading and forming a beautiful crown or top. Leaves oval-oblong, obtusely acuminate at apex, attenuate or narrowed at base, thinly coriaceous, with margins more or less reflexed, midrib depressed above, and delicate not contiguous lateral veins inconspicuously raised on both sides; of a yellowish green colour, alternate, glabrous, 3—6 in. long by 1—2 in. wide; petioles i4—} in. long. 3. Flowers 2—4—8 together on very short axillary somewhat pubescent cymes, glabrous, yellowish green, slender, 5 in. long; bracts small, pubescent; pedicels very short. Calyx tubular, somewhat inflated in middle, 4-toothed at apex, Jj in. long by iin. thick; corolla tubular, narrowly conical in bud, 4-fid (2), with acuminate lobes; stamens 11—12—14, glabrous, at base of corolla or on disk; filaments short; ovary obsolete. ?. Flowers 1—2 together, }—} in. long, on peduncles about 54; in. long, axillary, as long as the petioles. Bracts caducous. Calyx glabrous with 4 very wide retuse reflexed Mr HIERN, ON EBENACE. 173 (short?) lobes; corolla twice the length of the calyx, 4-fid, with acute, patent, pale-yellow, lobes white at the base; staminodes 8—10; styles 4 connate at the base. Fruit glabrous, 8-celled, 4 in. thick, globose, shining; fruiting calyx forming a shallow 4-cornered cup for base of fruit, with 4 reflexed undulate-plicate retuse lobes; 4—# in. in diameter. According to Moritzi this tree resembles D. Ebenwm, Linn. fil, and has even in the young twigs indications of black wood which becomes quite black in the older branches. The male flowers open in March. In woods, Java. De Vriese! 6 fl.; Zollinger! n. 1156; Dr Horssield! Ebenacez, n, 4, in fruit; Binnendyk! $ f1.; Hasskarl! ; Blume! 23. DiIosPYROS HALESIOIDES, Grisebach Cat. Pl. Cubensium, p. 168. n. 2 (1866). D. foliis alternis, obovato-ovalibus, apice acutatis, basi obtuse cuneatis, subcoriaceis, pellu- cido-punctatis, subtus fulvo-velutinis, robusté venosis, supra pubescentibus ; floribus masculis in eymis brevissimis axillaribus dispositis, 1—3-nis, velutinis, calyce breviter 4-fido campanulato, corolldé urceolato-oblongd, breviter 4-fidd, lobis ovatis acuminatis, staminibus 12, glabris ; fruc- tabus solitariis, subsessilibus, depresso-globosis, ferrugineo-sericeis, 8-locularibus ; calyce fructi- fero ampliato, lobis erecto-patentibus. Terminal buds ferruginous-hairy; young shoots pubescent at apex, glabrescent shining and terete below. Leaves alternate, somewhat obovate, acute and apiculate at apex, cuneate to an obtuse base, subcoriaceous, fulvo-velutinous and conspicuously rich-veined beneath, darker nearly glabrescent and nitescent above, except veins; midrib depressed above; 1—2} in. long by 3—1} in. wide, pellucid-punctate; petioles »—,, in. long, ferruginous- pubescent; bracts ovate, small, fulvo-pubescent. é. Flowers 3 in. long, in (1—) 3-flowered sessile or subsessile fulvous-hairy short cymes, on short pedicels; calyx campanulate, fulvo-velutinous, unequally 4-lobed, 1—4 in. long; lobes deltoid, acute, less than half the length of the tube, unequal. Corolla hairy outside, glabrous inside, campanulate-oblong, 3—} in. long, 4-fid; lobes ovate-lanceolate. Stamens (11—) 12, glabrous, unequal, 8 in 2 rows opposite lobes of corolla, the inner 4 of which are on shorter filaments than the outer 4 and inserted above them (or united with them in 4 pairs) and 4 alternate with the corolla-lobes, and inserted on corolla near its base; filaments slender ; ovary rudimentary, fulvous-hairy. é- Flowers unknown. Fruit solitary, sessile, depresso-globose, 1 in. thick by ? in. high, ferruginous-silky, 8-celled; fruiting calyx accrescent, deeply 4-fid, somewhat spreading, 12 in, across the top; lobes 4-ellipsoidal, with margins somewhat recurved ; albumen not ruminated, Eastern Cuba, Wright/ 2936 @ f1., 2937 in fruit. 24, DIOSPYROS BORNEENSIS, sp. nov. D. foliis alternis, oblongis, apice breviter acuminatis, basi angustatis, subglabris, tenuiter coriaceis, breviter petiolatis; floribus femineis secus ramos vetustos glomeratis, pedunculatis, pubescentibus, calyce tubuloso subintegro, corollé 5-fidd, lobis ovalibus reflexis obtusis; stami- nodiis 10, uniserialibus, glabris, basi corolle insertis; stylo 4-lobo; ovario conico, Serrugineo- tomentoso, 8-loculari, loculis 1-ovulatis ; fructibus magnis, globosis, tomentosis. 174 Mr HIERN, ON EBENACE. A tree; wood yellow, tough and stringy with black streaks. Terminal buds ferruginous- tomentose; young shoots puberulous; branches dark, glabrescent, terete. Leaves oblong, alternate, thinly coriaceous, shortly acuminate at apex, somewhat narrowed at base, puberu- lous when young, glabrescent, 5—64 in. long by 14—2 in. wide, including petiole {—} in. long; canaliculate on upper side, midrib depressed above, lateral veins about 12 on each side, distinct on under side, indistinct above, forming (by anastomosing) a marginal vein clearly marked beneath; tertiary veins oblique. Q. Flowers clustered, greenish white, large, on distinct fulvo-tomentose peduncles ;3—? in. long inserted on tubercles on older branches. Bracts small, obtuse, j;—y5 in. long, at base of peduncles. Calyx about 2 in. long by } in. thick, finely fulvo-tomentose outside, appress- edly silky inside, tubular, irregularly and shallowly toothed at apex. Corolla about 8 in. long when straightened, nearly glabrous but with ciliate margins to the lobes, 5-fid; lobes oval, imbricated, reflexed. Staminodes 10, glabrous, in one row, inserted at base of corolla. Style 4-lobed; ovary conical, ferruginous-tomentose, 8-celled; cells 1-ovuled; dissepiments thin. Fruit large, with a sweet black pulp, globose, fulvo-tomentose, crowned with short style, surrounded half-way up by burst calyx. Native name malam kuning. Tamgong Vinbong, Labuan, rather uncommon, Mr Motley / 7. 25. Or DIOSPYROS BATOCANA, sp. nov. D. foliis alternis, ovalibus, utrinque rotundatis, coriacers, glabris, supra nitentibus, subtus pallidis albidis, margine refleco, nervis tenuibus, petiolis rugosis ; floribus masculis sessilibus glomeratis secus ramos annotinos in nodulis dispositis, pentameris, fuligineo-hispidis; calyce apice lobato, corollé crassd, ovoided ; staminibus circiter 12, glabris, inequalibus, ovartt rudi- mento hispido. A large bush, quite or nearly glabrous except the buds and inflorescence; branches nigro-cinereous, spreading at about 45°. Leaves alternate, oblong-elliptical, shiming above, whitish beneath, firmly coriaceous, glabrous or with a few minute black setz beneath, more or less rounded at both ends, with reflexed margins and veins in delicate relief on both sides, 2—2} in. long by 3—1 in. wide; petioles 3—} im. long, thick, angular, obliquely and in other directions wrinkled, often twisted or recurved. &. Inflorescence arranged on nodules, covered with fuliginous and ferruginous hispid hairs, on the branches of previous season; flowers sessile, several together, clustered ; bracts at base of calyx; calyx fuliginous and ferruginous-hispid on both sides, 5-lobed at apex; corolla fuliginous-hispid outside, 5-fid, pale and glabrous inside, crass, ovoid; lobes imbri- cated sinistrorsely, obtuse; stamens 12—16 (?), glabrous, unequal, on short filaments; ovary wanting, represented by ferruginous hairs. Tropical Africa, Setoka, “Mikumbo,” Batoka country, Dr Kirk/, fruit eaten, ¢ fl. July. 26. Diosprros qu&siTA, Thw. Enum. Ceyl. Pl. p. 179. n. 7 (1860). D. foliis alternis, ellipticis vel oblongis, apice breviter et abrupte acuminatis, obtusis, bast parum angustatis, glabris, coriaceis, petiolatis, nervis reticulatis gracilibus; floribus masculis 3—9-nis, breviter cymosis, 4—-5-meris, hirsutis, calyce tubuloso apice dentato; staminibus 16, Mrz HIERN, ON EBENACEA. 175 glabris, ovarii rudimento hirsuto; floribus femineis solitariis, pedunculatis, pentameris, calyce winter lobos marsupio-dilatato, lobis acutiusculis; fructibus subglobosis, externe rugosis, subgla- bris, seminibus compressis, albumine non ruminato. Bedd. Ic. Pl. Ind. Or. (Pt. vu.) p. 26. t. 128 (1871). A large tree, nearly glabrous except the buds and flowers; branches terete, dark, spreading at about 40°. Leaves alternate, coriaceous, elliptical or oblong, abruptly and shortly acumi- nate, somewhat narrowed at base, glabrous except a few scattered weak appressed hairs beneath ; 3—7 in. long by 13—3 in. wide, turning fuscous in drying, with petioles canalicu- late and about } in. long, midrib depressed on the upper surface, lateral veins numerous, not conspicuous. $. Cymes 3—9-flowered, pilose, about equalling the petiole; flowers (closed) 4—% in, long. Calyx + in. long by 5% in. wide, obscurely 4—5-lobed at apex, pubescent, tubular, somewhat inflated about middle; lobes depresso-deltoid; corolla about }in. long (closed), oblong, clothed outside with subferruginous felt, 4—5-lobed; lobes ovate, about 2ths depth of corolla-tube; stamens 16, hypogynous, glabrous, not united in pairs, } in. long; anthers longer than the filaments; ovary rudimentary, hairy. 2. Flowers solitary, pentamerous; corolla shortly 5-lobed; fruiting calyx 5-fid, mar- supio-dilatated with ovate cordate lobes having reflexed sides and base, hairy inside, 12 in. across ; fruiting peduncle stout, patent, 2 in. long; fruit subglobose, 2 in. in diameter (imma- ture), rugose, nearly glabrous, 8-celled (?); seeds 1 in. long, shining, compressed; albumen not ruminated, According to Dr Thwaites this tree is the true Calamander of the Cinghalese; in Ceylon it is called Kaloomidereya-gass. Ceylon, Zhwaites! C. P. 3010. 27. DIosPYROS TOXICARIA, sp. nov. D. foliis alternis, elongato-ovatis, apice acuminatis, basi rotundis, glabris, nitentibus, coriaceis, subtus reticulatis ; petiolis robustis ; floribus masculis ferrugineo-tomentosis, sessilibus, aggregatis, e pulvino convexo surgentibus, bracteis basi obtectis, calyce apice lobato, corolla breviter 4-fidd, staminibus 11—13, glabris, ovarit rudimento dense piloso; fructibus solitariis, subsessilibus, ferrugineo-tomentosis, subglobosis, 8-locularibus; calyce fructifero, cyathiformi, breviter 4-lobo, aucto. A tree, 20—80 feet high, glabrous except the fruit and inflorescence, bark rather rough, gum sometimes exudes from it. Leaves elongate-ovate, rounded at the base, acuminate at the apex, alternate, coriaceous, shining, 2}—5 in. long (besides robust petiole about } in. long) by 1—2 in. wide, midrib depressed above, finely reticulated as in D. tessellaria. $. Flowers 5—12 together sessile or axillary, very short ferruginous hairy dense nodular cymes; bracts imbricated, unequal, ferrugineo-tomentose outside, nearly glabrous inside, oval, some (outer ones) } in. long; buds ovoid, ferrugineo-tomentose, } in. long; calyx lobed at apex, hairy outside, glabrous inside. Corolla shortly 4-fid, hairy outside, glabrous inside. Stamens 11—13, glabrous; ovary wanting; receptacle hairy. Native names 176 Mr HIERN, ON EBENACE. Sifatatu, Alacainist. Madagascar, Tranomaro, sands near the sea, July 1862, Dr Meller / Natives say that birds die soon after eating the fruit. ¢. Bracts caducous, imbricated. Fruit solitary, ferruginous-tomentose, subsessile; young fruit scarcely the length of the accrescent calyx, ovoid or subglobose, } in. high, umbilicate- depressed at apex, 8-celled, 8-ovuled; pericarp thick; fruiting calyx ferruginous-tomentose on both sides, cup-shaped, shortly 4lobed. Native name Vorongi. Madagascar, Tranomaro, July 1862, Dr Meller / The following may belong to this species: (1) Specimen with ovoid fruit ferruginous-tomentose 2 in. high, fruiting calyx 1} in. across appressed to base of fruit, leaves 44—6} in. long by 13—2} im. wide; Madagascar, Lastelle! 1841, seen in the Paris Museum. (2) Fruit nearly 1 in. long, rufous-tomentose. Fruiting calyx spreading } in. across; Madagascar, Chapelier/ Seen in the Paris Museum. (8) Fruit 1}in. long by 1} in. thick, ferruginous-tomentose, 8-celled; fruiting calyx spreading. Without leaves. Said to come from 8. Africa, but probably this is a mistake; Gerard! n. 190, seen at the Kew Museum. 28. DrIospyros TESSELLARIA, Poir. in Enclyc. Méth. v. p. 430, n. 5 (1804). D. ramulis fusco-cinereis; foliis alternis, ovalibus vel ovatis, apice rotundatis, bast rotundis, glabris, nitentibus, coriaceis, tenuiter reticulatis, petiolatis; floribus masculis sessilibus, sepius aggregatis, e pulvino convero ferrugineo piloso surgentibus, tetrameris, tomentosis, bracteis imbricatis ovato-rotundatis extus sericeis intus glabris; calyce apice lobato, staminibus 12—13, glabris; floribus femineis aggregatis, tetrameris; fructibus subglobosis, ferrugineo- sericeis vel subglabratis, 8-locularibus, edulibus. Alph. DC. Prodr. vim. p. 225. n. 12 (1844). Ebenus tessellaria, Commers. ex Poir. 1. c. D. Ebenum, Poir. l.c. p. 429. n. 4, non Koenig. D. reticulata, Willd. sp. pl. tv. p. 1109, n. 6 (1805), Alph. DC. Lc. n. 11 excl. 8 timoriana, non Decaisne. A tree or shrub with dark-cinereous glabrous branches. Leaves’ alternate, oval or ovate, rounded at both ends, especially at base, where they are sometimes slightly cordate, glabrous, coriaceous, shining, finely reticulated, 3—6 in. long by 1}—3} in. wide; margins slightly reflexed; petioles stout, J—} in. long. Flowers densely clustered, sessile, arising from lateral nodules on the young branches, fulvo-sericeous, tetramerous; calyx tubular, lobed at the apex, }—} in. long; ¢. stamens 12—13, glabrous, mostly in pairs, inserted on the receptacle. Fruit globular or ellipsoidal, 8-celled, nearly glabrate or sericeous, edible, 8-celled; fruiting calyx hemispherical or rarely flat, thickly coriaceous, sericeous, lobes short, rounded. Wood valuable; this species probably yields the ebony of Mauritius. Mauritius, in the forests of the highest parts of the island, Bouton/; Ayres!, Telfair!; shrub 6—8 feet high, fruit good to eat, sweet, fruiting calyx flat, Bouton! Mr HIERN, ON EBENACEAL 177 D. rubra, Gaertn. fil. Fruct. et Sem. Pl. mr. p. 138. t. 208 (1805), differs by a- flat 5-lobed fruiting calyx and 10-celled fruit; it may however belong to D. tessellaria, Poir., or. if not to D. chrysophyllos, Poi. 29. DIOSPYROS HAPLOSTYLIS, Boiv. MSS. D. foliis alternis, ovalibus, apice anguste acuminatis, basi angustatis, coriaceis, nitidis, subglabris, subtus tenwiter reticulatis, breviter petiolatis ; floribus masculis 3—6-nis, brevissime eymosis vel aggregatis, ferrugineo-sericeis, 4—5-meris, calyce breviter lobato, staminibus O19, glabris, biserialibus; floribus femineis solitartis, brevissime pedunculatis, Serrugineo-sericeis, 4—5-meris, staminodiis 4, glabris, ovario sericeo, depresso-globoso, 8-loculari, stylo apice 4-lobo ; fructibus subglobosis, glabrescentibus, 8-locularibus. A shrub of 12 feet or an erect tree 22 feet high or more; heart-wood black, very hard; young parts puberulous; branches glabrescent, terete, subcinereous, smooth. Leaves alternate, oval, narrowly acuminate at apex, somewhat narrowed at base, coriaceous, glabrous with depressed midrib above, highly and minutely reticular beneath with scattered appressed inconspicuous hairs, somewhat undulated, 2—3} in. long by 1—12 in. wide; petioles } in. long. é. Flowers clustered, 8—6 together, subsessile on young branches, ferruginous-pubescent, % in. long by { in. thick; bracts caducous, smaller than the flowers; calyx hairy on both sides, }in. long by # in. thick, lobes 4, erect, deltoid, one-third the depth of the calyx; corolla hairy outside, glabrous inside, lobes one-third the length of the corolla; stamens 10, 12, glabrous, hypogynous, biseriate, nearly equal, 2—} in. long, anthers 2, in. long linear; ovary rudimentary, ferruginous-sericeous. Q. Flowers solitary, ferruginous-sericeous, nearly } in. long, very shortly pedunculate ; calyx 5; in. long by } in. thick, campanulate, lobes 4—5, ovate deltoid, one-third depth of calyx; corolla 4-fid, lobes somewhat spreading; staminodes 4, glabrous, short, alternate with corolla-lobes; ovary densely sericeous, fleshy, depresso-globose, ;4; in. high by } in. thick, 8-celled, terminated at apex by style; style ;, in. long, 4-lobed at apex. Fruit ferruginous-sericeous when young, glabrescent, subglobose, 1 in. long by 1 in. thick, 8-celled; cells 1-seeded; fruiting peduncle } in. long, sericeous; fruiting calyx 3 in. long and wide, hairy on both sides, campanulate or nearly flat, spreading. Madagascar, Nossi Be, Boivin! 2108 bis, Pervillé! 439, 505; mountains at Diego, Suares, Bernier! 259 (excl. fr.). 30. DIOSPYROS MELANIDA, Poir. in Encycl. Méth. v. p. 431. n. 7 (1804). D. foliis alternis, ovalibus, utrinque rotundatis vel obtuse angustatis, glabris, coriacers, petiolatis, mediocriter reticulatis ; floribus masculis 1—38-nis, aggregatis, sessilibus, 5—6-meris, calyce subglabro apice lobuto, corollé 5—6-fidd, extus sericed, staminibus 22—24, glabris, basi corolle insertis; floribus femineis solitariis, fructibus subglobosis, sessilibus, 10-locularibus, calyce fructifero aucto, 5—6-lobo, tubo concavo, lobis recurvis sepe undulatis ; seminibus oblongis, albumine cartilagineo, non ruminato. Vou. XII. Parr I. 23 178 Mr HIERN, ON EBENACE, Alph. DC. Prodr. vin. p. 227. n. 22 (1844). Ebenus melanida, Commers. ex Poir. 1. ¢. (2?) D. pterocalyx, Boj. Hort. Maurit. p. 200. n. 7 (1837), Alph. DC. 1c. p. 225. n. 142 A tree with glabrous stem and branches. Leaves alternate, oval, rounded or obtusely narrowed at either end, glabrous, coriaceous, 1—8 in. long by 4—38 in. wide, petioles 4—4 in. long; margins slightly recurved, net-veins delicate, often coloured beneath. 6. Flowers 1—3 together, sessile, 5—6-merous, } in. long; calyx tubular, cup-shaped, tin. long, subentire or 5—G-lobed at apex, subglabrous; corolla 5—6-fid, silky outside, glabrous inside, lobes oval, rounded, spreading and reflexed; stamens 22—24, glabrous, inserted at the base of the corolla; ovary rudimentary, hairy, ¢. Flowers solitary. Fruit sessile, subglobose, as large as a moderate-sized apple, glabrous, shining, 10-celled, surrounded one-third way up by tube of calyx which has 5—6 wide reflexed and often undulated lobes. Seeds oblong, albumen not ruminated. Mauritius, Bouton’ The following localities are less certain; Bourbon, Richard/, Boivin! ; Round Island, Mauritius, Sir H. Barkly!; Rodriguez, Bouton/ 31. Drosrrros noposa, Poir. Encycl. Méth. v. p. 432 n. 9. (1804). D. foliis ovalibus vel oblongis, alternis, utrinque rotundatis vel obtuse angustatis, glabris, coriaceis, petiolatis, mediocriter reticulatis; floribus masculis 1—3-nis, subsessilibus, scepius 5-meris, calyce glabro, apice lobato, staminibus 20—82, glabris; floribus femineis solitariis, subsessilibus, staminodiis 12, ovario hirsuto, stylo 5-lobo ; fructibus subglobosis, glabratis, calyce Fructifero aucto, tubo cyathiformi, lobis erectis. Alph. DC. Prodr. vu. p. 226. n. 18 (1844). D. angulata, Poir. le. p. 434. n, 16, Alph. DC. lc. p. 226. n. 16. D. mauritiana, Alph. DC. lc. p. 226. n. 15 (1844). D. macrocalyx, Alph. DC. lc. p. 226. n. 17, non Kl, (2?).D. capensis, Alph. DC.! Lc. p. 226. n. 19. (2)D. Neraudii, Alph. DC. le. p. 227. n. 23. (?) D. Boutoniana, Alph. DC. lc. p. 236. n. 72. A glabrous shrub or tree; branches especially of the male plants often nodose at the inflorescence. Leaves oval or oblong, alternate, more or less rounded at both ends or occasionally cuneate at base, coriaceous, 14—Gin. long by 1—3in. wide; petioles {—}in. long. &. Flowers axillary, subsessile, about 1—3 together; calyx glabrous or glabrescent, subtruncate or shortly 4—6- usually 5-lobed at apex, cup-shaped, about }in. long; corolla about }in. long, sericeous outside, glabrous inside, deeply 4—6- usually 5-lobed; lobes oval, stamens 20—32, glabrous, hypogynous or inserted at the base of the corolla, spreading ; 2, g somewhat combined at base; filaments short; ovary rudimentary, hairy. 9. Flowers solitary, axillary, subsessile; bracts imbricated, caducous; calyx shortly 5-lobed, nearly glabrous, cup-shaped; corolla short; staminodes 12, separate, inserted at base Mr HIERN, ON EBENACEZ. 179 of corolla; ovary hairy; style 5-lobed. Fruit globular or ovoid, glabrate, 14—2in. high, resting at base on cup-shaped nearly glabrous calyx which in some cases reaches half-way up the fruit and has erect lobes. Mauritius, Boivin!, Gardner!, Duport!, Commerson!, 299, 301. Madagascar, Boivin! D. capensis, Alph. DC. is reported from the Cape of Good Hope, probably by mistake. Perhaps ought to be united to D. melanida, Poir. 32. Diospyros ANON#EFOLIA, Alph. DC. Prodr. vi. p. 227. n. 21 (1844). D. foliis elliptico-oblongis, alternis, obtusis, basi subacutis, glabris, submembranaceis, petio- latis; floribus masculis aggregatis, subsessilibus, calyce elongato-cyathiformi, basi acuto, glabro, obscure 5-lobo; corolla profunde 5-fidd, extus sericed ; staminibus 20—24, geminatis, glabris, corolle basi insertis. Branches and buds glabrous. Leaves alternate, elliptic-oblong, obtuse, glabrous, submem- branous, subacute at the base, 5—7in. long by 2—3in. wide, paler beneath; petioles }in. long. &. Flowers fascicled, 5—15 together, subsessile; bracts ovate, glabrous, caducous; calyx elongate, cup-shaped, acute at the base, smooth, glabrous, obscurely 5-lobed at the apex, tin. long; corolla deeply 5-lobed, silky outside, rather longer than the calyx. Stamens 20—24, united in pairs at base, glabrous, inserted at base of corolla. Mauritius (or Bourbon?) ex Alph. DC. Lc. Perhaps ought to be united to D. nodosa or to D. melanida or to both. 33. DIOSPYROS LEUCOMELAS, Poir. Eneycl. Méth. v. p. 432. n. 8 (1804). D. foliis alternis, ovalibus vel orbicularibus, apice rotundis, basi cordatis, subamplexi- caulibus, coriaceis, glabris, nitidis, subsessilibus ; floribus diccis, 1—3-nis, awillaribus, sessi- libus, 6—5-meris; calyce tubuloso-campanulato, apice lobato, extus sericeo; corolld profunde lobatd ; staminibus 30—40, glabris, receptaculo insertis; fructibus solitariis, glubris, calyce cyathiformi duplo et ultra longioribus, 8—12-locularibus. Alph. DC. Prodr. vu p. 236. n. 70 (1844), Ebenus leucomelas, Commers. MSS. n. 149 Ie. ex Poir. Lc. Diospyros reticulata, Sieb.! Pl. Maurit. n. 114. non Willd. nee Decaisne. Diospyros amplexicaulis, Lindl. et Paxt.! Flower Garden, 11. p.11. n. 271. fig. 139 (1851). Diospyros Commersoni, Gaertn. fil. Carp. ur. p. 136. t. 208 (1805). D. melanida, Neraud ex Alph. DC. Prodr. vur. p. 236. n. 70 (1844), non Poir. Cfr. D. Hebenaster, Gaertn. Fruct. et Sem. Plant. m. p. 478. t. 179. f. 9 (1791), non D. Ebenaster, Retz. A lofty tree with white wood but with black lines in the heart; trunk with a dark bark, much branched; branches glabrous, pale-cinereous, spreading at about 40°. Leaves oval or orbicular, alternate, subsessile, cordate at base, rounded at apex, subamplexicaul, coriaceous, quite glabrous and shining; often marked by coloured net-veins and occasionally 25-2 180 Mr HIERN, ON EBENACE®. by black blotches; 2—5}in. long by 14—3}in. wide; petioles j;—{in. long. Bracts im- bricated, subtomentose with grey hair, rounded, j;in. high by }in. wide, surrounding the base of the calyx. i 3. Flowers 1 or few together, axillary on young shoots or clustered on the shoots of previous season, sessile; calyx tubular, somewhat campanulate, with usually 6 short teeth at apex, covered outside with short brown or cinereous tomentum, 4—}in. high; corolla campanulate, 6—5-lobed, shortly ferruginous-sericeous outside, glabrous inside; tube }—5 in. long; lobes }in. long, spreading and recurved at extremities. Stamens 30—40, glabrous, inserted on the receptacle, nearly equal; anthers linear fin. long; filaments about yin. Tong, often somewhat combined at base; ovary only represented by a trace of hair. Q. Flowers arranged as in &. Fruit sessile, solitary, on branches deprived of their leaves (in the dry state), very glutinous, ex Poiret, glabrous ex Alph. DC., depresso-globose, umbilicate at the apex, about 1 in. high by 1} in. thick, 8—12-celled; fruit-calyx cup- shaped, about 4 the height of the fruit which it receives, 6-lobed at apex, coriaceous; ‘seeds cinereous, ?in. long; albumen not ruminated, white. | Mauritius, Commerson!; Sieber! 114; on the crest of the mountain to the left of the second Fenetre, above the French fort, Sept. ¢ fl, Ayres!; in forests on mountains at Savane and at Trois Ilots, Fl. May, Dec. Ayres MSS.; a specimen from Round Island near Mauritius without flower or fruit by Sir H. Barkly! probably belongs to this species; Madagascar, Gaertner, Chapelier / 34. DIOSPYROS CHRYSOPHYLLOS, Poir. Encycl. Méth. v. p. 433, n. 12 (1804). D. ramulis flexuosis, folits lanceolato-oblongis, apice utrinque obtuse angustatis, glabris, coriaceis, petiolatis; floribus diacis, 1—3-nis, subsessilibus, 5—4-meris, asxillaribus; calyce cyathiformt, eatus pubescente, apice lobato, corolld profunde lobatd, staminibus 11—135, glabris ; ovario in floribus femineis glabro, 10-loculari, stylis 5; fructibus solitariis, globosis, nétidis, 7—10-locularibus, calyce fructifero subtruncato, pateriformt. Alph. DC. Predr. vin. p. 225. n. 13 (1844). A shrub or tree with glabrous flexuous branches, subscandent (?). Leaves lanceolate-oblong, somewhat narrowed at base, usually more or less narrowed towards apex, obtuse, 2}—5}in, long by }—14 in. wide, besides petiole 4—3,in. long; somewhat paler and brilliant beneath golden-coloured) ; coriaceous, margins reflexed. Flowers subsessile, axillary, pentamerous or tetramerous. Calyx ferruginous-pubescent outside, cup-shaped, dentate at apex. Corolla ferruginous-sericeous outside, deeply lobed. 6. Flowers 1—3 together. Stamens 11—15. 2. Flowers solitary, more than in. long. Calyx 3,—% in. long; lobes 3; in. deep, widely ovate, wavy at margins, obtuse; tube very crass especially at base, felted outside, glabrous and shining inside. Corolla with 5 short imbricated rounded lobes, constricted at top of calyx, hairy outside in upper part, often remaining at top of young fruit. Staminodes 9, glabrous. Ovary glabrous but surrounded at base with a ring of hairs, 10-celled; styles 5, sericeous at base; stigmas lobed at apex. Fruit glabrous, globose about lin. in diameter, shining, green, 7—10-celled. Fruiting calyx subtruncate, Jin. across at top, glabrescent, Mauritius, Bojer !, Gardner /, Bouton !, Commerson/ Mr HIERN, ON EBENACE, 181 35. DIOSPYROS SENENSIS, Klotzsch in Peters Mossamb. I. p. 183 (1862). D. foliis alternis, obovato-oblongis, apice breviter acuminatis vel rotundatis, basi cuneatis vel subrotundatis, submembranaceis, subtus flavido-pubescentibus, breviter petiolatis; floribus 1—5-nis, breviter cymosis, axillaribus, tetrameris, pedicellis brevissimis; calyce anguste tubu- loso, apice lobato, corolla 4-fidd, staminibus 16, geminatis, glabris; in flore femineo stamino- diis 0, ovario glabro (2), 8-loculari ; fructibus solitartis, glabris, 2—8-locularibus. A shrub from 10 feet high to a tree 30 feet; occasionally subhermaphrodite or polygamous; branches terete, pale-cimereous or smooth and reddish; young shoots softly flavido-pubescent. Leaves membranous, alternate, obovate-oblong, suddenly narrowed or acuminate or occasionally rounded at apex, cuneate or nearly rounded at base, subglabrescent and deep green above with depressed midrib, somewhat flavido-pubescent or subglabrescent beneath, 2—74 in. long x 1—3} in. wide, besides hairy petiole =;—} in. long. Inflorescence axillary, in short 1—5-flowered cymes, flavido-pubescent, with small caducous bracts at base of very short pedicels; flowers greenish-yellow, fragrant; g peduncles not exceeding } in. long. é. Calyx +2 in. long by +4 im. thick, tubular, subtruncate or with 4 short rounded lobes at apex, flavido-pubescent outside and hairy inside. Flowers greenish-yellow, fragrant. Corolla tubular, about twice the length of the calyx, 4-fid, glabrous, except 4 hairy lines down the middle of the lobes; lobes oblong, obtuse; stamens 16, glabrous, in pairs, partly inserted at base of corolla and partly hypogynous; ovary usually rudimentary or wanting, occasionally 5-celled in subhermaphrodite flowers. @. Flowers shorter than in 6; staminodes 0; ovary glabrous (2), 8-celled; calyx hairy on both sides. Fruit solitary, glabrous, but hairy around base of style, acorn-shaped, 1 in. long by % in. thick; half inclosed in subtruncate calyx, 2—4—8-celled, not eaten (Dr. Kirk); style making a short conical projection; fruiting calyx shortly pubescent especially inside; albumen horny; seeds with green vittze on surface (Avrk); cotyledons cordate, ‘acute, foliaceous. Tropical Africa, Mozambique, Rios de Sena, Dr. Peters!; Lower Shire Valley, Dr. Kirk! 6 fl. January; Lupata, Dr. Kirk!, young fruit, January; Forest below Strigogo, left bank of Zambezi, Dr. Kirk/, fruit, April; North of Shire, Banks of Zambezi, Dr. Meller! , fl. January; Abbeokuta, Dr. Irving! 141; Abbeokuta &c., Niger expedition, Barter! 290, 3251, 3390; Eppah, Barter! 3250 (peduncles 1-flowered, 5>—,; in. long, in bud). 36. DIOSPYROS ROTUNDIFOLIA, sp. nov. D. foliis alternis, obovato-rotundis, utrinque rotundatis, glabris, coriaceis, breviter petiolatis, margine revolutis, nervis subtus inconspicuis; floribus solitarvis, axillaribus, glabris, diecis, breviter pedunculatis; calyce apice 5-lebo; corolla 5-fidé; staminibus 30, glabris, receptaculo insertis ; fructibus globosis, apice wmbilicatis, nitidis, uncialibus, 8-locularibus (2); calyce fructifero aucto, apice 5-lobo, plicato, pateriformt. Young parts puberulous; branches pale-cinereous. Leaves obovate-rotund, alternate, coriaceous, with recurved margins in the dry state, rounded at both ends, {—1}in. long by —1} in. wide, besides petioles ;j—} im. long, glabrous; veins inconspicuous beneath, 182 Mr HIERN, ON EBENACE. Peduncles axillary, solitary, crowded in upper axils, puberulous, recurved, ;;—} in. long, 1-flowered; bracts caducous. &. Flowers glabrous, about 2in. long; calyx ;3;im. long, hemispherical-campanulate, with 5 shallow apiculate lobes; corolla 5-fid, with oval spreading lobes; stamens in one case 30!, glabrous, nearly equal, inserted on the glabrous receptacle; filaments short, straight ; anthers about } in. long, 2-celled, dehiscing laterally from apex; ovary 0. @. Fruiting calyx acerescent, exceeding the young fruit, } in. high and broad, really 5-lobed at apex but apparently 5-fid by calyx being plicate and reflexed downwards and out- wards into 5 sides; quasi-lobes broadly ovate dilated at base and plicate so as to make the calyx 5-winged; styles 5, connate at base, jin. long, glabrous; stigmas bifid; young fruit depresso-globose, } in. high, glabrous, 8?-celled; ripe fruit globose, umbilicate at apex, shining, 3—1l in. in diameter; fruiting calyx pateriform, {—lin. across, ;5 in. high, with raised border above, plicate; seeds compressed, } in. long by ~; im. wide. S. Africa, Delagoa Bay, Forbes! 34. 37. DiospyRos ATTENUATA, Thw. Enum. Ceyl. Pl. p. 182. n. 18 (1860). D. foliis alternis, anguste ovatis vel oblongis, apice acuminatis, bast cuneatis, tenuiter coriaceis, glabratis, breviter petiolatis, creberrime venulosis; floribus masculis 3—10-ns, subsessilibus, oblongis, 4—5-meris, staminibus 4—5; floribus femineis solitariis, ovario 4-lo- ceulari, fructibus ovoideo-conicis, acuminatis, subglabrescentibus, 2—3-spermis, albumine non rumanato. Bedd. Ic. Pl. Ind. Or. (Part viz.) p. 28. t. 139 (1871). A moderate-sized tree; young shoots appressedly puberulous, quickly glabrescent; leaves alternate, narrowly ovate or oblong, acuminate at apex, more or less narrowed at base, quickly glabrescent, thinly subcoriaceous, 2—4in. long by 3—1} in. wide; petioles ;;—% in. long; midrib depressed above; net-veins very close together, in relief on both sides, delicate. &é. Flowers clustered, 83—10 together, sessile or subsessile on }in. long axillary nodules, strigose with black and subferruginous mixed hairs, 4—5-merous; calyx ;/; in. long, 4—5-fid, hairy on both sides, lobes narrowly deltoid. Corolla slender in bud, much ex- ceeding the calyx, 4—5-lobed, 1} in. long, lobes rather shorter than the tube; stamens 4—5, in one row, anthers glabrous, connective, prolonged at apex, filaments short, without or with light brown hairs; ovary 0 or rudimentary, conical, with light ferruginous hairs. . Flowers solitary, sessile; calyx }—} in. long, lobes more or less reflexed at the margin; corolla but little exceeding the calyx; staminodes 4—5; stigmas 2, short; ovary hairy, 4-celled, ovoid; cells l-ovuled. Fruit conical, with an ovoid base and acuminate apex, pale, softly hairy or nearly glabrescent, 2—38-seeded; fruiting calyx loose, deeply 4—5-lobed, not accrescent; seeds oblong, shining, acuminate; albumen not ruminated. Ceylon, Pasdoon Corle, Thwartes! C.P. 3478. 38. Diospyros acuta, Thw. Enum. Ceyl. Pl. p. 182. n. 17 (1860). D. foliis alternis, lanceolato-oblongis, apice acuminatis, basi suhrotundatis, cortaceis, glabris, robuste petiolatis, nervis inconspicuis; floribus masculis aggregatis, sessilibus, 4—5- Mr HIERN, ON EBENACEZ, 183 meris, calycis lobis lanceolatis, staminibus 4—5; floribus femineis 1—4-nis; fructibus ovoideis acuminatis, 2—3-spermis, seminibus acuminatis, albumine non ruminato. A moderate-sized tree, glabrous except the buds and inflorescence; branches terete. Leaves alternate, lanceolate-oblong, acuminate at apex, more or less rounded towards base, coriaceous, 5—12 in. long by 1}—4 in. wide, turning reddish beneath (when dry); petioles 34—1 in. long, stout, channelled above; midrib depressed above; lateral veins inconspicuous. Inflorescence appressedly fulvous-hairy, dicecious or sometimes moneecious, in which case the female capitula are towards the top of the branches, and the male ones beneath. &. Inflorescence dense, many-flowered, axillary, sessile; calyx } in. long, 4—5-lobed beyond the middle, lobes lanceolate, acute, hairy on both sides; corolla 4 in. long, 4—5-fid; stamens 4—5, short, glabrous; ovary rudimentary, very small. @. Flowers 1—4 together; calyx 4—4 in. long, lobes more or less reflexed at the margin; corolla about as long as the calyx; stigmas 2—3, j,in. long, spathulate; ovary 4- or 6-celled; fruit acuminate, 1} in. long, resting on a scarcely increased calyx, usually 2—3-seeded; seeds shining, oblong, acuminate, lin, long; albumen not ruminated. Ceylon, Pasdoon Corle, Thwaites! C. P. 3476. 39. DIOSPYROS TRICOLOR. D. fruticosa, foliis alternis, ellipticis, utrinque obtusis, supra subglabris viridibus, subtus albido-sericeis, costa ferrugined; floribus azillaribus, sessilibus, 1—4-nis, tetrameris, pubes- centibus, calyce quadrifido, corolld tubulosd, staminibus 6—8 vel pluribus, inequalibus ; floribus Femineis solitariis, staminodiis T—8, ovario ovoideo, sericeo, in stylum subulatum attenuato ; Sructibus subpyramidatis, glabris, junioribus 4-locularibus ; seminibus 2—4. Noltia tricolor, Schum. et Thonn. Plant. Guin. p. 189 (1827), in Kong. Danske Vidensk. Sel. Phys. og Mathem. Skr. m1. p. 209 (1828). A much-branched shrub, 2—4 feet high; branches terete, ferruginous-tomentose, diverg- ing, sometimes flexuous, procumbent. Leaves alternate, distichous, elliptical, obtuse, nearly rounded at base, with few lateral veins, green and glabrescent above, white-silky with the midrib and margin often ferruginous beneath, 1—3in. long by 3—2in. wide, the young ones silvery-silky on both sides; petioles =,—1in. long, tomentose. Flowers solitary or 3—4 together, axillary, sessile. 3. Calyx 4-fid, lobes acute, silky-tomentose, ferruginous; corolla tubular, 3 times the length of the calyx, scarcely dilated below, sub-4-lobed, subcoriaceous, “red,” silky outside, Zin. long; lobes acute, erect, inflexed at the margin; filaments 6—“8 or more, unequal, 4 often double the length of the rest, half the length of the corolla, pubescent below, inserted on the receptacle, either distinct or 2—8 together at the base,” anthers subulate, erect ; ovary rudimentary. @. Flowers solitary; corolla rather inflated at the base; staminodes 7—8, distinct; ovary ovoid, silky, attenuated into a subulate style; stigma acute; fruit conical-oblong and ferruginous-silky when young, afterwards conical, obsoletely tetragonal, yellow, quite glabrous, lin. long by }in. wide, 1-celled, 4-seeded; seeds oblong; pulp sweetish; calyx of young 184 Mr HIERN, ON EBENACE. fruit lin. high, 4-fid with acutely deltoid lobes, erect; young fruit 4-celled, 2 cells of which are each 1-seeded. Local name, Aumbe. West tropical Africa, Guinea, Thonning!, common in the vicinity of the shore; Cape Coast, Brass ! I have without doubt referred this plant to Diospyros, thus following the suggestions of Messrs Bentham and Planchon. See Niger Flora p. 442 (1849) and Annal. Sc. Nat. ser. Iv. vol. 3. p. 293 (1855). | Plate v. fig. 1. A fruiting branch, from Brass’ specimen in Hb. Mus. Brit. natural size. 40. DroSsPYROS FULIGINEA, sp. nov. D. foliis ovali-oblongis, apice anguste et valde acuminatis, basi sepius rotundatis, glabris vel subtus subglabris, coriaceis, costd superne depressa, venis inconspicuis, margine tenuiter revoluto, petiolo tereti, robusto, fusco; fructibus terns, 8-locularibus, 8-spermis, in cymis distinctis axillaribus fuligineo-hispidis dispositis ; calyce fructifero aperte campanulato, 4-fido, fuligineo- hispido, lobis deltoideis, erecto-patentibus, non plicatis. Branches cinereous, scattered more or less with small fuliginous spots, glabrescent ; young shoots fuliginous-hispidulous; leaves alternate, oval-oblong, narrowly and usually suddenly acuminate at apex, usually rounded at base, glabrous or scattered with short appressed inconspicuous hairs beneath, coriaceous, 43—7 in. long by 13—2} in. wide, midrib depressed above, veins inconspicuous, margin finely revolute; petioles stout, terete, fuscous, with short dark hairs, 2 in. long. 9. Cymes many-flowered (?), fuligious-hispid, in. long exclusive of the flowers, bearing in one case 3 fuliginous-hispid fruits with firm pedicels }—1 in. long; young fruit globose. with conical apex, exceeding the calyx, 8-celled, 8-seeded; fruiting calyx widely campanulate, about Jin. in diameter, 4-tid, thickly coriaceous, not plicate; lobes deltoid, spreading. Borneo, O. Beccari/ n. 2486. 41. Diospyros BRANDISIANA, Kurz in Journ. Asiat. Soc. Beng. vol. xu. Pt. 1. p. 72. n. 93 (1871). D. foliis ovalibus, alternis, apice acuminatis, basi rotundatis vel acutis, chartaceis, adultis glabris vel secus costas sparse appresse hirsutis, breviter petiolatis ; floribus cymosis e ramis ortis vel awillaribus, 5—4-meris, calyce 5-fido, lobis lineari-lanceolatis, acutis, corolld 5-fidd, lobis obtusis, staminibus circiter 16, filamentis brevissimis, pubescentibus, antheris glabris ; tn floribus femineis staminodiis 5, ovario dense fulvo-pubescente, 10-loculari, fructu immatura ovoideo, acuwminato. Flora, 1871, p. 342. A tree with young parts shortly pubescent. Leaves alternate, oblong to elliptic-oblong and oblong-lanceolate, acuminate, rounded or acute at base, entire, chartaceous, 4—6—8 in. long, the adult ones glabrous or usually sparsely and appressedly hirsute on the midrib; petioles qs—t in. long, puberulous, somewhat depressed above. Flowers 4—{yin. long -in the bud, pentamerous or tetramerous, in rather dense much-branched minutely-bracteated black-brown cymes springing from the branches or axillary; pedicels ;—j in. long, afterwards elongated) Mr HIERN, ON EBENACE. 185 tomentose; bracts minute, oblong-lanceolate, tomentose; calyx covered with slight black or dark brown tomentum, ;4;—3 im. long, deeply lobed, lobes linear-lanceolate, acute; corolla tube appressedly pubescent, ;,in. long, rather widened towards the base and commonly 5-sided, lobes equalling the tube, oblong, obtuse. 3. Stamens 14—16; filaments very short, pubescent; anthers linear, mucronulate, glabrous. Receptacle hairy. 2. Staminodes 5; ovary densely fulvo-pubescent, 10-celled, terminated by the rather long simple crass style. Very young fruit, ovoid-conical, acuminate, shortly pubescent. Burma, Martaban, Dombamee forests, Dr Brandis ! 42, DIOSPYROS SUBACUTA, sp. nov. D. fruticosa, foliis ovato-oblongis, distichis, apice acuminatis, basi rotundatis vel subcor- datis, sub-glabris, margine subciliatis, nitidis, subsessilibus, nervis inconspicuis ; fructibus soli- taris, axillaribus, oblongis, apice conicis, appresse pubescentibus, pedunculatis ; calyce fructifero 4-fido, pateriformi, pubescente. Shrub; young parts rufous-pilose-hispid with scattered hairs; branches dark, terete, glabrescent. Leaves ovate-oblong, distichous, subcoriaceous, glabrous or nearly so and shining, acuminate at apex, rounded or subcordate at base, subsessile, rich brown beneath in the dry state, darker above with elevated midrib; veins conspicuous; 14—3 in. long by 3 wide, spreading; petioles ;4— 4 in. long, thick, dark, subpilose; margins of leaves subciliate with pilose long hairs. ?. Fruit solitary, axillary, im. long by tin. thick, oblong, conical at apex, covered with short appressed brown pubescence, with several (?) cells; flowering peduncles lin. (or more?) erect-patent, rough; bracts caducous; fruiting calyx 4-fid, pubescent, tin. across by din. high, shallowly cup-shaped. Madagascar, S® Marie, Boivin / 43. DIosPpYROS PRURIENS, Dalz. in Kew Journ. Bot. vol. rv. p. 110. n. 2 (1852). D. foliis alternis, ovali-oblongis, apice breviter acuminatis, basi rotundatis vel subcordatis, tenuiter coriaceis, supra nervo excepto subglabrescentibus, subtus presertim secus nervos piloso-hirsutis, breviter petiolatis; jfloribus masculis avillaribus, pedunculis confertis, 1—2 floris, calyce 4-partito, utringue piloso, coroll@ profunde 4-fidd, extus sericed, lobis obtusis, staminibus 13—14, glabris, hypogynis; floribus femineis 4—5-meris, staminodis 4—5, glabris, ovario ferrugineo-hispido, 4-loculari, loculis 1-ovulatis; fructibus piloso-prurientibus, ovoideo-conicis, 4-locularibus, loculis 1-spermis. Bedd. Ic. Pl. Ind. Or. (Pt. vit) p. 26. t. 129 (1871); (?) Thw. Enum. Ceyl. Pl. p. 423, (1864). Young shoots, peduncles, petioles and underside of leaves, especially on the veins, softly pilose-hirsute, fulvous; branches terete, dark, glabrescent. Leaves oval-oblong; shortly and usually obtusely acuminate at apex, rounded or subcordate at base, thinly subcoriaceous, alternate, 2—4 in. long by 4—12 in. wide, with petiole 1,—1 in. long, subglabrescent above, except the depressed midrib; lateral veins not strong. Vou, XII. Part I. 24 186 Mr HIERN, ON EBENACEZ. é. Peduncles near together in the upper axils, #—§,in. long, 1- or 2-flowered; flowers } in. long or more; bracts rounded, caducous, glabrous inside; calyx 4 in. long, 4-partite, fulvo-pilose on both sides, lobes linear-oblong, lax; corolla appressedly sericeous outside, glabrous inside, }—% in. Jong, deeply 4-fid, constricted at top of tube, lobes ovate- oblong, obtuse, imbricated sinistrorsely. Stamens 13—14, glabrous, unequal, hypogynous, connate at base, shorter than the corolla-tube, surroundmg the hairy rudiment of the ovary; filaments about as long as the anthers. @. Flowers solitary, crowded in the upper axils on peduncles ;—} in. long; calyx iin. long with oblong spreading lobes, hairy on both sides, 4—5-partite; corolla 3 in. long, 4-fid, constricted about middle; staminodes 5 (in one case), inserted at base of corolla, glabrous, linear; ovary ferrugimous-hispid, 4-celled, cells 1-ovuled; styles 2, short, almost concealed by the long hairs on the ovary, glabrous, bifid at apex. Fruit ovoid-conical, 21 in, long, 4celled, 4-seeded, densely clothed with fulvous stinging hairs. Fruiting calyx spreading or reflexed, not accrescent. Bombay, Chorla Ghaut, Dr Ritchie! 1833; Dalzell!; Bababoodun hills, Mysore, Mr Law! ; (2) Ceylon, Saffragam district, 2000 ft. alt., Dr Thwaites, C.P. 2836. 44, DIosPpYROS APICULATA, sp. nov. D. foliis alternis, oblongis, apice acute acuminatis, bast cordatis, tenuiter coriaceis, supra glabrescentibus, subtus presertim secus nervos hispidis, breviter petiolatis; floribus masculis sub-3-nis, subsessilibus, axillaribus; calyce 4—5-partito, piloso-pubescente, corolld tubulos@, glabra, 4-lobd, staminibus 6—7 vel 12, inequalibus, glabris, ovarit rudimento hirsuto ; floribus femineis 1—3-nis, brevissime cymosis; fructibus solitariis, subsessilibus, ferrugineo-setosis, ovoideo- conicis, apice apiculatis, 4-locularibus; albumine seminum non ruminato. A tree with slender stem, about 4 feet high in the specimen seen; young parts ferruginous-hispid. Leaves oblong, alternate, thinly subcoriaceous, much acuminate at apex, cordate at base, hispid beneath, especially on the clearly marked veins, glabrescent above, with depressed midrib, of the same colour on both sides except the hairs, 4—7} in. long by 1}—23 in. wide, margins just reflexed; petioles ;;— 4, in. long, hispid. Bracts finely hispid. $. Flowers about 3 together, subsessile, axillary, 4; in. long; calyx 4—5-partite, about } in. long, pilose-pubescent on both sides except near the base inside, lobes lanceolate- linear; corolla glabrous, 2 in. long, tubular, 4-lobed, lobes spreading, oval, obtusely pointed at apex, contorted sinistrorsely in bud, 54; in. long; stamens glabrous, 6—7 or 12, unequal, anthers linear-oblong, pointed at apex; filaments often geniculate, dilated and connate at base, inserted in a very short tube at the very base of the corolla; ovary rudimentary, small, hairy. @. Flowers 1—3 together, on very short axillary finely hispid cymes. Fruit solitary, subsessile, finely ferruginous-setose especially upwards but not densely so and subglabrescent in lower part, ovoid-conical, about 1 in. long by $—% in. wide, apiculate at apex, ovoid at base, with indications inside of 4 cells, terminated by 2 (?) adjacent styles; seeds 4 (2), § in. long; albumem somewhat farinaceous (in dry state), not ruminated. Penang, Goot hill, Dr Maingay! no. 1514. Mr HIERN, ON EBENACEA. 187 45, Diospyros BARTERI, sp. nov. D. fruticosa, foliis alternis, ovali-ovatis, apice apiculatis acuminatis, basi cordatis, firmiter membranaceis, supra nervo excepto glabris, subtus pallidis hispido-sericeis presertim secus nervos, breviter petiolatis; floribus femineis solitarws, subsessilibus, hispidis, calyce 4—5- partito, lobis lineari-lanceolatis, corolld extus hispidd, 5-fidd, lobis acutis, staminodiis 11, brevibus, uniserialibus, pilosis, ovario glabro (apice eacepto), 4-loculari, loculis 1-ovulatis; fructibus conicis, acuminatis, glabris sed apice hirsutis, seminibus oblongis, albumine non ruminato. A shrub with young shoots rufous-hispid or afterwards fuscous-hispid; older branches dark, terete, glabrate, spreading at about 50°. Leaves alternate, oval-ovate, acuminate, apiculate, at base cordate, firmly membranous, dark green and glabrous except the depressed midrib and with depressed veins above; paler with hispid-pilose ferruginous hairs, especially on the veins beneath, 2—3 in. long by 1—1} in. wide; petioles hispid, =;—+ in. long. 2. Flowers solitary, subsessile, axillary, with narrow rufous-hispid-pilose caducous bracts. Calyx }in. long, rufous-hispid-pilose, 4—5-partite with linear-lanceolate lobes somewhat spreading in flower and sub-horizontal not accrescent in fruit; hispid inside. Corolla conical in bud, as long as the calyx, ferruginous-hispid outside, glabrous inside, 5-fid, lobes acute, imbricated. Staminodes 11, short, in one row, distinct (except 1 pair), pilose. Pistil conical; ovary glabrous except apex, 4-celled, cells 1-ovuled; styles 2, bilobed at apex, pilose below, as long as the young ovary. Fruit oblong-conical, 14 in. long, glabrous (except apex), shining, with shortly ferruginous-pubescent remains of styles, 2-celled; cells 1-seeded; seeds 2 in. long; albumen not ruminated. W. Africa, Guinea, Lagos. Niger Expedition. arter/ 20194. 46. DIosPYROS MICRORHOMBUS, sp. nov. D. foliis distichis, rhomboideo-ovalibus, ad apicem emarginatum angustatis, basi cuneatis, interdum sub-obliquis, subglabris, coriaceis, subsessilibus; jfloribus fenineis solitariis, graciliter pedunculatis, glabris, calyce profunde 4-lobo, lobis rotundatis, erecto-patentibus, corolla breviter 4-fidd, staminodiis 4, glabris, corolle basi insertis, ovario glabro, ovoideo-conico, 8-loculari. Of a dark colour when dry; branches covered with short patent pale pubescence, terete ; wood very good. Leaves subsessile, distichous, rhomboid-oval, narrowed to an emarginate apex, cuneate at base and sometimes slightly oblique, glabrous or very nearly so, coriaceous, } in. long by }in. wide, dark slatish green above, brownish beneath; veins indistinct. 9. Flowers solitary, on long slender glabrous peduncles which measure }—%in. long and bear appressed oblong glabrous bracts about middle and near base; flowers + in. long, glabrous; calyx 4in. long, deeply 4-lobed; lobe }-oval, }in. wide, rounded, erect-patent; corolla erect, 4 in. high, glabrous, 4-sided and shortly 4-fid; staminodes 4, glabrous, alternate with the lobes of the corolla and inserted at its base; ovary glabrous, 8-celled, ovoid-conical, terminated at apex by a 4-lobed conical style; divisions of the style emarginate at apex. “Ebenier de Madagascar, son bois est superbe; Iles de France et Bourbon,’ Hb. Mus. Paris. ! 24—2 188 Mr HIERN, ON EBENACEA!. 47. DIOSPYROS FOLIOLOSA, Wall. List n. 4143 (1828—32). D. glabra, foliis alternis, oblongo-lanceolatis, apice attenuato-acwminatis, basi obtusis, nitidis, tenuiter coriaceis, reticulatis, petiolatis; jfloribus masculis laxe cymosis, ovoideis, tetra- meris, calyce parvo, corollé ovoideo-urceolatd, breviter lobatd, staminibus 12—16, geminatis, connectivo et filamentis leviter pubescentibus, ovarii rudimento acuminato; floribus femineis solitarus, axillaribus, pedunculatis, 4- rarius 3-meris, staminodiis nullis, stigmatibus 4—3, sessilibus, ovario 4-loculart, loculis 1-ovulatis; fructibus globosis, junioribus pubescenti-squa- mosis, senioribus glabratis ; calyce fructifero fructum equantibus vel excedentibus, lobis cordato- ovatis, foliacers, nervosis. Alph. DC. Prodr. vit. p. 234 n. 58 (1844). Diospyros calycina, Bedd., Ann. Rep. Forests, Madras Pres. for 1867—68, p. 26 (1868), Flora Sylvatica, Madras, t. 68 (1870), Ic. Pl. Ind. Or. (Part vii.) p. 25. t. 123 (1871), non Audib. D. auriculata, Wight! (MS. in Hb. Kew), Hb. Wight!, Kew List n. 1716, non Stiehler. A good sized tree, glabrous in all parts except the stamens ovary and young fruit. Leaves alternate, oblong-lanceolate, thinly coriaceous, attenuate-acuminate at apex, narrowed or rounded at base, shining, green on both sides, 2—4}in. long by 4—1l1}in. wide, mid- rib depressed on upper side; net-veins delicate in relief on both sides ; petioles 3—1in. long, é. Cymes axillary, lax, about half the length of the leaves, 3—9-flowered; flowers } in. long, ovoid; calyx small, about ;4;in. high by 4in. across, 4-fid, with deltoid or ovate lobes; corolla urceolate, often gibbous at base, 4-fid, bright yellow in colour, much contracted at the top of the tube, lobes short, pointed, spreading; stamens 12—16, inserted on the recep- tacle and united in pairs by their short compressed more or less hairy filaments; anthers equal, lanceolate, dehiscing from the base, converging at the apex above the rudimentary 5-lobed ovary which terminates with a long acumen; connective somewhat hairy. @. Flowers solitary, axillary, on peduncles 2—1in. long; calyx with 4 or rarely 3 cordate imbricated veined accrescent partitions; corolla urceolate, gibbous; tube nearly globose, lobes 4 or rarely 3, short, reflexed; staminodes 0; stigmas 4 or 3, sessile; ovary 4-celled; cells l-ovuled. Fruit globose, covered when young with hairlike scales, glabrescent, 3 in. in diameter ; fruiting calyx about as long as the fruit or longer, sometimes lin. long, some- what glandular at base within around base of fruit; lobes cordate-ovate, foliaceous. Very abundant in the ghat forests from bottom to 3000ft. alt. in the Tinnevelly district and southern portions of Madura; it is called Vellay Toveray, and yields a valuable light-coloured wood, Beddome; Courtallum, Wallich / 48, Drospyros prLosuLa, Wall. List. n. 4132 (1828—82). D. folvis alternis, obovato-oblongis vel anguste ellipticis, apice acuminatis, basi obtusis, tenuiter coriaceis, supra glabris, nitidis, subtus secus nervos pubescentibus, petiolatis; floribus masculis pedunculatis, staminibus 12, glabris, inequalibus ; floribus femineis solitariis, pedun- Mr HIERN, ON EBENACE. 189 culatis, calyce 4-partito, lobis lanceolatis acutis, staminodiis 0, ovario rufo-hispido, 4-loculart, loculis 1-ovulatis. Gunisanthus pilosulus, Alph. DC. Prodr. vim. p. 220 (1844). A tree or shrub; branches terete, fulvo-pubescent when young, afterwards glabrescent and cinereous. Leaves narrowly elliptical or obovate-oblong, acuminate at apex, somewhat narrowed at base, alternate, thinly coriaceous, glabrous and shining above with depressed eae oe eae rae beneath and ciliate when young, glabrescent except the tin. long by 1—l}in. wide; petioles about fin. long, pubescent when young ; Tateral yeins not conspicuous. é. Flowers on the young shoots, tetramerous, pilose, about 3 in. long, on slender peduncles about $ in. long; calyx } in. long, lobes deep lanceolate acute lax; corolla rather slender, tube aie upwards {—} in. long, lobes lanceolate acute rather longer than the tube, at length spreading; stamens 12, glabrous, very unequal, ,,—1 in. high, inserted on the receptacle, filaments often geniculate, anthers about , in. rec y . Flowers solitary, on the young shoots, rather slender; peduncles 1—} in. long, pubescent, articulated at the apex to the flower, without bracts, tapering downwards in fruit; calyx of the young fruit 4-partite, pubescent outside, glabrous inside, lobes lanceolate, }—1in. long, spreading, acute; corolla deeply 4-fid, silky outside, tube cylindrical, narrowed upwards, shorter than the pe lobes acute; young fruit rufous-hispid, 4-celled; staminodes 0; style very short, covered by the hairs of the ovary; stigmas 2, glabrous; ovary 4-celled, cells 1-ovuled. Among the mountains of Silhet, Wallich!; Pegu, Dr Brandis /, local name Gjut. 49. DIOSPYROS SUBERIFOLIA, Decaisne MSS. in Hb. Mus. Paris. D. folits alternis, ovalibus vel obovato-oblongis, apice rotundatis emarginatis vel apiculatis, basi obtusis, subtus subtomentosis, margine minute repando-crenulatis, subsessilibus ; floribus masculis pubescentibus, pedunculis axillaribus, 1—2-nis, 1-floris, basi e bractearum nidulo ex- orientibus, calyce 5-partito, corollé urceolatd, breviter 5—6-dentatd, stuminibus circiter a) antheris hispidulis, filamentis glabris, ovarii rudimento hirsuto. Stems dark-cinereous, rough, glabrescent, softly subtomentose when young. Leaves oval or obovate-oblong, alternate, coriaceous, subsessile, softly sub-tomentose at least beneath, slightly convex from above, rounded emarginate or apiculate at apex, rounded or somewhat narrowed at base, margins minutely repand-crenulate, 1—3 in. long by }—1}in. wide, net- veins not very conspicuous; petioles very short. é- Flowers pedunculate, axillary; peduncles solitary or 2 together, arising from a nest of bracts at base, pubescent, tin. long or more; calyx 5-partite, pubescent outside, glabrous inside, ;4in. long, lobes ovate; corolla urceolate, shortly or irregularly 5—6-lobed, puberulous outside, glabrous inside, 3;in. long; stamens 21 (one of which is very thin) in one case, inserted on the receptacle or some at the very base of the corolla, some in pairs; anthers hispidulous upwards, lanceolate-linear, apiculate; filaments very short, slender, glabrous; ovary rudimentary, hairy. Cultivated in hort. Paris.!; supposed to have been brought from Chili. 190 Mr HIERN, ON EBENACEZ. 50. Drospyros sQUARROSA, Klotzsch in Peters Mossamb. I. p. 184 (1862). D. foliis alternis, ovalibus, utrinque rotundatis vel obtusis, tenwiter coriaceis, breviter pubescentibus presertim secus nervos vel subglabrescentibus, petiolatis; floribus femineis acxil- laribus, solitariis, pedunculatis, tetrameris; calyce profunde 4-fido, corolld 4-partitd, parti- tionibus obtusis, patentibus, subglabris, staminodiis 0, ovario subgloboso, glabro, 8-loculart ; stylis 4, bifidis; fructibus subglobosis nitidis, calycis fructifert lobis dependentibus, seminibus compressis. A tree, or much branched shrub, with young shoots delicately hispid, virgate; branches glabrescent, terete, spreading at about 80°. Leaves elliptical or somewhat obovate, alternate, thinly coriaceous, rounded at both ends or sometimes narrowed; with scattered patent pubescence or subglabrescent, subnitescent above; paler, with patent pubescence, rufous and denser on midrib and lateral veins beneath; patent, delicately reticulated, 14—3} in. long by 3—2 in. wide; petiole ~,—+in long, pubescent. ?. Flowers axillary, solitary, drooping, tetramerous; peduncles recurved, }—1 in. long, patently pubescent; bracts caducous, at about middle of peduncle, lanceolate, #,in. long; ealyx covered with short appressed tawny hairs on both sides, loosely hemispherical, } in. long, with 4 deep oval or ovate lobes; corolla 4-partite, openly cup-shaped or rotate nearly glabrous, but with scattered pale appressed hairs along middle of lobes; lobes reflexed, about 3 in. long, obtuse; stamens 0; ovary glabrous, somewhat 4-sided, qb in. high, 8-celled, cells l-ovuled; styles 4, glabrous, bifid to about middle, not persistent on fruit; fruit glabrous, somewhat 4-sidedly globular, about 2 in. high; fruiting calyx with pendent lobes, not accrescent. Africa, R. Zambezi at Senna (left bank), and Rivoque near Tette, January, in @ flower and fruit, local name “Mutshenje tuna tuna,” Sechuana dialect, Dr Kirk !; Sena, Dr Peters !, in hedges near water-courses. 51. Drospyros PANICULATA, Dalz. in Kew Journ. Bot. rv. p. 109. n. 1 (1852), Bedd. Ie. Pl. Ind. Or. (Pt. vit.) p. 25. t. 125 (1871). D. foliis oblongis, alternis, utrinque obtusis, glabris, subcoriaceis vel submembranaceis, reticulatis, petiolatis ; floribus masculis numerosis paniculatis pentameris fuligineo-pubescentibus, calycis lobis foliaceis reticulato-venosis, staminibus 20 geminatis glabris; floribus femineis solitariis pedunculatis pentameris; fructibus ovoideis glanduloso-hirsutis, 4-locularibus, calyce aucto plicato. A middle-sized or large tree with glabrous somewhat angular branches. Leaves oblong, alternate, thinly subcoriaceous or submembranous, narrowed rounded or obtusely acuminate at apex, but little narrowed at base, highly reticulated, with veins, except the midrib, in relief on both sides, 4—9 in. long by 14—3}in. wide; petioles }—}in long; net-veins pellucid when young. g. COymes paniculate, many-flowered, in axils of fallen leaves, pubescent with fuliginous hairs, 1—1} in. long; flowers 2—}in long. Calyx 5-partite shortly nigro-puberulous on both sides, } in. long, lobes foliaceous, widely oval, obtuse, net-veined, with a callous internal keel Mr HIERN, ON EBENACE. 191 and margins widely reflexed. Corolla pentagonal, fuliginous-hairy outside, glabrous inside, 5-fid, constricted in the middle; lobes oval, spreading in flower or reflexed. Stamens 20, glabrous, in pairs, the inner ones rather shorter, inserted on the disk or on the corolla; filaments short; ovary 0. @. Fruit solitary, axillary, on strong peduncles 4—3in. long, erect-patent; bracts caducous, large, ovate, about middle of peduncle; calyx glabrescent; fruit ovoid, 3—1}in. long, rounded at apex, tipped with remains of style, with mixed fuliginous and ferruginous hairs and glands, 3—4-celled ; fruiting calyx 5-lobed, accrescent, 5-partite, }—2in. high, more or less plicate, umbilicate below, lobes much widened auricled and imbricated at base, forming 5 dependent processes. Tallewarru, Canara Ghauts, Dr Ritchie! 1884, a large tree in fruit, May; Syhadree mountains, near Chorla Ghat, Bombay, Dalzell / 2—3000 ft. alt.; Anamallays, Major Beddome/ 285 (young fruit 4-celled, cells 1-ovuled or -seeded, style $in. long, glabrous above, lobed at apex). - 52. DI0sPYROS GRACILIPES, sp. nov. D, foliis alternis, ovalibus vel ovatis, apice seepius acuminatis, obtusis, bast angustatis, glabris, coriaceis, reticulatis, breviter petiolatis; jloribus femineis lateralibus, secus ramos vetustiores vel ramulos dispositis, tetrameris, pedunculis gracilibus, aggregatis, 1-floris, calyce 4-fido, pubescente, ovario breviter pubescente, S-loculari; fructibus oblongis obtusis, calyce fructi- Jfero aucto patente coriaceo. From a shrub 10 feet high to a large tree, glabrous except the extremities and inflo- rescence; branches at 25"—35°. Leaves alternate, oval ovate or nearly oblong, obtuse, usually acuminate at the apex, more or less narrowed at base, glabrous, coriaceous, of the same (metallic) colour on both sides, reticulated, shining, 2—5in. long 3—3in. wide; petioles =4|—1 im. long. @. Peduncles slender, on young branches or clustered on the old wood, 1—1% in. long, puberulous or glabrescent, 1-flowered, with small deciduous bracts below the middle; calyx Zin. long, coriaceous, covered on both sides with close pale tawny pubescence, deeply 4-fid, with ovate-deltoid lobes dilated towards the base undulating at the margin and shortly acuminate; ovary shortly hairy, ovoid-tetragonal, 8-celled, cells l-ovuled; styles 4, short; fruiting calyx spreading, with very short pubescence on both sides, whitened within, 1}—11in. across, 4-fid; fruit oblong, rounded at apex, ;5in. long, 4in. thick, nearly glabrous, whitened in parts, 8-celled. Madagascar, Bojer’; Forest Lomoumé, Nossi Be, Pervillé/! 275; East side, Chapelier! 82; native name Ozou-matana. 53. DIoSPYROS GRACILIFLORA, sp. nov. D. foliis ovalibus, alternis, apice anguste acuminatis, subcaudatis, basi cuneatis, firmiter submembranaceis, costdé utrinque puberuld, ceterum glabris, breviter petiolatis ; floribus masculis solitariis, gracillimis, gracillime pedunculatis, tetrameris ; staminibus 8, glabris, ovaria rudimento glabro. 192 Mr HIERN, ON EBENACE. Branches slender, terete, puberulous, leafy; leaves oval or somewhat obovate, alternate, narrowly acuminate or subcaudate at apex, cuneate at base, shining, firmly submembranous, glabrous except the midrib which is puberulous on both sides and depressed above, 14—44 in. long by 3—1}in. wide; veins inconspicuous above, lateral veins few; petioles 3{—;5 in. long, puberulous. g. Flowers solitary, very slender, 3in. long, on very slender remotely setulose peduncles }2 in. long, which arise from small bracts on the young branches; calyx jin. long, cam- panulate, 4fid, puberulous outside, glabrous inside, ciliate, lobes rounded; corolla narrowly tubular (in bud), 2in. long by 3; in. thick, deeply 4-fid, glabrous, somewhat constricted below lobes ; lobes obtuse, much contorted; stamens 8, biseriate, glabrous, unequal, anthers oblong, filaments more or less connate at base into hypogynous ring; ovary rudimentary, glabrous. Borneo, O. Beccari/ n. 1560. 54. Diospyros PERVILLEI, sp. nov. D. foliis anguste ovalibus, alternis, apice acuminatis, basi cuneatis, glabris, coriaceis, unicoloribus, petiolatis, nervis gracillimis; fructibus 1—3-nis, rigide cymosis, subglobosts, subglabris, nitidis, plurilocularibus, calyce aucto, reflexo, coriaceo, 4-partito, nervoso. A tree 40 feet high, very nearly glabrous in all its parts; branches at about 50°. Leaves alternate, narrowly elliptical, acuminate at apex, narrowed at base, glabrous, coria- ceous, of same (metallic) colour on both sides, shining, about 6in. by 2—2#in. wide; petioles about jin. long, strong; veins numerous, slender. @. Cymes about 3-flowered, rigid in fruit, common peduncle 4—in. long, fruiting pedicels 1—4in. long; fruiting calyx 4-partite, coriaceous, veined, lobes reflexed, oblong, rounded at apex, 3—lin. long by }—%in. wide. Fruit subglobose, 1 in. long by in. thick, nearly glabrate but with a few scattered short appressed weak hairs, shining, with remains of 4 styles at apex, S-celled and -seeded (?); seeds Zin. long, albumen not ruminated (?). Madagascar, Nossi Be, Perwillé/ 525. 55. DIospyROS DICTYONEURA, sp. nov. D. foliis ovali-oblongis, alternis, glabris, apice, acuminatis, basi parum angustatis vel sub- rotundatis, coriaceis, utrinque reticulatis, nitentibus, petiolatis; floribus masculis pentameris, cymosis ; cymis uncialibus, multifloris, axillaribus; calyce partito, bast plicato ; corolld tubulosd, carnosa ; staminibus 20, plerisque binis, antheris linearibus glabris, filamentis brevibus hispidis. Shoots terete, softly puberulous. Leaves alternate, oval-oblong, acuminate at apex, slightly narrowed or slightly subrotundate at base, glabrous, coriaceous, shining, with raised well- marked net-veins on both sides, 6—7in. long by 24—23 in. wide; midrib depressed above ; margins recurved ; petioles stout, wrinkled, 3—} in. long. é. Cymes axillary, lin. long exclusive of the flowers, many-flowered, pubescent ; flowers pubescent, pentamerous; calyx about Zin. broad and high, partite, plicate at base, lobes ovate- deltoid, sides sometimes plicate towards base, subobtuse, shortly pubescent on both sides; Mr HIERN, ON EBENACE. 193 corolla glabrous inside, }in. long, shortly 5-fid, lobes rounded ; stamens 20, mostly in pairs, subequal; anthers linear glabrous, filaments very short, hispid, more or less combined at base. Ovary rudimentary, represented by a bunch of hairs. Borneo, O. Beccari/ 2542, 2615. 56. DIOSPYROS ASTEROCALYX, sp. noy. D. foliis alternis, ovalibus, apice breviter acuminatis, bast obtusis, glabris, coriaceis, subtus conspicue reticulatis, petiolatis; floribus femineis racemosis tetrameris, racemis 5—7-floris, bast bracteatis ; calyce profunde 4-lobo, ferrugineo-velutino, stellato, lobis margine revolutis, corolld urceolatd 4-jida ; staminodiis 3—4; ovario velutino, 8-loculari, loculis 1-ovulatis. Buds and inflorescence ferruginous-velutinous, in other parts glabrate; leaves alternate, oval, shortly acuminate at apex, obtuse at base, coriaceous, conspicuously net-veined beneath, 24—74in. long by 1£—3}in. wide; margins recurved; petioles }—?in. long. Q. Flowers racemose; racemes 1—2} in. long, pedicels patent, unequal, ranging up to $in. long, the lower ones the longer. Calyx thickly coriaceous, deeply 4-lobed, stellate, 3— in. in diameter; lobes widely ovate but much revolute. Corolla widely urceolate, under } in. high, 4-fid, lobes hairy on both sides, obtuse; staminodes 3 (in one case), glabrous, inserted at base of corolla, alternate with its lobes. Ovary velutinous, ovoid, conical at apex, 8-celled, cells 1-ovuled ; style very short, lobed at apex, velutinous; stigmas glabrous. Borneo, O. Beccari/ n. 2612. 57. Diospyros HORSFIELDI, sp. nov. D. foliis alternis, ovalibus vel oblongis, apice acuminatis, basi subrotundatis vel obtusis, glabrescentibus, tenuiter coriaceis, supra nitentibus depresso-venosis, subtus reticulatis, breviter petiolatis ; cymis lateralibus vel azxillaribus, fuligineo-hispidis, calyce plicato, 4-lobo, corolld urceolatd 4-lobd, staminibus 14—16 (in fl. fem. 12, sterilibus), antheris glabris, filamentis hispidis, ovario in floribus femineis dense hispido, 8-loculari ; fructibus globosis. Diospyros frutescens, Hasskarl, Plant. Javan. Rar. p. 467 (1848), non Blume. Branches numerous, terete and glabrous, spreading at about 70°, green when young, afterwards turning black. Leaves oblong or elliptical, alternate, soon quite glabrous, acuminate at apex, somewhat narrowed or nearly rounded at base, with veins plainly depressed on upper surface and in conspicuous relief beneath, shining above, thinly coriaceous, 4—9} in. long by 1$—4} in. wide; petioles }—} in. long. &. Cymes chiefly in the upper axils, fuliginous-hispid, bearing 3—5 flowers, drooping ; peduncles ;3,—# in. long; pedicels j;—1in. long; bracts oval, leaf-like; flowers }—} in. long; calyx 41—1in. long, 4-lobed, lobes ovate, plicate-connivent, thickened and fuliginous- hispid on both sides over a lanceolate area proceeding from base to above the middle and with broad membranous everted glabrous and green margins; corolla urceolate, tetragonal, fuliginous-hispid outside, straw-coloured and glabrous inside, 4-lobed, lobes ovate, rather obtuse, Vou. XII. Parr I. 25 194 Mr HIERN, ON EBENACE. reflexed; stamens 14—16, inserted at the base of the corolla or on the disk, often in pairs united by their short hairy filaments; anthers glabrous; ovary rudimentary, minute. 9. Cymes corymbose, many-flowered, 1—3 in. long, frequently on older branches, bracteate, fuliginous; flowers 2—4in. long; calyx 4—}in. long, like g but occasionally 5-partite; corolla tetragonal, 4-partite; staminodes 12, in one row, attached by their hairy filaments to base of corolla, anthers glabrous; styles 4, short, spreading; ovary densely hispid, with black and rufous mixed hairs, 8-celled; cells l-ovuled. Fruit globose, with a central pit at apex around remains of styles, about 4— in. in diameter, black-hairy or nearly glabrescent; fruiting calyx reaching about }in. up fruit, lobes auricled at base. Malacca, Griffith! 3620; Java, Dr Horsfield! Eben. 1 (1182) drawings n. 128 (pt.) im Hb. Kew. ; Leschenault/ 1669; Perrottet ! 58. Drospyros Borvini, sp. nov. D. foliis alternis, ovato-lanceolatis vel-oblongis, apice obtuse acuminatis, basi cordatis, sub- glabris, subcoriaceis, breviter petiolatis; floribus masculis laxe cymosis, tomentoso-pubescentibus, tetrameris, calyce campanulato, corolla 4-fidd, lobis late rotundatis, staminibus 12—14, glabris, plerisque geminatis, ovariu rudimento pubescente. Young branches and inflorescence ferruginous-pubescent ; shoots terete, shining, rather dark. Leaves alternate, ovate-lanceolate or -oblong, rather obtusely acuminate at apex cordate at base, subcoriaceous, shining brown and nearly glabrous above with somewhat sunken veins, rather paler and nearly glabrous beneath with somewhat ruddy raised mid- rib and clear but not close net-veins, 24—6} in. long by 1—22 in. wide; petiole ;,—$ m. long, thick, pubescent. 6. Cymes lax, many-flowered, near ends of branches, }—2 in. long, shortly hispid- pubescent, ferruginous ; bracts lanceolate; flowers campanulate, $ in. long, tin. wide, tetra- merous, tomentose-pubescent; calyx nearly 2 in. long, campanulate, shortly 4-lobed or occasionally deeper, lobes depresso-deltoid, somewhat wavy; pubescent on both sides; corolla just exceeding the calyx, 4-fid, ferruginous-velutinous outside, glabrous within, lobes widely rounded, contorted sinistrorsely ; stamens (12 ex Baillon in note) 14! (in 2 flowers), glabrous, mostly in pairs, nearly equal, inner ones rather shorter, 4 in. long, anthers oblong-linear, din. long, dehiscing laterally ; ovary rudimentary, pubescent. Madagascar, Voyage of DL Boivin! 1847—1852. 59. Ditospyros Lourerrtana, G. Don, Gen. Syst. Gard. and Bot. iv. p. 39. n. 22 (1837). D. foliis alternis, oblongis vel obovato-oblongis, apice plus minus acuminatis, basi rotun- datis vel subcordatis, glabrescentibus, ciliatis, submembranaceis, supra saturate- subtus flaves- centi-viridibus, petiolatis ; pedunculis axillaribus sub-3-floris, glanduloso-pubescentibus, pedicellis bast bracteis foliaceis ovatis glandulosis deciduis suffultis; calyce 4-fido in fructu aucto, corolla urceolatd 4-lobd, staminibus 8 uniserialibus pilosis in fl. fem. effetis, ovario in jl. fem. 8- Mr HIERN, ON EBENACE. 195 loculari, tomentello, stylis 4; fructibus globosis uncialibus, seminibus oblongis, albwmine non ruminato. Alph. DC. Prodr. vit. p. 239. n. 95 (1844). Diospyros Lotus, Lour. Fl. Cochin. p. 226. n. 1 (1790), non Linn. nec Blanco. Diospyros macrocalyx, Klotzsch in Peters Mossamb. p. 182 (1862), non Alph. DC. A shrub 2—8ft. high or small tree with young parts and inflorescence glandular- puberulous and with a few scattered pilose hairs. Leaves alternate, oblong or obovate- oblong, submembranous, weakly pubescent on the veins and ciliate on the margins when young, glabrescent, obtuse rounded or subcordate at base, more or less acuminate at apex, 1} to 4im. long by 3—2,3,in. wide, besides petiole }—%,in. long; flowers subhermaphrodite or polygamous, drooping; calyx foliaceous. é. Cymes 3- or few-flowered, glandular-hairy ; peduncles +—},in. long, twice the length of the pedicels, bearing ovate cordate sessile bracts at apex; flowers about 1 in. long; calyx green, about }in. long with 4 deltoid lobes about #;in. deep, glandular-pubescent (closed in specimen), valvate in estivation; corolla deeply 4-lobed, somewhat pubescent outside, urceolate, white ; lobes contorted in zestivation’; stamens 8, in one row, inserted at base of corolla, subsessile, pilose, lanceolate ; ovary ovoid-conical or subglobose, puberulous, abortive or 8? -celled, sur- mounted by a 4-lobed style. 9 Cymes about 3- or many-flowered, about 2—1 in. long, glandular-hairy; peduncle about iin. long; flowers like the g ; staminodes 8, puberulous; ovary globose, shortly tomentose, 8-celled, cells l-ovuled; styles 4, included in the corolla; fruit globose, about lin. in dia- meter, puberulous or glabrate, 4-celled, 4-seeded. Fruiting calyx accrescent, deeply 4-lobed, more or less covering the fruit, about lin. long; lobes ovate, subglabrate, dilated and widely subcordate at base. Fruiting peduncle strong, {—}in. long; pedicels about 1in. long; seeds lin. long, oblong, embryo fin. long; cotyledons narrow, rather longer than the radicle; albumen cartilaginous, not ruminated. Local name in Sena (Mozambique) nhamodéma, according to Dr Klotzsch. The natives use the roots to clean and dye their teeth red; fruits in January and February; grows in the neighbourhood of Sena, Dr Peters/; Senna, Kirk! ; Rovuma River, Shiramba, Kirk! ; between Lupata and Tette, Airk/; Quiloa, Kirk! ; Congo, Burton /; Angola, district Golungo Alto, Welwitsch! No. 2535, frequent in thickets throughout the whole district, especially in mountainous woods, fruit said to be edible; var. vernalis, leaves $—2in. long by 1—3 in. wide, flowers solitary on shorter peduncles, fruiting calyx smaller, less foliaceous, a shrub 2—6 ft. high, Angola, district Golungo Alto, Welwitsch 25356. The characters approach those of the genus Royena. A specimen in the herbarium of the British Museum without flowers from Sierra Leone gathered by Afzelius! may possibly belong to this species. 60. Driospyros DENDO, Welw. MSS. D. foliis alternis, ovali-oblongis, apice acuminatis, basi leviter angustatis, tenuiter coriaceis, glabrescentibus, nitido-virentibus, persistentibus ; floribus brevissime cymosis, axillaribus, 5—6- 25—2 196 Mr HIERN, ON EBENACEAS. meris, diacis, calyce campanulato, utringue pubescente, 3 5—6-fido, 2 profunde lobato ; corolla aperte campanulatd, glabra, g 5—6-fidd, lois reflexis, 9 profunde 5—6-fida; 6 sta- minibus 20 vel 24, exsertis, subequalibus, geminatis, corolla medio inserts, pubescentibus ; Q staminodiis 0, ovario ovoideo, glabro, 4-locularibus, loculis 1-ovulatis ; fructibus subglobosis, glabris, 2-spermis ; seminibus sub-hemisphericis, albumine non ruminato ; calyce fructifero aucto, patente. Plate X. a. a male flowering branch, natural size. b. a male flower, magnified 3 dia- meters. c. a male corolla laid open, shewing the stamens, magnified 3 diameters. d. a pair of stamens, magnified 6 diameters. e. a female flowering branch, natural size. f. a female flower, magnified 3 diameters. g. the same after the removal of the corolla, mag- nified 3 diameters. h. a vertical section of the last, shewing ovules inside the ovary, magnified 4 diameters. i. a fruiting branch, natural size. k. a fruit, natural size. l.m. a seed, natural size, n. transverse section of a seed, natural size. o. embryo, magnified 6 diameters. A tree 25—35 feet high, valuable as timber. Wood very black and hard in the centre. Trunk 1—2 ft. in diameter. Branches terete, smooth, of dark brown colour, glabrescent ; young parts shortly and closely fulvo-pubescent. Leaves alternate, elliptic-oblong, shortly and obtusely acuminate at apex, slightly or scarcely narrowed at base, thinly coriaceous or sub- membranous, darker above, shining, glabrescent or midrib and sometimes principal veins puberulous on both sides, midrib depressed above; evergreen, 2—5}in. long by 1—23in. wide; petioles 1—1in. long, puberulous; principal lateral veins distant, clear and slender beneath, inconspicuous above, arching; tertiary veins transverse, slender. Internodes much shorter than the leaves. Inflorescence axillary or slightly supra-axillary, shortly and closely fulvo-pubescent, in short clustered several-flowered cymes. Flowers 5—6-merous, dicecious ; pedicels short. g. Flowers tin. long; calyx j;in. long, campanulate, 5—6-fid, shortly pubescent on both sides, lobes ovate; corolla glabrous, 5—6-fid; tube campanulate; lobes jin. long, elliptical, wholly reflexed, rounded at apex, contorted sinistrorsely in wstivation, Stamens 20, 24, ap- pearing at the mouth of the open corolla, equal or subequal, biseriate, distinct, one pair inserted alternate and another pair opposite to each corolla-lobe; inner series inserted slightly below the outer about the middle of the corolla, that is, about the top of its tube; anthers linear, erect, hairy, sessile or subsessile; pollen globular, smooth. Ovary rudimentary, glabrous. 2. Flowers }in. long. Calyx campanulate, deeply 5—6-lobed, shortly pubescent on both sides; lobes ovate-lanceolate; accrescent in fruit. Corolla openly campanulate, glabrous or nearly so, deeply 5—6-fid; lobes oblong, erect or spreading, obtuse. Staminodes 0. Ovary glabrous, obtusely conical, 4-celled, bilobed at apex; cells l-ovuled. Style 0; stigmas 2, com- pressed, with thin margins. Fruit subglobose, glabrous, about }in. in diameter, 2-seeded. Seeds sub-hemispherical, }in. in diameter; albumen white, not ruminated, cartilaginous; embryo axile, }in. long, nearly straight; radicle j5in. long, bent near upper end; cotyledons ovate, equal, thin, not veined. Fruiting calyx spreading, 1—1}in. across, puberulous; lobes ovate or lanceolate, subobtuse. See Welwitsch, Synopse das Amostras de Madeiras &c. p. 10 (1862). Mr HIERN, ON EBENACE. 197 W. Tropical Africa, Angola, Distr. Golunto Alto, frequent in dense primitive woods, flowers from December to February, fruits in March, Dr Welwitsch! nos. 2537, 2588. Native name Dendo or N-Dendo. 61. Drospyros (?) Cunaton, Alph. DC. Prodr. vi. p. 237 n. 79 (1844). D. foliis alternis, late lanceolatis, apice obtusis, glabris, brevissime petiolatis, margine revolutis ; floribus breviter racemoso-cymosis, calyce campanulato, lobis 4 rarius 5 rotundatis, corolle lobis 4 profundis acutis, staminibus 8, corolle adnatis, 4 basi, 4 medio loborum; ovario globoso, stylis 2; baccis globosis, 4-locularibus, loculis monospermis. (Cunalon), Blanco, Flora de Filipinas pp. 304, 305 (1837). A tree with erect and branching trunk. Leaves alternate, broadly lanceolate, obtuse at apex, glabrous; the margins entire and reflexed; petioles very short. Flowers in small racemose panicles. Calyx free, persistent, campanulate, with 4 or rarely 5 rounded lobes. Corolla longer than the calyx, with 4 deep acute lobes. Stamens 8, inserted on the corolla, 4 at the base and the other 4 at the middle of the lobes; filaments shorter than the corolla, compressed; anthers erect, acute. Ovary globose, enclosed within the flower; styles 2, linear, compressed; stigmas simple. Fruit baccate, globose, juicy, 4-celled; cells 1-seeded ; seeds oblong, convex and canaliculate outside, angular inside, very hard and horny, and “covered with a thin aril.” Cebu, Philippine Islands, Blanco, loc. cit. The leaves and fruit turn very black at maturity and are used by the islanders to dye cloth. The black colour produced is good and fast and without notable smell. Flowers in October. Called Cunalon in Bisayas, Philippine Islands. 62. DtIospyROS TETRASPERMA, Sw. Prodr. p. 62 (1788). D. foliis alternis, anguste obovatis, apice obtusis, bast cuneatis, glabris, subcoriaceis, breviter petiolatis; floribus masculis 3—4-nis, breviter cymosis, calyce campanulato, subglabrescente, 4- rarius 5-fido, corollé tubulosd, extus sericed, breviter 4-fidd, staminibus 8, glabris, gemi- natis ; floribus femineis solitariis, staminodiis 4, ovario conico, pubescente, 4-loculari, loculis l-ovulatis, fructibus globosis, glabris, seminum albumine “radvato-striato quasi fibroso, car- noso, albo.” Fl. Ind. Occ. p. 678 (1800), Gaertn. f. Carp. iii, p. 138. t. 208 (1805), Alph. DC. Prodr. Vill. p. 222. n. 1 (1844). D. obovata, Jacq. Hort. Scheenbr. iii p. 34 t. 312 (1798), non Wight. A shrub glabrous except the inflorescence and young parts; stem lin. thick; branches pale, at about 40°; shoots slender, subvelutinous. Leaves alternate, oblanceolate-oblong or obovate, subcoriaceous, the younger ones sometimes pellucid-punctate, cuneate at base into short petiole, rounded or obtuse at apex, deep green above, paler beneath; veins raised on both sides; 14—3in. long by }—1lin. wide; petioles =;—1in. long. g flowers in 3—4-flowered cymes; cymes recurved, }in. long, with short appressed hairs. Flowers about 41in. long. Pedicels very short. Bracts small, caducous. Calyx about 198 Mr HIERN, ON EBENACEA. in. long, green, nearly glabrescent, campanulate, 4—5- usually 4-fid; lobes deltoid or rounded. Corolla tubular, pale with appressed short hair outside, with 4 spreading obtuse lobes half the length of the tube. Stamens 8, distinct, 2 alternating with each corolla-lobe, the inner ones being shorter and inserted at very base of corolla-tube, or hypogynous, the outer ones longer with filament and anther about equal and inserted rather above base of corolla tube, or hypogynous; all glabrous. Ovary rudimentary, with short hairs. 2 flowers solitary, on erect peduncles about j4in. long; calyx and corolla as in ¢ ; staminodes 4, alternating with corolla-lobes and inserted at base of its tube; ovary conical, hairy, +-celled, 4-ovuled, continuous with hairy style which is 4-lobed and glabrous at apex. Fruit globose, about tin. thick, pale, glabrous, 4-celled, 4-seeded. Fruiting calyx 4—5-fid, not or scarcely accrescent, concave or somewhat spreading, glabrous. Fruiting peduncle j,—1in. long, patent; seeds tin. long; testa rather rough; albumen not ruminated, but somewhat striated in a radiated manner. Jamaica, Mr March! No. 1190; Purdie! (g and @ fl. and fr., October); Swartz, é fl. July; St Domingo, Jacguin, 3 fl. May; Cuba, teste Grisebach (the specimen Pl. Cub. Wright, n. 348, has a somewhat different foliage and fruit-calyx). 63. Diospyros CARTHEI, sp. noy. D. foliis alternis, elliptico-oblongis, utrinque obtusis, glabris, coriaceis, petiolatis ; floribus masculis sub-5-nis, subsessilibus, confertis, axillaribus, tubulosis, ferrugineo-pubescentibus, 4—6- fidis, calyce campanulato, corolld gracili; lobis obtusis, staminibus 8, inequalibus, ovarit rudimento piloso. Glabrous and dark except inflorescence and buds; branches terete. Leaves elliptic- oblong, alternate, coriaceous, not pellucid-punctate, of same colour on both sides, 4—4 in, long by 18—1Zin. wide; petioles 3in. long, spreading. g. Flowers about 5 together, subsessile, crowded, axillary, tubular, slender, }—?in. long, ferruginous-pubescent, the colour greenish beneath the hairs; calyx {in. long, cam- panulate, 4—6- (5—6!) -fid; lobes lanceolate. Corolla 4-fid, slender, {—{in. long, ferru- ginous-hairy outside, constricted in midrib; lobes imbricated, obtuse. Stamens 8, unequal by shorter or longer filaments, glabrous, ;4;—tin. long, anthers dehiscing longitudinally along their sides; pollen ellipsoidal. Ovary rudimentary, represented by hairs. Manila, Philippine Islands, Carthe/ 64. DIoSPYROS POLYALTHIOIDES, Korthals MSS. in Hb. Ludg. Batay. Eben. nn. 5—9, 12—14, D. foliis alternis, oblongis, apice acutd acuminatis, basi obtusis, tenuiter coriaceis, supra glabris, subtus subglabris ; floribus masculis, aggregatis, breviter cymosis, axillaribus, oblongis, sericeis, calyce campanulato, 4—5-fido, corolld tubulosd, breviter 4-fidd, lobis obtusis patentibus, staminibus 8, glabris, receptaculo insertis, ineequalibus ; floribus femineis axillaribus, breviter cymosis; fructibus subsolitariis, breviter pedunculatis, globosis, pubescentibus, 8-locularibus ; calyce fructifero aucto, profunde 4-lobo, ampliato, lobis undulatis, latis, erectis. Mr HIERN, ON EBENACE. 199 Diecious. Shoots ferruginous-pubescent, terete. Leaves oblong, alternate, obtusely nar- rowed or nearly rounded at base, acutely acuminate at apex, thinly coriaceous, glabrous and rather shining above except the depressed midrib, nearly glabrous beneath except the midrib and weak siender lateral veins, 6—8 in. long (besides hairy petiole -3,—% in. long) by 13—2} in. wide; margins just recurved; lower surface somewhat red; not pellucid- punctate; a few dark depressed glands usually exist on the lower surface, especially near the base and in the fruiting specimens. ¢. Cymes axillary, many-flowered, sericeous-ferruginous, ;;— 3; in. long (excluding the flowers; pedicels about ; in. long; bracts small. Flowers sericeous, about }in. long in bud, crowded. Calyx nearly } in. long, campanulate, 4—5-fid; lobes deltoid or oval, hairy on both sides; corolla shortly 4-fid, tubular; lobes obtuse, much imbricated in bud, oval, 4 in, long; glabrous inside, spreading; tube constricted at the top. Stamens 8, glabrous, inserted on the receptacle, unequal, combined more or less by their filaments at base; anthers linear, acute (when young), longer than their filaments. Ovary 0. Rarely a flower is trimerous. 2. Cymes axillary, about } in. long, sericeous-ferruginous, bearing 3—many flowers; bracts caducous; pedicels }in. long. Calyx plicate, 2 im. high, longer than the corolla. Flowers 4—5-merous. Fruit subsolitary, on peduncles }—{ in. long, enclosed when young by accres- cent deeply 4-lobed calyx; fruit globose, ferruginous-hairy, about } in. in diameter (perhaps not mature), 8-celled, (8-ovuled), 8-seeded. Pericarp rather thick ; dissepiments thin. Fruiting calyx ?in. high, deeply 4-lobed, hairy on both sides; ample at the sinuses; lobes widely ovate with margins wavy, wide at base. Borneo, Korthals / Plate VII. A branch in male flower, natural size. a. Calyx laid open and stamens, the corolla having been removed, magnified 3 diameters. 06. A branch in young fruit, natural 8vZé. 65. Diospyros KiRki, sp. noy. D. foliis ovalibus, alternis, utrinque rotundatis, coriaceis, velutinis, petiolatis; floribus masculis axillaribus, breviter cymosis, 4- rarius 5-meris, calyce campanulato, sepius 4-fido, corolla tubulosd, breviter 4-lobd, staminibus 9—10, glabris, inequalibus ; floribus femineis soli- tariis, breviter pedunculatis, staminodiis 8, ovario globoso, 4-loculari, fulvo-tomentoso, loculis l-ovulatis ; fructibus edulibus. A fruit-tree with young shoots ferruginous-tomentose-puberulous; branches cinereous glabrescent, terete. Leaves elliptical or oval-oblong, alternate, coriaceous, rounded at both ends; velutinous-puberulous, sub-nitescent above with delicate slightly raised veins; velutin- ous-pubescent fulvous beneath with raised rufous midrib and lateral veins; 14—4 in. long by $—21 in. wide; petioles hairy +—1} in. long. g. Inflorescence axillary, in several-flowered cymes, rufous-tomentose, raised on pe- duncles about 3 in. long, with short pedicels, bracteate; flowers -3, in. long, tetramerous or rarely pentamerous; calyx , in. high ferruginous-velutinous outside, appressedly hairy inside, 4-fid, campanulate, rarely with 5 unequal lobes; corolla inflated-tubular, with 4 short ovate patent lobes, glabrous inside; stamens 9, 10, glabrous, inserted at base of corolla or on recep- tacle, unequal, on short filaments; ovary 0. 200 Mr HIERN, ON EBENACE. : @. Flowers solitary, on short peduncles, ,% in. high; fulvo-velutinous ; calyx 4—5-lobed, 2in. long; with lanceolate erect lobes ;3—,; in. deep, hairy on both sides; corolla trun- cately conical, with 5 (or 4?) very short spreading obtuse lobes, glabrous inside; staminodes 8, inserted at base of corolla and 1 on receptacle (in flower examined), glabrous; ovary fulvous-velutinous, globular, 4-celled with 2 styles hairy at base; cells l-ovuled; stigmas glabrous, lobed; young fruit fulvo-velutinous, with calyx-lobes appressed or erect; pulp of fruit good when made into a cake. Africa, Zambesia, aboye Tette, common. Dr Kirk! 66. DrIospyROS VELUTINA, sp. nov. D. foliis alternis, ovalibus vel oblongis, coriaceis, subtus fulvo-velutinis interdum pubescen- tibus et pellucido-punctatis, petiolatis; floribus masculis ternis, breviter cymosis, 3—4-meris, ferrugineo-hirsutis ; calyce campanulato, 3—4-fido, lobis obtusis, corolld tubulosd, 3—4-lobd, staminibus 12, glabris, inequalibus; floribus femineis solitariis breviter pedunculatis, calyce 3—5-lobo, corolld 4-lobd, ovario dense fulvo-sericeo, subgloboso, 8-loculari; stylis 4; fructibus globosis, albumine non ruminato. A diccious shrub about 6 feet high or small tree; shoots, leaves especially on the under-side, and inflorescence ferruginous-velutinous; branches glabrescent, terete, shining, spreading at about 60°. Leaves oval or oblong, somewhat narrowed (sometimes acutely), obtuse, rounded or even cordate at either or both ends, coriaceous, shining and comparatively gla- brescent above with (in some specimens) more or less depressed veins, densely ferruginous- velutinous beneath, or in some specimens becoming less hairy and then with small pellucid dots, alternate, 1}—6 in. long by {—2} in. wide; petioles }—4 in. long, ferruginous-velutin- ous. Inflorescence short, axillary, ferruginous-velutinous; bracts narrow, caducous. " g. Flowers usually 3 together, on peduncles ;4,— , in. long, trimerous or the central ones tetramerous, about } in. long; calyx 1 in. long, ferruginous-velutinous outside, glabrous inside, 3- or 4-fid, with obtuse lobes; corolla ferruginous-sericeous, 3- or 4- lobed, tubular, nearly } in. long, lobes 1—1 in. deep, oval, glabrous inside, spreading; stamens 12, glabrous, some in pairs, unequal in the pairs, the inner ones the shorter; filaments short, anthers linear-oblong; ovary ferrugineous-hairy, rudimentary. ©. Flowers and peduncles solitary, ferruginous, hairy; peduncles $—,5;in. long; flowers 2in. long; calyx 4, im. long by } im. wide, 3—5-lobed; lobes } in. deep by } in. wide, fer- ruginous-tomentose on both sides, rounded or deltoid, cordate at base, with undulating sides, often emarginate at apex, with central boss inside near base; corolla shortly tubular with 4 short acute spreading lobes; staminodes 2 (in one case), glabrous; ovary densely fulvo- sericeous, subglobose, 8-celled, with a short neck terminated by 4 styles; cells 1-ovuled; stigmas emarginate. Fruit globose, shining, pale, glabrate, except at the apex, about 5-celled and 5-seeded, pulpy, }—} in. thick; seeds about gin. long, enveloped in pulp considered by Mr Miers, in Ann. Mag. Nat. Hist. ser. ii. vol. vitt. p. 164 (1851), to be of the nature of an aril, not however in the dried state suggesting such an origin; fruiting calyx 3—4- lobed, spreading, tomentose, ;4—} in. across; lobes more or less emarginate, especially in the trimerous ones; albumen horny, not ruminated, but (in some specimens) obscurely striate in a radiating manner. Mr HIERN, ON EBENACEA, 201 Brazil, Rio de Janeiro, Jurujuba Bay, Mr Miers! 3709; Serra de Araripe, Gardner ! 1512 (g fl. Sept.); between Franqueira and Canariera, Gardner! 2284 (albumen radiately striate, fruit in March); New-Granada, Prov. Mariquita, Piedros, banks of Magdalena, 1300 ft. alt. Triana! 2612; Mexico, Carmen and neighbourhood, Dr Warra! 226 (plant in young fruit with acute leaves and calyx 3-fid having pointed lobes). Possibly 2 or 3 different species are here described together. Cfr. Maba inconstans, Griseb. which is like this plant in some states. 67. DIOSPYROS PLECTOSEPALA, sp. nov. D. folis alterns, ovalibus, apice acuminatis, bast angustatis, subglabris, tenuiter coriaceis, breviter petiolatis ; floribus masculis brevissime cymosis, axillaribus, pentameris, hirsutis, brac- teatis, campanulato-oblongis, calyce profunde lobato, lobis rotundis valde contortis, corolle lobis ovalibus obtusis, stanvinibus 12 glabris inequalibus, ovarti rudimento hirsuto. Branches terete, sparsely hispid with mixed brown and black short hairs. Leaves alternate, oval, acuminate at apex, somewhat narrowed at base, thinly coriaceous, scattered especially beneath with a few inconspicuous appressed short stiff hairs, 1}—44 in. long by 4—12 in. wide, dark green above; lateral veins few, delicate; petioles }—+4 in. long, hispid. 6. Flowers few or several together, im very abbreviated hispid axillary cymes, penta- merous, $ in. long, campanulate-oblong; bracts small. Calyx deeply 5-lobed, scarcely half the length of the flower, hirsute outside, glabrous inside, lobes round, much imbricated, cordate at base. Corolla densely hirsute outside with pale appressed hairs, glabrous inside, 5-fid; lobes oval, obtuse. Stamens 12, glabrous, unequal, hypogynous or inserted at very base of corolla. Ovary minute, rudimentary, hairy. Borneo, O. Beccari/ n. 3225. 68. Dtospryros stricta, Roxb. Cat. Pl. Fl. Ind. p. 93 (1818). D. trunco stricto, apice tantum ramoso; foliis alternis, ovato-oblongis, apice valde acumi- natis, basi subrotundis, submembranaceis, ciliatis, subtus sparse pubescentibus, breviter petiola- tis; cymis masculis brevissimis, 3—6-floris, bracteatis, floribus subsessilibus, 4-meris, hirsutis, calyce parvo, profunde lobato, corollé urceolato-oblongd, staminibus 14—16, glabris ; fructibus solitariis, breviter pedunculatis, obovoideis, bast conicis, glabris; seminibus oblongis, albumine non ruminato. Roxb. Hort. Beng. p. 40 (1814); Fl. Ind. edit. 1832, 11. p. 589. n. 14; Drawings no. 2507 in Hb. Kew; Wall. List n. 4121 (1828—32); Alph. DC. Prodr. vim. p. 232. n. 47 (1844). A tall slender conical tree with a trunk perfectly straight, as in firs, to the very top; branches spreading at 40°, terete; young shoots subtomentose, covered with dull tawny patent , short hairs, glabrescent. Leaves ovate-oblong, much acuminate at apex, obtuse at base, sub- membranous, alternate, erect-patent, pubescent beneath, ciliate, glabrous above except on the midrib, 2—34 in. long by about 1 in. wide; petioles about } in. long, pubescent; veins incon- spicuous especially on upper face. Vou. XII. Part I. 26 202 Mr HIERN, ON EBENACE. $. Flowers } in. long, 3—6 together, crowded and subsessile on short pubescent cymes about the length of the petioles, tetramerous. Bracts numerous, hairy, at base of very short pedicels. Calyx tawny-hirsute outside, small, ;4, in. long, with 4 deep ovate apiculate lobes, glabrous inside. Corolla salver-shaped, 3, in. long, tawny-hirsute, much contracted towards top of tube; tube inflated below, 3 in. long; lobes oval, patent or reflexed, shorter than the tube. Stamens 14—16, glabrous, single, about half the length of the corolla-tube, most in- serted in one row at base of corolla and nearly equal, some inserted on the disk; filaments about as long as the anthers. Receptacle convex. @. Fruit solitary, on patent peduncles which are about } in. long and thicker towards the apex and continuous with sinall tawny-hairy shortly 4-lobed calyx. Fruit egg-shaped but somewhat conical towards base, 1} in. long by ;% in. thick, unequally 4?-celled, glabrous. Seeds oblong, albumen not ruminated. East Bengal, Tipperah, Roxburgh (4 fl. March); Griffith! 3624 (in fruit); Chittagong, Drs J. D. Hooker and T. Thomson!; Silhet, &e. Roxburgh, Hort. Beng. p. 40. 69. DrospyRos ERIANTHA, Champ. in Kew Journ. Bot. Iv. p. 302 (1852). D. foliis distichis, oblongo-lanceolatis, apice acuminatis, basi obtusis, tenuiter coriaceis, supra nitidis, subtus secus venas pilosis, breviter petiolatis; floribus masculis 1—3-nis, axil- laribus, subsessilibus, bast bracteatis, tetrameris, hirsutis, calyce profunde lobato, corolld hypocra- ; teriformi, lobis lanceolatis, acuminatis, patentibus, staminibus 14—16, glabris; floribus femineis solitariis, staminodus 8, wuserialibus, glabris, ovario villoso, 4-loculari, loculis 1-ovulatis ; fruc- tibus oblongis, subglabratis, monospermis, albumine non ruminato. Benth. Fl. Hongkongens. p. 210. n. 2 (1861). A small tree, with young shoots; margins, mid-rib and lateral veins of underside of leaves and inflorescence covered with stiff appressed rusty pubescence; branches spreading at about 35°, glabrescent, terete. Leaves oblong-lanceolate, much acuminate at apex, obtuse or nearly rounded at base, distichous, thinly coriaceous, shining and with slight depressed inconspicuous midrib and lateral veins above; ruddier and with raised and rather conspicuous midrib and lateral veins beneath; 24—44 in. long by 3—1} in. wide; petioles };—% im. long, pubescent when young. Bracts much imbricated, numerous, especially in @, concealing the very short peduncle and young flowers, pubescent when young, wide, rounded or obtusely narrowed. g. Flowers subsolitary, 1—3 together, axillary, not nodding, subsessile, tetramerous, in. long. Calyx deeply 4fid, } in. long, with lanceolate hirsute lobes. Corolla tubular, salver-shaped, hirsute outside, glabrous inside, 4-lobed, white; tube ;';in. long; lobes #; in, long, spreading, acuminate, lanceolate, imbricated sinistrorsely. Stamens 14—16, inserted in pairs at base of corolla, glabrous; anthers acuminate; the interior filaments shorter, the outer ones longer. Ovary rudimentary, small. 2. Flowers solitary, subsessile, tetramerous; calyx 3; in. long, like ¢. Corolla equal- ling the calyx; lobes acute. Staminodes 8, glabrous, in one row. Ovary hairy, 4-celled ; cells 1-ovuled; style bifid to the middle with contiguous emarginate lobes, glabrous except at base. Fruit glabrate or nearly so, oblong, about 4 in. long, shining, 1-seeded. Fruiting Mr HIERN, ON EBENACEA:. 203 calyx 2 in, long, with apiculate lobes, somewhat spreading. Albumen not ruminated ; embryo straight. Hong Kong, C. Wright! 64; in the Happy Valley woods, Champion! 133, 147; Borneo, Korthals / D. Teysmanni, Mig. in Fl. Ind. Bat. Suppl. 1. pp. 250, 583 (1860), belongs to the above species; it however differs by rather smaller leaves with nearly or quite glabrous lateral veins and with the upper surface paler than in the above species. Local name Kajoe-ngingeh. Near Kabagoesan on the coast in Lampong, 8S. Sumatra, Zeijsmann / 70. Dtospyros VARIEGATA, Kurz in Journ. Asiat. Soc. Beng. vol. XL. pt. ii. p. 73. n. 95 (1871). D. foliis oblongis, acutis vel acuminatis, tenuiter coriaceis, glabris, petiolatis ; floribus mas- culis tetrameris, ternis vel paucis, in cymis axillaribus breviter pedicellatis, calyce puberulo, lobis late oblongis obtusis, corolle tubo quam calyce paulum longiore, lobis ovatis acutis tubi longitudine, staminibus circiter 16 inegualibus, antheris glabris. Flora, 1871, p. 342. A moderate-sized tree, quite glabrous except the buds. Leaves varying from elliptic- oblong to oblong, usually rather unequal and but little narrowed at base, acute or acuminate, entire, 5—10 in. long, thinly coriaceous, glabrous; petioles 1—} in. long, crass; lateral veins prominent below; net-veins rather distant and conspicuous beneath. g. Flowers yellow, tetramerous, in bud 1—,% in. long, elongated, very shortly pedicelled, 3 or few together, in axillary shortly-stalked minutely puberulous bracteated cymes, on young usually leafless shoots, simulating racemes; bracts wide, rather acute, puberulous. Calyx puberulous; lobes widely-oblong, obtuse, about }in. long. Corolla urceolate (-oblong?); tube a little longer than the calyx; lobes ovate, acute, equalling the tube. Stamens about 16, unequal, inserted at the base of the corolla; filaments short; anthers linear, cordate at the base, acuminate, glabrous. Peeu, Dr Brandis ! 71. DIospyROS DASYPHYLLA, Kurz in Journ. Asiat. Soc. Beng. vol. XL. pt. i. p. 71. n. 92. (1871). D. foliis oblongis vel ovali-oblongis, apice acutis vel breviter acuminatis, basi rotundatis vel subcordatis, chartacets, secus nervos puberulis, breviter petiolatis ; floribus masculis tetrameris, in cymis brevibus fulvo-pubescentibus axillaribus vel supra foliorum delapsorum cicatrices erum- pentibus dispositis, calyce partito, lobis rotundatis, corolla tubulosd, paulum ampliatd, staminibus circiter 16, filamentis valde inequalibus, ovarii rudimento fulvo-hirsuto. Flora, 1871, p. 333. A tree (?) with branchlets densely tawny-pubescent. Leaves varying from oblong to oval-oblong, on petioles #;—} in. long, densely tawny-pubescent, rounded or subcordate at base, acute or shortly acuminate, 4—6 in. long by 14—3 in. wide, chartaceous, with long cilia when young, afterwards softly puberulous on the veins above and below. 26—2 204 Mr HIERN, ON EBENACE. g. Flowers in bud nearly 4 in. long, tetramerous, shortly pedicelled, arranged in short tawny-pubescent cymes, axillary or above the scars of fallen leaves; bracts suborbicular, pu- berulous, ciliated, about ;4;in. long. Calyx ferruginous-pubescent, lobed almost to the base; lobes rounded, ciliated. Corolla-tube appressedly tawny- or ferruginous-pubescent, } in. long, widely tubular; corolla-lobes equalling the tube, acute, oblong, canescent-velutinous outside. Stamens about 16, inserted at the base of the corolla; filaments very unequal, some j;—} in. long, but mostly very short; anthers oblong, acute. Ovary rudimentary, with tawny hairs. Karen hills, Taipo mountains, Burmah (between Sitang Hills and Salween River), at 4000 ft. alt., Dr Brandis ! 72. Diospyros BECCARII, sp. nov. D. ramulis petiolis et inflorescentid ferrugineo-pubescentibus ; foliis alternis, ovali-oblongis, apice acuminatis, basi rotundatis vel rarius parum angustatis, tenwiter coriaceis, superne glabris, subtus ferrugineo-pubescentibus ; floribus femineis solitariis, subsessilibus, basi pluribracteatis, axillaribus ; calyce 4-partito, lobis margine revolutis vel undulatis; corollé 4-fida, lobis obtusis; staminodiis 8, glabris; ovario glabro, 4-loculari, loculis 1-ovulatis. Young parts, petioles, underside of leaves and inflorescence ferruginous-pubescent ; shoots longitudinally wrinkled. Leaves oval-oblong, narrowly acuminate, obtuse at apex, rounded or rarely slightly narrowed at base, thinly coriaceous, glabrous above with indistinct veins, flat, 2—6 in. long by 1—24 in. wide; petioles stout, terete, }—} in. long. . Flowers solitary, axillary, subsessile, with several caducous ovate bracts at base ; bracts unequal, shorter than the calyx; calyx campanulate, {—3in. long, hairy on both sides, 4-partite; lobes ovate, with reflexed or undulated margins; corolla (immature) 4-fid, glabrous inside; lobes obtuse; staminodes 8, glabrous, equal, in one row; ovary glabrous, ovoid, 4-celled, cells 1-ovuled. Borneo, O. Beccari/ nn. 2492, 2591. 73. DIospyRos OLEIFOLIA, Wall. List n. 4128 (1828—32). D. foliis alternis, ovalibus vel oblongis apice obtuse acuminatis, basi angustatis, subco- riaceis, glabrescentibus, utrinque levibus nitidisque, nervis subtilissimis impressis inconspicuis, petiolatis ; floribus masculis ternis, breviter cymosis, tetrameris ; calyce extus glabro, intus to- mentoso, lobis latis acutis, corolla urceolato-oblongd, lobis brevibus rotundatis, staminibus cir- citer 20, ovarii rudimento pubescente ; fructibus solitariis, subglobosis. DC. Prodr. vii. p. 239. n. 88 (1844); Kurz in Journ. Asiat. Soc. Beng. vol. xu. Pt. m1. p. 72. n. 94 (1871); Flora, 1871, p. 342. A moderate-sized tree with dark bark, glabrous except young parts, which are fer- ruginous-tomentose. Leaves alternate, oblong-elliptical or oblong-lanceolate, narrowed at both ends, 23—6—9 in. long by 1—2}—2%in. wide, subcoriaceous, pale, smooth and shining on both sides, the yellowish midrib and inconspicuous veins all slightly depressed on the upper surface ; petioles }—}—in. long; margins just recurved. &é. Cymes drooping, 4—1 in. long, axillary, slightly pubescent, usually 3-flowered; com- Mr HIERN, ON EBENACE. 205 mon peduncle 4—Zin. long; pedicels 1—}in. long, hispidulous; flowers tetramerous, white. Calyx nearly jin. long, glabrous outside, densely fulvo-tomentose inside; lobes wide, acute. Corolla more than twice the length of the calyx, fulvo-tomentose outside; tube wide and inflated, about }—,4in. long; lobes short, rounded; stamens about 20, inserted at the base of the corolla and on the receptacle; filaments very short; anthers linear, acuminate, about tin. long. Ovary rudimentary, minute, fulvo-pubescent. Q. Fruit solitary, on young branches, very shortly pedunculate, sub-globose, 2 diameter, more or less rufous-pubescent, yellowish, in one case 3-celled and 3-seeded. Fruiting calyx jin. long, 4fid (in one case 3-fid), tomentose inside, pubescent outside; lobes ovate- deltoid. Pegu, Dr Brandis, Kurz! no. 3012. Java, Wynkoopers Bay, Teijsmann (Malay name Kayu arang); Amherst, Wallich! 4128, Anderson!, H. Falconer!, Herb. Hort. Bot. Cale. No. 242. in. in 74. DIOSPYROS FLAVICANS. D. foliis alternis, ovali-oblongis, apice acuminatis, basi obtusis, tenuiter coriaceis, glabris, breviter petiolatis; inflorescentid axillari, brevissime cymosdé, pauciflord, bracteis longis im- bricatis, floribus 4—5-meris, calyce partito, corollé hypocrateriformi tetragond, lobis obtusis, staminibus in flore masculo geminatis, 14—20, corolle basi insertis, glabris; ovario in flore femineo glabro, tetragono-pyramidali, 4-loculari, loculis 1-ovulatis; fructibus oblongis, glabris. Guatteria? flavicans, Wall. List, n. 7295 (1828—32). A diccious shrub 8—10 feet high or small tree, with virgate terete and somewhat -flexuous branches, appressedly ferruginous-pubescent as well as the leaves when young, gla- brescent, spreading at about 50°. Leaves alternate, oval-oblong, usually much acuminate at apex into a long obtuse point, somewhat narrowed at base, thinly coriaceous, 2—5}in. long by {—2in. wide, besides petioles j,—1in. long; quickly glabrescent, somewhat shining 2 on both sides; midrib somewhat depressed and lateral veins not conspicuous on upper surface, the latter clear and slender and anastomosing near margin beneath. Inflorescence axillary, shortly cymose, ferruginous-pubescent, with long bracts, 1—several-flowered; flowers white. 3 . Cymes very short; flowers clustered (or solitary); with short pedicels bearing long lanceolate foliaceous bracts at base sometimes 4in. long. Calyx }—1in. long, pilose on both sides, 4-partite or deeply lobed rarely 5-lobed, lobes ovate acute foliaceous, with plicate- valvate sides, lax. Corolla salver-shaped, about double the length of the calyx, pubescent outside, glabrous inside; tube tetragonal, 4—5-fid or partite. Stamens 14—16—18—20, inserted at or near base of tube of corolla, in pairs, the inner shorter on bent filaments, glabrous; anthers apiculate, equalling or shorter than the filaments; ovary 0. 9g. Cymes 1—few-flowered, }—1 in. long; bracts pubescent outside, glabrous inside, vary- ing in size, leaf-like, at base of pedicels, {—tin. long. Calyx 3,—2in. long, pubescent on both sides, 4-partite; lobes widely ovate, cordate, with undulated and recurved sides and base, plicate, foliaceous. Corolla caducous. Ovary glabrous, tetragonally pyramidal, 4-celled, terminated at apex by an erect glabrous bilobed style 4, in. long or shorter; cells 1-ovuled. 206 Mr HIERN, ON EBENACE. 4 Fruit glabrous, oblong, {—1in. long by 4 2in. thick, obtusely tetragonal, rounded at apex and terminated by remains of style, 4-celled. Fruiting calyx loosely embracing base of fruit, in. high, deeply 4-fid ; margins wavy-reflexed. Mergui, Tenasserim, Grifith! (Cfr. Notulz, vol. Iv. p. 291. n. 2. 1854) n. 3639; Malacca, Grifith! Kew List 454, 3623; Penang, G. Porter! from the hills (Wall. List 7295); (2) Tenasserim and Andamans, Herb Helfer! 3640; Malacca Maingay! 972, “8 Feb. 19, 1868, stamens 17—15, 9? testa subosseous.” An instance of phyllomania occurs in a specimen probably of this species collected by Helfer! n. 423, Tenasserim or Andamans. 75. DIOSPYROS SAPOTOIDES, Kurz MSS. D. foliis alternis, obovato-ovalibus, apice breviter acuminatis, basi cuneatis, mox glabrescen- tibus, tenuiter coriaceis, breviter petiolatis ; floribus masculis aggregatis, subsessilbus, tetra- meris, urceolato-oblongis, calyce profunde lobato, utrinque pubescente ; corolla 4-fida, lobis obtusis, staminibus circiter 16, glabris, biserialibus, inequalibus, ovario rudimentario. Branches terete, smooth. Leaves alternate, obovate-oval, shortly acuminate at apex, cuneate at base, quickly glabrescent, thinly coriaceous, glaucescent (bluish green in dry state) above, 3—10 in. long by 1{—3} in. wide; lateral vems 12—15 on each side the midrib, arching and anastomosing near the margin; petioles }—3 in. long. $. Flowers }in. long, urceolate-oblong, tetramerous, clustered, several together, sub- sessile, in axillary nodose dense abbreviated cymes. Calyx about jim. long, openly campa- nulate, hairy on both sides, deeply lobed; lobes cordate-ovate. Corolla 4-fid, hirsute outside at least along 4 hairy lines on tube; lobes oval, rounded. Stamens 15—16, in two rows, glabrous; inner row shorter. Ovary wanting. Pegu; flowers in April, 8S. Kurz! n. 3013. 76. Drospyros AUREA, (?) Teijsmann et Binnendijk Pl. Nov. Hort. Bogor. in Nederl. Kruidk. Arch. 11. p. 405 (1855). D. ramis fastigiatis; foliis bifariis, elliptico-oblongis, breviter acuminatis, basi acute angustatis, glabris, nitidis, tenwissime coriaceis, petiolis crassiusculis; floribus masculis aggre- gatis subsessilibus tetrameris, calycis lobis deltoideis acutis, corolla tubulosd, lobis ovali-oblongis patentibus, stuminibus 16, glabris, antheris apiculatis; floribus femineis solitarvis 4—5-meris, staminodiis 10—11, “stigmate profunde 3-fido”; baccd globosd, aurantiacd. Walp. Ann. v. p. 478 (1858). A small tree; trunk 4feet high with fastigiate terete contiguous leafy branches which form a dense head; young shoots petioles and pedicels ferruginous-puberulous as well as the midrib of the leaves beneath. Leaves alternate, distichous, glabrescent, oval-oblong, acuminate at apex, narrowed at base into petiole, very thinly coriaceous, shining, with midrib depressed and lateral veins slightly raised above, 4—8}in. long by 1}—2}in. wide ; petioles }—}in. long, rather thick, Mr HIERN, ON EBENACEA 207 & Flowers in very short many-flowered dense nodular cymes with very short pedicels, in the axils of fallen leaves, }—3in. long, slender. Calyx }—1in. long, scattered with few inconspicuous short ferruginous hairs, 4-fid; glabrous inside; lobes narrowly deltoid, acute, spreading. Corolla tubular, 4-fid, glabrous except 4 lines of short hairs outside; tube }in. thick in middle where it is slightly inflated; lobes oval-oblong, spreading. Stamens 16, glabrous, unequal, inserted on the tube of the corolla a little above its base, gs—t in. 0 long; anthers ovate, apiculate, »,—,;in. long; the longer filaments exceeding the anthers, in length. Ovary rudimentary, glabrous. Q. Flewers axillary, glabrous, subsessile, of a golden colour, solitary; calyx 4—5-lobed, with shallow rounded wide plicate lobes, glabrous. Corolla 4—5-fid, constricted at the apex, scarcely twice the length of the calyx. Ovary 10-celled, glabrous. Staminodes 10—11. Stigma deeply 3-fid (?). Fruit globose, $—%in. in diameter, of orange colour, tipped by style, subsessile, with flat or reflexed calyx. Gum sometimes exudes from the young branches. Java, Dr Horsjield/ Ebenacee nos. 3, 6; Bantam, Teijsmann and Binnendijk. 77. Diospyros nicRIcANs, Wall. List n. 6351 (1828—32). D. foliis alternis ovali-oblongis, apice valde acuminatis, bast obtuse angustatis, firmiter membranaceis, glabris, nitidis, breviter petiolatis; floribus masculis 3—6-nis, axillaribus, bre- vissime cymosis, subsessilibus, tetrameris, corollé gracili, profunde lobatd, staminibus 32, in- equalibus, nonnullis minutis, glabris; fructibus solitariis, breviter pedunculatis, glabris, 4-locu- laribus, sub-globosis, loculis monospermis, albumine non ruminato, calyce fructifero 4-partito patente vel reflexo. \ Alph. DC. Prodr, vim. p. 239. n. 87 (1844), non Dalz. A tree 50 feet high, with many lax cinereous, glabrescent branches; young shoots and petioles minutely puberulous. Leaves oval-oblong, much acuminate at apex, somewhat nar- rowed at base, alternate, turning black when dry, firmly membranous, glabrous except on midrib which is puberulous and depressed on the upper surface; lateral veins and net- veins delicate, not conspicuous above; 3—5 in. long by 1—1} in. wide; petioles ~j—4+ in. long. 3. Flowers in few (8—6)-flowered short axillary puberulous cymes, subsessile, }—4 in. long; bracts small, imbricated. Calyx with scattered short ferruginous hairs outside shortly 4-lobed. Corolla with few scattered short hairs outside deeply (2rds) lobed, slender; lobes reflexed at apex. Stamens 32 in one case, very unequal, many minute, glabrous. 9 . Fruit glabrous, ovoid or globose, poimted at apex, about 3 in. long, 4-celled, 4-seeded, solitary. Fruiting calyx 4-partite, with scattered ferruginous hairs outside, nearly glabrous 2 inside; with oval, flat, spreading or reflexed lobes, tin. long. Seeds oblong, 2 in. long; albumen not ruminated, embryo nearly as long as the albumen. Fruiting peduncles shortly hispid, 4 in. long, patent, unilateral, bearing 2 small bracts. Khasia, Churra, 2000 ft. alt.; Drs J. D. Hooker and T. Thomson! 842, June, in fruit; East Bengal, Griffith! 3628; (Silhet), Wallich/ 6351. 208 Mr HIERN, ON EBENACE. 78. Drosprros Esenum, Koenig in Physiogr. Salsk. Handl. 1. p. 176 (1776). D. ligno duro in centro nigro, foliis alternis, ovalibus vel oblongis, apice obtuse acumi- natis, basi obtuse angustatis, tenwiter coriaceis, reticulatis, glabris, breviter petiolatis ; floribus masculis subsessilibus, breviter cymosis, sepius 3-5-nis, tetrameris, calyce campanulato, ciliato, breviter 4-lobo, corolld tubulosd, medio constricta, glabra, 4-fidd, staminibus 16—32, fila- mentis 8; floribus femineis solitarws, staminodis 16 geminatis vel paucioribus, ovario 8- loculari, glabro vel appresse pubescente, calyce fructifero aucto, tubo campanulato margine in- tus elevato, lobis patentibus vel reflexis, Fructibus subglobosis, glabris vel appresse pubescentibus, seminum albumine non ruminato. Alph. DC. Prodr. viii. p. 234, n. 63 (1844); Ettingsh. Blatt-skel. Dikot. p. 89. t. 37. £13 (1861); Linn. fil. Suppl. Pl. p. 440 (1781); Roxb. drawings; Beddome, Fl. Sylvat. Madr. t. 65 (1870); Wight. Ic. t. 188. (1840). D. glaberrima, Rottb. in Act. Hafn. 1783. vol. m1. p. 540. t. 5. D. melanoxylon, Willd. Hb. n. 19243; Sp. pl. tv. p. 1109. n. 8 (1805); non Roxb. D. reticulata, Wall.! List, p. 159. n. 4120 E. (1828—32), non Willd. D. Ebenaster, Spach, Hist. Végét. rx. p. 407 (1840), t. 135 (1846), non Retz. D. nigricans, Dalz. in Kew Journ. Bot. Iv. p. 110 (1852); Bedd. Ic. Pl. Ind. Or. (vir) 5, excl. t. 124 (1871); non Wall. D. assimilis, Bedd. Report Forests of Madras for 1866—67, p. 20. t. 1 (1867). A large tree with glabrous branches. Leaves glabrous, alternate, oblong or oval, obtusely communicate or retuse at apex, somewhat narrowed at base, thinly coriacious 2—7 in. long by }—2} in. wide, with petioles 4—} in. long; net-veined, of same colour on both sides. é. Flowers 3—15 together, subsessile, on short pubescent cymes which about equal the petioles, about #,in. long in bud; Bracts small, caducous. Calyx funnel-shaped, about tin. long, shortly 4lobed, nearly or quite glabrous outside with ciliated margins, hairy inside; lobes rounded. Corolla tubular, constricted at middle, glabrous, 4-fid, with imbri- cated lobes. Stamens 16, unequal, more or less in pairs, glabrous, inserted at base of corolla, or ranging up to 32 on § filaments; ovary rudimentary or wanting. °. Flowers solitary, with 2 bracts at base, shortly stalked. Calyx much longer than in the 6, deeply 4+fid with an elevated callous marginal ring round its mouth. Stami- nodes 16, in pairs, or fewer. Style 1; stigmas 4; ovary 8-celled, glabrous or appressedly pubescent. Albumen of seeds not ruminated. Fruit depresso-globose or subglobose, } in. long, or globose and 4—1 in. in diameter, glabrous or appressedly pubescent. Fruit-calyx about 3—l1in. across, with spreading or reflexed lobes, receiving the base of the fruit by the cup-shaped tube which has an elevated circular margin felted inside. East India, Koenig !; Chorla Ghaut, Dalzell (called Kardé mardé in S. Canara); Assam, Griffith! ; Ceylon, Columbo, Ferguson /, Thwaites! 1912, 1913, 2437, 2439; East Bengal, Griffith ! 3621; Malacca, Griffith! 3635 “Cayoo Arang, Ebony Wood,” Maingay! 971. “Flowers 4—5 merous; Satiny-black. Leaves shining above. Flower yellowish;” Wight 1714; Wallich/ List n. 4120; Sumatra and Molucca Isl. ex Miq. Fl. Ind. Bat. 11 p. 1048 (1856) ; New Caledonia, Vieillard ! 898, Thiebault / 344, p- 2 Mr HIERN, ON EBENACE. 209 This valuable tree is not uncommon in the mountain forests on both sides of the Presidency of Madras and in Ceylon; it yields the best kind of Ebony, generally jet- black but sometimes slightly streaked with yellow or brown; it is very heavy, close and even-grained, and stands a high polish; unseasoned it weighs 90 to 100 lbs. the cubic foot, and 81 lbs. when seasoned; it is used for inlaying and ornamental turnery and sometimes for furniture, but there is not much demand for it in Madras. The sap-wood is white, hard, close-gramed, and strong, but not durable; it is however used by the natives for various purposes; it is called Nalluti in the Cuppapah and Kurnool hill-forests where the tree is very common and well known. Beddome 1. c. D. reticulata, Decaisne, Herb. Timor. in Nouv. Ann. Mus. ur. p. 406 (1834), non Willd.; D. reticulata, 8. timoriana, Alph. DC. Prodr. vit. p. 225. n. 11 var. (1844); D. timoriana, Mig. FI. Ind. Bat. um. p. 1045 (1856), ought probably to be referred to D. Ebenum, Koen., but I have not seen an authentic specimen. D. hebecarpa, A. Cunn. ex Benth, Fl. Austr. Iv. p. 286 (1869) is probably the same species; the fruit is #—lin. in diameter, covered with short hairs or glabrescent. Australia, Queensland, Cape York, W. Hill!; Endeavour River, A. Cunningham!; New Caledonia, Wagap, Vierllard / 2869. A specimen in Hb. Mus. Paris collected by Pancher/ in New Caledonia may be the same species (D. Ebenum, Koen.) but the leaves are more coarsely reticulated and the fruiting peduncles are longer (fin.). Cfr. D. samoensis, A. Gray. 79. DIospYyROS PELLUCIDA, sp. nov. D, foliis alternis, ovali-oblongis, apice acuminatis, basi angustatis, firmiter membranaceis, minute pellucido-punctatis, utrinque nitidis, glabris, breviter petiolatis ; floribus solitariis, axil- laribus, subsessilibus, polygamis, tetrameris, calyce profunde lobato, lobis acuminatis, leviter plicatis, corolle lobis profundis acutis, staminibus in jl. masc. 8, uniserialibus, glabris, fruc- tibus globosts subglabratis, 8-locularibus. Branches spreading at about 45°, terete, dark, glabrous, or minutely puberulous at the extremities. Leaves oval-oblong, alternate, firmly membranous, glabrous, of nearly same dark colour and shining on both sides, minutely pellucid-punctate, acuminate at apex, some- what narrowed at base, 44—6}in. long by 14—211in. wide, including petiole tin. long; midrib depressed and veins inconspicuously reticulated above, lateral veins anastomosing within the margin beneath. Flowers solitary, axillary, very nearly sessile; polygamous (a male flower and a young fruit growing on the same specimen), tetramerous. Calyx ,%, in. long, spreading, puberulous, but glabrescent outside, deeply 4-lobed, lobes }in. long, ovate, cor- date and dilated at base, acuminate at apex, spreading, with margins reflexed outwards, especially near base, somewhat plicate; tube thickened and hairy inside, cup-shaped, the thickened portion extending upwards a short distance up the middle of the lobes. 6. Corolla conical in bud, } in. high, glabrous above, puberulous below outside, deeply lobed; lobes acute. Stamens 8, equal, in one row, glabrous, tin. long; anthers com- pressed, 4, in. long. Style ,%, in. long, straight, erect, slightly puberulous below the lobed apex, receptacle (rudimentary ovary) puberulous. @. Young fruit 1 in. high by 53, in. thick, bluntly pointed at apex, pubescent ; 3 Wore 0 Teh 27 210 Mr HIERN, ON EBENACE. fruit globose, subglabrate, 4 in. in diameter, unequally S8-celled. Fruiting calyx not lengthened, spreading, about } im. high, supporting base of fruit; tube with raised rim within. Philippine Islands, Cuming / 1496, 1506. 80. DiospyROS TETRANDRA, sp. nov., non Span. D. foliis alternis, elliptico-oblongis, acuminatis, basi angustatis, tenwiter coriaceis, glabris, graciliter reticulatis, petiolatis; floribus masculis 3-nis, brevissime cymosis, tetrameris, tubu- losis, extus hispidis, calyce late campanulato, 4-fido, corolld breviter 4-fidd, staminibus 4, equalibus, antheris hispidis, ovarii rudimento hirsuto; floribus femineis, 1—3-nis, subsessilibus, stylis 4; fructibus solitariis, subsessilibus, globosis, nitidis; calyce fructifero aucto, concavo- plicato. A tree (2), shining and quite glabrous except buds, inflorescence, &e.; young branches terete, with smooth bark. Leaves alternate, elliptic-oblong, acuminate, somewhat narrowed at base, thinly coriaceous, 4—8 in. long by 14—3 in. wide; midrib narrowly depressed above; lateral veins clear and slender beneath, arching and anastomosing within the margin, inconspicuous and very delicate as well as the net-veins above; petioles }—2 in. long, with bladdery tumours on the under-side (especially on the younger ones of the male plants) extending from the top downwards and disappearing from the older petioles. é. Inflorescence axillary, very short, 3-flowered, with short rufous sete; flowers sub- sessile, $in. long, slender, with short rufous hairs. Calyx }in. long, 4-fid; lobes acute, somewhat spreading. Corolla tubular, shortly 4-fid; lobes spreading, rounded, +}, in. long. Stamens 4, inserted on the receptacle or at very base of corolla, equal, distinct ; anthers linear, with reddish short hairs, apiculate, as long as the glabrous filaments. Ovary ru- dimentary, rufous-hairy. ?. Inflorescence axillary, 1—3-flowered, shortly pubescent, without the flowers about equalling the petiole; bracts ovate, shortly pubescent; pedicels 3, in. long; flowers nearly tin. long, 4—5-, usually 4-, merous, with short appressed hairs. Calyx }in. high by }in. wide, rather larger in fruit, 4-lobed, lobes cordate, acuminate or emarginate, roundly plicate. Corolla elongate-urceolate, with reflexed ovate lobes. Staminodes.. Ovary... Styles 4, hairy. Fruit solitary, globose, #in. in diameter, shining, with short inconspicuous ap- pressed hairs, or subglabrate; fruiting calyx }—in. wide, }—jin. high; lobes forming below dependent hollows, ascending above. Guiana, Martin !, Rudge! s.D. 1806, Poiteau ! Plate VI. A branch in male flower-bud, natwral size. a. A piece of a male branch with more advanced flowers, natural size. b. A male flower on branch, magnified 3 dia- meters. c. A male calyx, magnified 6 diameters. d. The andreecium with rudimentary ovary in centre, magnified 6 diameters. e. A female branch with empty calyx, natural size. f. A piece of a fruiting branch, the fruit fractured, natural size. 81. Drospyros SPRUCEI, sp. nov. D. foliis alternis oblongis, apice valde acuminatis, basi subrotundis, coriaceis, supra glabris nitidis, subtus ferrugineo-tomentosis, nervis manifestis, petiolatis ; floribus masculis aggregatis, Mr HIERN, ON EBENACEA. 211 dense cymosis, ferrugineo-tomentosis, tetrameris, calyce campanulato, lobis deltoideis, corolla tubulosd, lolis rotundatis patentibus, staminibus 16, glabris, geminatis, inequalibus, corolle tubo brevioribus, ovarii rudimento rufo-tomentoso. A slender straight tree, 60 feet high, with ferruginous-pubescent branches. Leaves ob- long, nearly rounded at base, much acuminate and sub-caudate at apex, coriaceous, glabrous and with depressed veins on the upper side, ferruginous-tomentose with strong veins beneath, alternate, about 1 ft. long by 3—3}in. wide, edges recurved; petioles }—3 in. long, thick, “recurved” (Spruce). é. Flowers ferruginous-tomentose outside, in many-flowered ferruginous cymes; cymes about }in. long (excluding the flowers); pedicels about 3, in. long, stout. Calyx tin. long, campanulate, shortly tomentose on both sides, 4-fid with deltoid lobes. Corolla about } in. long, tubular, with 4 patent lobes, glabrous inside, tube 2in. long; lobes tin. long, rounded, pale green. Stamens 16, nearly or quite glabrous, in 8 pairs, sub-equal in those pairs which are opposite the corolla lobes and unequal in the alternate pairs; the longer ones 1in. long with the anthers about equalling the filaments; inserted at base of corolla; anthers with very few hairs on the back or glabrous; filaments glabrous. Ovary rudimentary, rufous- tomentose. South America, Columbia, San Carlos, frequent in the woods near river Guasié, $ fl. October. Spruce! 3138. Plate VIII. A branch in male flower, natural size. a. A male flower-bud, magnified 2 diameters. 6. A male flower expanded, magnified 2 diameters. c. A male corolla laid open shewing the stamens, magnified 3} diameters. d, e. Contiguous pairs of stamens, magnified 34 diameters. 82. Diospyros MARITIMA, Blume, Bijdr. Fl. Ned. Ind. p. 669 (1825). D. foliis alternis, ovalibus vel oblongis, utrinque obtusis, coriaceis, glabris, petiolatis, flo- ribus masculis aggregatis, 3—7-nis, subsessilibus, elongato-campanulatis, pubescentibus, calyce campanulato, apice 4—5- rarius 3-dentato, corolla tubulosd, 4-fidd, staminibus 15—18, ine- qualibus, plerisque geminatis, antheris glabris, filamentis basi hirsutis brevissimis; floribus femineis solitartis vel binis, staminodiis 4—10, glabris, ovario 8-loculari, ferrugineo-pubescente, fructibus subglobosis, glabrescentibus, senunum albumine non ruminato. Alph. DC. Prodr. vim. p. 234. n. 62 (1844), Decaisne in Nouv. Ann. Mus. m1. p. 406 (1834). Cargillia laxa, R. Br. Prodr. p. 526. n. 1 (1810), Alph. DC. Prodr, vir. p. 243, n. 2 (1844), Benth. Fl. Austr. Iv. p. 287 (1869). Cargilia maritima, Hassk. Cat. Pl. Hort. Bot. Bogor. 1. p. 159 (1844). Cargillia megalocarpa, F. Muell. Fragm, y. p. 163 (1866). Maba megalocarpa, F. Muell. Lc. Diospyros tetrandra, Spanoghe! in Linnea Xv. p. 336 (1841), non mihi. Diospyros megalocarpa, F. Muell. Austral. Veg. in Intercolonial Essays, 1866—67, p. 35 (1867). A small tree S—10 feet high with moderately thick trunk, dense head and drooping branches, or a handsome tree attaining 50 feet, glabrous except the buds and inflorescence ; 27—2 212 .Mr HIERN, ON EBENACE. branches and shoots terete, rather slender. Leaves oblong or oval, coriaceous or thinly so, of nearly same colour on both sides, shining above, alternate, usually rounded or obtuse near base, obtuse at apex, 2—10}in. long by 1}—3tin. wide, often with 2 glands at base near the petiole; petioles 1—}in. long; midrib depressed above; lateral veins rather clear beneath, raised and not conspicuous above. Bracts several, rather small, on very short stalks. 3. Flowers 3—7 together, crowded, subsessile, 2in. long in bud, elongate-campanulate. Calyx campanulate, 4—-5- rarely 3-toothed at the apex, silky-puberulous on both sides, iin. long, coriaceous; lobes } depth of calyx, depresso-deltoid. Corolla 4-fid, silky outside, 2—4 times the length of the calyx, tubular, Zin. long. Stamens 15—18, inserted at base of corolla, mostly in pairs, unequal; filaments very short, hirsute at base; anthers lanceolate- subulate or oblong, glabrous; pollen white, globose. Ovary rudimentary, hairy. @. Flowers 1—2 together, subsessile, about fin. long in bud. Calyx like g but thicker especially in fruit. Corolla 3 rds 4-fid. Staminodes 4—10, glabrous. Styles 4, short. Ovary ferruginous-pubescent, 8-celled; cells l-ovuled (4-celled, cells 2-ovuled according to R. Brown). Fruiting calyx broadly cup-shaped or flatly appressed to base of fruit, 4—5- lobed, coriaceous, about in. across, often }in. high. Fruiting peduncle very short and much thickened and continuous with calyx. Fruit depresso-globular, glabrescent, 3—1in. high by —1in. thick, 4 (?) -celled and seeded, marked at the apex by remains of short style. Seeds nearly Lin. long, somewhat compressed, brown and shining; albumen white, not ruminated. Radicle longer than the ovate cotyledons. N. Australia, Gulf of Carpentaria, opposite Groote Island, R. Brown! ; Escape Cliffs, Hulls ; Queensland, Cape York, W. Hill! ; Timor, Zippelius, Decaisne!, Gaichenol!, Spanoghe ! ; S. Java, Blume!, Zollinger! n. 1833; Java, Leschenault/; Straits of Sunda, Ld. Macartney ! Java, Hasskarl!; De Vriese and Teijsmann! 1859—60. Menado, Celebes, poisonous tree. Teijsmann and De Vriese/. 83. Diospyros PHILIPPINENSIS, Alph. DC. Prodr. vim. p. 231. n. 43 (1844). D. foliis alternis, ovalibus, apice obtuse acuminatis, bast angustatis, tenwiter coriaceis, glabrescentibus, breviter petiolatis ; floribus femineis 1—3-nis, breviter cymosis, bracteatis, tetra- meris, pubescentibus, calyce profunde lobato, corolld tubulosd, 4-fidd, staminodiis 6, leviter pubescentibus, ovario ovoideo-conico, fulvo-pubescente, 4-loculari, loculis 1-ovulatis. Young shoots buds inflorescence and underside midrib and margin of young leaves covered with short tawny tomentum; branches glabrescent. Leaves oval, rather shortly and obtusely acuminate at apex, obtusely narrowed at base, thinly coriaceous, alternate, gla- brescent, shining above, 24—5}in. long by 14—24in. wide; petioles {—1in. long; midrib depressed above; lateral veins distant, slender, inconspicuous especially above. @. Flowers in axillary 1—3-flowered bracteated cymes with several imbricated scales at the base, or solitary near the base of the young shoots of the year; peduncles or pe- dicels #;—}in. long; bracts rounded, tawny-pubescent; scales at the base of the young shoots several, much imbricated; flowers 3? in. long, tawny-pubescent outside, erect. Calyx ¢ in. long, loose, glabrous and shining inside, deeply 4-fid with rounded or sometimes api- Mr HIERN, ON EBENACE. 213 culate imbricated lobes. Corolla glabrous inside, 4-fid; tube 4in. long by din. thick; lobes oval, spreading and recurved, somewhat cordate at base, round at apex, imbricated. Sta- minodes 6 (in one flower), equal, somewhat tawny-hairy. Style very short, cut at apex. Ovary ovoid-conical, tawny hairy! 4-celled!; cells 1-ovuled; according to Alph. DC. lc. the ovary is glabrous and 6- (or 6—8-) celled. Manila, Philippine Islands, Cuming/ 1142. 84. DIOSPYROS PILOSANTHERA, Blanco, Fl. Filipin. p. 304 (1837). D. caule arboreo, foliis alternis, lanceolatis, coriaceis, glabris, basi 2—38-glandulosis, bre- vissime petiolatis; floribus [ femineis?] axillaribus sessilibus, 6-nis vel ultra, calyce 4—5-lobo, lobis revolutis, corollé calyce longiore pilos@, 5-lobd, staminibus 5—6, antheris medio pilosis (sterilibus ?), stylis 4, baccd 10-spermd. Alph. DC. Prodr. vin. p. 287. n. 77 (1844). A tree with hard wood. Leaves alternate, lanceolate, glabrous, coriaceous, with 2 or 3 glandular depressions at the base beneath; petioles very short. Q (2) Flowers axillary, sessile, 6 or more together; calyx with 4 or 5 large teeth recurved and bordered at maturity; corolla longer than the calyx, covered with hair outside, naked at the throat, 5-lobed; stamens 5—6; filaments short; anthers with a line of hairs along the middle; stigmas 4; fruit baccate, 10-seeded, edible; like a small guava; seeds horny, semicircular and thin at the two sides, and convex on the exterior. Philippine Islands, Blanco. 85. DIosPYROS LANCEZFOLIA, Roxb. Cat. Pl. Fl. Ind. (1813). D, foliis alternis, oblongis vel lanceolatis, apice acuminatis, basi angustatis, coriaceis, glabris ; floribus masculis fasciculatis, dense cymosis, 8—5-nis, pubescentibus, tetrameris, calyce campanulato, corolld tubulosd, staminibus 14—16, geminatis, inequalibus, subglabris ; floribus femineis solitariis, subsessilibus, axillaribus, 4—5-meris, staminodiis 8—10, ovario pubescente, 8-loculart, fructibus subglobosis, tomentosis, seninum albumine non ruminato.* Fl. Ind., Edit. 1832, vol. 1. p. 587; Roxb. drawings no. 2508; Alph. DC. Prodr. virr. p. 282. n. 46 (1844). D. multiflora, Wall. List, n. 4144 (1828—1832), Alph. DC. Prodr. vi. p. 231. n. 45, non Blanco. (2) D. amena, Wall. List. n. 4139 (1828—32), Alph. DC. lc. p. 281. n. 44 (1844), Ettingsh. Blatt-skel. Dikot. t. 41. f. 11 (1861). Goolal or Goolul is the vernacular name in Sillet, ex Roxb. lc. A pretty large tree, furnishing hard durable timber suitable for the construction of houses; glabrous except the buds under side of young leaves inflorescence and fruit. Leaves oblong oblong-lanceolate or -ovate, more or less narrowed at base, acuminate at apex, with midrib depressed on upper side, coriaceous, alternate, rather pale on both sides, with veins not conspicuous above, 2—3—6—94in. long by }—3—2in. wide, besides petioles 5 aes i—1Hin, long. 214 Mr HIERN, ON EBENACE. g. Flowers fascicled on very short dense cymes, 3—5 together, densely ferruginous- pubescent, 2—}in. long, tetramerous. Calyx campanulate, #j—} in. long, hairy on both sides, 4-fid or more shortly 4-lobed, with deltoid lobes. Corolla tubular with inflated tube, glabrous inside; lobes spreading, shorter than the tube. Stamens 14, 16, united in pairs by thin filaments and inserted at base of corolla, or hypogynous; inner ones shorter, 1—3in. long, glabrous except base of anthers or apex of filaments; common filaments jy im long; con- nective apiculate. Ovary 0; receptacle hairy. 9. Flowers solitary, subsessile, axillary, near together, 4—5-merous, }in. long, densely tawny-pubescent; bracts short, pubescent, imbricated. Calyx jin. high, 4—5-lobed; lobes with sides sometimes reflexed. Corolla-lobes cordate (ex Roxb.), imbricated. Staminodes 8—10, short, inserted at base of corolla. Style very short, about 8-lobed; ovary 8-celled, hairy. Fruit ovoid or globose, usually pointed at the apex, tawny-tomentose or appressedly silky, lin. or more long. Fruiting calyx pubescent on both sides, lin. across, with crass somewhat concave tube and 4 or 5 lobes spreading or recurved and much thinner towards the margins. Albumen not ruminated. East Bengal, Grifith! 3631, 3634; Sillet, Wallich! 4144, 4139 (2); Khasia, Churra, foot of hills; Drs J. D. Hooker and T. Thomson! 20 June 1850, in young fruit. In Khasia? or Cachar? it is called Sot-lo and is a poison for fish, Drs J. D. Hooker and T. Thomson! 86. Driospyros GARDNERI, Thw. Enum. Ceyl. Pl. p. 181. n. 12 (1860). D. foliis alternis, oblongis, apice acuminatis, basi leviter angustatis, tenuiter coriaceis, gla- bris, breviter petiolatis; floribus masculis 1—4-nis, subsessilibus, tetrameris, pubescentibus, calyce campanulato, corolla hypocrateriformi, staminibus 16, pubescentibus ; floribus femineis solitariis, ovario 8-loculari, fructibus depresso-globosis, subglabratis. Beddome, Ic, Pl. Ind. Or. (Pt. vi.) p. 27. t. 132 (1871). Patonia Walkerwi, Wight, Ill. 1. p. 19 (1840). A moderate-sized tree; young shoots puberulous, quickly glabrescent. Leaves alternate, thinly coriaceous or submembranous, glabrous, shining above with inconspicttous veins and channelled midrib, oblong, acuminate at apex, somewhat narrowed at base, 3—7 in. long by 1—2} in. wide; petioles }—2in. long; lateral veins depressed on the upper surface in the thinner-leaved specimens. 8. Flowers pubescent, 1—4 together, subsessile, on very short axillary pubescent cymes. Bracts small. Calyx 4 in., campanulate, 4-fid, covered with short hairs on both sides; lobes deltoid. Corolla about 4—4 in. long, conical in bud, salver-shaped in full flower, covered outside with appressed ferruginous silky shining hairs, glabrous inside, tube somewhat inflated below, with 4 spreading lobes about half the length of the tube. Stamens 16 (or about 12 according to Dr Thwaites), in pairs; filaments short, pilose; anthers linear, glabrous or some- what hairy. Ovary 0 or represented by a bunch of hairs. Mr HIERN, ON EBENACE, 215 @. Flowers solitary, erect, axillary, } in. long; peduncles 4 in. long. Calyx } in. long, covered with short tawny pubescence, openly campanulate, 4-fid; lobes with undulated and recurved margins. Corolla-lobes lanceolate, about 4 the length of the tube. Ovary 8-celled. Fruit depresso-globose, about 1 in. long (unripe), glabrate or with remains of ferruginous pubescence. Fruiting calyx accrescent, about } in. high by 14 in. across at top; lobes pointed and patent at apex; tube hemispherical. The timber of this tree is valuable for building and for cabinet-work, Dr Thwaites loc. cit. Ceylon, Thwaites! C. P. 1908, Macrae! 30, Walker !, Gardner! 532, up to 2000 ft. alt., called Kadoombaireya-gass. 87. Ditospyros HEUDELOTH, sp. nov D. foliis alternis, ovato-ovalibus, apice breviter acuminatis, basi obtusis, tenuiter coriaceis, subglabratis, breviter petiolatis ; floribus masculis aggregatis, 4—6-nis, subsessilibus, pubescenti- bus, calyce breviter 4—5-fido, corolla tubulosd, lobis obtusis, staminibus 13—15, filamentis bre- vibus hirsutis. Bushy tree 3—4 metres high; young parts puberulous; branches terete, dark, at about 35°, quickly glabrescent. Leaves ovate-oval, alternate, obtusely narrowed at base, shortly acuminate at apex, thinly coriaceous; dark green, glabrous and with depressed veins above; paler with few weak scattered appressed whitish hairs and with raised veins beneath; 2—3 in. long by 1—1} in. wide; petioles {—} in. long, wrinkled, glabrous; margins of leaves just recurved. 6. Cymes very short, 4—6-flowered, ferruginous-hairy; bracts short, hairy. Flowers (closed in specimen) shortly and appressedly pubescent, whitish, sweet-scented, subsessile. Calyx {4 in. high, campanulate, 4—5-fid, with ovate lobes. Corolla oblong, inflated in middle, 4—5-lobed at apex, glabrous inside, } in. long; lobes emarginate. Stamens 15, or in a tetra- merous flower 13, inserted at very base of corolla or on receptacle, nearly equal, ;2 in, long; filaments pubescent, very short, more or less connate at base; anthers linear, narrower towards apex, with a few hairs on back; dehiscing laterally by slits. Ovary rudimentary, hairy. Africa, Senegambia, Heudelot! 638, October, January. Plate V. fig. 2. A male flowering branch, natural size. a. Male flower-bud, magnified 4 diameters. 6. Half the corolla laid open, shewing some of the stamens, magnified 4 dia- meters. c. A pair of stamens, magnified 4 diameters. 88. DIosPpyROS UNDULATA, Wall. List, n. 4136 (1828—32). D. foliis lanceolato-oblongis, alternis, apice acuminatis, basi angustatis vel subrotundatis, glabris, nitidis, firmiter membranaceis, petiolatis; floribus masculis breviter cymosis, 3—9-nis, ferrugineo-pubescentibus, tetrameris, calyce 4-fido, corolla tubulosd, breviter 4-lobd, lobis obtusis, staminibus 11—14, pubescentibus ; floribus femineis 1—3-nis, breviter pedunculatis vel subsessi- libus, fructibus subglobosis, appresse pilosis, plurilocularibus, seminibus compressis, albumine non ruminato, calyce fructifero aucto, crasso, fructus basim amplectente. 216 Mr HIERN, ON EBENACEA. Alph. DC. Prodr. vir. p. 233. n. 55 (1844); G. Don, Gen. Syst. Gard. and Bot. tv. p. 40. n. 38 (1837). Var. 8B (2). D. macrophylla, Wall n. 4141 (1828—32), non Blume ; foliis fructuque majoribus. A tree; branches glabrous or young shoots puberulous. Leaves oblong or lanceolate- oblong, more or less acuminate at apex, acute or more or less rounded but not subcordate at base, firmly membranous, glabrous, shining, alternate, 3—15 in. long by 1—5 in. wide, besides petioles }—} in. long, thinly coriaceous; margins reflexed ; midrib depressed above ; lateral veins inconspicuous or depressed above. Inflorescence axillary, ferruginous-hairy. g. Flowers }—} in. long (in bud), conic-oblong. ferruginous-hairy, sessile on 83—9-flowered cymes not exceeding them in length, except in var. 8; bracts ovate. Calyx short, 4-fid, with deltoid acute lobes, less hairy inside except near the margins. Corolla tubular, glabrous inside, shortly 4-lobed, with obtuse spreading lobes. Stamens 11—14, inserted on the receptacle or at base of corolla, some in pairs, unequal except var. 8; anthers linear, hairy, sub- sessile, filaments short, hairy. Ovary rudimentary, hairy. 2. Flowers solitary or 3 together; peduncles or cymes short, not exceeding 3 in. long. Fruit subglobose, about 1 in. long by nearly the same width, flat at the top and slightly umbilicate at base of style, appressedly brown-hairy, about 6-celled and 6-seeded; pericarp thick; pulp mucilaginous; seeds compressed, about $ in. long; albumen not ruminated; embryo 1 in. long; cotyledons foliaceous, lanceolate, about as long as the radicle; fruiting calyx erect, embracing about half the fruit, very crass, hairy inside; 4-fid, with the sinuses nearly filled on the inner side; lobes deltoid, occasionally spreading at the tips. Amherst, Wallich ! 4136; Moulmein, Parish/; Malacca, Grifith/ 3619, 3636, Maingay! 977. Var. B. Tavoy, Wallich! 4141; Mergui, Grofith!; Malacca, Maingay! 974. 89. DIOSPYROS MULTIFLORA, Blanco, Fl. Filipin. p. 303 (1837), non Wall. D. foliis alternis, lanceolato-oblongis, apice obtusis, basi cuneatis, coriaceis, subtus puberulis, petiolatis ; floribus masculis 8-nis, aggregatis, brevissime cymosis, pubescentibus, calyce 4—5-fido, corollé tubulosd, apice lobatd, staminibus 15—18, filamentis hirsutis, antheris glabris ; fructibus venenosis. Diospyros Canomoi, Alph. DC, Prodr. vin. p. 287. n. 78 (1844). D. Lotus, Blanco, Fl. Filipin. edit. 11. p. 210 (1845), non Linn, A tree, glabrous except the buds, inflorescence and underside of leaves; branches terete, dark; leaves lanceolate-oblong, alternate, coriaceous, obtusely lanceolate or rounded at apex, cuneate at base and often with 2 glands on the upper side, glabrous with depressed and not conspicuous veins above, tomentose-puberulous, subglabrescent beneath, 6—8 in. long by 14—2} in. wide; besides petioles }—} in. long; margins revolute. é. Flowers ferruginous-pubescent, 2 in. long, axillary in clusters of about 8 each, sessile, in very short ferruginous-pubescent cymes, tetramerous or pentamerous. Calyx 7; in. long, 4—5-fid, ferruginous-tomentose on both sides; lobes deltoid, spreading in flower. Corolla glabrous inside, lobed at apex, rather fleshy, } in. long, tubular, Stamens 18 (in one case), Mr HIERN, ON EBENACE. 217 15 or more, hypogynous or at base of corolla, more or less combined at base by their hairy filaments; anthers linear, apiculate, glabrous. Ovary 0. 2. Fruit poisonous; reported to intoxicate fish; “even the crocodile it causes to rush from the water hurriedly.” Flowers sweet-scented. By rubbing the bark and leaves on eruptions, it is said that the latter disappear. Local names Canomot, Canomai. Philippine Islands, Cuming / 1829, Blanco. 90. Diospyros BIFLORA, Blanco, Fl. Filipin. p. 308 (1837). D. foliis alternis, lanceolatis, glabris, subcoriaceis, breviter petiolatis; floribus masculis azillaribus, binis, calyce campanulato, 3—4-lobo, corolld carnosd campanulato-oblongd, 4-lobd, staminibus 17—30, corolle basi insertis, filamentis brevissimis lanuginosis, ovarii rudimento pubescente. Alph. DC. Prodr. vit. p. 287. n. 76 (1844). A tree of 30 feet high. Leaves alternate, lanceolate, quite glabrous, entire, subcoriaceous, with only 2 glands at the base below; petioles very short and without glands. 3. Flowers axillary, 2 together, with a strong smell. Calyx campanulate, 3—4-lobed. Corolla fleshy, double the length of the calyx, inflated in the middle and narrowed above, forming a throat, with 4 reflexed lobes. Stamens 17—380, inserted on the corolla and not reaching the throat; filaments very short, woolly; anthers very long. Ovary hairy; style very short; stigma and fruit wanting. Philippine Islands, Blanco, Tagatog name Talang; flowers in June. 91. Diosprros (?) PARVIFOLIA, sp. nov. D. foliis aiternis, obovatis, apice rotundatis, basi cuneatis, coriaceis, glabrescentibus, nitidis, parvis, breviter petiolatis, venis inconspicuis ; floribus masculis solitariis, subsessilibus, awilla- ribus, pubescentibus, calyce campanulato, trilobo, corolla 4-fidd, staminibus 12, glabris, corolle basi insertis, biserialibus, antheris apice dehiscentibus, ovarit rudimento ferrugineo-hirsuto. Branches cinereous, at about 30°, the younger ones rufous-hispid at first, subsequently whitish-hairy, ultimately glabrate. Leaves alternate, obovate or obovate-oblong, rounded at apex, cuneate at base, hairy beneath when quite young, quickly glabrescent, coriaceous, with margins just reflexed, without conspicuous veins, shining, 4—#in. long by 4{—+Hin. wide, including petiole .~— in. long. 6. Bracts rufous-hairy, ovate or lanceolate; flowers solitary, subsessile, rufous-hairy, axillary; calyx },—1in. long, campanulate, 3-lobed, rufous-hairy on both sides, lobes 3 depth of calyx, rounded; corolla openly campanulate, covered with silky ferruginous hairs out- side, glabrous within, tin. long (when straightened), 4-fid with reflexed and somewhat emarginate lobes; stamens 12, glabrous, inserted at or near base of tube of corolla, in 2 rows, distinct, the inner ones at a lower level, filaments ,,—3,in. long; anthers 3, in. long, dehiscing laterally by apical pores; ovary rudimentary, represented by a bunch of ferruginous hairs. Madagascar ! Vou. XII. Part 1 28 218 Mr HIERN, ON EBENACEZ. 92. DIOSPYROS BUXIFOLIA. D. foliis alternis, ovato-ellipticis, utrinque angustatis, coriaceis, supra lucidis, subtus se- riceo-pubescentibus, subsessilibus, confertis, nervis inconspicuis ; floribus axillaribus, subsessilibus, masculis 3—4-nis, confertis, femineis solitariis, calyce 4-fido, corolla 4-fidd, breviter et late campanulatd, intus glabrd, staminibus 10—16, geminatis, glabris, in flore femineo 0; antheris apice rimosis; ovario femineo 4-loculari superne pubescente inferne glabro, loculis 1-ovulatis ; fructibus oblongis, 1—2-spermis, albumine non ruminato. Leucorylum buaifolium, Bl. Bijdr. Fl. Ned. Ind. p. 1169 (1826); Choisy, Mém. Ternstr. p. 43. t. 2 (1855); Mig. Fl. Ind. Bat. p. 1049 (1856). Diospyros microphylla, Bedd. Ic. Pl. Ind. Or. (vm) p. 27. t. 183 (1871). A large tree with glabrescent terete branches and straight trunk. Young shoots and inflorescence covered with pale ferruginous pubescence. Leaves distichous, close together, easily falling (in dried state), firm, occasionally minutely pellucid-punctate, the younger ones silky beneath, without conspicuous veins, ovate-oval, narrowed at both ends, subsessile, 3—2}in. long by £,—12in. wide; midrib depressed and often puberulous above. Flowers dicecious. 3. Flowers 3 or 4 together, subsessile, very short axillary cymes; flower in. long, tetramerous. Calyx j;in. high, covered with short hairs, having 4 rounded imbricated lobes >in. deep. Corolla j~;in. high, with 4 rounded apiculate reflexed lobes y;in. deep, hairy along middle lines outside. Stamens 10—16 (16! in all the flowers examined), glabrous, united by their filaments in pairs, the inner ones the shorter; anthers ovate or oblong, dehiscing at apex; filaments slender, equalling or exceeding the anthers, inserted at base of corolla. Ovary rudimentary, hairy. @. Flowers solitary, subsessile. Calyx 4-fid, with rounded lobes much imbricated in bud, pubescent outside; corolla 4in. long, 4-fid, hairy outside; staminodes 0. Ovary 4- celled, ellipsoidal and glabrous below, conical and pubescent above, cells 1-ovuled; style bipartite, short. Fruit cylindrical or oblong, conical at apex, dry, I1-celled, 1- rarely 2- seeded, 4—tin. long by }—}in. wide, pointed, glabrous and shining or subglabrous or fulvous pubescent at apex resting at base on small spreading pubescent or ciliate calyx; albumen cartilaginous not ruminated; cotyledons about equalling the radicle. Malacca, Maingay! 966 “ovary rudimentary 4-lobed ;”. Java, Blume!, Zollinger! 3247, 3438; India, S. Canara, &e. Major Beddome!; Borneo, O. Beccari! n. 1973. Major Beddome l.c. states that the S. Canara plant has the habit of Leucorylum buxifolium, Miq., but he does not regard his plant as the same species with it. According to Zollinger in the Obs. Bot. Noy. p. 18 (1857) the flowers in both sexes are usually pentamerous, the stamens usually 10, free, and the ovary apparently 2-celled. 93. Diospyros VESCOI, sp. nov. D. foliis alternis, obovatis, apice rotundatis, basi angustatis, coriaceis, subtus puberulis, inconspicue reticulatis, confertis, petiolatis, margine revolutis; floribus masculis axillaribus, breviter cymosis, calyce laze hemispherico, 4-fido, extus tomentoso, corolla campanulatd utrinque Mr HIERN, ON EBENACE, 219 tomentosd, breviter 4- rarius 3- vel 5-fidd, lobis obtusis, staminibus 13—16, plerisque geminatis, corolle bast insertis, antheris glabris, apice rimosis, filamentis tomentosis, ovario rudimentario. Young parts ferruginous, shortly pubescent; branches pale, cinereous, terete. Leaves alternate, obovate, rounded or emarginate at apex, narrowed or nearly rounded at. base, coriaceous, puberulous with curved hairs on both sides especially beneath, crowded, 1—3 in. long by §—l%in. wide; petioles 4—4in. long, puberulous, margins revolute, reticulated with delicate inconspicuous veins in faint relief on both sides, midrib slightly depressed above. 3. Inflorescence axillary on young shoots, 4—?in. long, ferruginous, pubescent with short hairs; peduncle ;%—%in. long; pedicels j—}in. long; flowers 1—}in., openly cam- panulate; calyx }—{in. long, hemispherical, tomentose outside, 4-fid, sometimes unequally so, lobes widely ovate-deltoid ; corolla campanulate, shortly 4-fid, occasionally 3- or 5-lobed, tomentose on both sides, lobes ovate-oval, obtuse; stamens 13 16, all or mostly in pairs, inserted near base of corolla, inner ones shorter, anthers glabrous, equal, lanceolate, acu- minate, filaments tomentose; ovary rudimentary, receptacle tomentose. Madagascar, Port Leven, Vesco!, St Marie, Boivin/ 2539 bd. 94. Diospyros MorristaAna, Hance ex Walp. Ann. iii. p. 14 (1852). D. foliis ovalibus, alternis, apice acuminatis, basi angustatis, tenwiter coriaceis, glabris, petiolatis; floribus masculis 3-nis, breviter cymosis, tetrameris, calyce utrinque pubescente, 4-fido, corolla urceolatd, breviter 4-lobd, lobis obtusis, staminibus 16—25, sepius 20, plerisque geminatis, corolle basi insertis, antheris linearibus, pubescentibus, ovarii rudimento glabro ; fructibus glabris, nitidis, subglobosis, 4-locularibus, locus monospermis, seminum albumine non ruminato, calyce fructifero patente, subglabro. A shrub (or tree?) quite glabrous except the buds inflorescence and extremities ; branches dark, terete, spreading at about 30°—35°. Leaves oval, acuminate at apex, more or less narrowed at base, glabrous, alternate, thinly and firmly coriaceous, with recurved margins, 2—3Lin. long by 1—1}in. wide, besides petiole 4—tin. long; shining above; veins few and slight &. Flowers whitish, 1—}in. long, tetramerous, 3 together on short drooping ferruginous- hairy axillary cymes; peduncles and pedicels each about ;';in. long; calyx ferruginous-hairy on both sides, 4-fid, ;;in. long, erect-patent, with deltoid lobes; corolla about .3,in. long, tu- buloso-urceolate in flower, ovate-conical in bud, lobes jin. long recurved, obtuse; stamens numerous, 16—25, usually about 20, mostly united in pairs, outer ones the longer, inserted at base of corolla, about tin. long; anthers linear, apiculate, hairy, dehiscing from apex; filaments short, glabrous; ovary rudimentary, glabrous. @. Flowers unknown. Fruit glabrous and shining, yellow, nearly globular, }—2in. in diameter, 4-celled; cells 1-seeded. Fruiting calyx nearly flat, nearly glabrate, j5in. across; seeds ,,in. long, compressed, chestnut-coloured; albumen cartilaginous, not ruminated. The male flowers appear in May; the fruit gathered in December is edible. Hong Kong, Hance! no. 460, C. Wright! 313. 28—2 220 Mr HIERN, ON EBENACEZ. 95. Drospyros squamosa, Boj. ex Alph. DC. Prodr. viii. p. 232. n. 49 (1844). D. foliis alternis, ovalibus, utrinque obtusis, glaberrimis, coriaceis, petiolatis; floribus masculis, 1—3- sepius 3-nis, sessilibus, bracteis amplis ovato-rotundatis imbricatis calyce vix brevioribus, calyce campanulato 4—5-fido, corolld breviter 4-fidd infundibuliformi, staminibus 22, corolle basi insertis, filamentis pubescentibus ; fructibus cubico-globosis glabris, apice excepto calyce 4-fido aucto ferrugineo-sericeo occultis, stylis 4 brevibus glabris. Branches glabrous. Leaves alternate, oval, rather obtuse at both ends especially at base, coriaceous, quite glabrous, flat, 8{—5in. long by 1}—1}in. wide; petioles jin. long; venation delicate, in relief on upper surface. g. Flowers 1—3 together, sessile, rather more than }in. long, arising from points on the branchlets rather above the (scars of the fallen) leaves; bracts 5—6, imbricated, 7,—} m. long, the outer ones the shorter, roundly ovate, scarcely falling short of the calyx, ferru- ginous-tomentose at the margins. Calyx campanulate 4—5-fid or shortly lobed, 7j—j5 in. long, ferruginous-pilose outside, lobes widely ovate, erect-patent. Corolla funnel-shaped, shortly 4—5-fid (2), subglabrous, exceeding the calyx, lobes obtuse. Stamens 22, inserted at the base of the corolla; filaments short, pubescent, frequently united in pairs. Q. Fruit cubic-globose, glabrous, in. high, concealed except at apex by accrescent calyx; styles 4, short, glabrous. Fruiting calyx crass, ferruginous-sericeous, 4-fid, $in. across, tube tetragonal 3in. high, lobes shortly ovate spreading. Madagascar, near Foul-pointe, Helsonberg!; Chapelier/ Local name, Valanguran. 96. DIOSPYROS COMORENSIS, sp. nov. D, foliis alternis, ellipticis, apice sepius acuminatis, basi angustatis, coriaceis, glabrescen- tibus; floribus masculis 3—4-nis, breviter cymosis, tetrameris, calyce laxe cyathiformi, 4-fido, corolla urceolatd glabrd carnosd breviter 4-lobd, staminibus 16 geminatis glabris corolle bast insertis, ovarii rudimento glabro. Young parts pilosely pubescent ; branches brown, scarcely terete. Leaves alternate, ellip- tical, coriaceous, narrowed at base and usually acuminate at apex, bluish brown above with cleanly depressed midrib and inconspicuous lateral veins, brown beneath with inconspicuous veins, nearly or quite glabrous, 2—2} in. long by {—1+ in. wide including petiole } in., often conduplicate in specimen. 6. Cymes axillary 3—4flowered, about } in. long, pilose, subferruginous, recurved, pedicels }—1 in. long; flowers -3,—2 in. long, tetramerous, ovoid in buds; calyx }—} in. long, pubescent on both sides, 4-fid, lobes erect-patent, deltoid; corolla ;4—% in, long, narrowly ovoid in bud, glabrous, fleshy, lobes much imbricated; stamens 16, placed in pairs in two rows at base of corolla, glabrous, }—35 in. long; anthers linear longer than the filaments, pollen somewhat 4- (?) sidedly ellipsoidal. Ovary rudimentary, glabrous. Female plant at present unknown. Comoro Islands, Mayotte, Boivin / 97. Drospyros Montana, Roxb. Coromand. p. 387. t. 48 (1795). D. trunco ramisque interdum spinosis, foliis alternis, ovalibus vel ovatis, apice obtusis vel acutis, bast interdum cordatis, tenuiter coriaceis, pubescentibus vel glabrescentibus, Mr HIERN, ON EBENACEA, 221 decidurs, petiolatis; floribus masculis breviter cymosis, tetrameris, glabriusculis, calyce late campanulato, profunde 4-fido, lobis ovatis, ciliatis, corolla urceolatd, breviter 4-lobd, staminibus 16, geminatis, glabris vel subglabris, corolle basi insertis; jfloribus femineis solitariis, breviter pedunculatis, staminodiis 4—12, glabris, ovario glabro globoso, 8-loculari, loculis l-ovulatis, stylis 4, glabris, seminibus 2—8, albumine non ruminato; calyce fructifero paulum aucto, plus minus reflexo. Wall. List n. 4115, Alph. DC. Prodr. vit. p. 230. n. 34 (1844), Wight Ic. t. 1225 (1850). D. cordifolia, Roxb. Lec. p. 38. t. 50 (1795); Wall. List n. 4116; Alph. DC. lc. n. 36 ; Wight, Illustr. Ind. Bot. Vol. m. t. 148 (1850). D. rugosula, R. Br. Prodr. p. 526 (1810). D. bracteata, Roxb. Cat. Pl. Fl. Ind. (1813); Fl. Ind. edit. 1832, Vol. m. p. 539 ex specimine in Hb. Mart.!; Alph. DC. lc. p. 239. n. 93. D. heterophylla, Wall. Cat. Burm. 599, List n. 4138 (1828—82), Alph. DC. l.c. p. 230. n. 39. D. sylvatica, Wall. List n. 4117! (1828—82), 8 velutina, Alph. DC. /. c. p. 231. n. 41 var., non Roxb. D. punctata, Decaisne, in N. Ann. Mus. Hist. Nat. m1. p. 407 (1834) ; Herb. Timor. Discr. p. 79 (1835); Alph. DC. Zc. p. 230. n. 37. D. rugulosa, Alph. DC. Prodr. vim. p. 229. n. 82 (1844). D. Goindu, Dalz. in Kew Journ, tv. p. 111 (1852). D. Waldemarwi, Klotzsch in Waldemar Reise, p. 101. t. 55 (1862). Yerra-gada of the Telingas (R. montana, Roxb.) ex Roxb. l.c.; Kak-woolymera of the Telingas (R. cordifolia, Roxb.) ex Roxb. l.c.; Vakanoi, Neilygerry Mts., base, Leschanault ! 198 (large tree), seen in Hb. Mus. Paris; TZumala, the Sanscrit name, Bun-Gaub, in Bengal, ex Roxb. Fl. Ind. (edit. 1832) vol. 1. p. 538; Kala Gorndu in Canara, Kala Nuddi, teste Dr Ritchie; Makar Kend, Hindwi dialect of Behar, ex Hamilt. in Tran. Linn. Soc. xv. p. 113 (1827); Gavindw or Goindu, ex Graham, Cat. Pl. Bomb. p. 108 (1839) ; Jugalagunti (signifies scolding wife), ex Buchanan, Journey, vol. I. p. 183 (1807). A tree often with spines scattered over the trunk and larger branches; young branches softly pubescent, of a pale colour. Leaves oval, oblong, obovate, or ovate-oblong, alternate, sometimes cordate at base, thinly coriaceous, of nearly the same yellowish-green colour (in the dry state) on both sides, softly pubescent or glabrescent beneath, softly puberulous or glabrous above, with depressed midrib and weak veins, deciduous, 1—4— 5 in. long by 4—2—2} in. wide; petioles 3,—}in. long, pubescent or glabrescent. Flowers white, scentless. $. Cymes 8-flowered or panicled, }—?in. long, patent or recurved; bracts ovate, ciliate, ;;in. long; at base of the pedicels; flowers }+—, in. long. Calyx 4 or 4 length of flower, deeply 4-fid, on both sides pubescent or nearly glabrous, with deltoid or rounded ciliated lobes. Corolla urceolate, shortly 4-lobed; lobes rounded, recurved; glabrous or nearly so. Stamens 16, united at base in 8 pairs and inserted at base of corolla, glabrous or very nearly so, with very short hairs, appearing at mouth of open corolla. Ovary rudimentary, glabrous except apex. Q. Flowers solitary, }—2in. long, on recurved peduncles ;,—4 in. long, which bear small caducous bracts. Calyx puberulous or nearly glabrous, deeply 4-fid, {—} in. long, 222 Mr HIERN, ON EBENACEZ. with imbricated often ciliated lobes. Corolla rather exceeding the calyx, glabrous, 4-fid. Staminodes 4, 8, 12, glabrous. Ovary glabrous, globular, 8-celled, cells l-ovuled; styles 4, glabrous, bifid at apex. Fruit globose, }—1}in. in diameter, glabrous and shining; fruiting calyx more or less reflexed, somewhat accrescent; seeds 2—6—8, albumen not ruminated (in D. rugosula, R. Br., there are two contiguous slight intrusions of the testa along the outer side of the seed). The wood is dark-coloured or variegated, hard and durable. Dr Dalzell states that bees are very fond of the flowers. There are two principal forms: a. montana proper. D. montana, Roxb, D. Goindu, Dalz. D. heterophylla, Wall. Leaves oval, 3—4 in. long. ¢ flowers panicled, with calyx glabrous except ciliate margin. @ flowers with 4 staminodes. B. cordifolia. D. cordifolia, Roxb., D. punctata, Decaisne, D. rugosula, R. Br. D. Waldemarii, Kl. Leaves oblong, often cordate at base, 1—2}in. long. ¢ flowers 3 together with hairy calyx. 9 flowers with 8 (D. Waldemarii) or 12 staminodes. India, Madras, Shuter/; Othacalmundapum, Kew list 1724!; Patna, Dr Ritchie! 1240; Moradabad, Dr T. Thomson! 985; ottler! 361; Sirhind, Dr YT. Thomson!; Bengal, Edgeworth! 6006; Ambala, Edgeworth!; Pinjor Valley, Edgeworth!; Ceylon, Thwaites! C. P. 1909; sea coast, Tinnevelly district, Kew list 1717!; Pondichéry, Perrottet! Sikkim, Dr Hooker! ; Courtallum, Kew list 1713!, 1726!; Bombay, Dalzell/; Concan, Dr Stocks! ; Canara, Dr Ritchie! 970; Belgaum, Dr Ritchie 972; Himalaya, Hoffmeister, teste KI. l.c.; Ava, Wallich! ; India, Magadi, Hejuru,S. W. Mysore. Timor, Decaisne! N. Australia, Victoria River, #. Mueller/, Carpentaria, &. Brown/; Australia, Port Darwin, Schultz! na. 607, 608. The natives are prejudiced against this tree. Buch. Ham. Journey, vol. 1. p. 183, vol. m. 125. Cfr. Diospyros sp. Bedd. in Clegh. For. 259 (1861), Muchi tanki; a very hard light- coloured wood, Godavari forests, Madras. 98. DiIospyROS ZOLLINGERI, sp. nov. D. foliis alternis, obovato-oblongis, apice acuminatis, basi plerisque rotundatis, coriaceis, glabrescentibus, petiolatis; floribus masculis axillaribus cymosis tetrameris, fulvo-pubescentibus, calyce campanulato, lobis deltoideis, corolld breviter 4-fidd, campanulatd, staminibus 16, geminatis, glabriusculis, ovartii rudimento glabro. Young parts and inflorescence puberulous or pubescent. Leaves alternate, obovate- oblong, acuminate at apex, usually rounded at base, glabrescent, 4—8 in. long, by 1—24 in. wide; midrib and lateral veins depressed above and in clear relief beneath; petioles about } in. long. 8. Inflorescence in short cymes axillary or in the axils of fallen leaves. Abortive buds in some cases are arranged in a panicle. Flowers tetramerous, tawny-pubescent; calyx campanulate, jin. high by }in. wide, 4-fid, nearly glabrous inside, lobes deltoid; corolla (in bud) }in. long, ovoid, shortly 4-lobed, appressedly pubescent; stamens 16, united in 8 pairs at the top of the filaments, nearly equal, inserted at the base of the corolla, not quite glabrous, but with short hairs on the back of the anthers and on the Mr HIERN, ON EBENACE. 223 upper part of the filaments; anthers dehiscing widely on both sides downwards from apex, pollen subglobose, smooth; ovary rudimentary, glabrous. Java, Zollinger! n. 2651. A specimen from Assam, collected by Col. Jenkins, has also abortive buds arranged in a considerable panicle; it does not however appear to belong to this species, having a somewhat different foliage, resembling in this respect D. variegata; Kurz. 99. Drospyros cimt1aTa, Alph. DC. Prodr. vin. p. 229. n. 31 (1844), non Rafin. D. folus alternis, ovato-ellipticis, basi obtusis, apice acuminatis acutisve, ciliatis, membra- naceis; floribus femineis axillaribus, breviter pedicillatis, tetrameris, calyce partito, lobis ova- tis obtusis, corolld campanulatd. Branches glabrous. Leaves alternate, ovate-elliptical, obtuse at base, acuminate or acute at apex, ciliate, 2—3in. long (including petiole 55,in. long) by 1—1} in. wide, membranous, with the nervation of the leaves as in D. virginiana except that the margin is ciliate and the acumen is usually acute. 2. Flowers axillary, on glabrous pedicels much shorter than the petiole or flower, tetramerous or sometimes pentamerous, 4in. long. Calyx 4-partite, silky inside at base, with ovate obtuse reflexed lobes; corolla glabrous, campanulate, narrower above, 4-fid, with obtuse lobes. Styles 4, united to the middle, glabrous, longer than the calycine lobes. Fruit edible. S. Mexico, Pavon! 100. Drospyros Lotus, Linn. Sp. Pl. p. 1057 (1753). D. foliis alternis, ovalibus, utrinque sepius obtusis, submembranaceis, subtus sepe pallidiori- bus et pubescentibus, petiolatis; floribus masculis 2—3-nis brevissime cymosis, subsessilibus, urceolatis, 4- rarissime 5-meris, axillaribus, calyce campanulato, lobis acutis, corolld breviter lobatd, staminibus 16, geminatis, antheris glabriusculis, filamentis glabris; floribus femineis solitariis, staminodiis 8, ovario apice excepto glabro, 8-locularibus, fructibus subglobosis, edulibus. Pallas, Fl. Ross. t. 58 et t. 59 fig. inferior, tom. I pars. 11 p. 20 (1788). Poir. in Lam. Encycl. Méth. vol. v. p. 428 (1804), t. 858 fig. inf. (1823). Nouveau Duhamel, vol. vi. p. 83. t. 26 (1801—19). Turpin, Dict. Sc. Nat. Planch. vol. m1. t. 65 (1816—29). Alph. DC. Prodr. vu. p. 228. n. 28 (1844). Reichenb. Pl. Ic. Fl. Germ. et Helv. (xvi) t. 1079 (1855), non Lour. Fl. Cochinch. p. 226 (1790). Ermellinus, Cesalp. De Plantis, lib. m1. cap. XxI. p. 104 (1583). Pseudolotus, Camer. Epit. p. 156 (1586). Lotus africana altera, Camer. Epit. p. 157 (1586). Lignum Vite, Gerarde Herball, p. 1309 (1597). Guaiacum patavinum, Gerarde Herball, p. 1310 (1597). 224 Mr HIERN, ON EBENACEZ. D. Kaki, var. 8. Thunb. Fl. Japon. p. 158 (1784), var. y. glabra, Alph. DC. Prodr. vim. p. 229. n. 30 (1844); non Linn. D. microcarpa, Sieb. in Ann. Soc. Hortic. Pays Bas 1844, p. 28. D. japonica, Sieb. et Zuce. in Abh. Bayer. Acad. Iv. 3. p. 136 (1846). D. Umlovok, Griff. Itin. Not. p. 355 n. 137. (1848). Dactylus trapezuntinus, Forskal Fl. Agypt—Arab. p. XxXvi. (1775). A dicecious moderate-sized tree or shrub, from 15 ft. high upwards; bark dark, rough, scored, but less so than in D. virginiana, L.; young parts pubescent. Leaves alternate, submembranous, more or less elliptical, usually paler beneath and often pubescent, 2—6 in. long by 1—2}in. wide; petioles 1—$in. long. Flowers tetramerous, or by exception pentamerous, axillary. &é. Flowers subsessile, 2—3 together, about 1 in. long, urceolate. Calyx campanulate, about } in. long, shortly 4-fid; lobes ovate, acute. Corolla urceolate-oblong, nearly or quite glabrous, 1rd way 4-lobed; lobes ciliate, obtuse, recurved. Stamens 16, combined by their glabrous filaments in 8 pairs; two pairs opposite each corolla-lobe; each pair consists of a shorter inner and longer outer stamen; filaments inserted at base of corolla-tube; anthers not quite glabrous. Ovary rudimentary. @. Flowers solitary, subsessile, wider than in the g. Calyx ultimately spreading. Corolla often remaining at apex of the young fruit, urceolate, yellowish white. Staminodes 8, in one row inserted at the base of the corolla, hairy. Ovary glabrous, except at apex from which 4 hairy lines often descend down the fruit, 8-celled, cells 1-ovuled; styles 4, some- what pubescent below. Fruit subsessile or apparently sessile, often with a glaucous tinge, subglobose, 2—#, in. thick; fruiting calyx spreading, }—% in. across, with a ring of short dense appressed silky hairs on the inside below the fruit. Flesh of the fruit astringent. Naturalized in the countries on the shores of the Mediterranean Sea. Russia in Asia, Pallas, called Kurma by the Persians, Churma or Kard-churma in Tartary, Dikot Phenik in Astracan; Asia Minor, Zohrab/; Turkey in Asia, Lazistan, near Rhize, spontaneous, Boissier/ n. 1464; Caucasus; China, Hance/ n. 13753, Canton; Pekin Mountains, Bunge ; Zsing Yune Pass, along North river, about 120 miles from Canton, Hance; Affghanistan, Griffith / n. 1289; N. W. India, common on the Huzara from 3000—6000 ft. alt., male plant called Gwaladar, female Amlok, Dr Stewart! n. 424; Tsu-sima Island, Straits of Corea, Wilford/; Japan, Nagasaki, Oldham! n. 529, called Sinanokaki. 101. Drospyros vire@rntANaA, Linn. Sp. Pl. p. 1057 (1753). D. foliis alternis, ovalibus, utrinque obtusis, submembranaceis, pubescentibus vel glabrescen- tibus, petiolatis, floribus masculis 1—38-nis, breviter cymosis, axillaribus, 4- rarius 5-meris, urceolatis, calyce campanulato, lobis lanceolatis, corolld breviter lobatd, staminibus 16, geminatis, paulum pubescentibus; floribus femineis solitarvis breviter pedunculatis, staminodiis 8, ovario apice excepto glabro, 8-locularibus, fructibus subglobosis, edulibus. Gaertn. fil. Carp. (111) p. 188. t. 207 (1805). Michaux, Arb. Amer. Septr. 11. p. 195. t. 12 (1812), Collin, Férslag af nagra Nord-americas Triid, p. 28 (1823). bo bo or Mr HIERN, ON EBENACEA, Watson, Dendr. Brit. 1. t. 146 (1825). Rafinesque, Medic. Fl. N. Amer. i. p. 153. t. 32 (1828). Alph. DC. Prodr. vil. p. 228. n. 29 (1844). Belgique Horticole, Iv. p. 118. tab. (1854). Ettingsh. Blatt-skel. Dikot. p. 89. t. 38. f. 12 (1861). Pishamin, Parkinson, Paradis. p. 570 (1629), Theatr. p. 1523. f. 4 (1640). concolor, Moench, Meth. p. 470 (1794). guaiacana, Robin, Voyages, vol. 1. p. 417 (1807). pubescens, Pursh, Fl. N. Amer. p. 265 (1814), non Pers. caroliniana, Muhlenb. ex Rafin. Florul. Ludovic. p. 139 (1817). Persimon, Wikstr. Jahr. Schwed. 1830, p. 92 (1834). ciliata, Rafin. New Flora and Bot. N. Amer. part II. p. 25 (1836), non Alph. DC. fertilis, Lodd. Cat. ex Loud. Arb. et Frut. Brit. 1. 1197 (1838). calycina, Audib. Cat. Hort. Tonn. 9. ex Spach, Hist. Végét. rx. p. 405 (1840), non Wall. &e. angustifolia, Audib. ex Spach, Hist. Végét. 1x. p. 405 (1840). lucida, Hort. ex Loud. Gard. Mag. 1841, p. 394, non Wall. intermedia, Hort. ex Loud. Encycl. Trees and Shrubs, p. 627 (1842). tree attaining in favourable places 60 feet in height and 20 in. in diameter in the trunk, according to Michaux, from whom other details are taken. The trunk of full-grown FSSSSSESSESSESS trees is covered with much and deeply-cracked blackish bark; the sap-wood after drying keeps a clear greenish colour, and the heart is brown. The wood is hard, compact and tough, and is used for several mechanical purposes. The inner bark is said to be useful in intermittent fevers. Young parts pubescent. Branches spreading at 50°’—60°. Leaves alternate, submembranous, more or less oval, slightly narrowed, rounded or even slightly cordate at base, usually shortly acuminate at apex, paler beneath and often pubescent; 2—7 in. long (besides pubescent petiole }—Zin. long) by 1—3} in. wide. Flowers tetramerous or occasionally pentamerous, greenish. $. Flowers in short 1—8-flowered pubescent cymes which measure (excluding the flowers) about 4in. long. Calyx small, about 4, in. high, partite, hairy, with lanceolate lobes. Corolla tubular-urceolate, } in. long, or in subhermaphrodite flowers } in. long, lobes one-third the length of the corolla. Stamens 16, in pairs, somewhat hairy. Ovary glabrous, rudimentary. 2. Flower solitary, } in. long and wide, on peduncles =, in. long; ovary glabrous, pilose at apex, 8-celled, cells 1-ovuled; styles 4, pilose at base. Fruit solitary, on peduncles { in. long, subglobose, 1—1} in. in diameter, glabrous, edible, tipped at apex with remains of style; skin thin, of a pale orange-colour when ripe, often marked externally with 4 depressed lines running down from the apex, and with a slight pruinose bloom; pulp with a sweetish apricot- like taste when ripe but somewhat astringent; seeds 6—8, sometimes 3—5, about 3 in. long, 8 in. broad and } in. thick. Fruiting calyx spreading, 4-fid, occasionally 5-fid (in one case small and trifid in a cultivated specimen), ?—1} in. across, subglabrous; lobes broadly ovate, 13 in. broad, usually somewhat concave from below and not appressed to the fruit, with recurved margins; tube convex from above with a circular depression at its outer margin. The fruit, which is locally known by the name of Persimon, does not fully ripen north Vou. XII. Part L 29 226 Mr HIERN, ON EBENACE. of New Jersey; it is said to become better fit to eat after it has suffered frost, and then it becomes very sweet but mawkish. Though eaten by the negroes, and often brought to market, it is not a table-fruit. There is however a sweet variety (D. virginzana, L. var. dul- cis), which is said to yield a good table-fruit. “For an interesting account of the properties of the tree and its fruit, see the inaugural thesis of the late Professor Woodhouse, of the University of Pennsylvania.” Darlington, Florula Cestrica, p. 47 (1826). Two other inaugural essays have been devoted to the study of the fruit of this tree; one by Benj. R. Smith, printed in the American Journal of Pharmacy, October, 1846, pp. 161 —167, and the other by John E. Bryan, in the same journal, May, 1860, pp. 215—217. From these essays the following results are taken. The fruit contains tannin, pectin (or perhaps malic acid), sugar, lignin and colouring matter and neither vegetable albumen starch nor resin. Of 600 grains of green persimon there were found to be 119 grs. of insoluble resinous matter, 64 grs. of saccharine matter slightly acid, 22 grs. of ligneous matter, 1 of green colouring matter, and the remaining 394 grs. were supposed to be water. It is further supposed that in the young fruit lignin serves as a sort of frame-work and as a means of circulation for the juices of the plant; but as the fruit ripens the lignin is converted into sugar, 20 parts of lignin producing 21 parts of sugar. The astringent principle is tannin analogous to that of Cinchona, Catechu, &c., and different from that of galls and oak-bark; and the fruit retains its astringency when dried carefully. An astringent and styptic. The inner bark is used in intermittent fever, in diarrhea, and with alum as a gargle in ulcerated sore throat. An indelible ink can also be made from the fruit. (See Resources of the Southern Fields and Forests, by Dr Porcher, pp. 423—427, Charleston, 1869.) In the southern United States of N. America the fruit hangs during part of the winter on the tree a long time after the fall of the leaves; and when at length it too falls, it is eagerly eaten by both wild and domestic animals. In Virginia, Carolina and the western States, the fruit is gathered, kneaded with bran, made into cakes and baked. These cakes mixed with tepid water serve to make beer with the addition of hops and yeast to cause fermentation. Spirit is also distilled by further fermentation. Neither the beer nor the spirit is made for the purposes of commerce. This species with its varieties has a foliage exceedingly like D. Lotus, L.; it differs from the latter by the male cymes and female peduncles being rather longer and by the larger flowers and fruit. Some specimens with regard to which the native country is unknown, though clearly belonging to one of these species, are extremely difficult to assign to either one of them with certainty. Michaux also speaks of a variety with smaller fruit, compressed seeds, and leaves pubes- cent underneath: this is D. pubescens, Pursh. Fl. N. Amer. p. 265 (1814) and the var. 8. micro- carpa, Rafin. Med. Fl. 11. p. 153. t. 32. A variety is occasionally met with in Sumter district, S. Carolina, with fruit about twice the ordinary size (Dr Porcher). DD. intermedia, Hort. is a variety with more numerous (about 20) hairy stamens. Polygamous flowers occur in cultivated specimens of this species. North America, United States, St Louis, Drummond!, Riehl! n. 178; New Orleans, T. Drummond! n. 204 bis; “Woods and old fields, Rhode Island and New York to Illinois, Mr HIERN, ON EBENACE. 227 and southward,” Asa Gray; “Florida! to Mississippi and northward,” Chapman; Cumberland, Olney! ; Missouri, Buckley! ; Virginia, Portsmouth, Rugel/, A. Gray! ; Kansas, Engelmann. It is cultivated in British Guiana, and has long been introduced into Europe. 102. Diospyros Kaki, Linn. fil. Suppl. p. 439 (1781). D. foliis alternis, ovalibus, utrinque obtusis vel angustatis, submembranaceis, subtus pubescentibus, petiolatis ; floribus masculis axillaribus, ternis, cymosis, tetrameris, urceolatis, calyce campanulato, lobis ovatis vel lanceolatis, corolld extus pubescente, staminihus sepius 16, geminatis, leviter pubescentibus ; jloribus femineis sepius solitariis, pedunculatis, staminodiis sepius 8, ovario sepius 8-loculart, fructibus globosis edulibus sepe magnis. Wight Ic. t. 415 (1840); Alph. DC. Prodr. vit. p. 229. n. 30 excl. var. y. glabra (1844); Thunb. Fl. Jap. p. 157 excl. var. 8. (1784); non Blanco. Ficus hortensis, fructu ossiculato eduli, folio Pyri, Kempf. Amcenit. exotic. p. 805 (1712). (2) D. lobata, Lour. Flor. Cochinch. p. 227 (1790); Alph. DC. lc. p. 233. n. 53. D. chinensis, Blume, Catal. Buitenz. p.110 (1823); Flora, 1825, p. 254; Bijdr. Fl. Ned. Ind. p. 670 (1825). D. Schi-Tse, Bunge, Enum. Pl. Chin. Bor. n. 237. p. 42 (1832). Embryopteris Kaki, G. Don, Gen. Dict. Gard. and Bot. Iv. p. 41 (1837). D. costata, Carr. in Rey. Hortic. 1870, p. 134 (fig. p. 133). D. Kaki var. costata, André in L'Ilustration Hortic. vol. xvi. p. 176. t. 78 (1871). D. Roxburghi, Carriere in Revue horticole, 1872, p. 253. fig. 28, 29. Local names ; Ono Kaki, Kempfer Amenit. pp. 805, 807. fig. p. 806 (1712) ; Kakwe, Javan name ex Bl. Bijdr. l.c.; Ahi, Rumph, Herb. Amboin. vol. 1. p. 137 (1750). A small tree; young branches inflorescence and underside of leaves pubescent or sub- tomentose. Leaves alternate, submembranous, more or less oval and acuminate at apex, paler beneath, 2—7 in. long by 1—3}in. wide; petioles 1—2in. long. Flowers pedunculate, dicecious or polygamous, tetramerous. $. Cymes axillary 1—+in. long, 3-flowered; pedicels about jin. long; flowers usually drooping, variable in size, 1—}in. long; calyx slightly hairy, with 4 deep ovate or lanceolate lax lobes, shorter than the corolia; corolla hairy outside, urceolate, yellowish-white ; stamens 16 (1424), in pairs, more or less hairy, filaments short. " 9. Flowers usually solitary, or pubescent, axillary; 2-bracteate peduncles ;,—? in. long, dilated and articulated to fruit at apex; calyx large, hairy on both sides, deeply 4-fid, about lin. or more wide, with widely ovate spreading lobes, cordate at the base, much accrescent in fruit at least in most cases, with a thickened and hairy shallow tube in fruit; corolla pu- berulous, about 1—} in. high and wide, 4-fid, with 4-oval recurved lobes; staminodes 8 (or 16%); ovary 8—10-celled, glabrous or nearly so; style hairy, 4fid; fruit glabrous or nearly so, glo- bular, sometimes as big as an orange, reddish or yellow, 8—10-celled [?in D. lobata, Lour. 1 in. in diameter and lobed], in D. costata, Carr. 2in. in diameter and more or less deeply ribbed or lobed. The Chinese preserve the fruits with sugar. This species has been for a long time under cultivation in China, Japan, &c. and presents much variety in the size and shape of its fruit. By the kindness of M. Carritre I have been 29—2 228 Mr HIERN, ON EBENACEAL. supplied with specimens of his D. costata, and I am also indebted to M. Decaisne for the inspection of original drawings of other forms of this species. On the whole view of the case I prefer to consider all as belonging to one species, which has under cultivation assumed much perplexing variation. Some varieties are considerably more hardy than others; the foliage also in some forms is fine and shining, in others smaller and more pubescent. D. lobata, Lour. may very possibly belong to this species, but the fruit is described as only lin. in diameter. D. Kaki, L. £. is closely allied to D. Lotus, L. and D. virginiana, L., all of which species are very variable and indeed are likely to be confused amongst each other ; D. virgin- iana, L. holds a middle position in respect of the size of the fruit and the length of the inflorescence. Fruit [see Hasskarl in Bonplandia, vil. p. 255 (1859)] globose, with a diameter of 2 in. or very depressedly globose, 2}in. wide and 1$in. high, glabrous and shining, scarlet; skin thin, membranous; flesh of an orange-scarlet colour, edible, sweet, with yellow fibres joined at the base, and then forked and longitudinally dispersed towards the surface; fruiting calyx with a brick-coloured tube and green reflexed lobes. Seeds laterally compressed; in the globose fruits 6, oblong, one face nearly straight the other convex, blunt at the apex, acute at the base, a little produced laterally, lin. long, }in. wide; in the depresso-globose fruits 8, one face rather straight the other more than semi-orbicular, widely rounded at the apex, rather acute at the base, scarcely produced; all dark, smooth, well wrapt in the fleshy pulp of the mesocarp, and when carefully removed from the flesh rather shining, marked on the convex face along its whole length with an acute yellowish raphe jin. thick. Testa thin, coria- ceous; albumen milk-white, cartilaginous; embryo small in proportion to the albumen, straight ; radicle terete, very slightly curved or usually straight, white, }in. long; cotyledons thin, whitish, lying parallel side by side, in the globose fruits ovate acute jin. long by ;;in. wide, in the depresso-globose fruits subrotund, $ in. in diameter. Japan, Nagasaki, fr. ripe in Oct., Oldham! 528, C. Wright! ; Tsu-sima Island, Str. of Corea, C. Wilford! 756, @ fl. May; Formosa, Oldham! n. 299; Khasia, Dr Hooker!, fruit, Sept. For a discussion of D. costata, Carr., in addition to the above reference, see a note by M. Carritre on D. Kaki in Rev. Hortic. 1869, p. 284; a letter of M. Decaisne in the Gardener’s Chronicle for 1870, p. 39; a letter of M. Carrigre in Gard. Chr. 1870, p. 312; and a paper with woodcuts and coloured plate by M. Carritre in Rey. Hortic. 1871, p. 410; also André in L’Illustr. Hortic. loc. cit. where the same coloured plate is given, and the Gardeners Chronicle for 1872, p. 576, whence the accompanying figure has been obtained. M. Carritre describes his D. Roxburghi as a moneecious shrub, at times subdiccious by abortion; the male flowers very numerous in comparison with the female and bearing 15—20 stamens; the fruit 12 in. (2 in. in figure) in diameter, with numerous brown prominences especially towards the apex. He thinks it identical with the D. Kaki, Roxb. from India, but quite distinct from the D. Kaki, Thunb. from Japan ; he finds it considerably more tender and sensitive to cold and much less productive of fruit. It must however be borne in mind that M. Carritre has described his species from a cultivated specimen, and also that the whole group of Kaki has been long cultivated in Japan, China and elsewhere, and thus it may be expected Mr HIERN, ON EBENACEZ. 229 230 Mr HIERN, ON EBENACE. that many differences have been acquired under cultivation extended over a wide geographical area which, while worthy of distinct notice in a horticultural point of view, yet do not properly amount to specific importance. The allied species D. virginiana, L. and D. Lotus, L. are in a similar manner subject to great variation and from similar causes. I have examined the herbaria both of Linnzus and of Sir J. E. Smith (including that of Linnzus the younger) without finding an authentic specimen of D. Kaki; but I think there is no doubt but the D. Kaki of Thunberg is identical with that of Linn. fil. 103. DIospyros CHARTACEA, Wall. List n. 4135 (1828—382). D. glabra, foliis alternis, elongato-lanceolatis, apice acuminatis, basi obtusis, submembra- naceis, minute pellucido-punctatis, breviter petiolatis; floribus masculis ternis, subsessilibus, tetrameris, calyce elongato-cylindrico, 4-fido, lobis ovatis, ciliatis; corolld brevi, 4-fidd, lobis obtusis, staminibus 16—20, geminatis, antheris pilosis, ovarii rudimento glabro. Alph. DC. Prodr. vit. p. 282. n. 51 (1844). Glabrous or very nearly so. Leaves elongate-lanceolate, acuminate at apex, rounded or somewhat narrowed at base, submembranous, minutely pellucid-punctate, alternate, 2—93 in. long, by }—3 in. wide; petioles ;,—} in. long; lateral veins prominent below. 6. Flowers subsessile, subglobose (in bud), 1; in. in diameter, ternate, crowded, in small very short cymes; bracts 2 or 3 times shorter than the flowers, ovate, acute, subciliate. Flowers {in. long, quite glabrous except the margins of the ciliated calyx-lobes. Calyx shortly cylindrical, 4-fid, with ovate rounded much imbricated lobes. Corolla (in bud) scarcely longer than the calyx, 4fid; lobes ovate, obtuse. Stamens 16—20, in pairs; anthers shortly pilose, filaments short, ;in. long (in bud). Ovary rudimentary, glabrous. Burma, Trogla hills, Bank of Sallun, Wallich! 4135. 104, Diospyros VAccrINIoIDES, Lindl. in Hook. Exot. Fl. t. 139 (1825). D. fruticosa, foliis alternis, ovalibus, apice apiculatis sepe acutis, basi angustatis, supra glabris nitidis, coriaceis, subsessilibus; floribus masculis 1—3-nis subsessilibus aaillaribus 4-meris, bracteis ovatis ciliatis imbricatis, calyce 4-partito, corolla lobaté, lobis lanceolatis acutis patentibus, staminibus 16, geminatis, glabris, corolle basi insertis ; floribus femineis solitarvis, subsessilibus, staminodiis 4—8, glabris, uniseriulibus, ovario inferne glabro, 8-loculari ; Jructibus globosis vel ellipsoideis, albumine non ruminato, calyce non accrescente. Loddiges, Cab. t. 1549 (1829); Wall. List n. 4130 (1828—32). Rospidios vaceinioides, Alph. DC. Prodr. vu. p. 220 (1844), Benth. Fl. Hongk. p- 210 (1861), Hance in Ann, Sc. Nat. ser. V. vol. v. p. 227 (1866). Non Vaccinium Sprengelii, Wall. List 6296! Vide Voigt, Hort. Suburb. Calcutt. p. 333 (1845). Vaccinium fragrans, Wall. ex Voigt, l.c. p. 345. D. vaccinifolia, Ettingsh. Beitr. Kenntniss. Foss. Fl. vy. Sotzka in Untersteiermark, in Sitzungsberichte der Math,-naturw. Cl. Kais. Akad. Wissensch. Xxvul. p. 494. t. v. fig. 5—6 (1858). Mr HIERN, ON EBENACE. 231 A small, erect, twiggy, leafy and evergreen shrub, resembling Buaus sempervirens. Branches covered with shaggy rufous silky hairs when young, then puberulous and finally glabrate, spreading at about 30° Leaves oval, apiculate and often acute at apex, more or less narrowed at base, alternate, subsessile, coriaceous, appressedly pubescent beneath when young, shining and glabrous above; 3—14in. long by }—%in. wide; midrib depressed above ; without conspicuous veins. é. Flowers in. long, tetramerous, drooping, subsessile, in very short 1—3-flowered axillary rufous-hairy cymes; bracts ovate, ciliated, imbricated, caducous. Calyx 4-partite, with 4 lanceolate-subulate erect-patent lobes Lin. deep pubescent on both sides. Corolla scarcely longer than the calyx, 4-fid, with lanceolate spreading lobes, with 4 hairy lines arid. Stamens 16 (12 ex Alph. DC.), glabrous, in pairs, the iuner ones shorter, Gneented at the base of the corolla-tube; anthers rather shorter than the filaments, dehiscing laterally from apex. Ovary rudimentary, hairy. 9. Flowers solitary, subsessile. Bracts caducous. Calyx 4—2in. long, hairy, 4-partite, with linear-lanceolate lobes. Corolla shorter than the calyx, 4-fid, with 4 hairy lines outside and lanceolate acute recurved lobes. Staminodes 4—8, glabrous, in one row, inserted at the base of the corolla-tube. Ovary ovoid, prolonged at apex into a pubescent 4-lobed style, glabrous below; (3-celled according to Lindley) 8-celled; cells 1-ovuled. Fruit globose or ellipsoidal, shining, usually glabrous except at apex, “3-celled,” about }in. high; cells 1-seeded; albumen not ruminated, cartilaginous; embryo axile, straight, cotyledons large foliaceous. Fruiting calyx not accrescent. China; Hong Kong, Major Champion!; C. Wilford !; Millett!; Hance! 604; C. Wright! 312; S. China, Seemann! 2454; Malacca, Griffith! 3643; Singapore, Penang, &e. Walker !, Wallich ! Var. pellucido-punctata. Leaves pellucid-punctate, thinly coriaceous. Fruit with scat- tered hairs. South Andaman, S. Kurz! 105. D1ospyROS CAYENNENSIS, Alph. DC. Prodr. vii. p. 224. n. 8 (1844). D. foliis alternis, oblongis, obtusis, basi angustatis, supra nitidis glaberrimis, subtus glabrescentibus, coriaceis, petiolatis; floribus 1—3-nis brevissime cymosis subsessilibus ferru- gineo-velutinis, calyce turbinato profunde 4-lobo, corolla calyce sublongiore, staminibus 10—12, glabris, ovario in flore femineo ovoideo glabro 8-loculari. Danzleria axillaris, Bert. ex Alph. DC. 1. c. Young shoots and flowers ferruginous-velutinous. Leaves alternate, oblong, obtuse or obtusely acuminate, abruptly narrowed at base, coriaceous, shining and quite glabrous above, glabrescent beneath, green on both sides in the dry state, 3—5in. long by 1—1jin. wide, reticulated; more or less revolute at the margin, petioles $in. long or shorter. Flowers drooping, puberulous, tetramerous; peduncles much shorter than the calyx. &. Flowers solitary or 3 together. Calyx 4in. long; lobes sub-erect, widely ovate, undulated, silky on both sides, thickened within over a triangular space at base. Corolla ovoid, sub-tetragonal, contorted in bud, fleshy, somewhat hairy outside. Stamens 10—12, glabrous, distinct or in pairs ; anthers subulate. 232 Mr HIERN, ON EBENACE. ©. Flowers axillary, 1—3 together, }in. long; pedicels equalling the petioles. Calyx curbinate at base, deeply 4-fid; lobes wide, cordate. Corolla silky outside, rather longer than the calyx. Ovary ovoid, glabrous, 8-celled, fruiting calyx nearly fin. high by more than }in. wide, 4fid, with reflexed undulated lobes and shallowly cup-shaped crass tube having internal elevated rim, appressedly and shortly hairy inside. Cayenne, French Guiana!; cultivated in Jamaica, Berter/, but not mentioned in Grise- bach’s Flora of the British West Indies. 106. Dtospyros LXVIs, Boj. ex Alph. DC. Prodr. vim. p. 232. n. 50 (1844). D. glabra, foliis alternis, ellipticis, utringue angustatis, coriaceis, breviter petiolatis ; floribus masculis, 1—3-nis, subsessilibus, tetrameris, calyce campanulato, corolle lobis obtusis, staminibus 16, geminatis, glabris. Glabrous. Branches slender, black in dried state. Leaves alternate, elliptical, attenuate at both ends, rather obtuse at very apex, coriaceous, revolute at the margins, 3in. long by 14—1} in. wide; lateral veins scarcely conspicuous; midrib depressed above ; petioles } in. long. &. Flowers solitary or 3 together, subsessile, tin. long, glabrous; calyx campanulate, shortly 4-fid, with acute ciliated deltoid lobes; corolla shortly 4-fid, double the length of the calyx or less, lobes widely ovate, obtuse; stamens 16, in pairs, unequal, apiculate, glabrous. Madagascar, East coast, Bojer! , Helsonberg ! 107. Diospyros THOUARSH, sp. nov. D. glabra, foliis alternis, ellipticis, utrinque paulum angustatis, coriaceis, subsessilibus ; floribus masculis, aggregatis, brevissime cymosis, tetrameris, urceolatis, calyce parvo, 4-lobo, staminibus 12, glabris, in flore femineo paucis, ovario ovoideo, 8-loculuri. Dark, glabrous except the bracts. Leaves alternate, elliptical, subsessile, coriaceous, somewhat narrowed at both ends, obtuse; veins reticulated, inconspicuous; midrib somewhat depressed above; 1}—8in. long by }—1}in. wide, rich dark brown beneath; margins just thickened beneath. Cymes axillary, many-flowered, short; bracts small, ciliated. 6. Flowers }in. long by jin. wide, urceolate. Calyx stin. long by ;!,in. wide, shortly 4-fid, lobes depresso-deltoid, apiculate; corolla }in. long, jin. wide, barrel-shaped, } way 4-lobed; lobes imbricated sinistrorsely, depresso-ovate; stamens 12, mostly or all inserted at base of corolla (some in pairs?), glabrous; anthers much exceeding the filaments, de- hiscing laterally from apex; pollen ellipsoidal, smooth. Ovary 0. @. Calyx jin. long by {—2in. wide, deeply 4-fid; lobes widely ovate, sub-cordate at base, apiculate at apex, suberect, accrescent. Corolla equalling the calyx, deeply 4-fid, not spreading. Staminodes 2 or more, 4 (%), small, inserted at base of corolla. Ovary ovoid, terminated by short 4-lobed style, 8-celled, cells 1-ovuled. Madagascar, Hb. Petit-Thouars! in Mus. Paris. Mr HIERN, ON EBENACE, 233 108. DrospyRos CHLOROXYLON, Roxb. Coromand. 1. p. 38. t. 49 (1795). D. foliis alternis, ovalibus, basi sepius rotundatis apice mucronatis, tenuiter coriaceis, subtus tomentosis, breviter petiolatis; floribus masculis aggregatis, subsessilibus, 4—10-nis, te- trameris, calyce campanulato, dense pubescente, profunde 4-fido, corolld 4-fida, staminibus 16, bisertalibus, glabris ; floribus femineis solitariis, sessilibus, staminodiis circiter 8, glabris, ovario glabro, 8-loculari; fructibus globosis, glabris, edulibus, seminum albumine non ruminato. Wall. List n. 4118 (1828—82), Alph. DC. Prodr. vii. p. 230. n. 40 (1844). D tomentosa, Poir. Encycl. v. p. 436. n. 22 (1804), non Roxb. D. capitulata, R. Wight Ic. tt. 1224, 1588 bis (1850). Cfr. D. glauca, Rottler in Gesellschaft Naturforschender freunde zu Berlin, Neue Schrift. Iv. p. 221 (1803); Alph. DC. Prodr. viii. p. 288. n. 84 (1844). Nella-woolymera of the Telingas, Roxb. Corom. l.¢.; Neenye or Ninet in Surat, Dr Gibson. A tree of middle size with irregular trunk, or in low lands only a large bush; bark scabrous, dark rust-coloured; branches spreading, sometimes spinous; young shoots pubes- cent-tomentose, subfulvous. Leaves oval or oval-oblong, usually rounded at base and mucronate at apex, thinly coriaceous; pubescent or subglabrescent and dark green on upper side} more or less tomentose and sub-tawny beneath, alternate, 3—3 in. long by 2—1,3, in. wide; petioles j;—} in. long; midrib depressed above; lateral veins not conspicuous. Inflorescence tawny densely pubescent; flowers white. 6. Flowers clustered sessile or subsessile, 4—10 together, about }in. long, tetramerous ; on peduncles jz m. long, with very short pedicels; bracts oval, small, glabrous inside. Calyx about j;in. high, campanulate, densely tawny-pubescent, deeply 4-fid, with apiculate lobes, glabrous inside. Corolla 4-fid, glabrous except 4 lines of hairs outside. Stamens 16, in 2 rows, inserted more or less in pairs, receptacle or at base of corolla, glabrous, the inner ones shorter; longer filaments as long as anthers; anthers dehiscing laterally from apex; connective apiculate or prolonged. Receptacle glabrous; ovary rudimentary glabrous. @. Flowers solitary, sessile (or subsessile in Wight Ic. t. 1588 bis), about tin. long, tetramerous ; bracts longer than in 6, shorter than the calyx. Calyx +in. long, campan- ulate, pale tawny, densely pubescent; lobes 3rds depth of calyx, apiculate. Corolla erect, glabrous, except 4 hairy lines down middle of lobes; lobes 4—2rds depth of corolla, ovate- lanceolate. Staminodes 7—9, glabrous, in one row, hypogynous or inserted at base of corolla, Ovary glabrous, surmounted by 4 erect glabrous styles, 8-celled; cells 1-ovuled, often approximated in pairs. Fruit globose, glabrous, tin. or rather more in diameter, on nearly flat calyx +in. in diameter. When ripe it is eaten raw among the Orixa mountains and is very palatable. Seeds 2—3; albumen not ruminated, testa thick slightly irregular inside. The wood of the larger trees is yellowish, very hard and durable, and is used by the natives for various economical purposes. Tranquebar, Vahl; Madras!; Bombay; Surat and Nassick, Dr Gibson’; Canara and Mysore, Mr Law/; Kew List, n. 1712, 1719, 3617; Wallich!; East Bengal! Vou. XII. Parr I. 30 234 Mr HIERN, ON EBENACEZ. 109. Diospyros(?) PERGAMENA, sp. Nov. D. glabra, foliis obovato-oblongis, alternis, basi leviter angustatis, apice anguste et abrupte acuminatis, firmiter pergamenis, petiolatis, floribus masculis in ramis vetustis dense aggregatis sessilibus et breve-cymosis, pentameris pubescentibus parvis, staminibus 20 binis, ovarit rudi- mento hirsuto; fructibus pedunculatis globosis wncialibus glabratis 3-locularibus 3-spermis ; albumine radiatim striato. Glabrous, young shoots terete; leaves alternate, obovate-oblong, narrowly and suddenly acuminate at apex, slightly narrowed at base, of the texture of firm parchment, dark brown above with depressed veins, paler with raised veins loosely reticulated beneath, 8—9 in. long by 2,3—3in. wide; petioles }in. long. &. Flowers densely clustered on the older branches, sessile and in short cymes, penta- merous, hirsute, subglobose (?), jin. in diameter (immature); calyx 5-fid, glabrous inside, lobes ovate; corolla 5-fid, lobes obtuse; stamens 20 in 10 pairs hispid, ovary rudimentary hairy. ¢. Fruit solitary (?), glabrate, subglobose, about lin. in diameter, 3-celled, 3-seeded (in one case); peduncle nearly }in. long; calyx 5-partite, }im. in diameter, closely hairy on both sides, reflexed; lobes involute; seeds fin. long by 2in. thick; albumen radiately striate, not ruminated, Borneo, O Beccari! n. 1787. 110. Drospyros CAULIFLORA, Blume, Bijdr. Fl. Ned. Ind. p. 668 (1825). D. foliis alternis, ovalibus, utrinque attenuatis, nitidis, glabris, breviter petiolatis ; floribus masculis axillaribus tetrameris, staminibus 16, geminatis, inequalibus ; floribus femineis late- ralibus secus ramos vetustiores paniculatis, calyce profunde 4-5-lobo, lobis basi margine sinuato refleris nigrescentibus, corolla wrceolatd, 4-fidd, fauce constrictd, fructibus globosis, glabris, edulibus, 4—8-locularibus, albwmine radiatim striato. Alph. DC. Prodr. vit. p. 238. n. 81 (1844) ; Hasselt in Hasskarl Pl. Javan. p. 767. n. 351 (1848); non Mart. A lofty dicecious tree. Leaves alternate, elliptical or oblong, attenuate at both ends, 4—9in. long by 1—8,3,in. wide, shining, glabrous; midrib and lateral veins depressed above; petioles 4—}in. long. &. Flowers axillary; calyx 4-fid; corolla 4-fid; stamens 16, in pairs, unequal in the pairs. 9. Flowers crowded in dense lateral panicles on the older wood, racemose; racemes 3—5 flowered, with bracteoles at the ramifications; peduncles nearly lin. long, thickened at the apex, turning black. Calyx deeply 4—5-lobed ; lobes turning black, narrow, reflexed, at the base with wavy margin. Corolla urceolate, 4-fid, much constricted at the top of the tube; tube tetragonal, covered with black hairs especially at the angles; lobes pale yellow, horizontal. Fruit fleshy, globose, glabrous, edible, lin. in diameter, 4—8-celled, green; seeds soli- tary in the cells, some imperfect ; albumen radiately striate; embryo turning yellow. Java, Bantam, 500 ft. alt. Hasselt, Reinwardt/, Blume! Mr HIERN, ON EBENACEZ. 235 111. Dtospyros RAMIFLORA, Roxb. Hort. Beng. p. 40 (1814). D. foliis alternis, ovalibus vel oblongis, apice acuminatis, basi angustatis, coriaceis, glabris ; Floribus femineis dense fasciculatis secus ramos vetustiores, 4—6-meris, tomentosis, calyce cam- panulato irregulariter lobato accrescente, corolla wurceolato-oblongd obtuse lobatd, staminodiis 10—12, glabris, ovario ovoideo-conico, fulvo-tomentoso, 10- vel 12-loculari, fructibus globosis, magnis, subscabris, edulibus, 10—12-spermis. Roxb. Fl. Ind., edit. 1832, Vol. 1. p. 535. n. 7; Drawings in Herb. Kew; Wall. List n. 4119 (1828—32); Wight Ic. t. 189 (1840); Alph. DC. Prodr. vu. p. 233. n. 57 (1844). A large dicecious tree with glabrous leaves and branches and straight trunk. Leaves oval or oblong, acuminate at apex, somewhat narrowed at base, coriaceous, alternate, shining and of same colour on both sides, margins slightly undulated and recurved, 4—10 in. long by 1—3in. wide, veins not conspicuous above, petioles 1—2 in. long; midrib wide and channelled above. 9. Flowers urceolate-oblong, collected in small short fascicles on the thick woody branches, tetramerous pentamerous or hexamerous; the inflorescence is however sometimes on young shoots or in racemes or panicles. Pedicels and calyces clothed with olive-coloured down; calyx ?in. long, urceolate, with inflated tube and deltoid lobes 3,—1in. deep; corolla }—} in. long, white, covered with short felt outside, glabrous imside except on the obtuse imbri- cated lobes which are about 4 the depth of the corolla, at first spreading and then revolute; tube somewhat inflated. Staminodes 10 or 12, double the number of the parts of the flower, glabrous, in one row, shorter than the corolla-tube. Ovary about the length of the calyx, ferruginous-hairy, ovoid-conical, 10 or 12-celled; cells 1-ovuled; style short; stigmas 5 or 6. Fruit globular, 2—3 in. in diameter, slightly scabrous, resting on the very thick enlarged calyx which is about 1}in. in diameter, 10—12-celled; cells 1-seeded; seeds transversely lined outside; albumen somewhat ruminated (?) Native name Oori-gaub or Goolul on eastern frontier of Bengal, where, according to Dr Roxburgh, the tree grows wild to a great size, and supplies the natives with very strong hard wood. Silhet, Wallich! 4119; Tipperah, Roxburgh. The position of this species in the genus is uncertain in consequence of the want of knowledge of the male plant and of the nature of the albumen in the seed; thus when more intimately known, the species may require to be removed to Sect. 1. MELONIA or elsewhere. : 112. Diospyros Diepennorstu, Miq. Fl. Ind. Bat. Suppl. 1. pp. 250, 583 (1860). D. foliis oblongis, apice breviter acuminatis, basi obtusis, firmiter coriaceis, glabris, breviter petiolatis; floribus femineis secus ramos vetustiores aggregatis, ovario ovoideo, hepta- gono, glabro, 14-loculari, basi abrupte stipitato-constricto, calyce coriaceo cupulato grossificante, extus parce appresso-pubescente. Buds somewhat hirsute; branchlets glabrous. Leaves oblong or obovate-oblong, the upper ones sublanceolate, rounded or obtuse at the base, shortly acuminate, glabrous, of firm parchment-like texture, smooth above with the lateral veins usually depressed, pale 30—2 236 Mr HIERN, ON EBENACEA!. beneath with the lateral veins prominent mostly patent united within the margin and loosely reticulated, 9—10 in. long. g. Flowers densely crowded on the old branches, with short pedicels and hirtellous bracts; ovary rather thickened, resting on a coriaceous cup-shaped patent obtusely 5-lobed calyx sparsely covered outside with appressed puberulence, abruptly stipitate-constricted at the base, ovoid-heptagonal, glabrous, marked at the apex with 7 pits and teeth, 14— (?16)- celled. According to Miquel it is clearly related to D. ramiflora, Roxb. but quite a distinct species. Malay name Djantoe-dipo. I have not seen a specimen. West Sumatra in Province Priaman, Diepenhorst. 113. Drospyros SUMATRANA, Miq. Plante Junghuhniane, p. 203 (1851—55). D. foliis distichis, oblongis, apice anguste acuminatis, basi cuneatis, submembranaceis, breviter petiolatis ; floribus femineis laxiuscule cymosis, fructibus immaturis ovoideo-oblongis, subglabris, 4-locularibus, loculis monospermis, albumine non ruminato, calyce fructifero pro- funde 4-lobo appresse pubescente, aucto, foliaceo, lobis suberectis, ovatis, acuminatis, basi cor- datis. Young parts inflorescence petioles and midrib of leaves beneath covered with short stiff puberulence. Leaves firmly submembranous, oblong, alternate, with a long narrow acumen at apex, cuneate at base, distichous, 24—5in. long by $—2in. wide; petioles 4—1 in. long; veins slightly depressed above, in relief beneath; lateral veins 5—6; very minutely and vaguely pellucid-punctate. Q@. Cymes rather lax, about } in. long or shorter (excluding the flowers) ; peduncles } in, long, about 3-flowered ; pedicels with yellowish hairs, thickened upwards; bracts foliaceous, caducous. Fruit (unripe?) oblong, §—Z in. long, glabrous; style, erect, glabrous (broken), dis- tinct. Fruiting calyx deeply 4-lobed, appressedly pubescent, erect or sub-erect, nearly as long as the young fruit; lobes ovate, acutely acuminate, wide and cordate at base, foliaceous, wavy; seed } in. long; albumen not ruminated; young fruit glabrous, 4-celled; cells 1-seeded. Sumatra, Korthals !, distr. Angkola, Junghuhn/; Borneo, Korthals / 114. Drospyros PENDULA, Hasselt ex Hasskarl Pl. Javan. p. 468. n. 352 (1848). D. foliis oblongo-lanceolatis, utringue acuminatis, gla’ris, breviter petiolatis, jfloribus masculis solitariis, femineis 1—2-nis vel breve-racemosis, calyce 4-fido, nigro-piloso, corglle lobis revolutis, filamentis 8, villosis, antheras 2—3 gerentibus, ovario ovoideo, 4—8-loculari, fructibus carnosis. A lofty dicecious tree. Leaves oblong-lanceolate, acuminate at both ends, 2$—8} in. long by 1—24 wide, shortly petiolate, glabrous. 3. Flowers solitary. 9. Flowers solitary 2 together or collected in small racemes, pendulous, Calyx 4-fid, bright green, nigro-pilose. Corolla pale yellow, with reflexed lobes. Filaments 8, short, hairy, bearing 2—3 anthers. Ovary ovoid, 4—8-celled; style thick, attenuate at the apex (or 2—4 connate styles); stigmas 4, emarginate. Fruit fleshy, pilose when young, 4—S8-celled ; cells 1-seeded. Java, Bantam Province, Mt. Pulassarie, flowers in June, between 4000 ft. alt. and the crater, Hasselt. Mr HIERN, ON EBENACEA. 237 115. Dtospyros MAcRopHYLLA, Blume Bijdr. Fl. Ned. Ind. p. 670 (1825). D. foliis alternis, ovalibus vel ovali-oblongis, apice acuminatis, basi rotundatis vel interdum subcordatis, tenwiter coriaceis, supra glabris, subtus glabriusculis, nervis gracilibus, breviter petiolatis ; floribus masculis paniculatis, pedicellis brevibus, calyce breviter 38—5-fido, urceolato, corolla breviter 5-lobd, crasso-ligned, staminibus 12, geminatis, glabris, cynus femineis pauci- floris brevibus, fructibus tomentosis subglobosis, calyce fructifero aucto. Alph. DC. Prodr. vu. p. 228. n. 27 (1844), non Wall. D. phyllomegas, Steud. Nomencl. Bot. edit. ii. vol. 1. p. 514 (1840). A tree 60 feet high, with dark terete branches. Leaves alternate, oval or oval-oblong, acuminate at apex, rounded or sub-cordate at base, thinly coriaceous, nearly glabrescent above with clear slender arching lateral veins, glabrous above, 3—10 in. long by 14—43 in. wide ; petioles 3—}in. long. g- Flowers axillary, paniculate, } in. long, pubescent; panicles many-flowered ; 1—1} in. long, ultimate pedicels mostly short. Calyx shortly 3—5-fid, globose-urceolate, 7; in. long, lobes deltoid; corolla silky outside, ovoid in bud, shortly 5-lobed, tube very crass and hard; stamens 12, unequal, in pairs, glabrous. 2. Cymes few-flowered, short, calyx 4—5-fid, hairy on both sides, accrescent in fruit ; fruit tomentose, sub-globose, 1in. or more in diameter. Java, in mountainous places, Blume! Local name Kitjallung. \ 116. Diospyros ovaLiroiiA, R. Wight, Ic. t. 1227 (1850). D. foliis alternis, ovalibus, apice obtusis, basi angustatis, tenuiter coriaceis, glabris, petiolatis, floribus aggregatis, 3—6-nis, brevissime cymosis, 4—5-meris, urceolatis, calyce brevi, pubescente, corolld subglabrd, lobis obtusis, staminibus 13—20, glabris, subequalibus, plerisque geminatis ; in flore femineo staminodiis 0—7, ovario hirsuto 4- vel 6-loculari, loculis l1-ovulatis ; fructibus solitariis, globosis, glabratis, seminum albumine non ruminato. Thw. Enum. Ceyl. Pl. p. 181. n. 13 (1864). A moderate-sized tree, glabrous except the inflorescence. Leaves oval- or obovate-oblong, rounded or obtusely pointed at apex, more or less narrowed at base, thinly coriaceous, alternate, midrib depressed above, 14—6in. long by $—21in. wide, with petiole 4—4 in. long, turning yellowish when dry, paler beneath with reddish midrib. $. Flowers clustered, 3—6 together, on very short hairy cymes, in the axils of fallen or present leaves, 4—5-merous, }—}jin. long. Calyx jin. long, tawny-hairy on both sides, ‘openly campanulate, with rounded or somewhat deltoid lobes about half the length of the calyx. Corolla twice the length of the calyx or more, urceolate, glabrous or nearly so, 4—5-fid or less deeply lobed, with obtuse spreading or recurved lobes. Stamens 13—15—20, glabrous, mostly inserted on the receptacle and in pairs, nearly equal, }in. long, filaments yy in. long. Ovary rudimentary, hairy. g. Flowers clustered, 3—6 together, on very short cymes, 4—5-merous, thicker than in g. Staminodes 0—7, glabrous, hypogynous or at base of corolla. Ovary conical, tawny 238 Mr HIERN, ON EBENACEA. hairy, 4- or 6-celled (2-celled ex Wight J. ¢): cells l-ovuled; stigma 2—3-lobed. Fruit solitary, subsessile, glabrate, globose, Zin. in diameter, usually I-seeded. Fruiting calyx reflexed, tomentose, thickened but not dilated or but slightly so. Seeds with albumen not Tuminated. Ceylon, 2000—4000 ft. alt., Thwaites ! 1815, 1816, 2533, Trincomalee, Moon/; Madras, Coimbatore, Wight! n. 1720; Anamalay hills, Beddome ! 117. Diospyros TEXANA, Scheele in Linnea xxu. p. 145 (1849). D. foliis alternis, obovatis, apice rotundis, basi cuneatis, submembranaceis, subtus pube- scentibus, subsessilibus ; floribus masculis 1—3-nis, breviter pedunculatis, pubescentibus, calyce 5—6-partito, corolld urceolatd, 5—6-fidd, lobis obovatis, staminibus 16—20, biserialibus, glabris ; floribus femineis solitariis, staminodiis 0, ovario sub-8-loculari, ovoideo, dense sericeo, fructibus globosis. A tall much-branched shrub, 12—15 feet high, with fastigiate branches spreading at 60°—70°, cinereous, verrucose, leafy, softly pubescent and pale at the apex; warts subrotund, of dark reddish colour. Leaves alternate, oblong-obovate, wedge-shaped at base, rounded or emarginate at apex, submembranous, softly pubescent, pale, glabrescent on upper side; veins inconspicuous above; nearly flat or with revolute margins, }—2in. long by }—1 in. wide; petioles ;1,—,1, in. long, hairy. Flowers with scent of vanilla. g. Flowers 1—3 together, in axils of present or fallen leaves, drooping, {—}in. long, softly pubescent, pale, crowded on young shoots; peduncles ${—}-3 in. long, pubescent; bracts caducous. Calyx with 5 or rarely 6 deep ovate or lanceolate lobes, shorter than the tube of the corolla, Jin. long, pubescent on both sides. Corolla urceolate, with 5 or per- haps rarely 6 recurved lobes about half the length of the corolla-tube, glabrous inside. Stamens 16—20, distinct, in 2 rows, glabrous; anthers longer than the filaments, dehiscing from the apex. (In one case a stamen is abnormal, an anther having two filaments.) Ovary rudimentary, with grey hairs. Q. Flowers solitary, pentamerous, fin. high by }in. wide. Calyx large but not accre- scent; peduncles 1—1 in. long, recurved, bearing caducous small bracts. Corolla with spread- ing lobes. Staminodes 0. Style 4-fid; stigmas dilated. Ovary 8 (?)-celled, densely pilose. Fruit globose, 4in. in diameter, dark, covered with scattered hairs, fleshy, sweet-tasted, ultimately shining. Albumen not ruminated. Fruiting calyx spreading or reflexed, 5-partite ; lobes #in. long, oblong, pubescent on both sides. North America, Texas, Galveston Bay, Drummond !, Fase. 11. n. 329. (¢ £1.) ; Drummond !, Fase. 11. n. 201. (Fr.); Lindheimer !, Flora Texana exsiccata, Fase. m1. n. 451 (6), 452 (? fl), 453 (Fruit); Mexico, between Laredo and Bejem, Feb. 1828, Berlandier! (¢ fl.); collected in expedition from Western Texas to El Paso, New Mexico, May—Oct. 1849, by Charles Wright! n. 423; Texas, Trécul, Oct. 1849, n, 1249, in woods by the sides of streams; Herb. Berlandierianum Texano-Mexicanum, n. 3030! (D. mewicana, Scheele MSS.), Ann. 1828; “ Hill sides, Fort Inge to Escondido Creek, and near Eagle Pass, Western Texas, flowers in March, fruit ripe in August about lin. in diameter,” Torrey. Mr HIERN, ON EBENACEA). 239 118. Drtospyros MABACEA, F. Muell. Austral. Veg. in Intercolonial Exhibition Essays, 1866—67, p. 35 (1867). D. foliis alternis, ovalibus, apice breviter acuminatis, basi cuneatis, chartaceis, costis exceptis subglabris, breviter petiolatis ; floribus masculis 5—7-nis, dense cymosis, tetrameris, calyce campanulato, 4-fido, lobis deltoideis acutis, corolld extus sericed, campanulatd, profunde lobatd, lobis ovalibus, staminibus 15—16, glabris. Cargillia mabacea, F. Muell. Fragm. Phytogr. Austr. v. p. 162 (1866), Benth. Fl. Austral. Iv. p. 287. n. 2 (1869). Maba quadridentata, F. Muell. Fragm. l.c. A tree, 20 feet high; young branches strigose-pubescent. Leaves oval or oblong, char- tacecus, alternate, cuneate at base, narrowed or shortly and obtusely acuminate at apex, pubescent on midrib and principal veins beneath and on the depression of the midrib above, nearly glabrous on the rest of the leaf, somewhat shining beneath, of same dark green colour on both sides, 3—4in. long by 11—12in. wide; petioles }—4in. long, hispid- pubescent; lateral veins slightly depressed on upper surface of leaves. $. Flowers 5—7 together, in short axillary hairy cymes; peduncles very short; pedicels do—iy In. long; bracts ovate; flowers tetramerous. Calyx campanulate, dark green, puberu- lous outside, glabrous inside, 4-fid; lobes deltoid acute. Corolla pale silky outside, about twice the length of the calyx, deeply 4lobed; lobes oval, emarginate, imbricated sinis- trorsely in bud. Stamens 15!, 16, many in pairs and inserted at base of corolla, a few hypogynous, all glabrous, tin. long; anthers narrowly lanceolate; filaments short. Ovary rudimentary, represented by a bunch of hairs. ¢. “Fruit a scarlet berry.” Australia, Tweed River, C. Moore / 119. Diospyros PENTAMERA, Woolls and F. Muell. ex F. Muell. Austral. Veg. in Inter- colonial Exhibition Essays, 1866—67, p. 35 (1867). D. foliis alternis, ovali-lanceolatis, apice obtuse acuminatis, bast attenuatis, coriaceis, gla- bris, breviter petiolatis ; jfloribus masculis 3—5-nis, pubescentibus, brevissime cymosis, sepius pentameris, calyce hemispherico, lobis deltoidets, corolla brevt, profunde 5-lobd, staminibus 15— 20, jilamentis brevissimis glabris, antheris villosis, ovarii rudimento pubescente ; floribus fe- mineis 1—3-nis, fructibus solitardis, subsessilibus, globosis, apice excepto glabratis, 1—4-locu- laribus, seminum albumine non ruminato. Cargillia pentamera, Woolls et F. Muell. in F. Muell. Fragm. Phytogr. Austr. Iv. p. 82 (1864), Benth. Fl. Austr. Iv. p. 288. n. 4 (1869). Maba pentamera, F. Muell. l.c. v. 163 (1866). C. arborea, A. Cunn. MSS. A large tree attainmg S80—100 feet in height and 2—3feet in diameter, glabrous except the young shoots and inflorescence, somewhat mgid in habit. Leaves alternate, lan- ceolate or oval, cuneate or attenuate at base and usually as much so at the apex, coria- 240 Mr HIERN, ON EBENACE. ceous, shining above, pale and yellowish at least beneath, 1{—3in. long by $—lin. wide, petioles jin. long; midrib slightly either raised or depressed above; net-veins nume- rous rather prominent above, rather inconspicuous beneath. Leaves usually marked beneath by some small dark glands arranged along 2 straight lines equally distant from the midrib. é- Flowers 8—5 together, tin. long, on short silky drooping cymes which measure without the flowers about ,—1in. long; pedicels very short. Calyx hemispherical, pubes- cent outside, glabrous or shining inside, half the length of the flower, 5-fid, with deltoid lobes; rarely 4—6-fid. Corolla sub-globose, open at the mouth, pale and shortly pubes- cent outside, glabrous inside, deeply 5-lobed, imbricated sinistrorsely in bud; lobes oval. Stamens 15—20 ex Bentham lc, usually 20 in pairs or in groups; anthers “tetragono- linear, rostellate, dehiscing laterally from apex downwards,” silky; filaments very short, glabrous. Ovary rudimentary, hairy. @. Flowers subsessile, 1—8 together, usually solitary (?). Fruit solitary, subsessile, globose or spheroidal, about }$in. long, glabrate except the apex, tipped by the remains of the hairy style, 2—4-usually 4- or rarely l-celled; cells 1-seeded; pericarp thin, crustaceous; dissepiments membranous; seeds fin. long or rather more; albumen white, cartilaginous; embryo }in. long; radicle clavate-cylindrical, slender, equalling the flat narrowly or linear- lanceolate cotyledons. Fruiting calyx 1in. long, receiving the base of the fruit, puberulous or pubescent on both sides, 5- rarely 4-fid; lobes often somewhat spreading. Australia, Moreton Bay, Leichhardt; Brisbane river, A. Cunningham!; New South Wales, C. Moore’, Paris Exhibition, Sydney woods 30, W. Macarthur! 49. Plentiful on the mountain brushes of the Hastings River, C. Moore; Clarence River, C. Moore!, Beckler, J. Wilcox; Queensland, Hill! ; Ash Island, Hunter’s River, Mrs Forde and Miss Scott ex W. Woolls, Contr. Fl. Austr. p. 192 (1867). Fruit eaten by the Carpophaga magniyica, Selby. Called Black Myrtle by the colonists. Timber soft when fresh, but exceedingly tough. 120. Diospyros PARALEA, Steud. Nomencl. Bot. edit. ii. vol. 1. p. 514 (1840). D. foliis ovalibus, alternis, apice acuminatis, basi obtusis supra nitidis glabrescentibus, subius costd margineque tomentosis vel subglabrescentibus, coriaceis, breviter petiolatis ; floribus masculis aggregatis, subsessilibus, ferrugineo-tomentosis, tetrameris, calyce 4-fido, corolld cam- panulatd, 4-fidd, staminibus circiter 16, antheris lineari-lanceolatis, hirsutis, filamentis brevibus glabris ; floribus femineis subsolitariis vel aggregatis, subsessilibus, staminodiis 8, ovario tomentoso, 8-loculari ; fructibus globosis, subglabratis, seminum albumine non ruminato. Alph. DC. Prodr, vin, p. 224. n. 10 (1844), Mig. in Mart. Fl. Bras. vir. p. 6. t. 3 (1856). Paralea guianensis, Aubl. Pl. Guyan. 1. p. 576 (1775). P. guyannensis, Aubl. l. c. t. 231 (stam. char. et fig. excl.); Paralia guianensis, Desv. ex Hamilt. Prodr. Pl. Ind. Oce. p. 45. n. 89 (1825). D. ferruginea, Spltgbr. in Vriese Ned. Kruidk. Arch. p. 327 (1848). D. longifolia, Spruce in Journ. Proceed. Linn. Soc. Lond. v. p. 7 (1861), Pl. Bras. exsice. n. 1516 (1851). A small moderate-sized or lofty tree of hard white wood; young shoots buds and inflorescence ferruginous-tomentose. Leaves oval-oblong or ovate-oblong, alternate, more or less rounded occasionally somewhat narrowed at base, acuminate at apex, coriaceous, Mr HIERN, ON EBENACE, 241 shining and glabrous or nearly so with depressed midrib above, pubescent puberulous or nearly glabrescent beneath especially on the marked midrib and recurved margins (hairs rufous in dried state; leaves bordered when young with white hairs which fall off, according to Aublet); lateral veins several, slender; leaves 3—8 in. long by 1—2} in, wide; petioles }—% in. long, glabrescent. Bracts rufous-hairy. 3. Flowers in axillary subsessile clusters, ;%,in. long, rufous-hairy. Calyx 4 in. long, acutely 4-fid; lobes deltoid. Corolla fleshy, campanulate or campanulate-oblong, sweet- scented, quadrangular, with a short inflated tube and 4 short lobes, glabrous inside (ferruginous on both sides ex Vriese), 4-fid. Stamens 16!, united by their filaments in 8 pairs (18 ex Aublet, about 13 ex Alph. DC, 8 ex Vriese), the inner ones the shorter; anthers linear-lanceolate, hairy at the back of the outer ones and at the front of the inner ones; filaments short, glabrous. Ovary rudimentary, ferruginous-hairy. 2. Flowers (subsolitary ex DC.) few together in axillary subsessile clusters. Staminodes 8. Ovary 8-celled; cells 1-ovuled. Fruit solitary or 2 together, subsessile, shining and glabrate or with some persistent ferruginous hairs, globose, about 1 in. in diameter, pericarp at length splitting from apex, 3—4-seeded. Fruiting calyx with 4 lobes cordate at base, rufous-hairy especially on the undulating margins, on the centre of the back and inside, suberect or spreading, 3—1 in. across. Seeds oblong, }in. long; albumen not ruminated. A decoction of the bark is said to be useful in case of fever in Guiana, where the plant is known by the name of Parala. French Guiana, Cayenne, Sagot! n. 1253, Martin!; Guiana, Mrs Parker! ; British Guiana, Schomburgk! 1492; Surinam, Hostmann! 547, Splitgerber, 541; S. Venezuela, near the rivers Casiquiari, Vasiva and Pacimoni, Spruce! 3159 ¢ flower (arbor gracilis 18-pedalis, ramulis longis pendulis. Flores flavo-virides. In ripis inundatis per totum Casi- quiarem, necnon in Orinoco superiore, Nov. 1853); Brazil, by the south bank of the Rio Negro close to its junction with the Solimoes, Spruce! 1516, in fruit (small tree with subverticillate subsimple branches, fruit green, seeds immersed in flesh, D. longifolia, Spruce). According to Mr Spruce, his D. longifolia has the branches arranged in whorls of five (very rarely three or four), while in D. Paralea the branches are alternate. The branches however in D. Paralea are sometimes verticillate. 121. Diosprros rHoDOCcALYX, Kurz in Journ. Asiat. Bengal, Vol. xu. Part 1. p. 71. n. 91 (1871). D. foliis oblongis vel ovali-oblongis, apice obtusis, basi angustatis, chartaceis, supra glabris, lucidis, subtus secus costam pubescentibus, breviter petiolatis; floribus masculis tetrameris, brevissime cymosis, calyce dense fulvo-pubescente, lobis oblongo-lanceolatis obtusiuscutis, corolla glabra, urceolatd, staminibus circiter 16 corolle basi insertis ; floribus femineis solitartis, sta- aminodiis 8—10, ovario dense fulvo-tomentoso, 4-loculari (2). Flora, 1871, p. 332. A small tree with young parts appressedly pubescent. Leaves oblong or oval-oblong, rarely obovate-oblong, retuse or rarely (on the same stock) obtusely apiculate, on slender and short petioles, acute or obtuse at base, chartaceous, of variable size 1—2 or 3—4in. long, Vou. XII. Part I. 3l 242 Mr HIERN, ON EBENACE®. glabrous and shining above, for the most part slightly pubescent beneath on the midrib ; veins conspicuous, net-veins lax. Flowers tetramerous, small, sessile or subsessile, axillary ; bracts linear, densely fulvo-tomentose, short. g. Cymes very short, tomentose, calyx densely tawny-pubescent; lobes oblong, lanceo- late, rather obtuse, corolla glabrous, scarcely }in. long; tube inflated; lobes short, oblong. Stamens about 16, inserted at the base of the corolla; filaments short, bearded; anthers linear, acuminate. Ovary rudimentary. ¢. Flowers solitary, sessile or subsessile. Calyx larger than in the ¢; lobes widely oblong, obtuse, at the base with margin plicate-dilatated and tinged with red. Corolla }in. long. Staminodes 8—10. Ovary oblong densely fulvo-tomentose, 4-celled (2). Siam, Rédbiri and Ka4nbiri, Tecjsmann 6000, 6007 in Herb. Bogor. According to Kurz, somewhat resembling in general habit “ D. heterophylla, Wall. and best placed near D, tomentosa,” Poir. I have not seen a specimen. 122. DIOSPYROS MACROCARPA, sp. Nov, D. foliis alternis, oblongis, apice acutis vel subacuminatis, basi cuneatis, coriaceis, subtus pubescentibus, subglabrescentibus, breviter jpetiolatis; floribus masculis asillaribus, breviter cymosis, subsessilibus, pubescentibus, tetrameris, calyce campanulato, 4-fido, corollé 4-fidd campanulatd, staminibus 16, geminatis, filamentis dense pilosis; fructibus solitariis, subsessilibus, ovoideis, glabratis, seminum albumine non ruminato. Cargillia macrocarpa, Vieill. Hb. Young parts shortly densely and softly pubescent. Branches dark, glabrescent. Leaves oblong, acute or slightly acuminate at apex, somewhat narrowed at base into petiole, alternate, coriaceous, dark-cinereous glabrous and shining above with slightly depressed midrib, paler appressedly pubescent and sub-glabrescent beneath with duller veins; margins more or less undulated; 2—5 in. long (besides petiole 7,—} in. long) by 3—1} in. wide. é. Cymes axillary, short, bearing about 3—5 subsessile flowers, about equalling the petiole, softly pubescent, subferruginous ; common peduncle in bud about ;4in. long. Bracts minute. Flower-bud ovoid, about 2 in. long, scarcely exceeding the calyx. Calyx cam- panulate, erect, 4-fid, with deltoid acute lobes; glabrous except near margin and shining inside. Corolla 4-fid, with rounded or mucronate sinistrorsely imbricated lobes; hairy outside, glabrous inside. Stamens 16 in 8 pairs, the pairs arranged in one row; filaments short, dilated and united in pairs at the base and (in young state) almost forming a short tube at base of corolla, densely setose-pilose especially the outer ones; anthers lanceolate- linear apiculate comparatively glabrous, but the outer ones surrounded by the dense long hairs of the filaments. Ovary rudimentary, small, pubescent, surmounted by 2 styles, Q. Fruit solitary, subsessile, ovoid, glabrate, about I}in. high by 1 in. thick or more, apparently 4-celled, fleshy, with rather thin pericarp. Bracts caducous. Fruiting calyx fiat, 4-fid, 8— in. across, shortly sub-pubescent outside, pubescent inside; lobes widely ovate, obtuse or mucronate. Seed rather more than 1 in, long, oblong, albumen not ruminated. New Caledonia, Balade, Wagap, Vieilard! n. 890; Pancher! n. 251, Mr HIERN, ON EBENACEZ. 243 123. DIosPYROS PERFORATA, sp. nov. D. glabra, foliis ovali-oblongis, alternis, apice acuminatis, basi angustatis, firmiter mem- branaceis, perforato-punctatis, breviter petiolatis, nervis patentibus; floribus masculis aggregatis, subsessilibus, pubescentibus, campanulatis, calyce profunde 4-fido, corolld urceolatd (?), 4-fidd, lobis latis, staminibus 16, geminatis, receptaculo insertis, interioribus brevioribus, antheris hispidis, filamentis superne hispidis inferne glabris, ovario 0, receptaculo leviter hispido. Glabrous except the inflorescence and buds; branches cinereous, longitudinally wrinkled. Leaves alternate, oval-oblong, acuminate, narrowed at base, firmly membranous, scattered with small dark glands especially alongside the midrib beneath, in places perforated, 6—74 in. long by 13—2}in. wide, dark and shining above with depressed patent lateral veins, pale brown beneath with rather distinctly marked lateral veins; petioles channelled above, }in. long. é. Flowers clustered, subsessile, few or several together on axillary or lateral nodules, tawny-pubescent, in bud about tin. long; bracts small, imbricated, on very short pedicels, dark-cinereous; calyx deeply 4-fid, lobes deltoid, glabrous inside; corolla urceolate (?), 4-fid, glabrous inside, appressedly silky outside, contorted sinistrorsely in estivation, 4-fid, slightly exceeding the calyx, lobes rounded; stamens 16, united by their filaments in 8 pairs, inner ones rather shorter, anthers longer than the filaments, with hairs on the back and front especially on the back of the outer ones and on the front of the inner ones, connective apiculate, filaments with spreading hairs above, glabrous beneath; ovary 0 or minute, repre- sented by a few short hairs on the receptacle, Ceram Island, Moluccas, De Vriese/ 1857—61. 124, Drospyros oBLoncA, Wall. List n. 4124 (1828—32). D. foliis oblongis, alternis, apice breviter acuminatis, basi obtusis, subcoriaceis, glabris, petiolatis; floribus femineis 1—3-nis, brevissime cymosis, confertis, pentameris, calyce profunde lobato, hispidis, lobis undulatis bast auriculatis, corolld calycem cquante, carnosd, ovario Serrugineo-pubescente, 10-loculart; fructibus subglobosis, subglabratis, seminum albumine non ruminato. Alph. DC. Prodr. vil, p. 228. n, 26 (1844); G. Don, Gen. Syst. Gard. and Bot. ry. p. 40 (1837) excl. synon. A tree or shrub; branches terete or warty, glabrous, puberulous at the extremities, dark. Leaves oblong, subcoriaceous, alternate, rounded near base, glabrous, shortly acu- minate at apex, 5—9}in. long by 2—4in. wide, with petioles 1—2in. long; midrib strong and lateral veins numerous clear parallel and spreading, both depressed on upper surface. @. Flowers crowded, subsessile or in very short 1—3-flowered cymes, on short young shoots or axillary, pentamerous. Calyx covered on both sides with a mixture of black and ferruginous short hairs, #;in. long, deeply 5-lobed, rather crass; lobes erect-patent with wavy margins auricled at base. Corolla 5-lobed, ferruginous-hairy outside, glabrous inside, fleshy, not exceeding the calyx (in bud). Staminodes 5, glabrous. Ovary 10-celled, ferruginous- hairy; cells l-ovuled. Fruit subglobose, nearly glabrate, Zin. long; surrounded at base by blackish hispid calyx jin. across with appressed or somewhat spreading lobes having wavy margins and auricled pouting bases; albumen horny, not ruminated. Penang, Wallich! n. 4124; Singapore, Maingay/ n. 967. 31—2 244 Mr HIERN, ON EBENACE. — 125. Driospyros EBENASTER, Retz. Obs. Bot., fasc. v., p. 31 (1789). D. glabra, foliis alternis, ellipticis vel oblongis, apice plerisque obtusis, basi angustatis, firmiter membranaceis, nitidis, petiolatis ; floribus 4—6-meris, pubescentibus, azxillaribus» polygamis, pedunculis brevibus unifloris vel masculis plurifloris, calyce 4—6-fido, lobis ovatis margine revolutis, corolla wurceolatd, apice lobatd, staminibus 8—20, leviter pubescentibus, ovario pubescente, 4—10-loculari; fructibus magnis edulibus, 4—10-spermis, albumine non ruminato. Alph. DO. Prodr. vit. p. 235. n. 64 (1844); non Spach; nec D. Hebenaster, Gaertn. Fruct. et Sem. Pl. m. p. 478. t. 179. £ 9 (1791). D. digyna, Jacq. Hort. Scheenbr. vol. U1. p. 35. t. 313 (1798); Alph. DC. lc. p. 238. n. 80; non Hort. D. revoluta, Poir. in Encycl. Méth. v. p. 435. n. 18 (1804); Alph. DC. lc. p. 234. n. 60. D. obtusifolia, Humb. et Bonpl. ex Willd. Sp. Pl. tv. p. 1112 (1805) ; Humb. Bonpl. et Kunth, Nov. Gen. et Sp. Pl. uu. p. 253. t. 247 (1818); Alph. DC. lc. p. 227. n. 24; non Bert. D. Sapota, Roxb. Hort. Beng. p. 40 (1814), Fl. Ind. edit. 1832, vol. 11. p. 535; Bot. Mag. LXIX, t. 3988 (1843); Alph. DC. Le. p. 228. n. 25. D. sapotanigra, DC. Ess. Prop. Med. Pl. p. 200 (1816). D. edulis, Lodd. ex Sweet, Hort. Brit. p. 270 (1827); Alph. DC. Le. p. 239. n. 90. D. decandra, Boj. Hort. Maurit. p. 200 (1837), non Lour. Sapota nigra, Blanco, Fl. Filip. p. 409 (1837). D. membranacea, Alph. DC.! lc. p. 227. n. 20 (1844). D. nigra, Blanco, FI. Filip. edit. i. p. 211 (1845). D. laurifolia, A. Rich. Fl. Cub. in Ramon de la Sagra, Hist. de Cuba, vol. x1. p. 86. t. 55 (1845—55), ex Walp. Ann. Bot. v. p. 480 (1858). D. brasiliensis, Mart. Fl. Bras. vit. p. 5. t. 2. f. 2 (1856). Hebenaster, Rumph. Amb, ut. lib. Iv. p. 13. t. 6 (1750). Sapotte Negro, Sonnerat, Voy. & la Nouv. Guin. p. 45. tt. 14—16 (1776). A tall shrub or even lofty tree, quite glabrous except the inflorescence; branches dark. Leaves alternate, elliptical or oblong, usually obtuse at apex, somewhat narrowed at base, firmly membranous, shining, evergreen, 83—12in. long by 14—34in. wide; midrib depressed above; net-veins not conspicuous; petioles ranging up to in. long. Flowers polygamous, 4—6- merous, }—lin. long, pubescent; peduncles axillary, pubescent, solitary, those producing male flowers with several flowers, those with hermaphrodite or female flowers 1-flowered, 1—1in. long, bracteate. Calyx ample, }—-}in. long, somewhat hairy on both sides, 4—6-fid, lobes ovate, with revolute margins and sinuses. Corolla urceolate, twice the length of the calyx, yellowish white or greenish, thick and fleshy, 4—6-lobed at apex, silky or nearly glabrescent. Stamens 8—20, slightly hairy, often some or all in pairs; filaments some- what pilose. Styles 2—5; ovary pubescent, 4—10-celled. Fruit globose, 14—4in. in dia+ meter, glabrous, shining, of olive yellowish-green colour when ripe, filled with a dark soft and paste-like pulp, edible; towards the centre of the pulp are 4—10 cells, each con- taining a large oval compressed seed; albumen cartilaginous,’ not ruminated. Fruiting calyx spreading, much thickened in middle, 5—6-fid, 1—1} in. in diameter or less, puberulous on both sides, lobes undulated. Fruiting peduncles about jin. long. - Mr HIERN, ON EBENACEA: 245 Local names, Kaus Magostan in Mauritius, Lolin in Amboina, Sapotte negro, &c. Philippine Islands, Sonnerat, Blanco, flowers in July; Celebes, Jacquin; Amboina, Rumf. Cultivated in Mauritius, at Calcutta, and Malacca, Maingay! 975; introduced into England and France &c., where it requires a hot-house for protection. Occurs also in cultivated places in tropical America, perhaps introduced; Mexico, Orizaba, Botteri! 909; Vera Cruz, Galeotti / 4609 (2000 ft. alt.); Cuernavaca, Humboldt and Bonpland! 3984 (5000 ft. alt.); Lizardo, Wawra! 249; Miradon, Wawra! 1029; Brazil, Rio Janeiro, Schott and Pohl! 4568; Cuba, Richard; Montserrat, Ryan! ex Hb. Vahl. "Blanco loc. cit, states that the tree in the Philippine Islands grows to a height of 24—30 feet and is carefully cultivated as well as indigenous. He says that the flesh of the fruit is blackish, and although it is eaten the taste is not well flavoured, that the leaves have caustic properties, and that the unripe fruit is reported to poison fish, An evergreen tree 30—50 ft.. high with light even-grained wood grown at Cordova, Mexico, and called Zapotillo, probably belongs to this species; a specimen exists in the Kew Museum. The type of this species cannot be found in Retz herbarium at Lund in Sweden. 126.. Diospyros sAMOENSIS, A. Gray in Amer. Acad. Vv. p. 326 (1862). D. foliis alternis, ovali- vel ovato-oblongis, apice obtuse angustatis, basi angustatis, coriaceis, glabris, petiolatis; jloribus masculis 3—9-nis, tetrameris, pubescentibus, calyce campanulato, 4-fido, lobis obtusis, corollé campanulatd brevi, 4-fidd, lobis obtusis ; staminibus 8—10, glabris ; floribus femineis solitariis, —petiolatis, ovario hirsuto, 8-loculari; fructibus globosis glabris, calycis fructiferi aucti tubo concavo depresso-cupuliformi, intus margine elevato ; seminum albu- mine non ruminato. : _ Branches glabrous or young ones scarcely puberulous. Leaves alternate, glabrous, oval or ovate-oblong, coriaceous, obtusely narrowed at apex, somewhat narrowed at base, 3—6 in. long by 1}—3in. wide; midrib depressed above; lateral and net-veins raised, slender; petioles 3—2 in. long. &. Peduncles 3—9-flowered ; flowers tetramerous, 1—}in. long, ovoid in bud. Calyx cam- panulate, tin. long, shortly puberulous, 4-fid; corolla silky outside, 4-fid; stamens 8—10, glabrous, unequal, some in pairs. @. Calyx-lobes rounded; calyx about equalling the corolla ; peduncles solitary, }—} in. long, puberulous, 1-flowered, equalling the flower; ovary hairy, 8-celled; fruit globose, }—1}in. in diameter, glabrous; fruiting calyx-tube flat or cupuliform with a raised border receiving the base of the fruit, and with 4 obtuse spreading or recurved lobes, glabrous, about 2 in. wide ;.seeds 23 in. long, closely packed together ; albumen not ruminated, white. Navigators’ Islands, South Pacific Exploring Expedition !; Friendly Islands, W. H. Harvey !, (caustic berry for burning ringworms, &c.) “Tutuna.” The foliage and fruiting calyx resemble ‘D. Ebenum, Keenig, but the plant is of a paler green colour and the flowers shorter. According to the Rey. Thomas Powell in Seemann’s Journal of Botany, vol. vi. p. 281 (1868), the wood of this large tree is hard and used for axe-handles and spear-points ; the fruit is used for poisoning fish ; and the secretion of the fruit is a vesicatory and turns the human skin black. Also the Samoan children are said to insert the midrib of the cocoa-nut leaflet into the fruit and apply the liquid thus obtained to their arms to produce blisters and eventually permanent pro- minences which they consider an ornament. Mr Powell describes the flowers as hermaphrodite. 246 Mr HIERN, ON EBENACEZ, 127. Diospyros OLEN, sp. nov. D. glabra, foliis alternis, ovalibus, apice breviter obtuseque acuminatis, basi cuneatis, coriacets, nitidis, utringue delicate reticulatis, breviter petiolatis ; floribus femineis solitariis, breviter pedun- culatis, axillaribus, bracteatis, 4- rarius 3-meris, calyce subglabro profunde lobato, lobis late ovatis acuminatis bast cordatis, tubo intus margine elevato, corolld 4—3-fidd, lobis acutis, staminodiis 0—6, glabris, ovario superne pubescente, inferne glabro, 8-loculari. Dark-cinereous, and except the inflorescence glabrous. Leaves oval, alternate, coriaceous, narrowed at base and usually with a short obtuse acumen at apex, of the same cinereous colour on both sides, somewhat shining on upper surface, with midrib depressed on upper side and net- veins delicately raised on both sides, 23—4 in. long by 1—1}in. wide ; petioles +,—1 in. long. ?. Peduncles solitary, 1-flowered, axillary, {—} in. long, patent, pubescent, bearing caducous patent lanceolate alternate bracts. Flowers usually tetramerous, rarely trimerous. Calyx about lin. across when spreading, deeply lobed, very nearly glabrous; lobes widely ovate acuminate, widened and cordate at base, 1—4in. long, spreading or reflexed, with numerous parallel slight Jongitudinal veins; tube with raised internal hairy border at top. Corolla about +in. high, glabrous in upper part, puberulous at least in places beneath, 3—4-fid, conical above in bud; lobes acute arching in flower outwards; tube urceolate in flower. Staminodes inserted on corolla, glabrous, about 6 (in a trimerous flower) or wanting (in a tetramerous one). Ovary glabrous beneath, pubescent and suddenly narrowed towards apex; terminating in a short 4-fid style glabrous at apex; ovary 8-celled; cells 1-ovuled. Indigenous name Olén, I. Lifu, Deplanche! No. 31, 128. Drospyros CareiiuiaA, F. Muell. Austral. Veg. in Intercolonial Exhibition Essays, 1866—67, p. 35 (1867). D. foliis alternis, ovalibus vel oblongis, apice obtusis, bast cuneatis, coriaceis, glabris, pallidis, breviter petiolatis, nervis subtus inconspicuis; floribus masculis breviter cymosis, pubes- centibus, tetrameris, awillaribus, campanulatis; calyce 4-fido, lobis deltoidets, corolle lobis obtusis, staminibus seepius 16, glabris, geminatis, corolle basi insertis; floribus femineis 1—3-nis, bre- vissime cymosis, axillaribus tetrameris; staminodiis 8, wni-serialibus, glabris, corolle basi insertis; ovario ovoideo, pubescente, 4-loculari, loculis bi-ovulatis; fructibus glabratis, ovoideis vel globosis, 1-spermis, albumine non ruminato. Annona microcarpa, Jacq. Fragm. Bot. p. 40. t. 44. f.'7 (1800—1809). Monodora microcarpa, Dunal, Monogr. Anon. p, 80 (1817). Cfr. Brown in Tuckey, Congo, p. 475 (1818). Cargillia australis, R. Br. Prodr. Fl. Nov. Holl. et Ins. Van-Diem. p. 527. n. 2 (1810); Alph. DC. Prodr. vit. p. 243, n. 1 (1844); Bot. Mag. t. 3274 (1833); Ettingsh. Blatt-skel. Dikot. p. 90. t. 35. f. 6 (1861); Benth, Fl. Austral. rv. p. 288, n. 3 (1869); F. Muell. Fragm. tv. p. 82 (1864). Maba Cargillia, F. Muell. Fragm. vy. p. 162 (1866). This species is cited on pages 30, 81, 86, 46, 54 by the name of Diospyros australis, Mr HIERN, ON EBENACE. 247 A large shrub or tree 20—40 or even 100 feet high, glabrous, or the young parts and inflorescence with short hairs; trunk sometimes 2 feet in diameter, Leaves oblong or oval, alternate, coriaceous, obtuse at apex, more or less narrowed at base, palish green especially beneath, 14—44in. long by %—1}in. wide including petiole }—}in. long; mid- rib flattish depressed above; lateral and net-veins in relief and not conspicuous; frequently with small black spots arranged in a row on each side of midrib beneath. Flowers dic- cious, tetramerous (or rarely trimerous 2). $. Flowers several together arranged on short axillary pubescent drooping cymes which without the flowers measure }—}in. long; calyx ; in. high, covered with pale ap- pressed short hispid hairs, shortly lobed, eamipanalie with deltoid lobes; corolla tin. long, deeply lobed, covered outside with pale short hairs, glabrous inside, ovoid in bud, campa- nulate in open flower; lobes erect or recurved at apex, obtuse; stamens 12—16, usually 16, in pairs, glabrous, inserted at base of corolla; anthers longer than the filaments, lan- ceolate linear, dehiscing by lateral slits near apex; ovary rudimentary, hairy. ?. Flowers 1—8 together, $ in. long, on very short cymes, campanulate, pubescent ; calyx ;2 in. high by }in. thick, 4-fid; eros deeply 4-lobed, lobes obtuse; staminodes 8, in one row, inserted at base of corolla, glabrous, with lateral slits; ovary ov Sid. hairy, 4-celled ; cells often with 2 ovules, without any trace of a dissepiment between them, alternate with the calycine lobes; style hairy, 2-lobed at apex; stigma 2-lobed and glabrous. Fruit glo- bular or ovoid, }—3in. thick; fuscous and glabrescent when ripe, edible, ultimately 1-celled and 1-seeded; albumen of seed not ruminated; fruiting calyx about }in, high, cup-shaped, shortly puberulous or nearly glabrous. Fruit called Grey plums. Slender-growing tree, with elongated trunk and elegant rigid foliage. Wood close, very tough and firm. In the forest regions towards the coast through New South Wales and Queensland. Australia, Hiigel!; Queensland, Brisbane River, Moreton Bay, F. Mueller; Rockhampton, Dallachy !; Crocodile Creek, Bowman; New South Wales, Port Jackson to the Blue Mountains, R. Brown!, F. Mueller! ; Berrima and Richmond River, C. Moore; Hastings and Mackay Rivers, Beckler! ; Illawarra, A. Cunningham !; Sydney, Bynoe! ; Sydney woods, Paris Exhibition No. 20, M. Macarthur !; New South Wales, Kiama, W. H. Harvey /; Cabramatta River, W. Woolls. 129. Driospyros Matacapal, Alph. DC. Prodr. vit. p, 237. n. 75 (1844). D. foliis alternis, ovalibus, glandulis sparsis; floribus axillaribus, 1—3-nis, calyce 4-lobo, baecd globosd, 4-loculari, loculis 2-spermis. A small tree having yellow wood, with some black spots; said to keep off bags when fresh. Leaves alternate, oval, with some scattered glands especially at the end. Flowers axillary, 1—3 together; calyx 4-lobed; fruit baccate, globose, 4-celled ; cells 2-ovuled. Local name Malacapai (Blanco, Fl. Filipin. p. 302, 1837); Tagatog, Philippine Islands, Blanco. 130. DrIospyROS SPINOSA, sp. nov. D. spinosa, foliis alternis, ovalibus, apice acuminatis vel obtusiusculis, bast rotundatis vel subcordatis, junioribus subtus pubescentibus; margine revolutis, breviter petiolatis; jfloribus 248 Mr HIERN, ON EBENACEA!. masculis brevissime cymosis, parvis, tetrameris, calyce hemispherico, corolld profunde 4-lobé, staminibus 16, glabris, ovarti rudimento glabro. 3 Dull, spinous ; young parts and inflorescence ferruginously tomentose-pubescent ; branches terete. Leaves alternate, coriaceous, oval, acuminate or pointed at apex, rounded or sub- cordate at base, with whitish loose hairs beneath when young, subglabrescent, dark green above, browner beneath, 1}—3 in. long by 3—1} in. wide; margins recurved; petioles { in. long, pubescent, terete. @. Inflorescence arranged in very short axillary cymes on the young branches; “flowers js in. long (in bud), subglobose, tetramerous. Calyx about half the length of the flower, hemi- spherical, appressedly hairy outside, glabrous inside, deeply 4-fid; lobes rounded, sinistrorsely contorted in bud. Corolla subglobose, glabrous except 4 hairy lines outside along middle of lobes, deeply 4-lobed ; lobes rounded, sinistrorsely contorted in estivation. Stamens 16, glabrous, subequal (2), in two rows(?), distinct, inserted at or near base of corolla; anthers lanceolate, acute, longer than the filaments, dehiscing laterally from the apex; ovary rudimentary, glabrous. Brazil, Martius! Herb. Reg. Monac., Ebenacee n. 144. 131. DrIosPYyROS OVALIS, sp. nov. D. fruticosa, foliis alternis, ovalibus, utrinque rotundatis, apice mucronatis, basi subcorda- tis, supra glabris nitidis, subtus villosis, subcoriaceis, breviter petiolatis; floribus masculis breviter cymosis, tetrameris, profunde lobatis, corolle lobis obovatis patentibus, staminibus circi- ter 20, glabris. A shrub, 2—3 feet high. Young parts underside of leaves and inflorescence and especially the buds subferruginous-pubescent. Branches terete, glabrescent and nitescent, numerous. Leaves (of the shoots of the current season) oval, alternate, subcoriaceous, mucronate at apex, subcordate at base, dark shining and glabrous above except depression of midrib, without conspicuous veins, shaggy underneath with ciliated margins, about 1 in. long by }—} in. wide; petioles about , in. long, pubescent. g. Inflorescence at the base of the shoots of the current season, cymose, with few or several flowers rather loosely arranged; cymes (excluding the flowers) }—3 in. long; pedicels about } in. long; bracts oval, densely pubescent.. Flowers 8, in. long, tetramerous, green. Calyx 4 in. long, partite with lanceolate erect-patent lobes, pubescent on both sides. Corolla 2 in. high (4 in. long when straightened), glabrous except 4 lines of hairs outside ; lobes #y in. deep, obovate, erect-patent and recurved at apex. Stamens 20 (18—20 ex Benth. MS. in Hb. Cantab.), equal, glabrous; anthers linear; filaments short, united almost in a short tube. Ovary rudimentary, glabrous (?), minute. Brazil, Pernambuco, sandy open places, Rio Preto, September. Gardner! 2813, Mr HIERN, ON EBENACE. 249 132. Diospyros Hisprpa, Alph. DC. Prodr. vu. p. 236. n. 68 (1844). D. foliis alternis, ovalibus vel ovali-oblongis, apice cuspidatis vel acuminatis, basi swepius obtusis, subcoriaceis, subtus ferrugineo-hispidis, breviter petiolatis ; floribus masculis 2—4-nis, breviter cymosis,. 4-meris, calyce hispido, 4-partito, lobis lanceolatis, corolld profunde lobatd, lobis oblongis, staminibus 18—24, subequalibus, glabris, ovarti rudimento pubescente; floribus femineis 4—5-meris ; fructibus solitariis, globosis, dense ferrugineo-hispidis, carnosis, 8-locula- ribus ; calyce fructifero 4—5-partito, patente, lobis lanceolatis. Miq. in Mart. Fl. Bras, vit. (Eben.) p. 4. n. 2 (1856). An arborescent shrub or tree, 10—30 feet high, with shoots underside of leaves and inflorescence ferruginous-hispid; branches spreading. Leaves oval or oval-oblong, cuspidate or acuminate, usually obtuse at base sometimes narrowed or in ¢@ subcordate, subcoriaceous, alternate, 2—5 in. long by 1—2} in. wide, darker and pubescent-velutinous above; petioles i—2 in. long, hairy. 8. Flowers 2 in. long, in 2—4 flowered distant or usually contiguous cymes (4—3 in. g, ferruginous-hispid on both sides, 4-partite, long); pedicels ;—1 in. long. with lanceolate lobes. Corolla green, deeply 4-lobed, pubescent along longitudinal stripes ; lobes oblong, somewhat narrowed at apex. Stamens 18—24, subequal, some or all in pairs, glabrous; anthers linear; filaments short. Ovary rudimentary, globose, hairy. 2. Flowers few together, in short cymes, tetramerous or pentamerous. Fruit solitary, on pedicels -,—1 in. long, densely ferruginous-hispid, globose, pointed at apex, about 1 in, in diameter, fete 8-celled, 8-seeded. Fruiting calyx 4—35-partite, ee 1} in. across ; lobes lanceolate. Seeds 8, oblong, compressed, } in. long. Brazil, between Goiavéira and Corrego de Jeragudé, Burchell! 7437, @ fil. Aug., tree 20—30 ft. high, corolla green; between Cérrego-findo and Pérto-Real, Burchell! 8396, in young fruit, November, tree 20 ft. high; Gozaz, 10 ft. high, Burchell! 6994; Minas Geraes, Claussen! 478. 133. Diospyros GOUDOTII, sp. nov. D. foliis alternis, ovato-oblongis, apice acuminatis, bast subcordatis, subsessilibus, sub- membranaceis, supra glabrescentibus, subtus puberulis ; fructibus globosis, solitarwis, axillaribus, pedunculatis, papilloso-verrucosis, pilis aspersis, calyce patente, 5-lobo, non aucto. Young parts tawny- or ferruginous-pubescent ; shoots terete, puberulous, glabrescent. Leaves alternate, ovate-oblong, widest near the middle, submembranous, acuminate at apex, subcordate at base, subsessile, glabrescent and dark green above with conspicuously de- pressed veins, puberulous and reddish brown below at least on veins, 6—10 in. long by Q9—Ain. wide; petioles {—1 in. long, ferruginous-pubescent. Fruit globose about lin. in diameter, scattered with pilose hairs, ferruginous-pilose at apex where is base of broken style; papillose-verrucose. Fruiting calyx not accrescent, hairy on both sides, spreading, % in. across, with 5 ovate or lanceolate lobes }—} in. long. Fruiting peduncle $—} in. long, ferruginous hispid-pubescent, thick, erect-patent, axillary, solitary, 1-fruited; bracts at base of peduncle, ovate, imbricated, caducous, ranging up to } in. long, New Granada, Muzo, Goudot! No, 3. Vout. XIL. Part L 82 250 _Mr HIERN, ON EBENACE. 134. DiosPYROS GAULTHERLEFOLIA, Mart. Fl. Bras. vu. (Eben.) p. 5. n. 5. t. 2. f. 1 (1856). D. foliis distichis, oblongis, apice obtusis, basi subcordatis, tenuiter coriaceis, subtus presertim secus nervos ferrugineo-hispidis, breviter petiolatis, marginibus in sicco late reflexis ; floribus masculis aggregatis, brevissime cymosis, 5-meris, calyce campanulato, 5-fido, corolla profund2 lobatd, staminibus cw o pilosis, floribus femineis subsessilibus aggregatis ; fructibus solitariis vel binis, globosis, apice abrupte conicis, setosis, papilloso-verrucosis, albumine non ruminato. A shrub or small tree 12—14 feet high; with rufous-hairy terete branches, spreading at 60°, glabrescent. Leaves oblong, distichous, obtusely lanceolate at apex, subcordate at base, thinly coriaceous, margins widely reflexed in the dry state; dark shining and glabrous except the midrib, with depressed veins above; ferruginous-hispid especially on the veins beneath ; 2—54 in. long by 1—2 in. wide; petioles ;; in. long. &. Flowers clustered in axils of leaves; cymes short, with oblong bracts glabrous inside, 3. in. long, pentamerous; calyx campanulate, ferruginous-hairy on both sides, {4 in. long, lobes ovate-oblong, +in. long; corolla glabrous outside except a few pilose hairs along 5 longitudinal lines outside, 5-sided in bud, deeply 5-lobed; stamens 45—75, anthers linear, slender, with long scattered ferruginous hairs, filaments short, combined at base and inserted at base of corolla or on the receptacle, nearly glabrous; ovary 0 or minute. 2. Flowers in subsessile clusters. Fruit solitary or 2 together, globose but abruptly pointed at apex, with long ferruginous stiff hairs that easily rub off, papillose-verrucose, scarcely lin. long, pulpy. Fruiting peduncle hairy, } in. long; testa thick; albumen not ruminated ; fruiting calyx with (4 or) 5 deep lanceolate lobes hairy inside, spreading, nearly 1 in. across. Brazil, Bahia, Blanchet 1886; common in sandy shrubby places near Maceio, Alagoas, February, 1838, Gardner/ 1412, in Q fl. and fr. The anthers in the figure quoted above are incorrectly drawn as glabrous except the apex. 135. DiI0spYROS SUBROTATA, sp. nov. D. foliis distichis, ovalibus, apice sepe acuminatis, basi subcordatis, tenuiter coriaceis, breviter petiolatis, costd eaceptd glabrescentibus ; floribus masculis aaillaribus, cymosis, 5—-6- meris, calyce aperte campanulato, corolla partitd, subrotatd, lobis obtusis patentibus, stamini- bus circiter 20, antheris pilosis, linearibus ; floribus femineis sub-6-nis, fructibus pubescentibus. A shrub of 8 feet high, or a small tree of 18—30 feet; young parts with pale appressed pubescence, glabrescent except the midrib of leaves and inflorescence. Leaves oval- or ovate- oblong, subcordate at base, more or less acuminate at apex, thinly coriaceous, with midrib depressed on upper side, distichous, with margins slightly reflexed, 3—7 in. long by 14— 3 in. wide ; petioles ;4;—1 in. long. 6. Inflorescence axillary, cymose, with several or numerous flowers and spreading pedicels, pubescent with short appressed hairs; cymes }—?in. long; pedicels ;—,§; in. long; flowers pentamerous or hexamerous; calyx openly campanulate, with short deltoid lobes, with shert appressed inconspicuous pubescence, ;4—+4 in. long; corolla subrotate, nearly } in. in diameter, Mr HIERN, ON EBENACEA, 251: with deep oval spreading convex lobes, } in. long, with longitudinal stripes of appressed hair outside, glabrous inside, rather thick; stamens about 20; anthers pilose, lmear; filaments consolidated, short, pistil 0. @. Flowers about 6 together in axillary cymes. Fruiting pedicels 1—}4 in. long or very short (sessile ex Burchell MSS.); fruit depresso-subrotund, 4-5-sided, yellow, shining, with scattered appressed short hairs, and nearly smooth skin, probably about lin. in diameter; fruiting calyx $in. across with acute deltoid spreading lobes and short appressed hairs inside. Brazil, at Para, Burchell! 9923, 9952, g fl. December; at Baido, Burchell! 9275. Fruit in June. 136. Diospyros POLYANDRA, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 7 (1861). D. foliis distichis, ovato-oblongis, apice acuminatis, basi subcordatis, tenuiter coriacets, subtus pubescentibus, breviter petiolatis; floribus masculis aaillaribus, cymosis, 4—7-sepius 6-meris, calyce hemispherico extus fulvo-pubescente, lobis acutis, corolle lobis profundis, patentibus, staminibus 40—50; antheris linearibus, pilosis, filamentis brevissimas, bast connatis. A tree 18—30 ft. high, with a trunk Qin. in diameter, and branches arranged in sub- terminal whorls, long, subsimple, leafy throughout, tawny-hairy at the extremities. Leaves ovate-oblong, acuminate at apex, subcordate at base, 35—6in. long by 14—3 in. wide, with petioles ;,—1in. long, distichous, thinly coriaceous, with recurved edges, with scattered appressed pubescence, glabrescent above; veins depressed on upper surface of leaf. $. Inflorescence in axillary not very crowded cymes which without the flowers measure about }in. long, densely hispid-pubescent, tawny; pedicels jj—,%, in. long; bracteoles de- ciduous; flowers 4—7- usually 6-merous, white, sweet-scented, about $in. long and cylin- dric-conical in bud; calyx hemispherical, about }in. long, with short acute lobes, glabrous inside, tawny-hairy outside; corolla with oval-oblong deep lobes spreading in flower, glabrous inside, with longitudinal stripes of hair outside; stamens 40—50; anthers linear, pilose ; filaments very short, connate at base; pistil 0. Brazil, south bank of Rio Negro at confluence with river Solimoes, Spruce/ 1528; fre- quent on the banks of the Casiquiare, Spruce! 3166. ¢ flowers in May and sparingly in November. According to Mr Spruce, the branches are arranged in whorls of five (very rarely three or four). 137. Drospyros coccoLOB£FOL1A, Mart. F]. Bras. vit. (Eben.) p. 6. n. 7. tab. 1. fig 1 (1856). © D. foliis alternis, ovalibus, utrinque obtusis, discoloribus, tenwiter coriaceis, subglabris, petio- latis; floribus masculis breviter cymosis, axillaribus, calyce sepius 4-partito, lobis ovatis vel lanceolatis, patentibus, ciliatis, corolla 4—6-partitd, lobis oblongis patentibus, staminibus 18—24, plerisque geminatis, hirsutis; floribus femineis 1—4-nis, 4-meris, staminodtis 4, ovario ovordeo- conico, piloso, 4-loculart, loculis 1-ovulatis. A small or moderate-sized dicecious tree, glabrescent in most parts. Shoots and lower surface of leaves pubescent especially on veins and margins, sometimes glabrous. Leaves oval, thinly 32—2 252 Mr HIERN, .ON EBENACE. coriaceous, or thickly membranous, somewhat or scarcely contracted and sometimes oblique at base, rounded obtuse or emarginate at apex, with about 8 lateral veins on each side at about 50°—60° with midrib, alternate, 2—44 in. long by 1—3} in. wide ; angular divergence 2; net-veins pellucid in Gardner's specimen, not so in Martin’s nor in Pohl’s; bluish green above, browner beneath; hairs ferruginous; petioles }— $,in. long, somewhat decurrent, leaving large scars on the branch; bracts transversely oblong, glabrous inside. é. Inflorescence in axillary, ‘sane ee usually 3- flowered ore cymes se in. long. Flowers } in. long, g ovate or lanceolate, ciliated, ae as aie as sake calyx, erect-patent. Corolla glabrous, or eee hairy lines on back, with 4—6 very deep oblong lobes much imbricated in the bud, erect-patent. Stamens 18—24, many or all united by their filaments in pairs, }—}in. long, nearly equal, inserted at very base of corolla (hairy either on the anthers or filaments), contiguous ; filaments short and with spreading hairs (not so in Gardner’s specimen), anthers linear-oblong, glabrous (pilose at base in Gardner's specimen), + in. long; pollen widely ellipsoidal. Ovary rudimentary, fulvo-sericeous, hemispherical, small; style 0. @. Inflorescence and outside of calyx fulvo-sericeous. Flowers axillary, solitary or 2—4 together; peduncles {—} in. long, thick, solitary or 2 together, articulated to the branches. Calyx } in. high, with 4 ovate-acute lobes. Corolla tubular, 4-fid, twice the height of the calyx, white, glabrous. Staminodes 4, inserted at the base of the corolla and alternate with its lobes, ’ filiform, included, with rigid hairs at base, glabrous above. Ovary ovoid-conical, covered with shining erect hairs, continuous with 4 linear oblong truncate-obtuse stigmas, “apparently 4- celled” with 1 ovule in each cell. A fruit, collected by Gardner from Brazil, where it is called Marmaleiro, and is said to be good to eat, probably belongs to this species; it is subglobose, rugose in the dry state, and nearly glabrous, but pointed and tawny-pubescent at apex, $in. thick; the calyx is spreading, slightly pubescent, with 4 deep ovate-oblong lobes, about din. across. Brazil, Serra de Araripe, Gardner! 1511 (é fl. October); in hot dry places near the river S. Francis in prov. Minas, eg. near Salgado and in the desert towards Vao do Paranan, @ flowers in August and September, Martius! ; near Oliveira, Pohl! 455. 138. Drospyros PEARCEI, sp. nov. D. foliis alternis, ovato-oblongis, apice acuminatis, basi obtusis vel rotundatis, tenuiter coriaceis, subtus appresse pubescentibus, petiolatis ; floribus masculis aggregatis, subsessilibus, sepius penta- meris, calyce campanulato, extus pubescente, 5-fido, lobis deltoideo-acutis, corolla subrotatd, lobis patentibus, staminibus circiter 30, receptaculo insertis, antheris linearibus, pilosis, jilamentis brevibus basi connatis. Young parts densely tawny-pubescent ; an evergreen (?) tree, 15 ft. high. Leaves ovate- oblong, rounded or slightly narrowed at base, alternate, acuminate at apex, thinly coriaceous, dark green and glabrous above except the depressed midrib and veins, with scattered appressed pubescence beneath, 6—7} in. long by 13—2¢in. wide: petiole }—4 in. long, pubescent. g. Flowers very numerous and crowded, subsessile, 4} in. long, conical in bud, white, pentamerous or occasionally hexamerous. Calyx campanulate, }in. long, pubescent, 5-fid; lobes Mr HIERN, ON EBENACEA. 253 deltoid-acute, glabrous inside. Corolla with hairy lines outside, twice the length of the calyx, deeply 5-lobed, subrotate, lobes spreading. Stamens about 30; anthers linear, pilose, with long terminal apiculus; filaments short, combined at base, inserted on receptacle; ovary 0, S. America, Peru (?), Monterico, 3000—4.000 ft. alt., rare, Pearce ! 159. DIOSPYROS PERUVIANA, sp. nov. D. foliis alternis, oblongis, apice acuminatis, basi subrotundis vel angustatis, coriaceis, subtus pubescentibus, petiolatis ; floribus masculis aggregatis, cymosis, 5—6-meris, calyce campanulato, extus pubescente, 5—6-fido, lobis lanceolatis vel ovatis, corolld profunde lobatd, lobis rotundatis patentibus, stamimbus 36—45, pilosis; floribus femineis aggregatis, subsessilibus, fructibus sub- globosis, papilloso-rugosis, setosis, calyce fructifero patente, non aucto. Young parts underside of leaves and inflorescence ferruginous-pubescent. Leaves alternate, more or less oblong, acuminate at apex, coriaceous, deep green, shining and glabrescent except the depressed veins above, pubescent beneath especially on the veins and recurved margins, 3—6 in. long by 1;—2in. wide; petioles }—} in. long. 3. Flowers cymose, several together, }—2in. long, crowded; cymes (excluding the flowers) yin. long ; calyx campanulate {—1 in. long, densely pubescent outside 1—2 in. long; pedicels 35 slightly so inside, 5—6-fid, lobes ovate or lanceolate, acute; corolla deeply 5—6-lobed, 4—2 in. long, lobes rounded, spreading widely in full flower, much imbricated sinistrorsely in bud, each with a longitudinal stripe of dense ferruginous silky hairs outside; stamens 36—45, appearing at the mouth of the open corolla, anthers linear, pilose, filaments glabrous or nearly so, combined at the base ; ovary wanting. Var. a. Sprucet. “A small tree, 15 feet high, not rarely pendulous at apex, with long subpinnate branches, 5 or occasionally 3 or 4 together.” Leaves ovate-oblong, nearly rounded at base. g flowers white, scentless, about 2in. long. Stamens about 45. @ flowers in sub- sessile clusters. Fruit sub-spheroidal, $ in. thick, $ in. long, papillose-rugose, covered with ferruginous sete, with remains of 4 styles at apex, yellow, rather fleshy, fruiting calyx not acerescent, 7-fid, spreading, about 41m. across, bearing remains of calyx at base of fruit, Tarapoto, E. Peru, in young woods, ¢ fl. in January, 1856, fruit in October, 1855, Spruce / n. 4411. Var. 8. ocanensis. Leaves oblong-lanceolate, narrowed at base. ¢ flowers greenish, dashed with rose-colour, about }in. long. Stamens 36. New Granada, Ocaiia, 3500 ft. alt., flowers in June, Schlim/ n. 698. Perhaps a distinct species, 140. Diospyros WEDDELII, sp. nov. D. foliis alternis, oblongis, apice obtuse acuminatis, basi cuneatis, coriaceis, glabris, breviter petiolatis, nervis inconspicuis ; cymis femineis puberulis, paucifloris; fructibus globosis, verru- cosis, breviter pubescentibus; seminum albumine non ruminato; calyce fructifero parvo, patente, 5-fido, utrinque puberulo, lobis ovato-deltoideis. Branches terete, young shoots puberulous, quickly glabrescent; bark of older branches pale. Leaves alternate, oblong, obtusely acuminate at apex, alternate at base, coriaceous, 254 Mr HIERN, ON EBENACE. glabrous, undulated in the dry state, 1}—5 in. long by }—1} in. wide; petioles }—} in. long, veins inconspicuous. Q. Cymes axillary, puberulous, few-flowered, 4 in. long. Fruit globular, verrucose, shortly pubescent between the rough points, 1} in. in diameter, tipped with remains of ferruginous-silky style. Albumen of seeds not ruminated. Fruiting calyx small, flat, 5-fid, puberulous on both sides, 4 in. in diameter, lobes ovate-deltoid. Brazil, near Rio de Janeiro, Weddell! 577. 141. Drospyros GLOMERATA, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 7 (1861). D. foliis alternis, ovato-oblongis, apice acutis acuminatis, basi rotundatis vel subcordatis, firmiter membranaceis, subtus pallidis appresse pubescentibus, breviter petiolatis ; floribus mas- culis aggregatis, axillaribus, sessilibus, sericeis, 5—6-meris, calyce campanulato, corolla profunde lobata, lobis patentibus, staminibus 26—33, sericeis; fructibus immaturis subglobosis, sub-10- locularibus. A slender tree 20—80 feet high; branches 5 together arranged in 3 subterminal whorls, very long (12 feet), simple or rarely forked, leafy and flowering to the base; terminal buds narrowly conical, covered with dense short yellowish hair; young shoots puberulous with short brown curly-patent hairs, terete, glabrescent, dark, smooth. Leaves alternate, ovate-oblong, firmly membranous, usually rounded or subcordate at base, acuminate and acute at apex, 6—12 in. long by 2—44 in. wide; dark green with few scattered weak pale hairs, glabrescent, and with depressed midrib above; pale and covered with appressed hairs and with raised and darker veins beneath ; petioles 1—} in. long, patent, slightly bent upwards at point of attach- ment of leaf. Flowers sub-polygamous, pentamerous or hexamerous. 6. Flowers numerous in crowded axillary sessile clusters, pale, silky, “white,” scentless, about } in. long, pentamerous or occasionally hexamerous ; bracts rounded, imbricated, hairy. Calyx campanulate, 5—6-fid, with acute deltoid or ovate lobes, glabrous or nearly so inside. Corolla deeply 5—6-lobed ; lobes oblong-obovate, glabrous inside, incurved near apex, erect- patent, distant upwards when in full Hower, imbricated sinistrorsely in bud; stamens nearly equal, 26—83, clustered and more or less united at base, inserted at base of corolla or on receptacle; anthers linear, with long straight silky hairs on back and front; filaments short, glabrous. Ovary 0 or in subhermaphrodite flowers ovoid pubescent 10(?)-celled terminated at apex by 5-lobed style. Young fruit subglobose, about 10-celled, N. W. Brazil, near Panuré by shady banks of Rio Uaupés, Spruce’ 2701, November; Martius!; French Guiana, Martin! 142. Drospyros CAPREHFOLIA, Mart. MSS. in Herb. D. foliis alternis, ovali- vel ovato-oblongis, apice acuminatis, basi angustatis, tenwiter cori- aceis, subtus pallidis, subglabris, breviter petiolatis ; floribus masculis subsessilibus, 4—5-meris, calyce campanulato, 4—5-fido, corolla subrotatd, staminibus circiter 45, pilosis, corolle basi insertis; floribus femineis solitariis, sessilibus, 5-meris, ovario dense hirsuto, stylis 4 (2) A tree 40 feet high; terminal buds small, rufous-hairy, lateral, often hard; young shoots Mr HIERN, ON EBENACEZ, 255 with scattered rufous hairs, glabrescent ; branches spreading at 40°—70°, terete, with a rather pale cuticle. Leaves oval or ovate-oblong, somewhat narrowed at base, acuminate at apex, thinly coriaceous, dark green shining and glabrous except depressed midrib and with depressed veins above, pale, subglabrous except the veins beneath, alternate, 2—3 in. long by #,—1,3; in. wide; petioles ;,—1 in. long. é. Flowers few together, in subsessile clusters, tetramerous or pentamerous; calyx tin. long, with scattered appressed hairs, campanulate, felted within, 4—5-fid, lobes deltoid acute +1, in. long; corolla J im. long, glabrous except longitudinal stripes of brown hairs outside, subrotate, lobes oval, spreading, 4 in. long; stamens 45 (in one pentamerous flower), inserted at base of corolla, anthers linear, with a few pilose erect hairs; filaments glabrous, combined at base ; ovary rudimentary. Q@. Flowers solitary, sessile, pentamerous, bracteate at base. Calyx 5-fid, with deltoid lobes, hairy on both sides; corolla spreading, 1} in. across or more, glabrous outside; ovary densely hairy, subrufous. Styles 4(), glabrous, erect, exceeding the ovary. Brazil, Cape Frio, Rio de Janeiro, Sello 1011!/; Maranhaéo, Don/; Guinea, Surinam, Martius! 1678. 143. Drospyros MANNH, sp. noy. D. foliis alternis, ovali-oblongis, apice acuminatis, basi angustatis, firmiter membranaceis, subtus pallidis, subglabris nervis exceptis, brenter petiolatis ; floribus masculis dense cymosis, axillaribus et secus ramos vetusios lateralibus, 5—6-meris, calyce profunde lobato, corolla sub- rotatd, staminibus 15—17, subequalibus, hispido-pilosis. A tree, with young shoots rufous-hispid or afterwards fuscous-hispid ; older branches dark, glabrate, spreading at about 50°. Leaves oval-oblong, narrowed at base, acuminate at apex, alternate, firmly membranous, glabrous and with depressed veins above, glabrous (except a few isolated erect hairs) and paler on the lamina and with rufous hispid hairs on the raised midrib and lateral veins beneath, flat, 5—7}in. long by 1$—2! in. wide; petioles fuscous, hispid, 1—3, in. long. . Inflorescence often on older branches, in several- or many-flowered dense short rufous- hispid cymes in the axils of present or fallen leaves; pedicels short; flowers 3 in. long, pen- tamerous or hexamerous. Calyx ferruginous-hairy on both sides, ;3;—2 in. long, deeply 5—6-fid, with lanceolate somewhat spreading lobes. Corolla subrotate in full flower, ovoid-conical in bud, } in. high, 5—6-partite, glabrous except patches of short pale hairs along exterior of lanceolate-oblong spreading lobes. Stamens 15—17, nearly equal, about } in. long, appearing at open mouth of corolla, hispid-pilose, with pale ferruginous hairs, on short filaments, not in pairs. Ovary wanting, represented by a few hispid hairs. West Equinoctial Africa, Gaboon River, ¢ fl. July, Mann! 924. 144. DrospyRos ARTANTHZFOLIA, Mart. Fl. Bras. vit. (Eben.) p. 7 (1856). D. foliis aliernis, oblongis, apice cuspidato-acuminatis, basi rotundatis vel angustatis, crassiuscule membranaceis, subtus fusco-hirtis, pallentibus, petiolatis; floribus femineis awilla- ribus, solitartis vel binis, calyce 5-partito, hirtulo, baccis depresso-globosis, 8-locularibus, dense rufo-setosis ; calycts fructifert lobis obtusis deltoiders. 256 Mr HIERN, ON EBENACE. Sinuous branches petioles and underside of leaves especially on the midrib and rather prominent veins villous with brown hairs. Leaves rather thickly membranous, oblong or ovate- oblong, 4—7 in. long by 2—4 in. wide, cuspidate-acuminate, rounded or contracted at the base, dark green, rather paler beneath, with 8—13 lateral veins on each side, alternate; petioles 4 in. long; veins depressed above. 9. Flowers axillary, solitary or 2 together, subsessile in fruit; calyx 5-partite, somewhat hairy; fruiting calyx divided beyond the middle; lobes triangular, rather obtuse, tawny- setulose especially in middle. Berry densely rufous-setose, 8-celled, depresso-globose, setze shining. S. America, N. Peru, Maynas, in woods, Péppig! 2266. 145. Diospyros PappiciaNa, Alph. DC. Prodr. vil. p. 224. n. 9 (1844). D. foliis alternis, ovali-lanceolatis, apice obtuse acuminatis, basi cuneatis, tenuiter cori- aceis, subtus appresse-pubescentibus, breviter petiolatis ; floribus masculis breviter cymosis, fulvo- pubescentibus, calyce aperte campanulato, breviter 4—5-fido, corolla tubulosd, apice obtuse lobatd, staminibus 12—20, subequalibus, filamentis brevibus glabris, antheris hispidis; fructibus glo- Losis, appresse papilloso-pubescentibus, calyce fructifero non aucto patente. Miq. in Mart. Fl. Bras. vit. (Eben.) p. 4, n. 4 (1856). A small bushy tree, rarely erect, 15—25 feet high; alternate branches and underside of leaves with scattered appressed hairs. Leaves oval- or oblong-lanceolate, obtusely acuminate or narrowed at apex, cuneate or abruptly narrowed at base, alternate, thinly coriaceous with very slightly reflexed margins, glabrescent above except the depression of midrib, 2—4 in. long by §—13 in. wide; petioles }—} in. long; lateral veins inconspicuous. 3. Inflorescence tawny-pubescent, cymose, bearing few or several flowers, in short cymes which measure about } in. long exclusive of the flowers; pedicels short, reflexed; flowers 2 in. long, tetramerous or pentamerous ; bracts ovate, acute, caducous; calyx +4 in. high, openly and shortly campanulate, shortly 4—5-fid with acute lobes, dark, with pale pubescence outside, glabrous inside; corolla 3% in. long, tubular, bright tawny-hairy outside, glabrous inside, shortly 4—5-lobed at apex, lobes obtuse; stamens 12—15 or 18—20, nearly equal; anthers hispid, linear, hypogynous; filaments short, glabrous, combined at base more or less in pairs ; ovary small, rudimentary, with short inconspicuous hairs. Fruit globular, nearly 1 in. in diameter, shining but with scattered appressed short brown hairs especially at apex arising from papillose bases, 6—8-celled. Fruiting calyx } in. in diameter, spreading but appressed to base of fruit, 4—5-lobed, not accrescent. Brazil, Amazon, Péppig! 2639; Povoagaé dos Juris, Martius! n. 3053; Rio Negro, fre- quent on margin of Gapé from Barcellos upwards, Nov., Spruce! 1938; St Hilaire’; Rio Uaupés, Gapd, October, Spruce! 2635, 146. DrIospYROS EMARGINATA, sp. nov. Plate IX. D. foliis alternis, obovatis, apice retusis vel emarginatis, basi cuneatis, coriaceis, costd ewceptd glabrescentibus, inconspicue reticulatis, breviter petiolatis; floribus masculis axillaribus, Mr HIERN, ON EBENACE®, 257 conferto-cymosis, fulvo-hirsutis, calyce 4—5-fido, corolld tubulosd, apice 4—5-lobd, stamini- bus 25—32, subequalibus, filamentis brevibus, antheris hispidis; fructibus globosis, subglabris, calyce fructifero vie aucto. A tall straight tree, 90 feet high, with a trunk 2 feet thick; shoots with a few inconspi- cuous appressed hairs. Leaves obovate, alternate, retuse or emarginate at apex, cuneate at base, coriaceous, quite glabrescent except the midrib beneath and its depression above, with highly reticulated but inconspicuous veins; 13—3 in. long by ;%—14 in. wide ; petioles about } in. long. g. Inflorescence axillary, tawny-hairy; cymes }—4 in. long, bearing several flowers on short pedicels; flowers 3 in. long, tetramerous or pentamerous, drooping, tawny; calyx 51, in. high, shortly and openly campanulate, 4—5-fid with sub-acute lobes, dark, with short scattered appressed hairs outside, glabrous inside; corolla tubular, with tawny-silky hairs outside, glabrous inside, 4—5-lobed at apex; stamens 25—32, nearly equal; anthers hispid, linear, filaments glabrous towards base, more or less combined at base in pairs or otherwise; ovary rudimentary, hairy. @. Fruit globular, about 1 in. in diameter, subglabrous but with a few scattered appressed short hairs. Fruiting ealyx about 4 im. in diameter, flat and appressed to base of fruit. Brazil, Rio Negro, Gapé below Barcellos, November. Always within (and not on) the skirts of inundated forests, nearly related to D. Péppigiana, Alph. DC. but less common, Spruce / 1913. Plate IX. A branch in male flower, natural size. a. a piece of a branch with male flower abnormally thickened by an insect, not magnified. . interior of male flower cut open, magnified 3 diameters. c. a stamen, magnified 10 diameters. d. a fruit, natural size. 147. DIOSPYROS RIGIDA, sp. nov. D. foliis alternis, oblongis, basi rotundis, rigide coriaceis, supra glabris, subtus pallide subvelutinis, costd robustd, nervis inconspicuis, petiolatis; fructibus cymosis, depresso-globosis glanduloso-pulverulentis, ceterum glabris ; calyce fructifero cyathiformi, fructum cequante, coriaceo, puberulo, profunde 4-lobo, lobis late ovatis erectis. Shoots shortly fuscous-hispid, terete; leaves alternate, oblong or oval-oblong, rounded at base, rigidly coriaceous, glabrous above, pale beneath and covered with thin velutinous tomen- tum, 5—14 in. long by 1}—33 wide, midrib stout, slightly depressed on the upper side, net-veins not conspicuous; petioles stout, wrinkled, puberulous, }—# in. long. ¢. Fruit about 3 together on the young branches, depresso-globose, 1 in. long, covered with reddish glandular pulverulence (as in D. Embryopteris), otherwise glabrous; peduncles 3—1} in. long, nigro-hispidulous, rigid ; fruiting calyx cup-shaped, as high as the fruit, 11 in. in diameter, coriaceous, puberulous, deeply 4-lobed; lobes widely ovate, erect. Borneo, O. Beccari/ n. 2285. 148. Diospyros EmpryoptTeris, Pers. Synops. I. p. 624. n. 6 (1807). D. foliis alternis, oblongis vel anguste ovalibus, apice sepius acuminatis, basi obtusis, coriaceis vel submembranaceis, glabris, petiolatis, reticulatis; floribus masculis avillaribus, racemose cymosis, 3—T-nis, 4- rarius 5-meris, pubescentibus, flavescentibus, calyce patente, 4—5-fido, corolld campanulatd, lobis obtusis, staminibus 24—, pubescentibus, antheris linearibus, fila- Vou. XI. Parr I. 33 258 Mr HIERN, ON EBENACEZ. mentis brevissimis; floribus femineis 1—5-nis, subsessilibus vel cymosis, 4-meris, staminodiis 1—12, pubescentibus, ovario farinaceo-glanduloso, sepius 8-loculari, stylis 4; fructibus globosis vel ellipsoideis. Excl. syn. Lam., Bot. Reg. t. 499 (1820), Alph. DC. Prodr. vu. p. 235. n. 65 (1844), Griff. Notule Iv. p. 289 (1854), Thw. En. Ceyl. Pl. p. 178. n. 1 (1860), Bedd. Fl. Sylv. Madras t. 69 (1870), non Boj. Embryopteris peregrina, Gaertn. Fruct. I. p. 145. t. 29. f. 2 (1788). Garcinia malabarica, Desrouss. in Encycl. Méth. m1. p. 701 (1789). Embryopteris glutinifera, Roxb. Coromand. I. p. 49. t. 70 (1795); £. globularia, ex Miq. Fl. Ind. Bat. m. p. 1048. n. 16 (1856). Diospyros glutinosa, Roxb. Hort. Bengal. p. 40 (1814); Konig ex Roxb. Fl. Ind., edit. 1832, I. p. 533. Diospyros glutinifera, Wall. List n. 4123 B (1828—82). Diospyros malabarica, Kosteletsky, Med. Pharmac. Flora (m1) p. 1099 (1834). Embryopteris gelatinifera, G: Don, Gen. Syst. Gard. and Bot. tv. p. 41 (1837). Diospyros citrifolia, Wall. ex. Alph. DC. l.c. Embryopteris glutenifera, Wight, Ic. Pl. Ind. Of. Vol. 11. pt. 2, p. 4, tt. 843, 844 (1843—47). Diospyros melanoxylon, Hassk. Cat. Pl. Hort. Bot. Bogor. 1. p. 159 (1844), Ettingsh. Blatt- Skel. Dikot. t. 41. £ 9 (1861), non Roxb. A middle-sized or large evergreen tree, glabrous and shining except the buds inflorescence and fruit; there is however occasionally a slight puberulence upon the petioles, &c. Branches straight, spreading. Bark scaly. Leaves oblong or narrowly oval, alternate, usually rounded at base, sometimes subcordate or slightly narrowed, acute lanceolate acuminate or obtuse at apex, highly reticulated with veins in relief on both sides with the exception of the midrib which is depressed on the upper side, coriaceous, of a pale green colour, persistent, 3—12 in. long by 3—33 in. wide; petioles }—#4 in. long, usually channelled above. Flowers yellowish- white, dicecious or polygamous. 6. Cymes about 3—7-flowered, tawny- or fuliginous-pubescent or puberulous, 1—# in. long (excluding the flowers) ; flowers ovoid, } in. long in bud, 2 in. long when open, tetramerous or occasionally pentamerous; calyx } in. long by 2 in. wide, 4-fid, pubescent, lobes pubescent inside; corolla 4 in. long, with pubescent patches of hair outside, glabrous inside, shortly cylindrical, lobes about + in. long, spreading, imbricated sinistrorsely in bud; stamens indefinite, 24—64 or more, nearly equal, inserted on the receptacle or at base of corolla, anthers linear, more or less hairy on back and front, filaments very short, hairy; ovary 0 or rudimentary; receptacle hairy. 9. Flowers 1—5 together, subsessile or cymose, tetramerous, larger than in the male plant, cymes ranging up to # in. long, glabrescent or pubescent; bracts caducous; calyx deeply lobed, pubescent or glabrescent, lobes dilatate-subcordate at base, erect-patent, ovate, 1—% in. long; corolla about 4 in. long, with short nearly erect lobes; staminodes 1—12, hairy (sometimes per- haps perfect stamens), inserted at base of corolla or partly hypogynous; ovary glabrous (normally), reddish-glandular, or with a basal ring of hairs (rarely hairy ?), 8 (—10) -celled; styles 4, hairy at base, dilated and lobed at apex, spreading; fruit usually solitary, subsessile or pedunculate, globular or ovoid, often large (1}—2 in. long), glandular or glabrate, 6—8—10-celled and Mr HIERN, ON EBENACE. 259 -seeded, of a yellowish rusty colour, covered with a rubiginous mealiness; fruiting calyx deeply 4-lobed, puberulous or glabrate, as wide as or wider than the fruit, spreading more or less or erect, with lobes dilatate-subcordate at base, imbricated sinistrorsely. An officinal preparation (Extractum Diospyri of the Pharmacopeeia of India) is a valuable astringent obtained from the fruit of this species, and is useful in diarrhoea chronic dysentery and leucorrhcea and as a local application to bruises and sprains. Of this variable species the following varieties may be noticed :— 8. atrata, Thy. l.c. Leaves thinly coriaceous; buds, peduncles and calyx fuliginous- pilose. y. nervosa, Thw. J. c. Veins on both sides of the coriaceous leayes very prominent; leaves rounded at the base. Buds, peduncles and calyx nigro-pilose. Fruiting calyx-lobes erect. Local names. Panitsjika-maram, Reede, Hort. Malabar. pt. mi. p. 45. t. 41 (1782). Malabarensibus; Tembiri, Brachmanis; Fruita da Grude, Lusitanis ; Lym-appel, Belgis. Tumika of the Telingas, ex Roxb. Corom. J. c. Mangostan-utan of the Malays. Tindooka, the Sanscrit name, ex Roxb. Fl. Ind. Gawb in Bengal. Kibaragma or Kledong in Java. Timberee-gass in Ceylon. Kasi in Banda, India. Gusvakendhu in Goomsur forests, Madras. The fruit when unripe contains a large quantity of tannin, and when ripe is eaten but is not very palatable. The astringent viscid mucus of the fruit is used in Bengal for paying the bottom of boats, and an infusion is employed to steep fishing-nets in to make them more durable. It is also used for book-binding since it preserves the books from insects. Masts and yards of country vessels are made from this tree in Ceylon. India, Silhet, Wallich ! 4123; Quilon, Hurdwar, Amherst, Tavoy, Wallich; N. W. India, Hb. Royle, M. P. Edgeworth!; Bengal, Behar, Hooker fil. and T. Thomson! (Cult. 2); Assam plains!; Upper Assam, Jenkins! 277; Ceylon, Thwaites! C.P. 1915, Walker !, Hb. Wight! 1711 bis, Gardner ! 531 (8 or y); Canara, Mangalor, Hohenacker! 869; Siam, Sir R. Schom- burgk! 115; Java, Dr Horsfield! Eben. 2, 7, 8; Zollinger! 3565; E. Doon, Dr Brandis! Var. 8. Ceylon, Thwaites! C.P. 2731; Mergui, Griffith! 3626, 3627; Tenasserim, Packmann ! Var. y. Ceylon, Thwaites! C. P. 1910. 149, DIOSPYROS CORIACEA, sp. nov. D. tota coriacea, glabrata; foliis alternis, oblongo-lanceolatis vel ovalibus, apice acumi- natis, basi fere rotundatis vel breviter angustatis, petiolatis; floribus femineis solitariis vel raro binis breviter pedunculatis axillaribus, calyce lato, plicato, 4—8-fido, lobis obtusis, corolla breviter semi-ellipsoided 4—3-fidd, lobis rotundatis valde contortis, staminodiis 5 glabris, ovario minute granuloso-glanduloso, subgloboso, 8-loculari, stylo apice lobato, fructibus subglobosis levibus, calyce fructifero ampliato longitudine fructus. Shoots dark-cinereous, glabrate, terete; leaves alternate, oblong-lanceolate or oval, acumi- nate at apex, nearly rounded or somewhat narrowed at base, coriaceous, glabrate, moderately reticulated, 2—4 in. long by $—1} in. wide; petioles +—}in long. 2. Flowers solitary or rarely 2 together, in upper axils, glabrous, coriaceous; peduncles 1 1in. long; calyx wide, plicate, }in. wide, jin. high, 4—8-fid, lobes obtuse; corolla shortly ovoid, as high as the calyx, }in. wide, 4—3-fid, lobes rounded, much contorted ; 33—2 260 Mr HIERN, ON EBENACEZ. staminodes 5 (in one case), glabrous; ovary subglobose, glabrous, covered with minute gland- ular pulyerulence, 8-celled, cells 1-ovuled; style lobed at apex, glabrous; fruit subglobose, 2in. high, glabrous, smooth; fruiting calyx jin. in diameter, widely plicate, about as high as the fruit. Borneo, O. Beccari! n. 1422, 3455. 150. DrospYROS CRASSIFLORA, sp. nov. D. foliis alternis, oblongis, apice anguste acuminatis, basi angustatis, glabris, unicoloribus, tenuiter coriaceis, patentibus, petiolatis, nervis inconspicuis ; floribus masculis crassis, 1—8-nis, brevissime cymosis, awillaribus, calyce depresso-hemispherico, 4—5-fido, utrinque puberulo, lobis rotundatis, corolla ellipsoided, carnosd, apice 4—6-lobd, staminibus «© ©, subequalibus, plurt- serialibus, dorso pubescentibus, hypogynis, ovario minuto, hirsuto. A tall tree, nearly glabrous except the inflorescence; branches dark, terete. Leaves alternate, spreading, oblong, narrowly acuminate at apex, narrowed more or less at base, of same green colour on both sides, very thinly coriaceous, shining above with depressed midrib and inconspicuous veins, with clear lateral and delicate tertiary veins beneath, 7—S8 in. long by 2—22in. wide; petioles 4—4in. long. $. Flowers }—? in. long, 1—8 together, on very short, shortly pubescent axillary peduncles or cymes. Calyx depresso-hemispherical, toughly coriaceous, }in. in diameter, shortly puberulous on both sides, 4—5-fid; lobes rounded. Corolla “ fleshy, of a light pmk colour and of the size and form of a pigeon’s egg,” shortly tomentose outside, nearly oom glabrous inside, 4—6-toothed at apex; teeth contorted sinistrorsely as regarded from within, ;—1 in. deep, obtuse. Stamens very numerous, about $in. long, inserted on the receptacle, subequal, in several rows; anthers linear, acute, 2-celled, somewhat hairy on the back; filaments very short. Ovary minute, hairy. Female flower and fruit unknown. West Tropical Africa, Old Calabar, Rev. W. C. Thomson !, 12 March, 1863. 151. Drospyros piscotor, Willd. Sp. Pl. Iv. p. 1108 (1805). D. foliis alternis, oblongis, apice acuminatis, basi rotundatis, coriaceis, supra nitidis glabris, subtus pallidis appresse pilosis vel glabrescentibus, petiolatis, nervis inconspicuis ; floribus masculis in cymis brevibus trifloris secus ramulos juniores terminaliter confertis, scepius tetrameris, sericeis, calycis lobis ovalibus, rotundatis, corolla infundibuliformi profunde 4-fidd, staminibus 24—28, subequalibus, glabris, geminatis; floribus femineis solitariis, axillaribus, sessilibus, staminodiis 4, 5, 10, glabris, corolle basi insertis, ovario dense piloso, 8-loculars ; fructibus subglobosis, carnosis, pilosis, 4—6-spermis, albwmine non ruminato, calyce fruotifero fructus bast appresso. Alph. DC. Prodr. vit. p. 235. n. 66 (1844). Cavanillea philippensis, Desrouss. in Lam. Encyel. 11. p. 663 (1789). C. Mabolo, Lam. Encycl. tab. 454 (1828). D. Mabola, Roxb. Hort. Beng. p. 40 (1814), Lindl. Bot. Reg. t. 1139 (1828). Mr HIERN, ON EBENACE, 261 D. Embryopteris, Boj. Hort. Maurit. p. 200 (1837), non Pers. Embryopteris discolor, G. Don. Gen. Syst. Gard. and Bot. tv. p. 41 (1837). Diospyros Kaki, Blanco, Fl. Filip. edit. i. p. 302 (1837), non Linn. f. D. Blancoi, Alph. DC. Prodr. vu. p. 237. n. 74 (1844). D. embriopteris, Blanco, Fl. Filip. edit. ii. p. 209 (1845). D. melanida, Sieber!, Fl. Maurit. Suppl. n. 29; non Poir. A tree of moderate size, 40 feet or more high; the trunk furnishes a hard compact ebony of an exceedingly deep black colour. Young shoots and inflorescence fulvo-sericeous. Leaves oblong, alternate, coriaceous, rounded at base, acuminate at apex, brown glabrous and shining above, pale and appressedly pilose beneath, with shining silvery hairs that penetrate the skin and cause it to itch, ultimately glabrescent, 5—8—12 in. long (including petiole }—{in. long) by 2—8—4in. wide, rigid; lateral veins delicate, inconspicuous; midrib depressed above, stout beneath, wrinkled when dry as well as the petioles and young shoots. Sometimes small glands are found on the under side of the leaves. é. Flowers about ;%;in. long, subsessile on short contiguous 3-flowered cymes, which are arranged in terminal or axillary racemes, sweet-scented, tetramerous or occasionally pentamerous. JBracteoles shortly deltoid, acute. Calyx turbinate-campanulate, coriaceous, wider than the corolla-tube, 2 long, deeply lobed, lobes oval, rounded or mucronate; silky out- side, glabrous inside. Corolla silky outside, glabrous inside, coriaceous, funnel-shaped ; lobes rather longer than the tube, spreading, oval. Stamens glabrous, 24—28, in pairs, nearly equal, hypogynous or inserted at the base of the corolla-tube, erect, more or less united at their base; filaments shorter than the linear laterally dehiscing anthers; ovary hairy, rudimentary. 9. Flowers solitary, axillary, bracteate at base, about # in. long, subterminal-spicate, tetramerous or pentamerous, sessile. Calyx open, about }in. high; lobes nearly }in. long and wide, }-oval, coriaceous, cordate at base, appressedly silky outside, glabrous and shining inside, imbricated in various ways. Corolla gin. long, shortly tubular, contracted about middle, silky outside except near base, glabrous inside; tube #in. long, truncate- ovate, lobes about as long as the tube, spreading, }-oval, obtuse, margins incurved, im- bricated sinistrorsely. Staminodes usually 4, occasionally 5 or even 10, much shorter than the corolla; filaments about as long as the barren (2) anthers; all glabrous, alternate with corolla-lobes ; ovary very densely pilose, large, 8- or more-celled, fleshy, 8 !-celled in specimen of Dr Maingay, depresso-conical, cells 1-ovuled; styles 4, distinct, hairy outside or glabrous, arched, converging at apex. Fruit thick, fleshy, globose or subglobose, densely hairy, reddish, like a quince, 4—6-seeded, with flesh rose-coloured, 8—4in. in diameter, pulp white; hairs ferruginous; albumen cartilaginous, not ruminated ; fruiting calyx flattish, appressed, rather more than lin. in diameter. The wood is very hard, of a dark flesh colour, which in time becomes black like ebony. The fruit has an agreeable smell like a quince (but sometimes not so), and is edible after removing the hairs and skin. Local names Mabolo in Tagalog, Amaga in Bisaya, Talang in Pampango, according to Blanco, l.c. Philippine Islands, Manila, Gaudichaud!; Blanco. Cultivated in Mauritius (Hb. Kunth !) and in the Calcutta and Paris Gardens; also introduced at Mahé I. Seychelles, Horne! 345; 262 Mr HIERN, ON EBENACE. Guadalope, Perottet! cultivated (?); Malaya, Pulo Ticus, “ Stem thin,” Dr Maingay! 970/2; Borneo, 0. Beccari! n. 1892, Wallich/ 4131. A form with leaves pale and having numerous inconspicuous veins on both sides, pro- bably introduced, is found at Rio de Janeiro, Brazil, Glaziow! 1560, 1561. 152. DIospyROS ARGENTEA (D. argenteus), Griff. Not. Iv. p. 288 (1854). D. foliis alternis, oblongis, apice acuminatis, basi rotundatis vel cordatis, coriaceis, supra glabris, subtus dense argenteo-pilosis, breviter petiolatis; floribus masculis breviter cymosis, sepius tetrameris, sericeis; calyce 4-fido, campanulato-cylindrico, lobis ovalibus; corolla bre- viter tubulosd, lobis ovalibus; staminibus 22—24, subcequalibus, hirsutis, geminatis, ovari rudimento pubescente ; floribus femineis solitariis, breviter pedunculatis, staminodiis 4—5, ovario dense hirsuto 4-loculari, loculis imperfecte divisis; fructibus ellipsoideis, strigoso-pilosis, 8-locularibus, seminibus 6—8, albumine non ruminato; calyce fructifero 4-partito, aucto ; lobis oblongis. Buds lanceolate-acuminate, with silvery silky hairs; branchlets somewhat compressed, covered as well as inflorescence and petioles with very brilliant silvery and silky hairs which at length become ferruginous-silvery. Leaves alternate, oblong, coriaceous, cordate or rounded at base, sharply acuminate at apex, glabrous above, densely velutinous-pilose beneath with silky shining silvery hairs, which afterwards become ferruginous-silvery and at length mostly fall off, leaving an appressed pubescence and the under surface of the leaf pale, 7—11 in. long by 2—3} in. wide; petioles }—4in. long; margins reflexed; midrib stout, depressed above ; lateral veins inconspicuous. 3. Cymes axillary, spreading, near ends of branchlets, 4—#in. long (exclusive of the flowers), bearing 3— flowers; common peduncle 1—2 in. long; ultimate pedicels short; bracts ovate, glabrous inside. Flowers (closed) nearly } in. long, silky outside, usually tetra- merous. Calyx %,in, long, campanulate-cylindrical, silky on both sides, 4- (in one case 3-) fid, lobes oval. Corolla 2in. long, shortly tubular, 4-lobed, silky on both sides especially outside, lobes Lin. deep, oval. Stamens 22—24, in pairs, nearly equal, very hairy, filaments much shorter than the anthers; ovary rudimentary, hairy. ¢@. Flowers solitary, in axils of upper leaves; peduncles 1—1in. long. Calyx about in. long, 4-fid, densely furred on both sides, campanulate; calyx-lobes ovate, apiculate. Corolla ;4;in. long, 4-fid, tomentose; lobes oval, apiculate, imbricated, hairy inside. Stami- nodes 4—5, alternate with corolla-lobes, hairy above; ovary globose, densely hairy, 4-celled; cells imperfectly divided ; ovules 8; styles 4, hairy, erect, }—4in. Fruit with 1 oval bract at the base 2in, long, ;4,in. wide, glabrous inside, egg-shaped, 21—3in. long by 14—2 in. thick, very strigosely pilose, greenish-white or yellowish, shortly cuspidate at the apex, 8-celled. Fruiting calyx 4-partite, sometimes 8 in. wide; lobes very large, oblong, concave, obtuse, with metallic lustre, very silvery-silky outside, veined inside, 1{—2in. long by { in. wide; seeds 6—8, subcylindrical, slightly attenuated at both ends; ranging up to 2in. long by fin. wide, imbedded in pulp; albumen cartilaginous or horny, white; embryo ;4—?in. long; radicle thick, clavate, about equalling or shorter than the cotyledons, Malacea, Griffith! 3625; Maingay! n. 968. Mr HIERN, ON EBENACEA), 263 153. Drospyros Toposta, Hamilt. in Trans. Linn. Soc. Vol. xv. p. 115 (1827). D. foliis alternis, oblongis ovatis vel lanceolatis, apice dcwminatis, bast obtusis, coriaceis, glaberrimis, crebre reticulatis, petiolatis ; floribus masculis axillartbus, cymosis, calyce initio clauso lobis connatis demum irregulariter apice rupto, corolla urceolatd, apice 4—5-lobd, staminibus @ , glabris ; floribus femineis solitariis, staminodiis 12—16, ovario 4- (rarius 6-) loculari, fructibus subglobosis vel ellipsoideis, glanduloso-pubescentibus vel glabrescentibus, seminibus 1—4; calyce Fructifero 3—4-lobo, pubescente. Ettingsh. Blat.-Skel. Dikot. t. 42. f 7 (1861); Bedd. Ic. Pl. Ind. Or. (Part. vir.) p. 25, t. 122 (1871); Alph. DC. Prodr. vu. p. 287. n. 73 (1844). D. racemosa, Roxb. Hort. Beng. p. 40 (1814); Fl. Ind., edit. 1832, vol. mu. p. 536; Wight, Ic. t. 416: D. lanceolata, Wall. List n. 4122 (1828—32), non Poir. D. incisa, Hamilt. ex. Wall. l.c. Embryopteris racemosa, G. Don, Gen. Syst. Gard. and Bot. tv. p. 41 (1837). Called Yopost in Bengal, where it is cultivated on account of the fragrancy of the flowers; Kahakaala-gass in Ceylon, see Thw. ‘Enum. Ceyl. Pl. p. 179. n. 4 (1860); Goolul in Silhet and Tipperah, see Roxb. Hort. Beng. p. 40 (1814). A large or middle-sized tree with glabrous terete branches. Leaves alternate, oblong ovate or oval, acuminate at apex, obtusely narrowed or rounded at base, coriaceous, closely and clearly net-veined, with midrib depressed on upper surface, shining above, quite glabrous, 3—8in. long by 1—3{in. wide; petioles 4—}in. long. Foliage like D. paniculata, Dalz. g. Cymes axillary }—1in. long, slightly hairy or glabrescent, usually 3-flowered, im cultivated specimens 3—12-flowered; flowers }in. long, yellow, pedicels shorter than the calyx; bracts caducous, at the top of peduncle: calyx at first closed in bud with con- nate lobes, afterwards irregularly broken from apex in unequal acute lobes, scattered with inconspicuous short sete, about in. high. Corolla urceolate, 4-lobed at apex, glabrous except a few short hairs outside along the middle lines of the lobes; Dr Hamilton states that the corolla is 5-lobed. Stamens numerous, indefinite, in one case 33, glabrous, mostly hypogynous ; filaments very short; ovary rudimentary. . Flowers solitary; fruiting peduncle {—}in. long, sometimes at base shortly adnate to the branch so as to become supra-axillary; bracts at top of peduncle, caducous. Calyx as in 6. Corolla tubular-urceolate, 4-lobed at apex. Staminodes 12—16. Ovary 4 rarely 6- celled. Style 0, stigma 4lobed. Fruit oblong or subglobose, 3—1lin. long; glandular and covered with short weak close tawny hairs or glabrescent. Fruiting calyx hairy, with 3—4 oblong or rounded lobes, $— in. across, spreading; seeds 1—4, albumen cartilaginous, not ruminated but with very faint radiating strie near the circumference. East Bengal, Griffith! 3622; Ceylon, not uncommon in damp forests up to an elevation of 4000 feet, Zhwaites! C.P. 1911, 2514, Gardner! 533; Silhet, Roxburgh, Wallich / 4122; ? Khasia, Dr Hooker! (part). A specimen from Borneo, collected by O. Beccari! n. 3052, with leaves 5—11in. long by 13—4in. wide, and subglobose 4celled 4-seeded fruit with deeply trifid calyx nearly lin. in diameter, probably belongs to this species. 9 264 Mr HIERN, ON EBENACE. THE FOLLOWING SPECIES OF DIOSPYROS ARE TOO IMPERFECTLY KNOWN TO BE PLACED IN THEIR POSITIONS IN THE SECTIONS. 154. Diospyros GraTa, Wallich, List n. 4142 (1828—32). D. folits alternis, oblongis, utrinque angustatis, obtusis, glabris, floribus femineis solitariis, subsessilibus, ovario fulvo-hispido ; fructibus globosis, subglabratis, calyce fructifero 5-fido, pen- tagono, utrinque pubescente. Alph. DC. Prodr. vit. p. 282. n. 48 (1844). Branches nearly glabrous, pubescent at the extremities. Leaves alternate, glabrous, oblong, narrowed at both ends, obtusely acuminate at apex, 3—6in. long by 1—2in. wide; midrib depressed above; veins slender, crowded, not conspicuous; petioles 4—{ in. long, glabrous. Q. Fruit solitary, subsessile, globose, about lin. in diameter, glabrate or with remains of ferruginous hairs ; fruiting calyx stellate, 5-fid and 5-cornered, hairy on both sides, tawny, 3in. across; peduncles very short, hairy, Nepal, Wallich! Cfr. D. lanceefolia, Roxb. 155. Diospyros orIXENSIS, Wight Hb.!, non Klein. D. foliis alternis, ellipticis, apice obtuse angustatis vel breviter acuminatis, basi obtusis, glabrescentibus, tenuiter coriaceis, breviter petiolatis; fructibus solitartis asillaribus subglo- bosis, breviter pedunculatis ; calyce fructifero profunde 4-fido, appresso vel leviter patente, extus piloso, lobis obtusis. Young shoots petioles and peduncles hirsute, afterward puberulous, ultimately glabrous, terete; leaves alternate, thinly coriaceous, elliptical, obtuse at base, obtusely narrowed or shortly acuminate at apex, glabrescent, brown on both sides, midrib slightly depressed above and veins inconspicuously raised above, more manifest beneath, subnitescent, 14—3} in. long by #—l} in. wide; petioles in. long, strong. @. Fruit solitary, axillary, dark, subglobose, about 2 in. in diameter, on peduncle about equalling the petiole; bracts caducous; fruiting calyx deeply 4-fid, appressed to base of fruit or somewhat spreading, +in. across, subpilose outside ; lobes obtuse; seeds 2—3, oblong, tin. long. Courtallum, Hb. Wight / 156. Diospyros popECcANDRA, Loureiro Fl. Cochinch. p. 228. n. 5 (1790). D. foliis alternis, late-lanceolatis; floribus aaillaribus; corolle tubo subgloboso, lobis 4, brevibus ; staminibus 18, corolle basi insertis; baccis compressis, lentiformibus, 8-spermis. Alph. DC. Prodr. vit. p. 238. n. 86 (1844). Embryopteris Loureiriana, G. Don, Gen. Syst. Gard. and Bot. rv. p. 41 (1837). Mr HIERN, ON EBENACEA, 265 A large tree with sub-patent branches. Leaves widely lanceolate, quite entire, alter- nate. Flowers hermaphrodite according to Loureiro, axillary, white; corolla 4-lobed, tube subglobose, large, lobes short; stamens 18, inserted at the base of the corolla, Fruit pallid, compressed, lentiform, 1-celled [?], 8-seeded, pulpy; pulp moderate, somewhat sweet, astrin- gent, edible, not good-tasted; seeds compresso-ovate, bony, large. Spontaneous and cultivated in Cochinchina, Loureiro. Local name, Cay Thi trdm. Wood like that of D. decandra, Lour., but without the very black veins in the heart; white and smooth and with dense fibres. Used in gardens to support black pepper plants. 157. Drospyros (?) prtosa, Alph. DC. Prodr. vii p. 219 (1844). D. caule arboreo, foliis alternis, ovato-lanceolatis, subtus tomentosis, breviter petiolatis ; floribus masculis racemosis, rubro-fuscis, calyce 5-lobo, lobis ovatis, corollé 5-lobd, tubo brevi, lacintis ovato-oblongis, crassis, patentibus, calyce sublongioribus, filamentis 15 brevibus, antheris oblongis. Euclea pilosa, Loureiro, Fl. Cochinch, p. 629 (1790). A large tree with ascending branches; dicecious. Wood fit for house-building. Leaves alternate, ovate-lanceolate, quite entire, tomentose beneath; petioles short. Flowers reddish- brown, “in terminal racemes.” é. Calyx 5-lobed, lobes ovate, pilose on both sides; corolla 5-lobed, tube short, lobes ovate-oblong, crass, pilose, patent, rather longer than the calyx; filaments 15, short, anthers oblong, erect. Cochinchina, Zoureiro. Vernacular name Cdy Whaoe. 158. Diospyros HasseLtu, Zoll. Obs. Bot. Nov. p. 15. n. 3 in Natuurk. Tydschr. Neer. Ind. Vol. xiv. (1857). D. foliis ovalibus, utrinque attenuatis, nitidis, glabris; floribus asillaribus, racemosis, racemis suberectis, calycis marginibus in axillis loborum deflexis, laciniis acutis, pedicellis subclavatis pilosis, corolle (fem. ?) tubo 4-gono, pilis nigris presertim ad angulos tecto, staminibus 8 [12], iisdem que lobis corolle alternant simplicibus longioribus, aliis brevioribus bicruris, stylis 4, bifidis; baccd glabra, 8-loculart. Java. Described by Zollinger from a drawing of Kuhl and van Hasselt No. 2b in the Buitenzorg botanical garden. 159. Drosprros Kunin, Zoll. Obs. Bot. Nov. p. 15. n. 1 in Natuurk. Tydschr. Neerl. Ind. Vol. xiv. (1857). D. foliis oblongis, utrinque acuminatis, integris ; floribus lateralibus axillaribus, pedicellis calycem cequantibus, staminibus 8 [12] alternatim bicruris (antheris interioribus brevioribus) aliis sinuplicibus, stylis 2 bifidis, baccd pilosd. Java. Described by Zollinger from a drawing of Kuhl and van Hasselt No. 3 in the Buitenzorg botanical garden, Vou. Xil. Parr I, 34 266 Mr HIERN, ON EBENACE. 160. Drospyros PENDULIFLORA, Zoll. Obs. Bot. Nov. p. 15.n.2 in Natuurk. Tydschr. Neerl. Ind. Vol. xrv. (1857). D. foliis oblongis, utrinque acutis, acuminatis; floribus masculis lateralibus pendulis, pe- dunculo bifido, pedicellis flores cequantibus, calyce nigro-piloso 4-lobo, corolla apertd, staminibus 8 [circiter 202], filamentis brevibus pilosis, alternatum 2- vel 3-cruris ; floribus femineis solitarns pendulis, corolle lobis erectis, staminibus 12 sterilibus, baccd pilosd 5—8-loculari. Java. Described by Zollinger from a drawing of Kuhl and van Hasselt No. 2 a in the Buitenzorg botanical garden. 161. Drospyros (?) cystopus, Mig. Fl. Ind. Bat. Suppl. 1. pp. 250, 584 (1860). D. ramulis teretibus presertim superne cum petiolis foliisque subtus maxime secus nervos rufo-pubescentibus, glabrescentibus, foliis alternis, oblongis, apice caudato-acuminatis, bast ro- iundatis, tenuiter subcoriaceis, supra glabris, subtus costulis patentibus utrinque 18—12 te- nuibus venulosis pertensis, in sicco glauco-fuscescentibus. Young parts rufous-hispid ; branches terete. Leaves alternate, oblong, caudate-acuminate at apex, rounded at base, thinly sub-coriaceous, glabrous above, rufous-hispidulous beneath especially on the raised midrib and lateral ves; about 9in. long by 2}—3 in. wide; petioles lin. long, channelled; lateral veins about 15 on each side, inconspicuous above, slender and more conspicuous beneath ; midrib much raised beneath, tapering towards the apex. Flowers and fruit unknown, and therefore the plant is of uncertain position. Sumatra; Lampong, near Kebang, Tezsmann/ Local name Daréhan-darehan. 162. Drospyros PYRRHOCARPA, Miq. Fl. Ind. Bat. Suppl. 1. pp. 250, 583 (1860). D. ramulis novellis cum petiolis costdque subtus pubescentibus glabrescentibus; foliis e basi rotundatd usque acutiusculd elliptico-oblongis plerisque breviter obtuso-acuminatis, coriaceis, glabris, supra secus costam canaliculatis, subtus pallidis costulis 9—7 tenwibus arcuato-patulis a margine leviter incurvo distanter unitis, dense tenereque reticulatis; floribus secus ramulos inferne lateralisus solitartis brevi-pedunculatis, cum calyce 4—5-partito (lobis acuminatis coriaceis) utrinque rufo-tomentosis ; baccis cerasi majoris mole depresso-globosis, calyce adaucto reflexo (lobis antice convewis) suffultis rufo-ochrascenti-tomentosis. West Sumatra, in province Priaman, Diepenhorst ; Malay name Hampadoe-Kajoe. 163. DiIospyroS PLATYPHYLLA, Welw. MSS. D. arborea, laxe ramosa, apice foliosa, foliis alternis, ellipticis rotundis vel obovatis, apice obtusis, basi rotundatis swpe inequalibus, valde coriaceis, supra glabrescentibus nitidis, subtus tomentosis reticulatis, breviter petiolatis; fructibus edulibus. Mr HIERN, ON EBENACEZD. 267 A moderate-sized tree, with lax tortuous dark-cinereous branches leafy and angular at the apex. Leaves alternate, elliptical rotund or obovate, rounded or obtuse at apex, rounded and often unequal at base, very coriaceous, glabrescent and shining above, more or less tomentose beneath, reticulated but inconspicuously so above, 3—6in. long by 141—34in. wide ; petioles }—1 in. long. Flowers monstrous in the specimen, the inflorescence consisting entirely of densely imbricated ferruginous-tomentose foliaceous scales. W. Tropical Africa, Angola, Pungo Andongo, in sandy woods from Calunda to Condo, fruit said to be edible, Dr Welwitsch! no. 2531; native name Musolveira, the same as that of Diospyros mespiliformis, Hochst, of which it may very possibly prove to be an aberrant form. 164. DIospyROS PLATYCALYX, sp. nov. D. foliis alternis, obovato-oblongis, apice rotundatis, basi cuneatis, coriaceis, utrinque pu- berulis vel glabrescentibus, petiolatis ; fructibus solitarius, subglobosis, glaberrimis, apice umbili- catis, nitidis, 10 (?)-locularibus, breviter pedunculatis ; calyce fructifero profunde 5—6-lobo, plicato, aucto, lobis late ovatis, cordatis, auriculatis ; seminibus conpressis, albumune non ruminato. Tree of 20 feet; young shoots with short patent whitish tomentum; branches glabrescent, terete, palish. Leaves obovate-oblong, alternate, coriaceous, undulated, rounded at apex, cuneate at base, brown, shining, with slight veins, puberulous or glabrescent on both sides, of nearly the same colour on both sides, 2—3in. long by }—1lin. wide; petioles puberulous, }—4 in. long ; midrib slightly depressed on upper surface. Fruit solitary, in axils of fallen leaves, on shoots of previous season, 7% in. long (besides the calyx) by in. thick, subglobose, quite glabrous, umbilicate at apex, 10 (?)-celled, with cells 1-seeded, shining, paler than leaves and calyx. Fruiting peduncle stout, with wide convex articulation, ;4—+ in. long, glabrate. Fruiting calyx 2in. deep by 14in. wide, concealing half the fruit, nearly glabrous, deeply 5—6-lobed ; lobes widely ovate, acute, cordate, much auricled at base, firmly membranous, with sides of lobes reflexed, folding with contiguous lobes and forming 5 dependent spurs the points of which are tin. below the level of the articulation of the fruit; seeds compressed, fin. long or more, albumen not ruminated. Seychelles Islands, Pervillé! 640. 165. DIOSPYROS LEUCOCALYX, sp. nov. D. fruticosa, glabra, foliis alternis, oblongis, apice obtuse acuminatis, basi rotundatis vel subcordatis, costd et nervis lateralibus subtus validis, petiolis validis tumido-crassis; calyce fructifero 4-partito, intus albido-pruinoso, lato, lobis late cordatis, acuminatis, foliaceis. A small shrub, glabrous, dark green but shining. Leaves alternate, subcoriaceous, oblong, obtusely acuminate at apex, rounded or subcordate at base, 1 foot long by 5 inches wide ; midrib and lateral veins strong beneath; petioles more than }in. long, strong, dark, tumid- crass. Fruiting calyx 4-partite, white-pruinose within, 2in. high by 3in. or more wide, erect-patent ; lobes widely cordate, ovate, acuminate at apex, foliaceous. Madagascar, Ambanivoule, Goudot! A.D. 1838. 34—2 268 Mr HIERN, ON EBENACE. 166. Diospyros BERNIERI, sp. nov. D. foliis alternis, ovali-lanceolatis, apice subacwminatis, basi angustatis, coriaceis, glabris, breviter petiolatis, nervis inconspicuis; fructibus solitariis, appresse hirsutis, breviter pedun- culatis ; calyce fructifero utrinque pubescente, 4-fido, tubo concavo, tetragono, incrassato, lobis reflexis, undulatis, late ovatis. Glabrous except the inflorescence; branches pale, terete. Leaves alternate, dark above, oval-lanceolate, somewhat narrowed at base, obtusely or sometimes acutely sub-acuminate at apex, coriaceous, veins indistinct, reddish brown beneath; midrib depressed above, blackish beneath ; 2—3}in. long by }—1in. wide; petiole in. long, black in the dry state. Fruiting peduncles very thick, }in. long and as thick, pubescent, solitary ; fruiting calyx tin. high by nearly 2in. across, pubescent on both sides, 4-fid; tube concave, 4sided, thickened; lobes reflexed, wavy, widely ovate. Fruit ferruginous, shortly and appressedly hairy. Madagascar, common in the forests of Tintingue; vernacular name Voane Silac, Bernier ! 113. Foliage of D. levis, Bojer. 167. DIOSPYROS PRUINOSA, sp. nov. D. bracteis exceptis glaberrima, folirs alternis, ovato-ovalibus, utrinque obtusis, viz coriaceis, brevissime petiolatis, nervis inconspicuis; floribus masculis axillaribus, brevissime cymosis ; fructibus solitariis, axillaribus, 8-locularibus, breviter pedunculatis, bracteatis, subglobosis, cum calyce 4—5-fido plicato patente aucto violaceo-pruiosis. Quite glabrous except the small shortly and slightly ciliated bracts; branches pale brown, terete. Leaves alternate, ovate-oval, more or less obtuse at both ends, submembranous or sub- coriaceous, of a rich brown colour when dry, rather paler beneath, 1—24in. long by 4—1} in. wide; petioles ;4,— ,;1n. long; veins indistinct, spreading; midrib flat above, darker beneath. g. Cymes axillary, 3—8-flowered, fin. long, dark. @. Fruit solitary, axillary on the young branches, shortly globose, }in. thick by 3 in. high, 8-celled, several-seeded, as well as the calyx violaceo-pruinose; peduncles dark i—} in. long; bracts several, ovate, about ;4,in. long; calyx plicate-patent, lin. in diameter, 4—5- (usually 4-)fid, undulated; lobes widely ovate cordate, apiculate or mucronate at apex. Madagascar, Ste Marie, Boivin! 2538; Port Leven, Vesco! 1850. 168. Diospyros cUNEIFOLIA, Hb. Delessert. D. foliis alternis, obovatis, apice rotundatis, basi cuneatis, breviter hispidis, subsessilibus, confertis ; fructibus solitarvis, subglobosis, pilis brevibus hispidis aspersis, pedunculatis ; calyce fructifero pubescente, 5(—6%)-partito, lobis oblongis, patentibus. Mr HIERN, ON EBENACEZ. 269 Shoots puberulous, glabrescent. Leaves alternate, obovate, subsessile, crowded, rounded at apex, cuneate at base, shortly hispid, }—1} in. long by about }in. wide. @. Fruiting peduncle axillary, solitary, ferruginous-hispid, 4in. long; fruiting calyx 5-(or 6-?)partite, pubescent; lobes oblong, spreading, 3 in. long by 4} in. wide; fruit solitary, somewhat depressedly globose, dark, covered with scattered pale short hispid hairs, about tin. long. Mexico, Pavon in Hb. Delessert! 169. DIospyRos APEIBACARPOS, Raddi, Quarante nuove del Brasile, in Atti Soe. Modena, Vol. xvull. p. 12. n. 10 (1820), D. foliis alternis, lanceolatis, acutis, supra glabris, subtus villoso-sericeis, brevissime petiolatis ; baccis depressis, papillis adspersis et setis crebris, subdecaspermis ; calyce 5-lobo. Alph. DC. Prodr. vim. p. 239. n. 96 (1844), Mart. Fl. Bras. vi. p. 8 (1856) excl. syn. A tree of about 30 feet high, with not very thick trunk, very slightly branched; the young branches rather setose at the extremity. Leaves alternate, lanceolate, elongated at the apex, entire, smooth above, scattered with yellowish hairs beneath which are closer along the midrib and round the margin. Calyx 5-lobed. Fruit depressed, scattered with papilla and short setule almost like the hairs with which the petioles of the leaves are covered, size and shape of the Apeiba of Aublet, 1—2 in. thick, S—10-seeded. Brazil, Estrella Mountains, Raddi, fruits in April; Minas, San Francisco River, in woods, Martius; near Borba by River Madeira, Riedel. Martius in Fl. Bras. vil. (Eben.) p. 8 states that this plant is the same as D. sericea, Alph. DC.; but the fruit appears to be different. 170. Diosprros(?) XYLOPIOIDES, Mart. in Fl. Bras. vu. p. 8. n. 4 (1856). D. ramulis subdistichis fulvo- et apices versus albido-sericeis ; foliis subcoriaceis lanceolatis acuminatis basi acutis (20—36"" long., 3—5" lat.) supra glabris, subtus sericeis pilis ap- pressis flexuosis albis, in nervo margine petioloque fulvulis; floribus aaillaribus geminis ternisve bractersque fulvo-sericeis. Arborea. Rami cortice tenui deductili, qualis in multis Diospyris obtinet. Ramuli presertim in extremitatibus dense albo- aut fulvulo-sericei. Folia, tam figura quam dispo- sitione et compage ea Aylopiae frutescentis et nonnullarum affinium assimilantia, supra saturate viridia nervo impresso, subtus pilis mollibus appressis in margine et nervo fre- quentioribus, venis vix conspicuis. Flower-bud sessile, narrowly campanulate, }in. long; calyx trifid or tripartite, glabrous inside, lobes ovate-lanceolate; corolla not exceeding the calyx, silky outside, puberulous inside, tripartite or tripetalous, valvate; stamens 3, erect, with hairy lines, filaments short; ovary rudimentary. 270 Mr HIERN, ON EBENACE:. S. America. Guiana, in woods (Martius !). Scarcely a true Diospyros and nearer to Maba, but probably neither, and perhaps the type of a new genus. The foliage is exceedingly like that of Maba sericea. EXCLUDED AND NOMINAL SPECIES OF DIOSPYROS. Diospyros acapulcensis, Kunth = Maba acapulcensis. Diospyros acuminata, Wall. List, n. 4129 (1828—32). Cfr. Laurine. Diospyros albens, Pres] = Maba albens. Diospyros ambigua, Vent. non Sap.,= Royena ambigua, Vent. Diospyros Berterii, Alph. DC.= Maba inconstans, Griseb. Diospyros cauliflora, Mart. in Fl. Bras. vit. p. 7 (1856), non Blume, = Maba caulilora. Diospyros cerasifolia, D. Don, Prodr. Fl. Nep. p. 144 (1825) = Hurya symplocina, Blume. Diospyros conduplicata, Kunth = Maba inconstans, Griseb. Diospyros cupulosa, F. Muell. = Maba rufa, Labill. Diospyros fasciculosa, F. Muell. = Maba fasciculosa, F. Muell. Diospyros feminina, Hamilt. ex Alph. DC. Prodr. vit. p. 238. n. 83 (1844), = Zurya symplocina, Blume. Diospyros frondosa, Wall. List, n. 4125 (1828—32) = Bocagea elliptica, Hook. fil. et Thoms. Diospyros geminata, F. Muell.= Maba geminata, R. Br. Diospyros grandifolia, Wall. ex Voigt, Hort. Suburb. Caleutt. p. 345 (1845). Mame only. Diospyros hewasperma, Hasselt = Maba elliptica, Forst. Diospyros hirsuta, Desf. non Linn. fil, = Royena hirsuta, Linn, Diospyros humilis, F. Muell. = Maba humilis, R. Br. Diospyros inconstans, Jacq.= Maba inconstans, Griseb. Diospyros lanceolata, Poir., non Wall., = Maba lanceolata. Diospyros lycioides, Desf. = Royena pallens, Thunb. Diospyros microcarpa, Span. in Hook. Comp. Bot. Mag. 1. p. 348 (name only, 1835), non Sieb. Diospyros myrmecocarpus, Mart.= Maba myrmecocarpa. Diospyros oblonga, G. Don, Gen. Syst. Gard. Iv. p. 40 (1837) =? D. venosa, Wall. Diospyros obovata, Wight, Icon. t. 1226 (1850), non Jacq., = Sapotacea. Diospyros obtusifolia, Bert., non Humb. et Bonpl., = Maba inconstans, Griseb. Diospyros psidioides, Kunth = Maba inconstans, Griseb. Diospyros pubescens, Pers., non Pursh, = Royena hirsuta, Linn. Mr HIERN, ON EBENACEA. 271 Diospyros punctata, Korth., non Decaisne, = Maba punctata. Diospyros salicifolia, Humb. et Bonpl.= Maba salicifolia. Diospyros sericea, Alph. DC. = Maba sericea. Diospyros sericocarpa, F. Muell. = Maba rufa, Labill. Diospyros serrata, Hamilt. ex D. Don, Prodr. Fl. Nep. p. 148 (1825) = Ewrya acumi- nata, DC. MMNospyros venosa, Wall. List, n. 4126 (1828—382) =? Anonacea. Diospyros virginica duleis = Carpodinus edulis, Don. Cfr. Alph. DC. Prodr. viii. p. 329 (1844). Diospyros (sp.), Salt! Voyage to Abyssinia, p. 14 (1814) = Zuclea multiflora, var. V. TETRACLIS, gen. nov. &. Flores dioect. Flores cymost, tetrameri, subglobosi. Calyx depresso-globosus ; lobis brevibus depresso-deltoideis, prefloratione valvatis. Corolla carnosa, 4-fida, extus puberula, intus hirsuta; lobis prejloratione valvatis. Stamina circiter 30, pleraque geminata, prope corolle basim inserta ; filamentis brevibus compressis pubescentibus ; antheris hispidulis oblongis liberis, lateraliter bilocularibus ; pollen globosum, leve. Ovarit rudimentum nullum. ¢. Bractee caduce. Fructus superus, solitarius, pedunculatus, subglobosus, subtomen- tosus, ferrugineus, carnosus, 8 (?)-locularis et -spermus; pericarpio crasso. Calyx profunde 4-lobus, accrescens, appressus, semina pendula oblonga, testa non nitidd, Arbor madagascariensis; foliis coriaceis alternis simplicibus integerrimis exstipulatis ; floribus axillaribus, apice 4 lineis cruciatis preefloratione notatis. 1. TETRACLIS CLUSLEFOLIA, sp. nov. T. foliis oblongis vel obovato-oblongis, apice rotundatis, emarginatis vel breviter acumi- natis, obtusis, bast cuneatis, subglabris, coriaceis, petiolatis, nervis tenuibus, crebris. Puiate XI. A fruiting branch, natural size. a. A male inflorescence, natural size. b. A male flower, magnified 4 diameters. c. The same after the removal of the calyx, magnified 4 diameters. d. A vertical section of a male flower, magnified 6 diameters. e. A transverse section of male flower, magnified 5 diameters. A large tree with young parts puberulous; branches dark, somewhat angular, gla- brescent, remotely rugose-verrucose. Leaves oblong or obovate-oblong, rounded emarginate or shortly acuminate at apex, cuneate at base, with recurved margins, subglabrous, yellowish green and shining on both sides; midrib depressed above; lateral veins close, widely spreading, very feeble, and in relief on both sides of the leaf; [the lateral veins on 272 Mr HIERN, ON EBENACE. the lower side of the leaves are too distinctly rendered by the lithographer in Plate XI]; 5—10in. long (besides channelled petioles 3—lin. long) by 1{—38in. wide. g. Cymes on young branches, shortly subferruginous-pubescent, bearmg 38—10 flowers, 3—Zin. long (not including the flowers); common peduncle }—}in. long; pedicels —+} in. long; flowers 1—}in. in diameter, pubescent. Calyx nearly as high as the corolla, 4-lobed at apex, at base somewhat 4-sided outside. Corolla puberulous outside, hirsute or hispidulous inside. Stamens 30 (in one flower), mostly united by their filaments in pairs; anthers hispidulous, filaments hairy inserted near the base of the corolla; pollen globular, smooth, about 7},in. in diameter. Ovary wanting. 2. Fruit (unripe?) solitary, }—{in. high, by 1—1}im. thick, crowned at apex by remains of 4-partite style; fruiting peduncles }—}in. long, thickened upwards, puberulous; fruiting calyx 4-sided, softly hairy on both sides; lobes widely ovate, acute, somewhat cordate and pouting at base, reaching half the height of the fruit, thickly coriaceous. Madagascar, Richard! 388, Nossi-bé; Pervillé! 6. Fosstn EBENACES. About 60 specific names of this family relating to fossils have been published; the first was published by Dr Alexander Braun, about 25 years ago, and the last by Prof. W. Ph. Schimper, in the present year (1872). All these fossils occur in Tertiary strata, with the exception of one, namely Diospyros primeva Heer from the beds of Nebraska in North America, which beds have been recently referred to the Cretaceous period, though they were formerly supposed from the facies of the contained flora to be Tertiary. The majority of the species have been founded on leaves alone; and the venation of these no doubt accords more or less closely with that of those species of Ebenacex, such as Diospyros Lotus, Royena hirsuta, Euclea lanceolata, &c., which fossil botanists seem to regard as the types of their respective genera. There is in fact much variety of venation amongst the recent species of the family; and with respect to recent plants it is quite impossible to assign to the family, with even a moderate amount of certainty, a given leaf of an unknown genus. A few of the fossil species have been described from the calyx fruit or seed, with or without leaves; and the best of these specimens, such as those which have been named Diospyros brachy- sepala, and Euclea relicta, present fair evidence of belonging to the structure of Ebenacex, while even in these instances the genus cannot be properly fixed, and other families are not absolutely excluded. With regard to many of the fossil species, the utmost inference founded on reasonable grounds which can be deduced, is a favourable suggestion of Ebenacee for the family to which the specimens may probably belong; and with regard to other specimens of the published species, it appears to me that Ebenacee is not a probable family for them. It would be much the better plan to refer all fossils, which have nearest affinities to Ebenacex, to a fossil genus ZHbenacites, as was done in the first instance by Saporta, but subsequently relinquished by him in favour of Diospyros. On the whole then, as I place but little confidence in the determination of the fossils, I wish in Mr HIERN, ON EBENACE. 273 no way to confirm them in their present places; but since they have been published as Ebenaceous, I quote them as they stand, with the addition, in some cases, of additional particulars and remarks; I have added the synonymy in accordance with the views of the principal authorities in fossil botany, and have drawn up artificial keys for the genera, and also for the species in each genus, in order to set forth the distinctive characters of the genera and species, so far as their published descriptions allow, and to found a basis for their systematic arrangement. Key To THE Fossi, GENERA, Leaves small, not exceeding lin. long, midrib alone robust. L Royena. Leaves exceeding 1in. long; lateral veins more or less clearly marked. Calyx 4—5-merous. Leaves narrowly elliptical, 3—4 in. long, narrowed at both ends, Il. Huciea. Leaves ovate lanceolate oval or oblong or exceeding 4in. long. III. Diospyros. Calyx 3-merous. TV. MAcREIGHTIA. Diospyros heringiana, Ettingsh. has narrowly elliptical leaves 2—3}in. long, but it was published previously to the reference of any fossil to the genus Luclea. Fossils with a trimerous calyx, especially if the foliage is Ebenaceous, have been referred by authors to Macreightia, a genus which I have merged in the older genus Maba; if then they still merit reference to a recent Ebenaceous genus, they must all be included under the genus Maba. I. Key To THE FossiL Species OF ROYENA. Leaves linear, }in. broad. 1. BR. Myosotis. am oblanceolate or wider, +—}in. broad. Leaves oblanceolate or oblong, lin. long. Leaves oblanceolate. By eR. greca. Leaves oblong. 3. &. Amalthee. Leaves oval or round, }—in. long. Leaves cuneate-orbicular. 4. R. euboea. Leaves oval. : 5. Rk. Pentelict. Vou. XII. Part I. 35 274 Mr HIERN, ON EBENACE. 1. Royena Myosotis, Ung. Foss. Fl. Eub. in Denkschrift. Kais. Akad. Wissensch. Math.- Naturw. Xxvil. p. 69. t. xIv. fig. 5—8 (1867). R. foliis lineari-lanceolatis minimis in petiolum brevem attenuatis integris coriaceis, nervo medio solo distincto; calyce quinquelobo quatuor lineas lato, laciniis mequalibus rotundatis. Diospyros Myosotis, Ung. Gen. et Sp. Pl. Foss. p. 436 (1850), Syll. Pl. Foss. m1. p. 28. t. x. f. 13, 15, (1866); Schimp. Pal. Vég. m. p. 952 (1872); Web. Paleontogr. 1. p. 190. t, XIV. fbb) non! fe 5)ae In Miocene formations, Kumi, Negropont; Eocene, in marly schist, Radoboj, Croatia. Leaves }—1 in. long by jin. wide. Calyx jim. in diameter. Cfr. Porana. According to Prof. Schimper, Unger has comprised several different plants under this species, and Ettingshausen in Sitzungsber. Math—Naturw. Akad. Wissensch. Xxxvitl. p. 492 (1858), has shewn that the leaf which Unger described and figured for this species in Foss. FI. v. Sotzka, p. 172. t. 43. f. 15 (1851), is a leaflet of Cassia phaseolites, while he thinks that Unger's fig. 16 may be a calyx of Celastrus. 2. ROYENA GR&CA, Ung. Foss. Fl. Eub. in Denkschrift. Kais. Akad. Wissensch. Math.- Naturw. XXvul. p. 68. t. XI. fig. 40—51 (1867). R. foliis lanceolato-lingulatis breviter petiolatis integerrimis coriaceis, nervo primario valido, nervis secundariis tenuissimis transversissimis ramosissimis; calyce firmo patente semiquinquefido deciduo, laciniis inequalibus ovato-acuminatis extus striatis 8-millim. longis, margine parum involutis ; drupd siced quadriloculari. Diospyros greca, Saporta in Bull. Soc. Géol, France, xxv. p. 321 (1868). Schimp. Pal. Vég. u p. 954 n. 1 (1872). In Miocene formations at Kumi, Negropont. Leaves lin. long by ;23,;in. wide, oblong, obtuse, cuneate at base; petiole very short. 3. Royena AMALTHES, Ung. Foss. Fl. Eub. in Denkschrift. Kais. Akad. Wissensch. Math.- Naturw. XXVIL p. 69. t. xiv. f. 1 (1867). R. foliis ovato-lanceolatis minimis obtusis in petiolum attenuatis integerrimis coriaceis, nervis secundariis crebris tenuibus ramosis reticulatim conjunctis. Schimp. Pal. Vég. 1. p. 955. n. 2 (1872). In Miocene formations at Kumi, Negropont. Leaves oblanceolate about lin. long by }in. wide. Cfr. R. greca, Ung. 4, ROYENA EUBOEA, Ung. Foss. Eub. in Denkschr. Kais. Akad. Wissensch. Math.- Naturw. Vol. xxvu. p. 69. tab. xiv. fig. 2—4 (1867). R. foliis minimis petiolatis cuneato-orbicularibus coriaceis integerrimis nervo primario valido, nervis secundariis inconspicuis. Schimp. Pal. Vég. 1. p. 955. n. 3 (1872). Mr HIERN, ON EBENACEZ, 275 In Miocene formations at Kumi in Negropont. Leaves 1—}in. long by 1—} in. wide; petioles 4;—75 in. long. Of quite uncertain family. 5. RoyENA Penrecict, Ung. Foss. Fl. Eub. in Denkschr. Kais. Akad. Wissensch. Math.- Naturw. Vol. xxvu. p. 70. t. xiv. f. 9 (1867). R. foliis minimis ovato-ellipticis petiolatis integerrimis coriaceis, nervis secundaris sub- simplicibus fere inconspicuis. Schimp. Pal. Vég. m. p. 955. n. 4 (1872). In Miocene formations at Kumi in Negropont. Leaf 3in. long by }in. wide; petiole ~,in. long. Not unlike a short leaf of 2. glabra 16 L., but placed in this genus on insufficient evidence. Ii. Key to THE Fossi. SPECIES OF EUCLEA. Leaves petiolate. Leaf 3in. long, acuminate at both ends. 1. £E. miocenica. Leaf 4 in. long, narrowed at both ends, not acuminate. 2. E, Apollinis. Leaf sessile. 3. LE. relicta. 1. Evuciea miocenica, Ung. Syll. Pl. Foss. 11. p. 25. t. vu. fig. 8, 8* (1866). E. foliis lanceolatis utrinque acuminatis petiolat’s integerrimis coriacets, nervo primario valido, nervis secundartis flecuosis ramosis rete nervorum tertiariorum laxo inter se conjunctis. Heer Mioc. Balt. Fl. p. 84 t. xxvii fig. 3—8 (1869); Schimp. Pal. Vég. m1 p. 956. n, 1 (1872). In marly schist, Croatia, Unger; Rixhéft, Samland, W. Prussia. Leaf 3in. long by Sin. wide; petiole ;3,in. long. Genus and family quite uncertain. 2. Euctra APOLLINiIs, Ung. Syll. Pl. Foss. m1. p. 26. t. vu. fig. 10, 10* (1866). E. foliis lanceolatis breviter petiolatis integerrimis coriaceis, nervo primario valido, nervis secundariis crebris flexuosis ramosisque rete nervorum tertiariorum laxo inter se conjunctis. Rhododendron Apollinis, Ettingsh. ex Ung. lc. In marly schist, Eocene; Radoboj, Croatia, Unger. Leaf 4 in. long by Sin. wide; petiole scarcely in. long, narrowed at both ends. Leaf very like EZ. miocenica Ung. but rather larger. Prof. Schimper unites this with E. miocenica. 3. Evciea rexicra, Ung. Foss. Fl. Eub. in Denkschr. Kais. Wissensch, Math.-Naturw. Vol. xxvul. p. 68. t. x1 f. 39 (1867). E. foliis lanceolatis utringue attenuatis sessilibus integerrimis coriacets, nervo primario valido, nervis secundartis angulo subrecto exorientibus flexuosis ramosissimis in retem nervorum tertiariorum laxum divisis. Schimp. Pal. Vég. 11 p. 956. n. 2 (1872). In Miocene formations at Kumi in Negropont. Leaf 3tin. long by -8;in. wide, narrowed at both ends; of quite uncertain family. 35—2 276 | | Mr HIERN, ON EBENACE, Ill. Key To THE Fossit SpEcIES oF DIOSPYROS. The species whose leaves have been described may be arranged as follows. | Leaves acute or subacute at apex (sometimes obtuse in D. brachysepala). Leaves more or less narrowed at base, or if sometimes rounded at base then acuminate at apex. Petioles short, not exceeding } in. in length. | Leaves reticulated, secondary veins manifest, branched. | Leaves membranous, often unequal at the base. 1. D. anceps. | Leaves coriaceous, equal at the base. | Leaves 24—3 in. long, ovate. 2. D. vetusta. Leaves about 6 in. long, oval-oblong. 3. D. Loveni. Secondary veins simple or subsimple, manifest. | Leaves about 6 in. long. 4. D. Wodant. | Leaves about 7 in. long. 5. D. Lignitum. Secondary veins obsolete. 6. D. Weberit. | Petioles long. Lateral veins rather distant. Tertiary veins transverse. 7. D. alaskana. Tertiary oblique or in various directions. Midrib very stout; net-veins subobsolete. 8. D. incerta. Midrib moderate ; leaves reticulated. | Calyx 4-fid, with widely ovate or rounded lobes. 9. D. brachysepala. | Calyx 5-partite, with linear lobes. 10. D. paradisiaca. Lateral veins crowded, 11. D. lotoides. Leaves obtuse or subobtuse (sometimes acute in D. varians). Leaves lanceolate or ovate. | Secondary veins serpentine. 12. D. primeva. Secondary veins not serpentine, subremote. | Leaves about 34—43 in. long, membranous. 13. D. Auricula. | Leaves about 2 in. long, subcoriaceous. 14. D. dubia. leat veins numerous, not serpentine. | Leaves 41—11 in. wide. | | Tertiary veins reticulated. 15. D. varians. | Tertiary veins less ramified. 16. D. obscura. Leaves 1{ in. wide. 17. D. paleogea. Leaves oval or elliptical. Leaves 24—3} in. long. | Leaves ;3,—,%, in. wide. 18. D. heringiana. Leaves 14 in. wide. 19. D. pannonica. Leaves 1} in. long by § in. wide 20. D. Royena. Leaves rounded, not cordate at base, subobtuse, not rounded at apex. | Calyx 4-partite with oblong lobes; leaves oval, subcoriaceous. 21. D. stenosepala. Calyx deeply 4-fid, with widely elliptical lobes; leaves oval-oblong, coriaceous 22. D. bilinica. Leaves rounded at both ends, not cordate 23. D. oblongifolia. Leaves cordate at base. 24. D. Parthenon: Mr HIERN, ON EBENACEAS. 277 25. D. obliqua with linear lobes, and 26. Hbenacites rugosus with wider lobes, are known only from the calyx; and 27. D. Zollikofert is described from a cluster of seeds only. 1. Drospyros anceps, Heer, Fl. Tert. Helvet. m1. p. 12. t. cr. fig. 15—18 (1859). D. foliis ovato-ellipticis, apice acwminatis, bast obtusis, membranaceis, hic illic incequilateris, integerrimis, petiolatis, nervis secundariis remotiusculis, sub angulo sat aperto egredientibus, curvatis, ramosis, ipsis et ramis arcuato-conjunctis, rete laxo. Gaudin et Strozzi Mém. Foss. Tose. 1 p. 51. n. 48. t. vit. f. 6 (1859); Heer, Mioe. Balt. Flora, p. 84 t. xxv. fig. 7—9 (1869); Schimper, Pal. Vég. m. p. 948. n. 12 (1872). Miocene; Germany and Tuscany. Leaves acute or subacute, 1}—3}$ in. long by §—1? in. wide; petioles }—+ in. long. 2. Drospyros veTusTA, Giebel, Fl. Sachs. Thiirmg. Braunkohl. in Zeitschr. xvi. p. 57 (1860). D. foliis alternis, ovato-ellipticis, apice acutis vel acuminatis, bast angustatis, coriaceis, nervis secundartis subtilissimis, areis reticulatis; fructu globoso, 5-angulato, 5-spermo; calyce fructifero patente, 5-fido, lobis rotundatis. Heer, Sichs.—Thiiring. Braunk. p. 10 [416]. n. 24. t. viz. f. 1—6 (1861). Schimp. Pal. Vég. 1. p. 946. n. 6 (1872). Eocene; Lignites of the Ligurian strata of Skopau in Thiiringe, Saxony. Leaves 21—3 in. long by ;%—1} in. wide; petiole about 3, in. long; calyx {in. in diameter; fruit 2 in. in diameter. 3. Drospyros Lovent, Heer, Fl. Foss. Arct. p. 118. t. vil. fig. 7 0, ¢, 8, t. xLVII. fig. 8 (1868). D. foliis firms, coriaceis, integerrimis, nervis secundariis remotis, sub angulo acuto egredientibus, valde camptodromis, ramosis, areis argute reticulatis. Schimp. Pal. Vég. 1 p. 949. n. 15 (1872). Miocene; Atanekerdluk, North Greenland. Leaves perhaps 6 in. long by 18 in. wide, areolate, elliptic-oblong. 4. Dtospyros Wopant, Ung. Gen. et Sp. Pl. Foss. p. 435. n. 2 (1850). D. foliis ovato-oblongis, apice acuminatis bast attenuatis petiolatis integerrimis membra- naceis, nervo medio valido, nervis secundariis remotis subsimplicibus sursum arcuatis tenuibus ; baccd globos@ exsuccd semipollicari, calyce quinquelobo deciduo patente, lacinus lanceolatis obtusis striatis pollicaribus. Ung. Syll. Pl. Foss., pug. iii, in Denkschrift. Kais. Akad. Wissensch. Math.-Naturw. XXV. p. 27. t. Ix. fig. 10—12 (1866), Ettingsh. Beitr. z. Foss. Fl. v. Radoboj, p. 55, Schimp. Pall Vég. um. p: 951. n. 22 (1872). Plumeria Flos-saturni, Ung. Gen. et Sp. Pl. Foss. p. 433 (1850); Syll Pl. Foss. mi. p. 27. t. IX. fig. 10—12 (1866). Anona macrophylla, Ung. Gen. et Sp. Pl. Foss. p. 442. n. 3 (1850). Eocene; in marly schist, Radeboj, Croatia. Fruit 2 in. in diameter; calyx deeply 5-lobed, 18 in. in diameter, patent; lebes narrowly 278 Mr HIERN, ON EBENACE. elliptical, obtuse, striate, }—Z in. long by {—}in. wide. Leaves about 6 in. long. Of un- certain family. 5. Drospyros Lignirum, Ung. Syll. Pl. Foss., pug. iii., in Denkschrift. Kais, Akad. Wissensch. Math.-Naturw. xxv. p. 30. t. 1x. f. 9 (1866). D. foliis ovato-oblongis utrinque attenuatis petiolatis integerrimis membranaceis, nervo primario valido, nervis secundariis distantibus simplicibus ramosisque ; seminibus suborbiculart- oblongis obtusis levibus compressis, chalazd parvd immersd. Anona Lignitum, Ung. lc. pug. i. p. 25. t. x. fig. 1—7 (1861), Gen. et Sp. Pl. Foss. p. 441 (1850). Miocene; lignite of Salzhausen in Wetterau and Frofajach in Styria. Leaves 7 in. long by 1$in. wide; seeds Zin. long by in. broad. The seed is not typical of the family. Not given by Schimper in his Traité de Paléontologie végétale among the Ebenacez. 6. Drospyros WEBERII, Massal. Syll. Pl. Foss. Tert. Venit. p. 77 (1859). D. foliis (2) ovatis acutis subpetiolatis integerrimis, nervo primario valido, nervis secundariis nullis ; calyce quinquelobo deciduo minimo patente, laciniis apiculatis. D. Myosotis, Web. Tert. Fl. Niederrhem. Braunkohl. in Dunker et Meyer, Beitr. Naturgesch. Vorwelt, Vol. m1. p. 19. t. xiv. fig. 5 a (1852), non Ung. In Tertiary formations, Italy, &c. Calyx 4 in. in diameter. The leaves (at least) probably belong to Royena Myosotis, Ung. 7, DIOSPYROS ALASKANA, Schimp. Pal. Vég. 1. p. 949. n. 17 (1872). D. foliis ellipticis utrinque acutis, subcoriaceis, nitidis, integerrimis, longe petiolatis, nervo medio valido, nervis secundariis subtilibus (validis ex Lesq.), superioribus alternantibus, inferioribus oppositis, omnibus valde curvatis camptodromis sub angulo acuto egredientibus, areis nervulis transversis obsoletis ramosis reticulatis. D. lancifolia, Lesquer. in Sill, Amer. Journ. ser. ii, vol. XXvII. p. 361. n. 13 (Maio, 1859); Heer, Foss. Pf. v. Van Couver u. Brit. Columb. p. 8 t. 1 f. 10—12, m f. 1—8; Fl. Foss. Alaskana, p. 35. t. mr. f. 12 (1869); non Al. Br. D. lanceolata, “Lesq.” ex Schimp. l.c., non Poir. British Columbia and Neniltschik (Alaska); Billingham Bay, Washington Territory, N. America. Leaf 4in. long by 1} in. wide; petiole } in. long. 8. Diospyros tNcertTA, Massalongo, Synops. Fl. Foss. Senigall. p. 76. n. 187 (1858). D. foliis ovato-lanceolatis utrinque attenuatis acuminatis petiolatis penninerviis integerrimis, costd. validissimd, nervis secundariis rectiusculis, sub angulo 45—50° exorientibus subequi- distantibus, subramosis, apice bifurcatis, arcuatim conjunctis, nervulis venisque subobsoletis. Massal. Fl. Foss. Senigall. p. 295. t. XxvI—xxvil. fig. 6, 29 (1859). Miocene ; Senigallia, Italy. Leaves 4—44 in. long (including petiole 3—#, in. long) by 14 in. wide. The specific name is more suitable than the generic. Mr HIERN, ON EBENACEAD. 279 9. DIosPYROS BRACHYSEPALA, Al. Br. in Leonh. et Bronn, Neues Jahrb. fiir Mineral. 1845, p. 170. D. foliis elliptico-oblongis, apice obtuse vel acute angustatis vel acuminatis, basi cuneatis vel obtusis, subcoriaceis vel submembranaceis, integerrimis, petiolatis, penninerviis, nervis secun- dariis alternantibus, remotiusculis, sub.angulo acuto vel recto egredientibus, eurvatis ramosis, tpsis et ramis dorsalibus marginem versus arcuato-conjunctis, brochiodromis ; calyce quadrifido, medio cicatrice annulart impressd notato, lobis brevibus, late ovatis vel rotundis, apiculatis ; baccd exsuccd, semipollicari. Ung. Bliitt. Swoszowice in Nat. Abh. Gesamm. t. xtv. f. 15 (1849); Heer, Miocene Baltische Flora, p. 84. n. 65. t. xxv. fig. 1—6, t. xxvut. f. 1. (1869), Fl Tert. Helvet. mr. t. cr. fig. 1—14 (1859), Fl. Foss. Arct. p. 117. t. xv. £ 10—12, t. xvm. f. 5 h, 7, t. xivu. fig. 4B, c, d, 5—7 (1868), Braunk. Bornstidt in Abh. Nat. Halle, t. mm. fig. 7, 8 (1869), Phil. Trans. vol. 159. pt. IL p. 475. n. 48. t. L. f 13, t. Lv. f. 8 (1870); Sismonda, Pal. Piém. p. 55. t. XI. f. 6, t. xvi. f. 5, t. xix. f. 3 (1865); Ettingsh. Foss. Fl. Bilin, in Denkschrift. Akad. Kais. Wissensch. Math.-Naturw. Bd. xxvin. p. 232. t. xxxvul. f. 28, t. xxxrx. f. 1 (1868); Schimp. Pal. Vég. 11. p. 949. n. 18 (1872). Tetrapteris Harpyiarum, Ung. Foss. Fl. v. Sotzka, p. 46, t. xxrx. fig. 9 (1850), in Denkschr. Bd. 11. t. L (1851). Getonia petreweformis, Ung. Foss. Fl. v. Sotzka, t. xxxur. f. 2—4, in Denkschr. Bd. 1. t. Liv. (G. petrewfolia ex Schimp. l.c.) G. macroptera, Ung. Foss. Fl v. Sotzka, t. xxxiu. fig. 8, in Denkschr. Bd. m1. t. Liv. G. truncata, Goepp. Tert. Fl. v. Schossnitz, p. 37. t. xxv. fig. 11 (1855). D. lancifolia, A. Br. ex Bruckm. Fl. Oening. Foss. in Jahr. Ver. Nat. Wiirtemb. p. 232 (1850), non Lesgq. D. langifolia, “ Al. Braun” ex Stizenb. Verst. Baden, p. 83 (1851). Arbutus diospyrifolius, Massal. Lett. Scarab. p. 29. n. 203 in Ann. Sc. Nat. Bologn. (1854); Fl. Foss. Senigall. p. 296. t. xxvI—xxvu. f, 3, t. xiv. f. 7 (1859). D. longifolia, “Stiz.” ex Heer, Fl. Tert. Helvet. mi. p. 11 (1859), non Spruce. D. latifolia, “ Al. Br.” ex Schimp. Pal. Vég. uo. p. 949 (1872). Upper middle and lower Miocene formations; North Greenland, W. Prussia, France, Switzerland, Italy. Leaves 11—5 in. long by 3—24 in. wide; petioles ranging up to Zin. long. Calyx $—}in. in diameter. 10. Drospyros PARADISIACA, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Wissensch. Math.-Naturw. xxv. p. 234. t. xxxvull. fig. 29—31, 34 (1868). D. foliis lanceolatis, utrinque angus'atis, basi acutis, integerrimis membranacers, petiolatis, nervo primario distincto recto, nervis secundariis tenuibus, inferioribus sub angulo 45°, medits et superioribus sub angulis obtustoribus, arcubus laqueorum maculis externis instructis, nervis tertiariis tenuissime dictyodromis; baccd ovided, exsuccd ; calyce 5-partito, patente, deciduo, lacintis linearibus obtusis, nervoso-striatis, via semipollicaribus. 280 Mr HIERN, ON EBENACEA. Schimp. Pal. Vég. 11 p. 946. n. 5 (1872). Miocene; Tripoli de Kutschlin, Bohemia. Leaf 3hin. long or more by ;%in. wide; petiole }in. long; calyx-lobes }—in. long by + in. wide; fruit (?) 2in. long. 11. Drospyros LoTomsEs, Ung. Syll. Pl. Foss., pug. m1, in Denkschrift. Kais. Akad. Wissensch. Math.-Naturw. xxv. p. 30. t. x. fig. 1—12 (1866). D. foliis lanceolato-oblongis utrinque attenuatis longe-petiolatis, margine plus minus undulato, integerrimis plurinervits, nervo primario valido, nervis secundartis crebris, sub angulo plus minus acuto emissis, marginem versus arcu brevi conjunctis, apice se conjunctis, nervis tertiariis trans- versalibus utplurimum obsoletis vel parum conspicuts, calyce minimo, 5-fido, patente, laciniis rotundatis. Ung. Foss. Fl, d. iilt. Braunk. d. Wetterau, p. 59 ; Schimp. Pal. Vég. 1. p. 951. n. 20 (1872). Borraginites myosotiflorus, Ludw, Palwontogr. in Mey. Beitr. Naturg. Vorw. vill. p. 116 (1860). Miocene; lignite at Wetterau, Germany. Leaves 3—5hin. long by {—1$in. wide; petiole ,{—1}in. long. Apparently not Ebe- naceous. Cfr. Juglans acuminata, Ludw. and J. ventricosa, Ludw. . 12. Diospyros primava, Heer, Phyll. Crét. der Nebraska, p. 19. t. 1. fig. 6, 7. D. foliis oblongo-ovalibus, apice obtusiusculis, integerrimis ; nervis secundartis serpentinis, ramosis, camptodromis. Schimp. Pal. Vég. m. p. 948. n. 14 (1872). Upper cretaceous deposits (?) in Nebraska, N. America. D. anceps of the Miocene of Europe and D. alaskana of the molasse of North America greatly approach this species from Nebraska. 13. Drospyros AurticuLA, Ung. Gen. et Sp. PI. Foss. p. 436 (1850), Syll. Pl Foss. 111. p. 26. t. rx. f. 1—4. D. foliis ovatis utrinque attenuatis integerrimis membranaceis, nervo primario valida, nervis secundariis subremotis, sub angulo acuto egredientibus, sursum repetito-arcuato-anasto- mosatis, subarcuatis, apice ramosis; calyce quadrifido vel quinquefido deciduo patente, laciniis subquadratis emarginatis basi callosis striatisque semipollicaribus. Schimp. Pal. Vég. 11. p. 947. n. 10 (1872). D. auriculata, Stiehler, Synops. Pflanz. Vorw. 1. 147 (1861), non Wight. Eocene; Croatia, in marly schist at Radoboj. Leaves somewhat narrowed at both ends, obtuse at apex, 38—4$in. long by 14—114} in. wide; petiole ;§,—;%,in. long; calyx-lobes }in. long. The characters given agree with Diospyros. Mr HIERN, ON EBENACEA. 281 14. Driospyros nusra, Goeppert, Tert. Fl. Java, p. 47. t. xin. f. 72 (1854). D. foliis ovatis subobtusis subcoriaceis integerrimis penninerviis, nervis secundariis alter- nantibus subremotis sub angulo acuto 60° circa exorientibus adscendentibus curvatis ramosis, ramulis ante marginem in maculas transeuntibus in rete solutis. Schimp. Pal. Vég. 11. p. 947. n. 9 (1872), non Wight. Pliocene ?; Pesawahan, Java, Junghuhn 353. Leaf (estimated from a fragment) probably about 2 in. long by 1 in. wide. Family quite conjectural. 15. DIOsPYROS VARIANS, Saporta in Ann. Sc. Nat. ser. v. vol. 1. p. 111. t. Iv. fig. 14, t. vi. fig. 4 (1865), vol. vim. p. 91. t. x. fig. 7, 8 (1867). D. foliis lanceolatis ellipticis oblongo-lanceolatis vel ovatis, apice obtusis quandoque breviter attenuatis, basi parum inequalibus, subcoriaceis, breviter petiolatis, integerrimis ; petiolo trans- verse rugoso ; nervo primario valde expresso ; nervis secundariis tenuibus nwmerosis reticulatis ; nervis tertiariis in rete flecuoso subtiliter venuloso coeuntibus. Schimp. Pal. Vég. um. p. 944. n. 1 (1872). Tertiary; S.E. France, frequent. Leaves 2}in. long by }in. wide, 3}in. by I4in,, 341n. by 2 in. 16. Drospyros opscurA, Sap. Etud. mu. p. 283 in Ann. Se. Nat. ser. v. vol. Iv. p. 188 (1865). D. foliis lanceolatis, coriacets, breviter lateque petiolatis; nervo primario valido, secun- dariis obliquis, secus marginem areolatis, inconspicuts. Schimp. Pal. Vég. u. p. 947. n. 8 (1872). Upper Tertiary; S.E. France, Armissan ; rare. Only differs from D. varians by thicker and little longer petiole, by the more regularly lanceolate leaves, and by the less ramified secondary ascending veins, which are united near the margin by very obtuse curves. 17. Diospyros PALZOGHA, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Wissensch. Math.-Naturw. XXvill. p. 233. t. xxxvill. f. 24—26, 32 (1868). D. foliis ovalibus, obtuse acuminatis, basi angustatis, integerrimis, coriacets, petiolatis, 4—5 pollices longis, nervo primario distincto, nervis secundariis crebris, tenuibus, flecuosis, ramosis ; baccd globosd, exsuccd, fere pollicari ; calyce jirmo, quinque-partito, patente, deciduo, semipollicari, laciniis ovato-lanceolatis, acuminatis. Schimper, Pal. Vég. vol. 11. p. 945. n. 4 (1872). Miocene; Tripoli de Kutschlin, Bohemia. Leaf 43 in. by 13in. wide; petiole in. long; calyx nearly fin. in diameter. Wo, SOM, 1a Jk 86 282 Mr HIERN, ON EBENACEAL. 18. DIOSPYROS H&RINGIANA, Ettingsh. Tert. Fl. Haring, p. 61. t. xx. f. 26, toexxiiot. L851): D. foliis lanceolatis vel elongato-lanceolatis, petiolatis, integerrimis, sub-coriaceis, utrinque angustatis, petiolo rugoso; nervatione dictyodroma, nervo primario valido, nervis secundariis tenuibus, sub angulo 60—80° orientibus, arcuatis, ramosis; calyce 4-fido, segmentis parum productis, acutis. Saporta in Ann. Sc. Nat. ser. iv. vol. xIx. p. 72. t. 1x. f. 1 (1863); Schimper, Pal. Vég. u. p. 945. n. 2 (1872). Tertiary ; in calcareous bituminous schist at Haring, Tyrol; marly beds, S.E. France. Leaves 14—3} in. long by 4—;5in. wide, narrowly elliptical, obtuse at apex; calyx } in. in diameter. 19. DrospyRos PANNONICA, Ettingsh. Foss. Fl. Wien, p. 19. t. mr. f. 8 (1851). D. foliis ellipticis, bast angustioribus, integerrimis, petiolatis, nervis secundariis undulatis, sub angulo 50—60° orientibus, apice ramosis et in rete abeuntibus, nervis reticulatis e nervo primario sub angulo recto, e nervis secundartis sub angulo acuto egredientibus, ramosis. Schimper doubtfully unites this to D. anceps, Heer. Vienna, in marly schist. Leaf 22 in. long by 14in. wide. Very like leaf of D. brachysepala, but apparently more obtuse. 20. Diospyros Royvena, Ung. Syll. Pl. Foss., pug. ii., in Denkschrift. Kais. Akad. Wissensch. Math.-Naturw. xxv. p. 29. t. 1x. fig. 18, 19 (1866). D. foliis ovalibus breviter petiolatis integerrimis sesquipollicem longis, nervo primario distincto, nervis secundariis crebris tenuibus ramosis ; calyce firmo quinquelobo patente deciduo semipollicari, laciniis acuminatis. Schimp. Pal. Vég. m1. p. 952 (1872). In marly schist; Radoboj, Croatia. Leaves 11in. long by 2in. wide; petiole -3;in. long. Calyx-lobes ;8;1n. long by in. wide. Family quite uncertain. 21. Drospyros STENOSEPALA, Heer, Fl. Foss. Alaskana, p. 35. t. vill. fig. 7, 8 (1869). D. foliis ovalibus, basi rotundatis, integerrimis, subcoriaceis ; calyce fructifero quadripartito, lobis oblongis, apice rotundatis. Schimp. Pal. Vég. u. p. 949. n. 16 (1872). Miocene; English Bay, Alaska, N. America, Fwruhjelm. The shape and size of the calyx correspond with those in D. brachysepala, but the lobes are longer and narrower. Calyx 3 in. in diameter; leaves 1} in. wide. Mr HIERN, ON EBENACE. 283 22. Drospyros BILINICA, Ettingsh. Foss. Fl. Bilin, m1. p. 45, in Denkschrift. Akad. Wissensch. Math.-Naturw. Xxvil. p. 233. tab. xxxrx. fig. 17, 18 (1868). D. foliis coriaceis, oblongo-ellipticis, crassiuscule petiolatis, basi rotundatis, apice sub- obtusis, integerrimis, nervo primario basi valido, apicem versus sensim angustato, nervis secundarwis sub angulis acutis orientibus, tenuissimis, subremotis, nervis tertiariis obsoletis ; calyce profunde quadrifido, deciduo, patente, minimo, laciniis ovalibus, longitudinaliter nervoso- striatis, basi coarctatis. Schimp. Pal. Vég. u. p. 947. n. 10 (1872). D. bohemica, Schimp. l.c. p. 945. n. 3. Miocene; menilite-opal of the valley of Schichow near Bilin, Bohemia. Leaf 43 in. long by 1$in. wide; petiole in. long. Calyx }—%in. in diameter, lobes rounded. The leaf much resembles that of D. Auricula, Ung., but differs by the more con- siderable thickness of the petiole and midrib. 23. DIoSPYROS OBLONGIFOLIA, Heer, Braunk. von Bornstiidt in Abhandl. d. Nat. Gessellsch. zu Halle. x1. Bd. p. 17. t m1. f. 9 (1869). D. folits oblongis, utrinque obtusis, integerrimis; nervis suprabasilaribus ultra medium productis, ceteris utrinque 4 remotis, patentioribus, apice cum nervis tertiarits transversis arcuato-conjunctis, nervulis e nervo medio et e nervis secundariis sub angulo recto emissis, inter se parallelis, reticulo minuto. Schimp. Pal. Vég. 1. p. 950. n. 19 (1872). Eocene; Bornstaédt near Eisleben, Saxony, about N. Lat. 514°. Leaf nearly 3 in. long by rather more than lin. wide, rounded at both ends. 24. Diospyros PARTHENON, Ung. Syll. Pl. Foss., pug. iii., in Denkschrift. Kais. Akad. Wissensch. Math.-Naturw. xxv. p. 29. t. ix. f. 8 (1866). D. foliis ovato-acuminatis basi subcordatis integerrimis membranaceis longe petiolatis, nervo primario valido, nervis secundariis crebris tenuibus subpatentibus apice diviso anasto- mosatis. Schimper, Pal. Vég. 11. p. 948. n. 13 (1872). Miocene; lignite at Wetterau, Germany. Leaf 4in. long by 12, in. wide; petiole 13in. long. The long petiole associated with a subcordate leaf-base is not suggestive of Ebenacez. 25. Drospyros oBLigua, Ung. Syll. Pl. Foss., pug. iii, in Denkschrift. Kais. Akad. Wissensch. Math.-Naturw. xxv. p. 29. t. 1x. fig. 17, 17* (1866). D. calyce quinquelobo deciduo minimo patente, laciniis e basi lata angustatis linearibus obtusis. 36—2 284 Mr HIERN, ON EBENACE. Schimper, Pal. Vég. m1. p. 951. n. 23 (1872). In marly schist, Radoboj, Croatia. Calyx about }in. in diameter; lobes about }in. long. Like Royena Myosotis, Ung., but calyx-lobes narrower. Cf. Porana. 26. EBENACITES RUGOsUS, Sap. in Sap. et Math. Exam. Anal. Fl. Tert. Provence, p. 31 (1861). E. foliis (2) ovatis, petiolatis, integerrimis ; nervis secundariis curvatis, nervis tertiariis sinuosis transversim reticulatis; floribus unisexualibus ; calyce 5-lobo lobis inequalibus, extus rugoso-sulcatis, intus levibus, estivatione imbricatis; masculorum corollé erecté breviter urceolatd calycibus breviore; feminorum segmentis calycinis primum erectis, ovarium 2—3- stylum foventibus, demum patentibus, indurato-persistentibus, baccam globosam ipsis breviorem stipantibus. Diospyros rugosa, Sap. in Ann. Sc. Nat. ser. iv. vol. xvii. p. 264. t. xi. f 3. A, B, C, D, E, F. (1862); Schimp. Pal. Vég. u. p. 946. n. 7. (1872). Tertiary ; beds of gypsum at Aix, S.E. France, common. The styles are not Ebenaceous in character. Male flower 3 in. diameter, calyx and corolla lobed half way, lobes ovate; female flower }im. thick, calyx deeply lobed, lobes ovate acuminate, styles } in. long. 27. Diospyros ZOLLIKOFERI, Ung. Syll. Pl. Foss., pug. iii, in Denkschrift. Kais, Akad. xxv. p. 27. t. ix. f. 6 (1866). D. seminibus ovoideis, compressis, distinctis, nwmero octo in orbem dispositis—residuis fructus baccati globularis. Schimp. Pal. Vég. 0. p. 951. n. 21. (1872). Miocene; Hengsberg, Styria. Seeds —2 in. long by {—,% in. wide. IV. Key to THE Fosstt SPECIES OF MACREIGHTIA. Peduncles not thickened upwards Leaves 13—2} in. long by 3—+, in. wide. 1. MM. germanica. Leaves more than 8 in. ioe i 1} in. wide, 2. M. microcalya. | Peduncles thickened upwards. Calyx-lobes ovate-acuminate. 3. IM. longipes. Calyx-lobes ovate or cuneiform, obtuse 4. M. miinzenbergensis. 1. MAcCREIGHTIA GERMANICA, Heer, Fl. Tert. Helvet. m1. p. 13. t. ciii. fig. 1, 2 (1859). M. foliis late-lanceolatis acuminatis in petiolum mediocrem attenuatis integerrimis vel margine inequali passim denticulatis coriaceies, nervo medio robusto, nervis secundariis e Mr HIERN, ON EBENACE. 285 nervo primario sub angulo acuto egredientibus subsimplicibus rectis parallelis; calyce firmo pedunculato tripartito, lobis bast ovato-acuminatis nervosis; baccd rotundd calyce basi cinctd. Ettingsh. Foss. Fl. Bilin, p. 234. (1868); Schimp. Pal. Vég. um. 953. n. 1 (1872); Ung. Syll. Pl. Foss., pug. iii. p. 26. t. viii. fig. 12, 13 (1866). Celastrus europeus, Ung. Gen. et Sp. Pl. Foss. p. 459 (1850), Syll. Pl Foss., pug. ii. p. 10. t. ii. fig. 10—15 (1864). Tertiary; Parschlug, Styria; Croatia; Oeningen, &c. Calyx only like Macreightia (Maba) in being trimerous; if the leaves are sometimes denticulate as stated, the plant cannot belong to Ebenacee. In Celastrus europeus the leaves measure 13—2} in. long by 3—, in. wide; petioles about 4 in. long. 2. MACREIGHTIA MICROCALYX, Ettingsh. Foss. Fl. v. Bilin in Denkschrift. Akad. Wissensch. Math.-Naturw. xxvii. p. 234. t, xxxix. f. 2—5. n. 2 (1868). M. foliis lanceolo-oblongis, basi angustatis, obtusis, apicem versus angustatis, margine integerrimis, nervis secundariis camptodromis, nervo primario valido, nervis tertiartis obsoletis ; calyce submembranaceo, pedunculato, tripartito, extus piloso, lobis ovato-acutis, basi latis, apice breviter cuspidatis, nervoso-striatis ; baccd rotundd, calycis basi cinctd. Schimper, Pal. Vég. 1. p. 953. n. 2 (1872). Miocene; Kutschlin, Bohemia. Leaf 14 in, wide by more than 3in. long. Calyx 4—3in. long; perhaps a fourth lobe at the back of the impression is concealed by the front ones. 3. MACREIGHTIA LONGIPES, Ettingsh. Beitr. z. Kenntn. d. Tertfl. Steierm. p. 58, t. Iv. fel Oli: M. calyce longe pedunculato, pedunculo sursum sensim incrassato, lobis ovato-acuminatis, acutis. Schimp. Pal. Vég. m1. p. 954. n. 3 (1872). Lignite at Leoben, Styria, Austria. 4. MACREIGHTIA MUNZENBERGENSIS, Ettingsh. Foss. Fl. d. alt. Braunk. d. Wetterau, p. 59. M. calyce tripartito lobis ovatis vel cuneiformibus, obtusis, nervosis. Schimp. Pal. Vég. 1. p. 954 n. 4 (1872). Hydrocharis ovata, Ludw. Palewontogr. in Mey. Beitr. Naturg. Vorw. vil. t. xxtv. f. 6 (1860). Tertiary ; sandstone at Miinzenberg, Darmstadt, S.W. Germany. Calyx (2)-lobes ?—Z in. long. As much like a calyx of Barringtonia or a split fruit of Viola as Maba. Peduncles thickened upwards. The following names of fossil species have been published apparently without descriptions or with extremely meagre ones. 286 Mr HIERN ON EBENACEA. Diospyros lawrina, Massal. Syllab. Pl. Foss. Tert. Venet. p. 77 (1859). Macreightia ttalica, Massal. 1. c. Macreightia? wmbellata, Massal. 1. c. Diospyros discreta, Saporta, Vég. Sud-Est France, Ep. Tert. in Ann. Sc. Nat., ser. v. vol. xv. p. 321 (1872). Diospyros ambigua, Saporta, l.c., non Vent. Diospyros rhodendrifolia, Saporta, l. ¢. Diospyros corrugata, Saporta, l. c. Diospyros styracifolia, Saporta, l.c. p. 333; D. tyracifolia, Sap. in. Bull. Soc. Géol. France, Xxv. p. 895 (1868). Diospyros raminervis, Saporta, l. c. p. 333; in Bull. Soc. Géol. France, xxv. p. 895 (1868). Diospyros Scheuzeri, A. Br. ex Ung. Pflanzenwelt, p. 233 (1851), is Labatia Scheuzeri, A. Br. ex Stiehler, Synops. Pflanz. Vorwelt, I. p. 147 (1861). ADDITIONS AND CORRECTIONS. During the time that the previous sheets have taken in passing through the press numerous new specimens of Hbenaceze have reached this country and been presented to my notice, containing indeed several new species and affording additional matter for the more complete knowledge of old ones. So far as circumstances allowed, I have incorporated the additional matter in its proper place, and such information as was not sufficiently early for that purpose, I now add at the end: I also take the same opportunity of correcting the misprints, mistakes, and omissions that have been noticed, and of making any slight additions that further research has rendered desirable. The estimate given on page 61 for the number of recent species in the family and in the genera Maba and Dviospyros will require a slight increase in each case; thus the whole Natural Order contains more than 260 recent species, of which about 100 are new or not previously described; and if the fossils are included the whole number will be at least 300. P. 27, 1.10. For Intctntz read ILictinea&. P. 28, 1. 5 from bottom. For Bertolini read Bertolont. P. 30, 1. 11. For Paralia read Paralea. » 1. 27. For Blum. read Blanco. P. 33, 1. 4. Add, Java, Sumatra and Borneo. » 118. Strike out Java (?). P. 37, 1. 10. Strike out Mart. » L 18, 24, 33. For caprefolia read capreefolia. P. 40, 1 19. For 4117. Diospyros sylvatica, Roxb. read 4117. Diospyros sylvatica, Wall. ], 24. For Roxb. read Wall. P. 41. Insert, among the numbers of Hs. GRIFFITH AND HELFER, 3620. Diospyros Horsfieldii. P. 44, 1. 7. For Hendelotii read Heudelotii. 288 Mr HIERN, ON EBENACE. P, 44. Insert BeccaRi. 1865—1868. Borneo, District Sarawak. No. 1399. Maba punctata. No. 2285. Diospyros rigida. 1422. Diospyros coriacea. 2486. Diospyros fuliginea. 1423. Maba punctata. 2492. Diospyros Beccarii. 1429. Maba (2) cordata. 2542. Diospyros dictyoneura. 1550. Maba Maingayi. 2591. Diospyros Beccarii. 1560. Diospyros graciliflora. 2612. Diospyros asterocalyx. 1600. Diospyros lateralis. 2615. Diospyros dictyoneura. 1670. Maba merguensis. 3052. Cfr. Diospyros Toposia, Ham. 1787. Diospyros pergamena. 3101. Cfr. Diospyros. 1822. Maba Teijsmanni. 3120. Diospyros (sp.). 1837. Maba (?) cordata. 3225. Diospyros plectosepala. 1892. Diospyros discolor, Willd. 3455. Diospyros coriacea. 1948. Maba Beccarii. 3567. Diospyros rotundiflora. 1949. Diospyros (sp.). 3568. _ Maba myrmecocalyx. 1973. Diospyros buxifolia. P. 44, 1. 20. For 1854 read 1852. » Under GERRARD AND M°KEN insert 18. Royena cordata, E. Mey. P. 45. 1606. Euclea daphnoides. 1. 10. ‘For sofia E. Mey. read glandulosa, Harv. » At bottom, add, 1740. Cape of Good Hope. Euclea lanceolata, E. Mey. P. 46. Under Botus add, 128. Graaf Reinet. Royena pallens, Thunb. » 527. 4 Royena cordata, E. Mey. 572. a Euclea ovata, Burch. 655. - Euclea undulata, Thunb. 50. Under BEerNIER add, 259 (excl. fr.). Madagascar. Diospyros haplostylis, Boiv. 51, 1. 9. For Maba capreefolia read Diospyros capreefolia, and prefix 1011. 57, 1. 4 from bottom. After lobis insert integris. 60, 1. 2. After equal insert entire. 67, 1. 32. For p. 38 read p. 37. . 69, Insert 1827. Ferreola guineensis, Schum. and Thonn, Plant. Guin. p. 448. Guinea, Africa. » lL 4& from bottom. For Sweet. read Sweet,. P. 72. Insert 1843. Rymia polyandra, Endl. Cat. Hort. Acad. Vindob. 1. p, 123. n. 4583. Cape of Good Hope. P. 73. Insert 1850. Anona macrophylla, Ung. Gen. et Sp. Pl. Foss. p. 442. Croatia. last two lines for 1850 read 1851. P. 75. Insert 1861. Diospyros longifolia, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 7. South America. 1861. Diospyros glomerata, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 7. South America, id FUare fos 0) se 1861. 1861. l. 6 from bottom. For Island read Islands. ” Mr HIERN, ON EBENACEZ. 289 Diospyros polyandra, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 7. S. America. Macreightia myristicoides, Spruce in Journ. Proc. Linn. Soc. Lond. v. p. 8. S. America. P. 76, insert 1865. Diospyros obscura, Sap. Etud. p. 283 in Ann. Se. Nat. ser. v. vol. Iv. p. 138. S.E. France. P. 77, insert 1868. Diospyros paleogza, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Kais. 1868. 1868. 1868. 1868. 1868. 1868. 1869. 1869. LE ee Be P. 94, L PP. 95, 96. Wissensch. Math—Naturw. XXVIII. p. 233. t. XXXVI. fig. 24—26, 82. Bohemia. Diospyros bilinica, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Kais. Wissensch, Math.—Naturw. XXVIII. p. 233. t. XXxXIX. fig. 17, 18. Bohemia. Diospyros paradisiaca, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Kais. Wissensch. Math. Naturw. XXVIII. p. 234. t. XXXVI. fig. 29—31, 34. Bohemia. Macreightia microcalyx, Ettingsh. Foss. Fl. Bilin in Denkschrift. Akad. Kais. Wissensch. Math.—Naturw. XXVIIL p. 234 t. xxxrx. f. 2—5. Diospyros greca, Saporta in Bull. Soc. Géol. France xxv. p. 321. S. E. France. Diospyros styracifolia, Saporta in Bull. Soe. Géol. France xxv. p. 895. S. E. France. Diospyros raminervis, Saporta in Bull. Soc. Géol. France xxv. p. 895. S. E. France. Diospyros oblongifolia, Heer Braunk. vy. Bornstadt in Abh. Nat. Gesell. Halle, xi. Bd. p. 17. t. m1. fig. 9. Saxony. Diospyros stenosepala, Heer Fl. Foss. Alaskana in Kongl. Svenska Vetenskaps— Akad. Handl., Ny Foljd, viii. p. 35. t. vu. fig. 7, 8. N. America. Macreightia longipes, Ettingsh. Beitr. z. Kenntn. d. Steierm. p. 58. t. Iv. fig. 10, 11, ex Schimp. Pal. Vég. m1. p. 954 (1872). Austria. Macreightia miinzenbergensis, Ettingsh. Foss. Fl. d. alt. Braunk. d. Wetterau, p. 59, ex Schimp. Pal. Vég. 11. p. 954 (1872). Germany. Diospyros primeva, Heer Phyll. Crét du Nebraska, p. 19. t. 1 fig. 6, 7. N. America. Diospyros bohemica, Schimp. Tr. Pal. Vég. 11 p. 945. n. 8. Bohemia. Diospyros alaskana, Schimp. Tr. Pal: Vég. m1. p. 949. n. 17. N. America. Diospyros Roxburghi, Carriére in Revue Horticole, p. 253. N.E. India. Diospyros discreta, Saporta in Ann. Sc. Nat. ser. v. vol. xv. p. $21 (sine descriptione). S.E. France. Diospyros rhododendrifolia, Saporta in Ann. Sc. Nat., ser. Vv. vol. xv. p. 321 (sine descriptione). S.E. France. . Diospyros corrugata, Saporta in Ann. Sc. Nat., ser. v. vol. XV. p. 321 (sine descriptione). S.E. France. For Stamens 16—17 read Stamens 16—22. For Stamens 16—17 read 16—22. Euclea rigida, E. Mey. must be removed from £. pseudebenus, E. Mey. to E. lancea, Thunb. P. 99, 1. 8 from bottom. For Stamens 16 read Stamens 12—16. P. 100. To the localities for EUCLEA DivinoruM add Basuta country, 7. Baines ! ”? Add as a synonym of EUCLEA MULTIFLORA Diospyros (Sp.), Salt, Voyage Abyss. p. 14 (1814) ; Vou. XII. Part L. 37 290 Mr HIERN, ON EBENACEE. And among the localities for this species, add Sofala Bay, Mozambique, 19 August, 1809, Salt! . 103, 1. 10 from bottom. For (1847) read (1851). . 104, 1. 18. After Fruit isert edible. 106. Zo EucLEA UNDULATA, Thunb. add Wooded chasms, Swellendam, a tree, Lichtenstein ; Basuta country, “Tolangoola,” 7. Bazines!. According to Thunberg, the berries when bruised and fermented yield a vinegar. P. 107, last line. After pilose insert except MM. (?) cordata. P. 141. In the character of MABA CORDATA insert floribus masculis tubulosis, 5-meris, lobis lanceolatis, calyce partito, staminibus 12, glabris, tnequalibus. rg td And in the description insert 3. Flowers tubular, # in. long, pentamerous. Calyx } in. long, partite; lobes narrowly lanceolate, glabrous inside. Corolla-tube equalling the calyx, pubescent outside above; lobes 2 in. long, oval-lanceolate, subacute, puberulous outside, glabrous inside. Stamens 12, unequal, glabrous, inserted on the receptacle; anthers apiculate ; filaments unequal, more or less combined at base. Ovary 0. Pedicels } in. long; bracts caducous, unequal. Borneo, O. Beccari! n. 1837. P. 142, 1. 16. For MSS. read in Journ. Proc. Linn. Soc. Lond. v. p. 8 (1861). P. 144, At the end of the descriptions of the species of MABA insert EXCLUDED SPECIES OF MABA. Maba Cargillia, F. Muell. Fragm. v. p. 162 (1866) = Diospyros Cargillia, F. Muell. Maba pentamera, F. Muell. Fragm. v. p. 163 (1866) =Diospyros pentamera, Woolls et F. Muell. Maba quadridentata, F, Muell. Fragm. v. p. 162 (1866) = Diospyros mabacea, F. Muell. P. 144, 1. 2 from bottom. Add non Solander ex Lowe Man. Fl. Madeira vol. 11. p. 34 (1872). P. 151, 1. 11. For glabrous read subglabrous. P. 159, 1. 19. Prefix the reference Kern. Hort. Semperv. t. 64. P. 161, 1. 16. Add ; non Wall. P. 197, 1. 1. For Golunto read Golungo. P. 201, 1. 7 from bottom. Add ; non Hort. P. 218, 1. 9. After Bat. insert II. 1. 17. Add D. dioica, Spanoghe in Hook. Comp. Bot. Mag. 1. p. 348 (1835). P. 222,. Among the localities for Diospyros MONTANA, Roxb., insert Pegu, Kurz! 3008, 3009. P, 225. At end of synonymy of Diospyros virGINIANA, Linn., add D. stricta, Hort. ex Loud. Encyel. l.c., non Roxb. D. digyna, Hort. ex Loud. Encycl. lc, non Jacq. P, 240, 1. 4 from bottom. Add non Heer. P. 244, ‘To the synonymy of Diospyros Epenaster Retz, add Cfr. Lolin, Valentyn, Oost-Ind. Deel mt. Stuk i. p. 223. t. LxIv (1726). P. 286, 1. 7. Jor rhodendrifolia read rhododendrifolia. ALPHABETICAL INDEX OF THE LATIN NAMES. The Numbers on the right-hand of the names denote the pages on which the names severally occur; the Numbers on the left-hand, in the case of adopted specific names of recent species, denote the number of the species under its genus. Aberia tristis, Sond., 90. Ecidium (sp.), 86. Amuxis (§), 146, 156. Annona microcarpa, Jacq., 68, 144, 246. Anona Lignitum, Ung., 73, 278. », macrophylla, Ung., 277, 288. Anonacex, 62, 63, 271, tab. 1, Arbutus (sp.), Linn., 78, 83. 59 diospyrifolius, Massal., 74, 279. Barberia (§) 107, 110. Barringtonia, Forst., 285. Bicornes, Giseke, 57. Bignoniacez, 28. Bixineex, 63, tab. 1. Bocagea elliptica, Hook. f. et Thoms., 270. Borraginites myosotiflorus, Ludw., 280. Brachycheila pubescens, Harvy., 73, 90, 93. Brachynema ramiflorum, Benth., 65, 75. Buxus (sp.), Linn., 78, 88. », sempervirens, Linn., 231. Cargilia Auct. Cfr. Cargillia. Cargillia, R. Br., 60, 64, 66, 144, 145, 146, 155. arborea, A. Cunn., 239. australis, R. Br., 56, 64, 68, 246. laxa, R. Br., 56, 64, 68, 211. mabacea, F. Muell., 76, 239. macrocarpa, Vieill, 242. maritima, Hassk., 73, 211. 5 megalocarpa, F. Muell., 76, 211. » pentamera, Woolls et F. Muell., 76, | 239. Carpodinus edulis, G. Don, 271. Carpophaga magnifica, Selby, 240. Casasia calophylla, Rich., 126. 38 17 Cassia phaseolites, Heer, 75, 274. Cavanilla, Auct. Cfr. Cavanillea. Cavanillea, Desrouss., 144, 146, 156. - Mabolo, Lam., 69, 260. 55 philippensis, Desr., 67, 260. Celastrinex, 63, tab. 1. Celastrus, L., 274. a crispus, Thunb., 69, 90, 99. 5 europeus, Ung., 73, 285. Chailletiacez, 63, tab. 1. Convolvulacex, 63, tab. 1. Cordia, Plum., 96. Cunalon, Blanco, 197. Cunalonia (§), 146, 150. Dactylus trapezuntinus, Forsk., 67, 144, 224. Danzleria (§), 146, 153. 55 axillaris, Bert., 72, 145, 231. Diclidanthera, Mart., 27. Diospiros, Auct. Cfr. Diospyros. Diospyracee, Voigt, 57. Diospyrex, 57. Diospyri, Tratt., 57. Diospyros, Linn., 30, 32, 58, 59, 60, 61, 63, 64, 65, 66, 107, 144, 165, 184, 264, 270, 272, 273, 276, 280, 287, 288, tab. 1. oe acapulcensis, Kunth, 69, 128, 270. 3 acuminata, Wall., 40, 70, 270. = acuta, Thw., 33, 41, 75, 149, 182. ss affinis, Thw., 33, 41, 75, 147, 169. 5 alaskana, Schimp., 276, 278, 280, 289. 3 albens, Presl, 70, 127, 270. » ° ambigua, Sap., 286. 53 ambigua, Vent., 67, 86, 270. mA amoena, Wall., 40, 70, 213. 5 amplexicaulis, Lindl. et Paxt.,74, 179. 37—2 144 56 37 76 Diospyros anceps, Heer, 75, 276 ”? ” ” Mr HIERN, 2 angulata, Poir., 68, 178. angustifolia, Lodd., 7 anonefolia, Alph. DC, 38, 72, 148, 179. apeibacarpos, Radd., 37, 60, 69, 269. apiculata, Hiern, 33, 42, 149, 186. argentea, Griff., 34, 41, 42, 74, 156, 265. Arnottiana, Miq., 76, 171. artanthefolia, Mart., 37, 53, 74, 155, 255. assimilis, Bedd., 76, 208. asterocalyx, Hiern, 150, 193, 288. atrata (var.), 259. attenuata, Thw., 33, 41, 75, 149, 182. aurea, Teijsm. et Binn., 34, 54, 74, 151, 206. Auricula, Ung., 73, 276, 280, 283. auriculata, Stiehl., 280. auriculata, Wight, 42, 188. australis, 30, 31, 36, 46, 54, 246. Barteri, Hiern, 35, 43, 149, 187. batocana, Hiern, 31, 35, 148, 174. Beccarii, Hiern, 151, 204, 288. Bernieri, Hiern, 38, 50, 268. Berterii, Alph. DC., 72, 127, 270. bicolor, K1., 75, 165. biflora, Blanco, 34, 70, 152, 217. bilinica, Ettingsh., 276, 283, 289. Blancoi, Alph. DC., 73, 261. bohemica, Schimp., 283, 289. Boivini, Hiern, 38, 150, 194. borneensis, Hiern, 33, 55, 148, 173. Boutoniana, Alph. DC., 72, 178. brachysepala, A. Br., 73, 979, 276, 279, 282. bracteata, Roxb., 69, 221. Brandisiana, Kurz, 33, 77, 149, 184. brasiliensis, Mart., 51, 74, 244. burmanica, Kurz, 34, 77, 147, 166. buxifolia, Hiern, 34, 42, 288. calophylla, Hiern, 38, 50, 147, 156. calycina, Audib., 71, 225. calycina, Bedd., 188. canarica, Bedd., 77, 164, 165. Candolleana, Wight, 41, 73, 164. Canomoi, Alph. DC., 73, 216. 55, 152, 218, 137 96 149 112 ON EBENACE, Diospyros capensis, Alph. DC., 72, 178, 179. capitulata, Wight, 73, 233. capreefolia, Mart., 37, 51, 155, 254, 287, 288. Cargillia, F. Muell., 77, 155, 246, 290. caroliniana, Muhl., 69, 225. Carthei, Hiern, 33, 151, 198. cauliffora, Blume, 34, 58,69, 154, 234. cauliflora, Mart., 142, 270. cayennensis, Alph. DC., 37, 72, 153, 231. cerasifolia, D. Don, 69, 270. chartacea, Wall., 34, 40, 70, 153, 230. chinensis, Blume, 69, 227. chloroxylon, Roxb., 30, 34, 40, 41, 42, 64, 67, 153, 233. chrysophyllos, Poir., 38, 53, 68, 148, 177, 180. ciliata, Alph. DC., 36, 153, 223. ciliata, Rafin., 70, cra citrifolia, Wall., 72, 2 coccolobeefolia, Mart., ae 51, 53, 74, 155, 251. Commersoni, Gaertn. fil., 68, 179. comorensis, Hiern, 38, 153, 220. concolor, Moench, 67, 225. conduplicata, Kunth, 69, 127, cordifolia, Roxb., 40, 41, 54, 6 222 coriacea, Hiern, 156, 259, 288. corrugata, Sap., 286, 289. costata, Carr., 77, 227, 228, fig. p. 229. crassiflora, Hise 35, 156, 260. crumenata, Thw., 33, 41,75, 147, 169. Cunalon, Alph. DC., 30, 34, 60, 73, 150, 197. cuneifolia, Hb. Deless., 36, 268. cupulosa, F. Muell., 77, 114, 270. cystopus, Miq., 34, 75, 266. dasyphylla, Kurz, 33, 77, 151, 203. decandra, Boj., 244. decandra, Lour., 31, 33, 65, 67, 147, 160, 265. Dendo, Welw., 195, tab. x. densiflora, Wall., 33, 40, 70, 147, 171. dictyoneura, Hiern, 150, 192, 288. Diepenhorstii, Miq., 34, 58, 60, 75, 145, 154, 235. 270. 1, 22 1; 29, 35, 48, 60, 150, 156 ~I (9.4) 69 47 19 40) 86 134 ALPHABETICAL INDEX OF THE LATIN NAMES. 293 Diospyros digyna, Hort., 290. ” digyna, Jacq., 67, 244. dioica, Span., 70, 290. discolor, Willd., 29, 34, 37, 40, 42, 46, 54, 59, 68, 156, 260, 288. discreta, Sap., 286, 289. dodecandra, Lour., 34, 60, 67, 264. dubia, Goepp., 276, 281. dubia, Wall., 40, 70, 159, 160. dulcis (var.), 226. Ebenaster, Retz., 29, 30, 31, 34, 36, 37, 42, 50, 67, 154, 179, 244, 290. Ebenaster, Spach, 208. Ebenum, Koen., 29, 34, 38, 40, 41, 42, 56, 59, 67, 151, 208, 209, 245. Ebenum, Poir., 176. ebenus, Auct. Cfr. D. Ebenum. edulis, Lodd., 69, 244. ebretioides, Wall., 33, 40, 70, 147, 162, 163. emarginata, Hiern, 37, 52, 156, 256, 1 ee © embriopteris, Blanco, 261. Embryopteris, Boj., 261. Embryopteris, Pers., 29, 30, 31, 34, 40, 41, 42, 43, 53, 54, 55, 68, 156, 257. erlantha, Champ., 31, 33, 52, 74, 151, 202. exculpta, Hamilt., 69, 158. exsculpta, Auct., 158. culpta, Hamilt. fasciculosa, F. Muell., 77, 135, 270. feminina, Hamilt., 73, 270. ferruginea, Spltgbr., 73, 240. fertilis, Lodd., 70, 225. flavicans, Hiern, 34, 40, 41, 49, 59, | 151, 205. foliolosa, Wall., 30, 33, 40, 70, 150, 188. frondosa, Wall., 40, 70, 270. frutescens, Blume, 34, 55, 69, 134, | 147, 170. frutescens, Hassk., 193. fuliginea, Hiern, 149, 184, 288. Gardneri, Thw., 29, 34, 41, 53, 75, | 152, 214. gaultheriefolia, Mart., 37, 53, 74, 155, 250. Cfr. D. ex- | 141 133 53 52 154 23 29 158 87 bo Diospyros geminata, F. Muell., 77, 119, 270. ” glaberrima, Rottb., 67, 208. glabra (var.), Alph. DC., 224. glauca, Rottl., 34, 67, 233. glomerata, Spruce, 37, 52, 155, 254, 288. glutinifera, Wall., 69, 258. glutinosa, Roxb., 69, 258. Goindu, Dalz., 74, 221, 222. Goudotii, Hiern, 37, 54, 155, 249. graciliflora, Hiern, 150, 191, 288. gracilipes, Hiern, 38, 50, 59, 150, 191. greca, Sap., 274, 289. grandifolia, Wall., 73, 270. grata, Wall., 34, 40, 70, 264. guiacana, Rob., 68, 225. heringiana, Ettingsh., 74, 273, 276, 282. : halesioides, Griseb., 37, 52, 76, 148, 173. haplostylis, Boiv., 29, 38, 50, 148, 177, 288. Hasseltii, Zoll., 35, 74, 265. hebecarpa, A. Cunn., 36, 54, 56, 77, 209. Hebenaster, Gaertn., 179, 244. heterophylla, Wall., 40, 70, 221, 222, 242. Heudelotii, Hiern, 35, 44, 152, 215, 287, tab. v. fig. 2. hexasperma, Hasselt, 73, 123, 270. hirsuta, Desf., 83, 270. hirsuta, Linn. fil., 29, 33, 41, 42, 43, 67, 147, 163, 164, 165, 166. hispida, Alph. DC., 37, 49, 51, 72, 155, 249. Horsfieldii, Hiern, 33, 54, 150, 193, 287. humilis, F. Muell., 77, 120, 270. incerta, Massal., 75, 276, 278. incisa, Hamilt., 69, 263. inconstans, Jacq., 66, 127, 270. insculpta, Hamilt., 69, 158, 159. insignis, Thw., 29, 33, 41, 75, 147, 157. intermedia, Hort., 71, 225, 226. japonica, Sieb. et Zuce., 54, 73, 224. Kaki, Blanco, 261. Kaki, Linn. fil., 30, 31, 34, 41, 52, 54, 67, 153, 227, 228, fig. p. 229, 230. 14 165 33 100 118 129 Mr HIERN, ON EBENACE. Diospyros Kirkii, Hiern, 31, 35, 151, 199. Korthalsiana, Hiern, 33, 147, 168. Kuhlii, Zoll., 35, 74, 265. Kurzii, Hiern, 33, 65, 147, 162. levis, Boj., 38, 72, 153, 232, 268. lancezefolia, Roxb., 30, 34, 41, 68, 152, 213, 264. lanceolata, Poir., 68, 131, 270. lanceolata, Schimp., 278. lanceolata, Wall., 40, 263. lancifolia, A. Br., 74, 279. lancifolia, Lesq., 278. langifolia, Stiz., 279. lateralis, Hiern, 147, 167, 288. latifolia, Schimp., 279. laurifolia, A. Rich., 37, 74, 244. laurina, Massal., 75, 286. laxa, Teijsm. et Binn., 74, 172. leucocalyx, Hiern, 38, 54, 267. leucomelas, Poir., 30, 38, 50, 68, 148, 7S: Lignitum, Ung., 76, 276, lobata, Lour., 67, 227, 22 longifolia, Heer, 74, 279. longifolia, Spruce, 240, 241, 288. lotoides, Ung., 76, 276, 280. Lotus, Blanco, 216. Lotus, Linn., 30, 31, 41, 52, 54, 66, 153, 223, 226, 228, 230, 272. Lotus, Lour., 195. Loureiriana, G. Don, 35, 44, 48, 63, 70, 150, 194. Loveni, Heer, 77, 276, 277. lucida, Hort., 225. lucida, Wall., 40, 70, 164. lycioides, Desf., 68, 85, 270. mabacea, F. Muell., 36, 77, 154, 239, 290. Mabola, Roxb., 69, 260. Mabolo, Wall., 70. macrocalyx, Alph. DC., 72, 178. macrocalyx, K1., 195. macrocarpa, Hiern, 38, 56, 154, 242. macrocarpa, Korth., 168. macrophylla, Blume, 34, 69, 154, 237. macrophylla, Wall., 40, 216. malabarica, Kostel., 70, 258. Malacapai, Alph. DC., 30, 33, 73, 155, 247. 143 82 30 46 97 94 89 77 31 124 Diospyros Mannii, Hiern, 35, 44, 155, 255. 2”? maritima, Blume, 34, 36, 52, 54, 55, 64, 69, 152, 211. mauritiana, Aiph. DC., 72, 178. megalocarpa, F. Muell., 77, 211. melanida, Neraud, 179. melanida, Poir., 38, 50, 68, 148, 177, 179. melanida, Sieber, 261. melanoxylon, Blume, 172. melanoxylon, Hassk., Ettingsh., 258. melanoxylon, Roxb., 29, 31, 33, 41, 42, 43, 67, 147, 159, 160, 161. melanoxylon, Willd., 208. membranacea, Alph. DC., 72, 244. mespiliformis, Hochst., 29, 31, 39, 43, 44, 48, 71, 147, 165, 267. mexicana, Scheele, 238. microcarpa, Sieb., 224, (var.) 226. microcarpa, Span., 35, 70, 270. microphylla, Bedd., 77, 218. microrhombus, Hiern, 29, 38, 150, 187. mollis, Wall., 30, 40, 71, 162. montana, Heyne, 159. montana, Roxb., 29, 34, 36, 40, 41, 42, 43, 67, 153, 220, 222, 290. Moonii, Thw., 41, 75, 164. Morrisiana, Hance, 31, 52, 74, 153, 219. multiflora, Blanco, 30, 34, 57, 152, 216. multiflora, Wall., 40, 70, 213. Myosotis, Ung., 73, 274. Myosotis, Web., 278. myrmecocarpus, Mart., 74, 141, 270. Neraudii, Alph. DC., 72, 178. nervosa (var.), 259. nigra, Blanco, 73, 244. nigricans, Dalz., 208. nigricans, Wall., 34, 40, 41, 70, 151, 207. nilagirica, Bedd., 77, 164. nodosa, Poir., 38, 53, 68, 148, 178, 179. oblonga, Wall. 34, 40, 42, 69, 154, 243, 127 ALPHABETICAL INDEX OF THE LATIN NAMES. Diospyros oblongifolia, Heer, 276, 283, 289. obovata, Jacq., 67, 197. obovata, Wight, 42, 270. obscura, Sap., 276, 281, 289. obtusifolia, Bert., 127, 270. obtusifolia, Humb. et Bonpl., 68, 244. ocanensis (var.), 253. octandra, Hiern, 33, 41, 167. oleifolia, Wall., 34, 40, 70, 151, 204. Qlen, Hiern, 38, 55, 154, 246. oligandra, Bedd., 77, 164. oocarpa, Thw., 33, 41, 60, 75, 145, 148, 171. oppositifolia, Thw., 30, 33, 41, 75, 147, 157. orixensis, Hb, Wight, 35, 42, 264. orixensis, Klein, 68, 161. ovalifolia, Wight, 34, 41, 42, 73, 154, 237. ovalis, Hiern, 37, 53, 155, 248. paleogea, Ettingsh., 276, 281, 289. paniculata, Dalz, 33, 43, 74, 150, 190, 263. pannonica, Ettingsh., 74, 276, 282. paradisiaca, Ettingsh., 276, 279, 289. Paralea, Steud., 30, 37, 52, 53, 71, 154, 240, 241, 287. Parthenon, Ung., 76, 276, 283. parvifolia, Hiern, 38, 152, 217. Pearcei, Hiern, 155, 252. pellucida, Hiern, 34, 57, 151, 209. pellucido-punctata (var), 231. pendula, Hasselt, 34, 73, 154, 236, penduliflora, Zoll., 35, 74, 266. pentamera, Woolls et F. Muell., 30, 36, 54, 77, 154, 239, 290. perforata, Hiern, 34, 154, 2438. pergamena, Hiern, 154, 234, 288. Persimon, Wikstr., 70, 225. peruviana, Hiern, 52, 53, 155, 253. Pervillei, Hiern, 38, 50, 150, 192. philippinensis, Alph. DC., 34, 57, 72, 152, 212. phyllomegas, Steud., 71, 237. pilosa, Alph. DC., 30, 35, 72, 106, 265. pilosanthera, Blanco, 30, 35, 70, 152, 213. pilosula, Wall., 33, 40, 70, 150, 188. platycalyx, Hiern, 38, 50, 267. 163 67 145 64 136 167 43 111 75 295 Diospyros platyphylla, Welw., 35, 48, 59, 266. ” ” ” ” plectosepala, Hiern, 151, 201, 288. Poeppigiana, Alph. DC., 37, 52, 53, 12; 156) 2565 25 polyalthioides, Korth., 33, 151, 198, tab. vil. polyandra, Spruce, 37, 52, 155, 251, 289. primeva, Heer, 272, 276, 280, 289. pruinosa, Hiern, 38, 59, 268. pruriens, Dalz., 33, 41,43, 74,149,185. psidioides, Kunth, 69, 127, 270. pterocalyx, Boj., 70, 178. pubescens, Pers., 68, 83, 270. pubescens, Pursh, 225, 226. punctata, Decaisne, 70, 221, 222. punctata, Korth., 136, 271. pyrrhocarpa, Mig., 35, 75, 266. quesita, Thw., 30, 33, 41, 75, 148, 157, 174, racemosa, Roxb., 69, 263. ramiflora, Roxb., 29, 34, 40, 58, 69, 154, 235, 236. raminervis, Sap., 286, 289. reticulata, Decaisne, 209. reticulata, Sieb., 179. reticulata, Wall., 208. reticulata, Willd., 68, 176. revoluta, Poir., 68, 244. rhodocalyx, Kurz, 34, 77, 154, 241. rhododendrifolia, Sap., 286, 289, 290. rigida, Hiern, 156, 257, 288. rotundiflora, Hiern, 147, 163, 288. rotundifolia, Hiern, 36, 46, 148, 181. Roxburghi, Carr., 227, 228, 289. Royena, Ung., 76, 276, 282. Roylei, Auct., 159=D. Roylii, Wall. Roylii, Wall., 40, 70, 159, 160. rubiginosa, Roth, 69, 159. rubra, Gaertn. fil., 68, 177. rugosa, Sap., 76, 284. rugosula, R. Br., 56, 68, 221, 222, rugusola, Alph. DC., 221. salicifolia, Humb. et Bonpl., 68, 128, 129; 271. samoénsis, A. Gr. 30, 34, 46, 75, 154, 209, 245. Sapota, Roxb., 40, 69, 244. gapotanigra, DC., 69, 244. sapotoides, Kurz, 151, 206. 296 107 ” ” Mr HIERN, ON Diospyros Scheuzeri, A. Br., 74, 286. Schi-tse, Bunge, 70, 227. senegalensis, Perr., 44, 72, 165. senensis, K1., 35, 43, 75, 148, 181. sericea, Alph. DC., 51, 53, 72, 140, 269, 271. sericocarpa, F, Muell., 77, 114, 271. serrata, Hamilt., 69, 271. speciosa, Wood, 77, 158. spinosa, Hiern, 37, 155, 247. Sprucei, Hiern, 37, 52, 152, 210, tab. vim. Sprucei (var.), 253. squamosa, Boj., 38, 50, 72, 153, 220. squarrosa, K1]., 35, 75, 150, 190. stenosepala, Heer, 276, 282, 289. stricta, Hort., 290. stricta, Roxb., 33, 40, 41, 60, 68, 151, 201. styracifolia, Sap., 286, 289. subacuta, Hiern, 38, 149, 185. suberifolia, Decaisne, 150, 189. subrotata, Hiern, 37, 49, 155, 250. sumatrana, Miq., 34, 74, 154, 236. sylvatica, Roxb., 29, 33, 41, 65, 67, 147, 161, 287. sylvatica, Wall., 40, 221, 287. Tallak (var.), 134. tessellaria, Poir., 29, 31, 38, 68, 148, 175, 176, 177. tetrandra, Hiern, 37, 152, 210, tab. v1. tetrandra, Span., 71, 211. tetrasperma, Sw., 37, 52, 53, 151, 197. texana, Scheele, 32, 36, 51, 52, 73, 154, 238. Teysmanni, Migq., 75, 203. Thouarsii, Hiern, 38, 153, 232. Thwaitesii, Bedd., 77, 164. timoriana, Miq., 35, 74, 176, 209. tomentosa, Poir., 68, 233, 242. tomentosa, Roxb., 40, 158. Toposia, Hamilt., 34, 41, 53, 59, 69, 145, 156, 263, 288, toxicaria, Hiern, 30, 38, 44, 148,175. tricolor, Hiern, 35, 149, 183, tab. v. fig. 1. truncata, Zoll. et Mor., 29, 33, 54, 55, 73, 148, 172. 67, EBENACE. 4 Diospyros Tupru, Buch., 29, 31, 33, 68, 147, 88 104 70 66 15 93 101 140 170 98 158, 159. tyracifolia, Saporta, 286. Umlovok, Griff., 73, 224. undulata, Wall., 34, 40, 41, 42, 70, 152, 215. vaccinifolia, Ettingsh., 230. vaccinioides, Lindl., 31, 34, 40, 41, 52, 54, 55, 69, 153, 230. varians, Sap., 76, 276, 281. variegata, Kurz, 34, 77, 151, 203, 223. velutina, Hiern, 36, 37, 50, 51, 53, 54, 151, 200. velutina (var), 221. - venosa, Wall., 40, 70, 270, 271. vernalis (var.), 195. verrucosa, Hiern, 35, 147, 167. Vescoi, Hiern, 38, 152, 218. vetusta, Giebel, 75, 276, 277. virginiana, Linn., 30, 32, 52, 58, 66, 153, 223, 224, 228, 230, 290. virginica, Auct. Cfr. D. virginiana. virginica dulcis, 271. Waldemarii, Kl., 75, 221, 222. Weberi, Stiehl. Cfr. D. Weberii. Weberii, Massal., 75, 276, 278. Weddellii, Hiern, 37, 51, 155, 253. Wightiana, Wall., 40, 70, 159, 160. Wodani, Ung., 73, 276, 277. xylopioides, Mart., 37, 74, 269. Zollikoferi, Ung., 76, 277, 284. Zollingeri, Hiern, 34, 43, 55, 59, 153, 222. (sp.) Bedd., 222. (sp.) Salt., 271, 289. ” Diospyrus, Roxb. Cfr. Diospyros. Diplonema, G. Don, 27, 90. ” ” ambigua, G. Don, 71, 86. elliptica, G. Don, 71, 92. Drebbelia subarborescens, Zoll., 65, 75. Dyospyros, Dumort. Cfr. Diospyros. Ebenacites rugosus, Sap., 75, 272, 277, 284. Ebenoxylon, Auct. Cfr, Ebenoxylum. Ebenoxylum yerum, Lour., 67, 122. Ebenus, Commers., 144, 146, 148. Ebenus, Rumph., 122. ” ” Cfr. Maba. leucomelas, Commers., 68, 179. melanida, Commers., 68, 178. tessellaria, Commers., 68, 176. 10 12 17 11 13 Mr HIERN, ON EBENACE. Ehretia ferrea, Willd., 67, 107, 117. Eleis guineensis, Jacq., 107. Embryopteris, Gaertn., 144. discolor, G. Don, 70, 261. gelatinifera, G, Don, 70, 258. glutenifera, Wight, 258. glutinifera, Roxb., 67, 258. Kaki, G. Don, 71, 227. Loureiriana, G. Don, 71, 264, peregrina, Gaertn., 67, 258. . racemosa, G. Don, 71, 263. Ericacex, 63, tab. I. Ermellinus, Cesalp., 144, 146, 151, 223. Erycibe glomerata, Wall., 40. Erycibee, 63, tab. 1. Euclea, Linn., 27, 32, 58, 59, 60, 61, 63, 64, 65, 66, 90, 144, 273, 275, tab. 1. acutifolia, E. Mey., 36, 71, 92, 94. angustifolia, Benth., 73, 96. Apollinis, Ung., 76, 275. bilocularis, Hiern, 35, 92, 102. Burchellii (wvar.), 105. coriacea, Alph. DC., 36, 46, 47, 72, 90, Wil GR) el Se) daphnoides, Hiern, 36, 45, 49, 92, 102, 288. desertorum, Eckl. et Zeyh., 73, 97. divinorum, Hiern, 31, 35, 36, 46, 92, 98, 99, 289. Dregeaua, Alph. DC., 72, 93. elliptica, Alph. DC., 72, 92. 2, 93 fructuosa, Hiern, 35, 92, 101. herbacea, Lour., 67, 106. humilis, Eckl. et Zeyh., 73, 105, 106. Kellau, Hochst., 35, 44, 71, 92, 103. Kraussiana, Bernh., 72, 93. lancea, Thunb., 36, 67, 92, 95, 96. lanceolata, E. Mey., 35, 36, 44, 45, 46, 47, 48, 49, 71, 92, 96, 97, 272, 288. linearis, Zeyh., 36, 47, 73, 92, 96. 71, 92, 102. macrophylla, E. Mey. d., 101. miocenica, Ung., 76, 275. multiflora, Hiern, 35, 36, 44, 45, 46, 47, 48, 49, 92, 100, 271, 289, tab. m1. myrtina, Burch., 49, 69, 105, 106. natalensis, Alph. DC., 36, 72, 92, 101. Vou. XII. Parr I. macrophylla, E. Mey., 36, 44, 45, 46, 9 18 297 Euclea ochrocarpa, E. Mey., 72, 97. ss ovata, Burch., 36, 48, 49, 58, 69, 91, 92, 98, 288. 3 pilosa, Lour., 67, 106, 265. 55 polyandra, E. Mey., 36, 46, 47, 49, 71, 90, 92. x pseudebenus, E. Mey., 30, 35, 36, 46, 48, 71, 92, 95. Fs pubescens, Eckl. et Zeyh., 73, 93. 3 racemosa, Linn., 30, 36, 46, 47, 48, 49, 52, 67, 92, 103, 104, 3 relicta, Ung., 77, 272, 275. s rigida, E. Mey., 71, 96. rufescens, E. Mey., 71, 99. 3 tomentosa, E. Mey., 36, 48, 71, 92, 93. 5 undulata, Thunb., 30, 31, 36, 45, 47 48, 49, 67, 92, 105, 288, 290. 5 n. 9140, E. Mey., 94. Euphorbiacez, 63, 106, tab. 1. Eurya acuminata, DC., 271. 5, symplocina, Blume, 270. Extractum Diospyri, 259. Ferreola, Roxb., 59, 107, 108. 5 buxifolia, Roxb., 67, 117. 5 guineensis, Schum. et Thonn., 117, 288. Ferriola buxifolia, Roxb. 69, 107, 117. Ficus (sp.), Keempf., 144, 227. Fornasinia ebenifera, Bertol., 28. Garcinia malabarica, Desr., 67, 144, 258. Getonia macroptera, Ung., 73, 279. petrezefolia, Schimp., 279. petreeformis, Ung., 279. », truncata, Goepp., 74, 279. Guaiaeana, Tourn., 144, 146, 150. Guaiacane, Juss., 57. Guaiacum (sp.) Gerarde, 144, 223. Guatteria flavicans, Wall., 70, 145, 205. Gunisanthus, Alph. DC., 63, 64, 66, 145,146,150. 5 pilosulus, Alph. DC., 72, 189. Hebenaster, Rumph., 144, 244. Highulaenda, Herm., 117. Holochilus, Dalz., 64, 107, 109. 55 micranthus, Dalz., 74, 133. Humiriacee, 63, tab. 1. Hydrocharis ovata, Ludw., 285. Hlicinese, 27, 62, 287, tab. 1. Juglans acuminata, Ludw., 280. 3 ventricosa, Ludw., 280. Juniperus communis, L., 96, ” ”? 38 298 Mr HIERN, ON EBENACE. Kellaua Schimperi, Alph. DC., 71, 90, 103. Labatia Scheuzeri, A. Br., 286. Laurinex, 63, 270, tab. 1. Leguminose, 28. Leucoxilon laurinum, E. Mey., 71. (Zzel.). Leucoxylon, Auct. Cfr. Leucoxylum. Leucoxylum, Blume, 64, 66, 144, 146, 152. buxifolium, Blume, 27, 69, 218. Lignum Vite, Gerarde, 144, 223. Loeselia, L., 78. Lotus (sp.), Camer., 144, 223. Maba, Forst., 30, 32, 59, 60, 61, 63, 64, 65, 66, 106, 270, 273, 285, 287, 290, tab. 1. abyssinica, Hiern, 35, 44, 109, 132. acapulcensis, Hiern, 36, 50, 109, 128, 129, 270. acuminata, Hiern, 32, 108, 112. albens, Hiern, 36, 50, 109, 126, 270. Andersoni, Sol., 32, 76, 108, 124. angustifolia, Miq., 75, 117. Beccarii, Hiern, 110, 140, 288. buxifolia, Pers., 29, 30, 31, 32, 35, 36, 38, 40, 41, 42, 44, 48, 50, 56, 64, 68, 108, 116, 123. Cargillia, F. Muell., 76, 246, 290. caribeea, Hiern, 37, 52, 109, 125. cauliflora, Hiern, 37, 58, 110, 142, 270. compacta, R. Br., 36, 57, 68, 108, 121,124. confertiflora, Hiern, 33, 55, 110, 136. cordata, Hiern, 110, 141, 288, 290. Cumingiana, Alph. DC., 57, 73, 117. cupulosa, F. Muell., 76, 114. diffusa, Hiern, 38, 50, 108, 111. Ebenoxylon, G. Don, 71, 123. Ebenus, Spreng., 69, 123. Ebenus, Wight, 117. elliptica, Forst., 32, 38, 56, 67, 108, 122, 125, 270. elliptica, Seem., 116, 118. fasciculosa, F. Muell., 36, 38, 55, 56, 76, 110, 135, 270. foliosa, Rich., 38, 56, 75, 108, 113. geminata, R. Br., 30, 36, 54, 56, 68, 108, 117, 119, 270. glabrescens (var.), 118. granatensis (var.), 128. Grisebachii, Hiern, 37, 52, 109, 125. guineensis, Alph. DC., 73, 116, 117. hemicycloides, F’. Muell., 36, 77, 108,111. 47 Lor) 38 17 Maba hermaphroditica, Zoll., 33, 55, 74, 110, ” 137. Hilairei, Hiern, 37, 54, 110, 143. Hillebrandii, Seem., 38, 55, 76, 108, 122. humilis, R. Br., 36, 56, 68, 108, 119, 120, 270. inconstans, Griseb., 37, 50, 51, 52, 53, 54, 76, 109, 127, 201, 270. interstans, F. Muell., 76, 121. intricata, Hiern, 32, 50, 109, 126. javanica, Zoll., 33, 74, 110, 138. lamponga, Miq., 33,59, 75, 107, 110, 133. lancea, Hiern, 35, 107, 108, 118. lanceolata, Hiern, 38, 109, 131, 270. laurina, R. Br., 36, 56, 68, 108, 115. laxiflora, Benth., 77, 135. littorea, R. Br., 56, 68, 117. madagascariensis, Alph. DC., 78, 117. Maingayi, Hiern, 33, 42, 110, 138, 288. major, G, Forst., 31, 32, 67, 108, 124, 125. Mannii, Hiern, 35, 43, 44, 109, 129. megalocarpa, F. Muell., 76, 211, Mellinoni, Hiern, 37, 110, 143. merguensis, Hiern, 33, 41, 110, 134, 288. micrantha, Hiern, 33, 109, 133. microphylla (var.), 117. Motleyi, Hiern, 33, 55, 110, 139. Mualata, Welw., 30, 35, 48, 108, 111. myristicoides, Hiern, 37, 52, 110, 142. myrmecocalyx, Hiern, 110, 139, 288. myrmecocarpa, Hiern, 37, 110, 141, 270. natalensis, Hary., 36, 44, 76, 109, 131. neilgherrensis, Wight, 73, 117. nigrescens, Dalz., 32, 43, 75, 108, 115. oblongifolia, Hiern, 32, 108, 112. obovata, R. Br., 36, 56, 68, 107, 108, 119. obovata (var.), 128. ovalifolia, Hiern, 32, 59, 108, 113. Pavonii, Hiern, 36, 109, 129. pentamera, F. Muell., 76, 239, 290. punctata, Hiern, 33, 55, 107, 110, 136, 271, 288, tab. tv. quadridentata, F, Muell., 76, 239, 290. quiloénsis, Hiern, 35, 109, 132. reticulata, R. Br., 36, 57, 68, 108, 121, 122, bo bo Mr HIERN, ON EBENACEZ. 299 Maba revoluta, Vieill., 56, 114. 123, 270, 271. ruminata, Hiern, 38, 56, 110, 135. salicifolia, Hiern, 109, 129, 271. ” 108, 116. 271. sericocarpa, F, Muell., 76, 114. 130. Smeathmanni, Alph. DC., 73, 117. sumatrana, Miq., 33, 55,74, 108, 123, 124, Teijsmanni, Hiern, 33, 110, 137, 288. vaeciniefolia, Benth., 73, 117. Vieillardi, Hiern, 38, 56, 108, 124. Mabie: Mason. Cfr. Maba. Macreightia, Alph. DC., 64, 66, 107,109,273, 285, acapulcensis, Alph. DC., 72, 128. acuminata, Thw., 41, 76, 112. albens, Alph. DC., 72, 127. andamanica, (Kurz), 77, 124. buxifolia, Griseb., 76, 125. caribeea, Alph. DC., 72, 126. caribea, Griseb., 126. conduplicata, Alph. DC., 72, 127. germanica, Heer, 75, 284. inconstans, Alph. DC., 72, 127. intricata, A. Gr., , 126. italica, Massal., 28 longipes, Hituingel -7 28 75 75 79, rufa, Labill., 36, 38, 56, 69, 108, 114, sandwicensis, Alph. DC., 38, 55, 73, sericea, Hiern, 37, 49, 110, 140, 270, Seychellarum, Hiern, 38, 50, 107, 109, Myrsine Kellau, Hochst., 71, 90, 103. Myrsineex, 63, tab. 1. Noltia, Schum., 64, 144, 146, 149. Noltia tricolor, Schum. et Thonn., 69, 183. Olacine, 61, 63, 65, tab. 1. Olax, L., 65. Oleacez, 63, tab. 1. Padus (sp.), Burm., 90, 104. Paralea, Aubl., 144, 146, 154. » guianensis, Aubl., 67, 240. 5 guyannensis, Aubl., 240. Paralia guianensis, Desv., 240. Patonia (§), 146, 152. A Walkerii, Wight, 71, 145, 214. Pishamin, Parkins., 144, 225. Pisonia buxifolia, Rottb., 67, 107, 117. Pistachia (sp.), Plukn., 78, 80. Plumeria Flos-Saturni, Ung., 73, 277. Porana, Burm., 274, 284. Pseudo lotus, Cam., 144, 223. Rhipidostigma, Hassk., 64, 107, 110. Teijsmanni, Hassk., 74, 137. 55 Zollingeri, Hassk., 74, 138. Rhododendron Apollinis, Ettingsh., 76, 275 Rospidios, Alph. DC., 64, 66, 145, 146, 155. 3 vaccinioides, Alph. DC., 72, 230. Royena, Linn., 28, 32, 58,59, 60, 61, 63, 65, 66, 78, 79, 84, 195, 273, tab. 1. Amalthee, Ung., 77, 273, 274. ambigua, Vent., 36,47, 67,79, 84,86, 270. angustifolia, Willd., 67, 83. brachiata, E. Mey., 71, 85. ” 6. 4, 285, 289. 284, 285, 10 » cistoides, Welw., 35, 48, 78, 79, 87. cordata, E. Mey., 35, 36, 44, 45, 49, 71, 79, 81, 82, 288. microcalyx, Ettingsh., 289. miinzenbergensis, Ettingsh., 284, bo 285, 289. myristicoides, Spruce, 142, 289. oblongifolia, (Kurz), 124. oblongifolia, Thw., 41, 75, 112. obovata, Mart., 74, 127. ovalifolia, Thw., 41, 76, 113. Payonii, Alph. DC., 72, 129. psidioides, Alph. DC., 72, 127 umbellata, Massal., 75, 286. Maaraisthice Stiehl. Cfr. Macreightia. Magnoliacee, 63, tab. 1. Marcreightia, Kurz, 124. Cfr. Macreightia. Melonia (§), 146, 147, 235. Monodora microcarpa, Dunal, 69, 144, 246. cuneata, Poir., 68, 83. cuneata, Specie ., 8d. cuneifolia, E. Mey., 71, 85. decidua, Burch., 48, 49, 69, 85. euboea, Ung., 77, 273, 274. falcata, K. Mey., 72, 88. glabra, Einn., 36, 46, 47, 48, 49, 66, 79, ~ 88. glandulosa, Harv., 36, 79,89, 288, tab. 1. greca, Ung., 77, 273, 274. hirsuta, Eckl., 81. hirsuta, Jacq., 85. hirsuta, Linn., 36, 45, 46, 47, 48, 49, 66, 79, 83, 270, 272. 38—2 300 Mr HIERN, ON EBENACE. Royena hirsuta, Sieb., 88. »” ~~ 12 5 EBENACES, 27 ; economic products, 28 ; geographical distribution, 31 ; lists of numbered collections, 39; description of the family, 57; affinities, 61, tab. 1; genera, 63; brief history of the specific names, 65 ; chronological list of published specific names with references and localities, 66; description of the hispidula, Harv., 85, 86. latifolia, Willd., 68, 84, 90. lucida, Linn., 30, 35, 36, 45, 46, 49, 66, 79, 80. lycioides, Desf., 69, 85. macrophylla, E. Mey., 72, 101. media, Hort., 71, 90. microphylla, Burch., 48, 49, 69, 83. Myosotis, Ung., 77, 273, 274, 284. myrtifolia, Wendl., 69, 88. nitens, Harv., 36, 45, 79, 87. oleifolia, Desf., 85. opaca, E. Mey., 71, 81. pallens, Thunb., 35, 36, 44, 45, 46, 47, 48, 49, 67, 78, 79, 85, 86, 87, 270, 288. parviflora, Hiern, 36, 45, 63, 78, 79, 88. Pentelici, Ung., 77, 273, 275. polyandra, Linn. fil., 67, 90, 92. polyandra, 8, Pers., 86. pubescens, Willd., 68, 85. ramulosa, E. Mey., 72, 85. rufescens, E. Mey., 71, 99. rugosa, E. Mey., 71, 83. scabra, Burm., 66, 82. scabrida, Harv., 35, 45, 79, 82. scandens, Burch., 49, 82. sericea, Bernh., 72, 85. sessilifolia, Hiern, 36, 78, 79, 84, 90. supra-cordata, Burch., 49, 81. 4 Royena villosa, Linn., 35, 36, 44, 45, 47, 49, 66, 79, 82. » (sp.) n. 15, Eckl. et Zeyh., 90. » (sp.) n. 9140, Drege, 90, 94. Royenia, Auct., 78. Cfr. Royena. Rymia, Endl., 90. 5 polyandra, Endl., 92, 288. Sapota nigra, Blanco, 70, 145, 244. Sapotacee, 62, 63, 270, tab. 1. Spheeria (sp.), 100. » Staphylodendron, Herm., 78, 80. Staphylodendrum, Commelin., 78, 80. Sterculiacee, 28. Styracere, 27, 62, 63, tab. 1. Tamarix, L., 96, 98. Ternstroemiacee, 62, 63, tab. 1. 1 Tetraclis clusiwfolia, Hiern, 27, 32, 38, 50, 60, 61, 63, 65, 271, tab. 1., tab. x1. Tetrapteris Harpyiarum, Ung., 73, 279. Tiliacex, 63, tab. 1. Trichanthera (§), 64, 107, 110. Vaccinia, Adans., 57. Vaccinium (sp.), 78, 145. 53 fragrans, Wall., 73, 230. = pensylvanicum, Mill., 66, 88. ~ Sprengelii, Wail., 230. Viola, L., 285. Vitis Idvea (sp.), Plukn., 78, 88. Ximenia americana, L., 98. Xylopia frutescens, Aubl., 269. INDEX OF SUBJECTS. genera and species exclusive of fossils, 78 ; Royena, 78, t. 1.; Euclea, 90, t. m.; Maba, 106, t. Iv.; Diospyros, 144, tt. v—x; imperfectly known species of Diospyros, 264; Tetraclis, 271, t. x1. ; fossil Ebenacer, 272; additions and corrections, 287; alphabetical index of the Latin names, 291. III. On the equation which determines the form of the strata in Legendre’s and Laplace's theory of the Figure of the Earth. [Received Oct. 16, 1871. Read Oct. 30, 1871.] 1. The equation to which the present memoir refers is the following: ve & 2 1 " d TF yn t3 a” a d Ve —, =o) patda— aay gen] Pag (Yat )da— 5 |p g (. This will vanish of course if A=0, which makes Y, also vanish. Tt will also vanish if (n +c) a ae" + 2 a, == GG ANE oa docecdseccsacsessseuseas (19). But Legendre says that if (19) be supposed to hold we shall find that = is not always negative; this is quite true and is easily verified. 39—2 308 Mr TODHUNTER, ON AN EQUATION For (19) may be thus expressed pe Pa. 1 nto? = Pte i” 3p But fae a ' a 7 We 3 so that o, is greater than = if “e is always negative. : ae pa... Ope And as n+c is greater than 3 we have a fortiort o, greater than ae if da 8 always negative. It is obvious that (19) might always be satisfied as it determines only the ratio of g to h; and Legendre implies that if we had not assumed p to decrease from the centre to the surface we should thus have found a value of Y, to satisfy 7(Y,)=0. Legendre however seems to overlook the equation (18); in virtue of this if ae changes sign it must be when o vanishes, and in his theory it is of course impossible that o, which is propor- tional to the mass, should vanish when a is not zero. Thus it will be seen that equation (19) is quite inadmissible in Legendre’s theory, in which p and o denote positive quantities, and o must increase with a. Therefore with this particular law of density we have consistently with the general proposition no solution admissible of Z(Y,)=0, except Y,=0. 12. Nevertheless it will be convenient for our purpose to examine the consequences which follow from (19). Assuming that we make H vanish by means of this relation we proceed to consider whether K will also vanish. Return to equation (12). Vi; a d f n+3 We have i Pa, 1h (a) a } da nt { nh nt d = pa" (a) = ad (a) i =parrs(a)— [erat & a | Pits A co da = p,a,""*¢ (a,) + Am (3 — m) | “gn aa = p.,"$(a,) + AB B—™ a, = pial PIG) tea gh = n+3 (a,) a,"o, Po, $(a,) + we m (3 3 m). Hance Toes «(i 3 alt ence ¢ (a,) a, 7 onci (Pi aE waaon 3 IN THE THEORY OF THE FIGURE OF THE EARTH. 309 To make this vanish we must have {(2n +1) (n +c) — m (B—mM)} a, = (MHC) pyA,s...ccceccsccsccevscenes (20). B ; 1 m Ne ut n +n4+g—m(8—m)=(0-5) ; thus 2n? + 2nc+n+¢—m (3—m) = (n+ 0c)’, so that (20) becomes (n+ 08a, =(n-+0) p,a,' that is (n+) o, = p,a,*, and this is in fact equivalent to (19), so that K does vanish. We have therefore the following result: if p=ga"+ha"%, and ig ee then I(Y,)=0, provided h and g be taken in the proportion assigned by (19). 13. It may however be conjectured that since Y, becomes infinite when o vanishes in the preceding Article we shall find our primary equation (1) is really not satisfied. But on examination this apparent difficulty will disappear. The first term in (1) is — so that provided the product of Y, into @ remains finite there is nothing imad- missible. Then by integration by parts the other terms of (1) may be transformed into 1 2 1 2 nt3 dp Bona Yt (2n+ 1) ral Rae: da ie it i Gaai (pee Y, dp wiprearmae os 2n+1 ce 7s al fo aa ep (EE ik ae =f. Y,, dp Qn+ {eet so i) Gee Ya Bae ae a” da da. i ee Seat so that the product Hae remains finite when o vanishes. a a that is into Now Hence the conclusion arrived at in Art. 12 remains undisturbed by the vanishing of o. 14. In the example of Art. 12 it will be seen that when by a certain relation between constants, we had secured the vanishing of H, then K also vanished. And this in fact might have been anticipated. For since (8) is true for all values of a, we may put a@=0; then since we assume that Y, is finite or zero when a=0, we see that AK must be zero. Thus any solution of (5) which is never infinite necessarily makes K=0, and reduces Ha® EW a) sea 310 Mr TODHUNTER, ON AN EQUATION 15. Let us now advert to Laplace’s discussion of the primary equation. It will be sufficient to touch briefly on this as the Mécanique Céleste is readily accessible, and Laplace’s method has been reproduced by others. Laplace’s discussion is essentially of the same kind as Legendre’s, the differences being not important. Laplace, however, considers that the density must be finite at the origin. This amounts to saying that the m im Art. 6 must be zero; and therefore the c of that Article becomes n+1. It seems to me that there is no reason for Laplace’s assertion, and that Legendre is correct in saying that all that is essential is that the mass should be infinitesimal when the volume is such. If we take the ordinary law of gravity we might at first consider it to be a serious difficulty that the force becomes infinite for particles in contact; but really there is no difficulty inasmuch as the attraction of a sphere on a particle at its surface is infinitesimal when the sphere is such. _ By assuming that the density must be finite at the origin, Laplace’s discussion is rendered somewhat less general than Legendre’s. Laplace gives a proposition like that of Legendre’s in Art. 7. Laplace’s final step, like Legendre’s, consists in shewing that the right-hand member of (12) does not vanish, except when ¢ (a) is always zero. For by integration by parts we have (2n +1) ¢$ (a,) a,” [patda - is p = \ (a) a} da E ; (2n — 2) p,a,"**¢ (a,) -| ; B (2n +1) $ (a,) a,"a* — $ (2) | zt Ue This is necessarily positive. For = is negative by supposition; and by Art. 7, or by Laplace’s equivalent demonstration, ¢(a@) increases with a, so that the expression : (2n + 1) ¢ (a,) a," —¢ (a) a” is necessarily positive. 16. Laplace’s main assumption is that p decreases from the centre to the surface. Also he assumes that at the origin the density is finite. It might seem on glancing at his process that he also assumes Y, to be always capable of expansion in powers of a; but in reality it is sufficient for him to assume that such is the case when a is very ay, da sign at the origin; and therefore we may if we please assume this if it appear more obvious instead of the assumption which Laplace makes. have the same small. The assumption is made to enable him to shew that Y, and On the whole although the difference is slight between the processes of Legendre and Laplace, the former appears to me rather the better. Laplace like Legendre confines himself to shewing that the equation J(Y,)=0 has no admissible solution except Y= 0. IN THE THEORY OF THE FIGURE OF THE EARTH. ole 17. I now proceed to the treatment of the equation which we find in O’Brien’s Mathematical Tracts published in 1840. As I am persuaded that this treatment is altogether illusory, I must present it fairly to the reader. I will therefore use very nearly the author’s words, but change the notation slightly to conform to that already adopted. Let v and v' be two quantities which satisfy the equations LOVES LC) Se oe re (21), and let O and C' be two constants: then Y,= Cv+ Cv’ will be a solution of (5); for performing the operation Z on both sides of the equation Y, = Cv + Cv, we have I(Y,) = 1(Cu+ Cv’) = CI (v) + CT (v’) (evidently from the nature of the operation J), or 1(¥,) = Car + & by (21). Now if we get rid of the integral signs ‘in this equation by the same process we have applied to the equation (1), the second side will evidently be made zero by the differentiations, and thus we shall arrive at the same differential equation as before; hence Y, = Cu+ Cv’ is a solution of (5). Now this is evidently true whatever be the values of C and C’; hence this solution contains two arbitrary constants, and is therefore the most general solution that (5) admits of. Hence all values of Y,, which satisfy (1), since they also satisfy (5), must be values of Cu+C'v'. To determine what values of Cv+C'v' satisfy (1) substitute Cu+C'v' for Y, in‘(1), and we find CI (v) + CL (v’') = 0, ni CO or Ca + oa = 0 by (21). Now this equation ought to be true for all values of a; hence C=0, and C’=0; hence it is evident that only one value of Cv+ C'v', namely zero, satisfies (1); and hence it follows from the equation (1) that Y,=0. This is the demonstration I propose to examine. 18. The first point to which I would call attention is the excessive generality of the proposition which is thus supposed to be established. It will be observed that no restriction is put on Y, or on p. Thus we are not compelled to have Y, finite when a is zero. And p is not such a quantity as would necessarily correspond to density in the physical problem, for here p may be positive or negative. Also n is not restricted to be a positive integer, but may be any number positive or negative, whole or fractional. In fact in this process 312 Mr TODHUNTER, ON AN EQUATION the equation Z(Y,)=0 is quite liberated from all its physical connexion; and it is main- tained that whatever p and n and Y, may be, the equation has no solution except Y,=0. 19. My next remark is that the highly general proposition thus maintained is certainly untrue. The assertion is that Y, must be zero. Now to overthrow this assertion it is suffi- cient to point to two examples which have already been discussed. We have shewn in Art. 10 that whatever p may be, when n=1 the value = satisfies the equation J(Y,)=0. And we have shewn in Art. 12 that whatever n may be, with the values of p and Y, there assigned we have 7(Y,)=0. These two examples as we have already shewn do not bear against Legendre’s demonstration of his duly restricted proposition; but they are decisive against the highly general proposition which is now under discussion. 20. The form in which the argument is put is certainly strange. In order to shew that the equation 7(Y,)=0 has no solution, it is assumed that the equations (Y,) =a" eh ay = n+l a can both be solved. But the points assumed appear as difficult as that which is to be demonstrated. For in fact out of the three equations here brought before us if we assume that any two can be solved the same kind of argument might be employed to shew that the third equation could not be solved. 21. The best method of shewing where the argument fails is to employ the reverse process of Art. 5. We may admit that the general solution of (5) will be of the form Gp (a) + Ex @), where G and £ are arbitrary constants, and y (a) and x(a) functions of a; these functions could not be specifically determined unless p were given explicitly. If we perform the : : . HO) Paks ation J on this expr he res i > + SH: operation J on this expression the result will be earl + oan The value of H will be given by He oat {5 any(a)t asad 5. ane (of : ae The value of K will be of the same form and we may denote it thus K= Gd(a,) + Eu(a,), where A (a,) and pw (a,) are certain functions of a, which we need not state explicitly. Now O’Brien supposes, and quite correctly as we see, that G@ and E may be so taken as to make H vanish. When G and £& are so taken he denotes Gy (a)+Ey(a) by v, assuming in fact that K cannot also vanish. But of course it is theoretically quite pos- ‘sible that the constants which occur may be so related as to make K vanish when H vanishes. In fact in the two examples we have brought forward we have made one of the two @ and £ zero, say G; and then we found that the coefficients of H in H and XK yanish, » IN THE THEORY OF THE FIGURE OF THE EARTH. 313 22. It might perhaps be suggested that the argument intends to shew that no general form can be found which will satisfy Z(Y,)=0 whatever may be the value of p. The answer is twofold. In the first place this is really not the problem to be solved; what we wish to shew is that whatever value may be assigned to p there is no admissible value of Y,. In the second place the example of Art. 10 shews that the argument is not valid even when the problem is changed in the manner suggested. 23. I shall now consider a proof given by Mr Pratt. In the first edition of Pratt’s Mechanical Philosophy Laplace’s process was adopted. In the second edition a new process was given to which attention is invited in the preface, where this passage occurs, “ Mr O'Brien in his Mathematical Tracts, Part [, has given an excellent demonstration, and shorter than Laplace’s, but the one now given is shorter and simpler even than his.” Mr Pratt's proof is reproduced in his separate publication on Attractions...and the Figure of the Earth. We may observe here that if O’Brien’s proof had been sound its excessive generality might have compensated for the want of almost any other qualities. 24. The proof given by Mr Pratt is interesting; it is founded solely on the primary equation without any use of the differential equation (5). But the assumptions on which it rests are rather large. It assumes that Y, and p can each be expanded in a series of ascending positive powers of a for all the values of a@ between 0 and a,. It does not assume that p decreases from the centre to the surface; but it does assume that p does not vanish at the origin. Both Legendre and Laplace suppose that p has a value at the origin which is not zero; but with them this is no new assumption, being in fact involved in their main assumption that p decreases from the centre to the surface. The assumed forms for Y,, and p are used in the primary equation which is developed in powers of a, and it is found impossible to make the terms separately vanish, 25. ‘The assumptions respecting the possibility of expanding Y, and p are certainly considerable assumptions. The assumption respecting Y, shortens the demonstration as it enables us to omit the process by which Legendre and Laplace shew that Y, and sia have the same sign at the origin. The assumption that p can be expanded im a series of ascending powers of a, is of course a large assumption analytically, for it excludes many conceivable forms for p. And in a physical point of view the problem is very much restricted ; for it is assumed that p is the same function of a throughout the imvestigation ; whereas it is obviously conceivable that p may be discontinuous in form and even in value. To take a simple example. Let us examine 77 by the method now under discussion we can shew that there is no value of Y, except zero, which will satisfy the primary equation under the following suppositions as to p; from a=0 to a=b let p=D+Ha"+..., and from a=b to a=a, let p= Ma*+Na’+..... Here we suppose m and the other expo- nents which occur in the first expression for p to be positive; in the second expression Vout. XII. Part I. 40 314 Mr TODHUNTER, ON AN EQUATION for p we will suppose the exponents yw, v,... to be arranged in ascending algebraical order of magnitude, but they need not be restricted to be positive. Let Y,= Wa‘ +... First suppose a to be less than 6. Then a Da? IBGE 2 =S | patda= 3 jee Soh a Po ——( Yea") da = Whar ene ex ) Ie das) = I, da (ee) WM (s+ 2—n) B+ts+2—n a d a = s—nt2 a—nt2 +] p5;( so that n(n +1) y,-@ Gate | (a) dart 26 (@)\ ad (a) eset tes coenee (25). The last expression cannot vanish simultaneously with ¢(a). For suppose that as a increases from zero $(a) first vanishes when a has a certain value. Since $(a) begins by being positive ¢'(a) must be negative or zero when ¢ (a) first vanishes; hence the expres- sion on the right-hand side of (25) is then necessarily positive and cannot vanish. Thus from (24) we should have «=0, which is contrary to the supposition. 318 Mr TODHUNTER, ON AN EQUATION, &e. 30. Thus I do not assume that p continually decreases from the centre to the surface, nor even that p is always positive; but only that o never vanishes except at the origin : so that if this demonstration be accepted it will form the most general which the proposition has yet received. It will be observed that the conditions which we have imposed are not satisfied by the two examples which are discussed in Arts. 10 and 12. In Art. 10 we see that aY, does not vanish with a; and in Art. 11 we have o vanishing for a certain value of a. Thus these two examples do not supply any objection to the new demonstration. 31. I may observe that from (24) if we supposed Y_ to be given we may by inte- gration theoretically determine the value of p. For the right-hand member of (24) would be thus a given function of a, say y(a); and by integration logo = | ¥ (a) da; this gives ¢, and then by differentiation we find p. In this way we might establish rela- tions between Y,, p, and a, which would satisfy (5); but by virtue of the demonstration in Art. (29) we cannot obtain admissible solutions of (1). 1. TODHUNTER. October, 1871. IV. On the Centro-surface of an Ellipsoid. By Professor Carney. [Read March 7, 1870.] THE Centro-surface of any given surface is the locus of the centres of curvature of the given surface, or say it is the locus of the intersections of consecutive normals, (the normals which intersect the normal at any particular point of the surface being those at the consecutive points along the two curves of curvature respectively which pass through the point on the surface). The terms, normal, centre of curvature, curve of curvature, may be understood in their ordinary sense, or in the generalised sense referring to the case where the Absolute (instead of being the imaginary circle at infinity) is any quadric surface whatever; viz. the normal at any point of a surface is here the line joining that poimt with the pole of the tangent plane in respect of the quadric surface called the Absolute: and of course the centre of curvature and curve of curvature refer to the normal as just defined. The question of the centro-surface of a quadric surface has been considered in the two points of view, viz. 1°, when the terms “normal,” &c. are used in the ordinary sense, and the equation of the quadric surface (assumed to be an ellipsoid) is taken to be ate + ae 1; 2°, when the Absolute is the surface X?+Y*+Z*+ W*=0, and the equation of the quadric surface is taken to be aX?+8Y*+7Z°+8W*=0:—in the first of them by Salmon, Quart. Math. Jour. t. U1. pp. 217—222 (1858), and in the second by Clebsch, Crelle, t. 62. pp. 64—107 (1863): see also Salmon’s Solid Geometry, 2nd Ed. 1865, pp. 143, 402, &. In the present Memoir, as shewn by the title, the quadric surface is taken to be an Ellipsoid; and the question is considered exclusively from the first point of view: the theory is further developed in various respects, and in particular as regards the nodal curve upon the centro-surface: the distinction of real and imaginary is of course attended to. The new results suitably modified would* be applicable to the theory treated from the second point of view; but I do not on the present occasion attempt so to present them. The Ellipsoid; Parameters §, n, &c. Art. Nos. 1—6. 1. The position of a point (X, Y, Z) on the ellipsoid 320 Mr CAYLEY, ON THE CENTRO-SURFACE may be determined by means of the parameters, or elliptic co-ordinates, & 7; viz. these are such that we have b.G 2 F Me waEe ae - | a SAT am Cty Fy pecan or, what is the same thing, & 7 are the roots of the quadric equation PG Ve Ze as Rap ee (In its actual form this is a cubic equation, but there is a root v=0, which is to be thrown out, and the quadric equation is thus, v +u(7+0+¢-—X’>- Y?-Z) + {P+ Co +a — (b+) X-(P +a) Y -(@+8) Z}=0, died P=@'+b +e, Q=Fe+ca+ab’, R=ab'e’, the equation is v+u(P—X?— V3-Z)4+ 9-040 XL - (e+e) Y?-(@+h)2=0). 2. It is convenient to write throughout —c =a, c*—a'=B8, a@—-b=y, (whence a+ B++y=0). As usual, a is taken to be the greatest and ¢ the least of the semi-axes, we have thus a, y each of them positive, and 8 negative, =—f’ where f’ is a positive quantity =a+y. A distinction arises in the sequel between the two cases a’ + c?> 2b’ and a’ + ¢?<2b’, but the two cases are not essentially different, and it is convenient to assume a’+ c?>20*, that is, a —b°>?-c* or y>a, say y~@ positive. The limiting case a’+c?=20" or y=a requires special consideration, 3. We have — By X? =a’ (a’ + &) (a + 7), -y2 VY? =P H+ &) (+7), =aB Z = (c? + &) (o +4). It is in fact easy to verify that these values satisfy as well the equation of the ellipsoid OF AN ELLIPSOID. 321 as the assumed equations defining the elliptic co-ordinates £ » We may also obtain the relations K+ Y24+ Raat H+e+E+n, UX*4 RY? 4+ eZ =a' +b'+c+b%c' + ca? + a°b? + (a? +8°+ 0°) (E+) + En. These, however, are obtained more readily from the equation in v, viz. the roots thereof being £, 7, we have —&-n=04+04+0—-X*- y?— Z* &n =0'e' + c'a® + a°b* — (0° +?) X°- (8 +47) Y?-(@ +0) Z which lead at once to the relations in question. 4. Considering € as constant, the locus of the point (X, Y, Z) is the intersection of the ellipsoid with the confocal ellipsoid xe iz ‘abs @+e Be e+e viz. this is one of the curves of curvature through the point; and similarly considering m as constant, the locus of the point is the intersection of the ellipsoid with the con- ile focal ellipsoid XC Nees Yn +n tn etn? viz. this is the other of the curves of curvature through the point. 5. If instead of € and » we write 2 and k, we may consider h as extending between the values —a’, —0°, and k as extending between the values —0*, —c’. h=const. will thus give the series of curves of curvature one of which is the section by the plane X=0, or ellipse semi-axes 0, c; say this is the minor-mean series. In par- ticular h=—a* gives the ellipse just referred to; and h=—4#*, or say h=—D?—e gives two detached portions of the ellipse semi-axes a, c; viz. each of these portions extends from an umbilicus above the plane of wy, through the extremity of the semi-axis @ to an um- bilicus below the plane of ay. And in like manner £=const. gives the series of curves of curvature one of which is the section by the plane Z=0, or ellipse semi-axes a, 6; say this is the major-mean series. In particular /=— c* gives the ellipse just referred to; and k=—b*, or say k=—B? +e gives the remaining portions of the ellipse semi-axes a, c; viz. these are two portions each extending from an umbilicus above the plane of zy, through the extremity of the semi- axis c, to an umbilicus above the plane of ay. The ellipse last referred to may be called the umbilicar section, the other two prin- cipal sections being the major-mean section and the minor-mean section respectively. In the limiting case h=k=—0*, we have the umbilici, viz. these are given by xe OY Sin a lature 3B" Y=0, Cone aie The two series of curves of curvature cover the whole real surface of the ellipsoid; so that at any real point thereof we have &=h, n=hk, or else =k, n=h, where h, k are Vou. XII. Parr I. 41 322 Mr CAYLEY, ON THE CENTRO-SURFACE negative real values lying within the foregomg limits — a’, —b’ and — 6’, —c* respectively. But observe that £ » taken separately may each extend between the limits —a’, —¢. 6. Suppose =», the equation in v will have equal roots, or the condition is (P—X*-Y?°-Z’yP =4{Q- (P+) X*-(e +a’) Y°- (7+) 7}, yiz. this surface by its intersection with the ellipsoid determines the envelope of the curves of curvature. This envelope is in fact a system of eight imaginary lines, four of them belonging to one of the systems of right lines on the ellipsoid, the other four to the other of the systems of right lines. For in the values of X*, Y*, Z* writing n= &, we find VBy= =a +6, Va qa5 =P+E Vi ePla e+e or representing for shortness the left-hand functions by + X’, + Y’, +7’, the eight lines are fe X' |= eS =e” ey’ Biaeies od¥’ dle eaerlicay tobe Geka 47 |= aoa, | = ae ghana Geral oR peee ax gt tla ail PES Sigh! aD i eh eek ZV ole 2) eee so that in the two tetrads each line intersects the four lines of the other tetrad, but it does not intersect the remaining three lines of its own tetrad. The intersections are four points corresponding to &=—a’, being the imaginary umbilici in the plane X=0, four to £=—/J being the real umbilici in the plane Y=0, four to §=—c* beg the imaginary umbilici in the plane Z=0, and four corresponding to =, which may be called the umbilici at infmity*. Sequential and Concomitant Centro-curves. Art. No. 7. - 7. Consider any particular curve of curvature ; the normals at the several points thereof successively intersect each other in a series of points forming a curve; and we have thus, corresponding to the particular curve of curvature, a curve on the centro-surface, which * According to Salmon, Solid Geometry, p. 229, the | But whether properly umbilici or not, the 4 points which I number of umbilici for a surface of the n™ order is | call the umbilici at infinity do in the present theory pre- =n(10n?—25n +416); viz. for n=2, this is=12, as in the | sent themselves in like manner with the 12 umbilici. ordinary theory, not recognizing the umbilici at infinity. OF AN ELLIPSOID. 323 curve may be called the sequential centro-curve. Again the same normals, viz, those at the several points of the particular curve of curvature, are intersected, the normal at each point by the consecutive normal belonging to the other curve of curvature through that point; and we have thus corresponding to the particular curve of curvature, a curve on the centro-surface, which curve may be called the concomitant centro-curve. If instead of a single curve of curvature we consider the whole series, say of the major-mean curves of curvature, we have a series of major-mean sequential centro-curves, and also a series of major-mean concomitant centro-curves; and similarly considering the series of the minor- mean curves of curvature we have a series of minor-mean sequential centro-curves and also a series of minor-mean concomitant curves; the configuration of the several curves will be discussed further on, but it may be convenient to remark here that the centro-surface may be considered as consisting of two portions, say, (A) locus of the major-mean sequential centro-curves; and also of the minor-mean concomitant centro-curves ; (B) locus of the minor-mean concomitant centro-curves, and also of the major-mean sequential centro-curves, Investigation of expressions for the Co-ordinates of a point on the Centro-surface. Art. Nos. 8 to 13. $. Consider the normal at the point (X, Y, Z). Taking in the first instance (a, y, 2) as current co-ordinates, the equations are a-X_ y-Y 2-2 _ ee ae ,>=A suppose, & ¢ Sa be or, what is the same thing, r\. «=X(1 +%), y= (1 +i), 2=Z(1 +%). Suppose now that the normal meets the consecutive normal, or normal at the point X+dX, Y+dY, 7+dZ; and let x, y, z belong to the point of intersection of the two normals; we must have 0=ax(1 +r)+Aa, a B dn, o=d¥(1 +h) + o=dz(1+h)+2n, which determine the direction of the consecutive point; the equations in fact give Canes ee a a > ay Ya one daz Z dZ, a) e 41—2 324 Mr CAYLEY, ON THE CENTRO-SURFACE or, what is the same thing, 0=| wdX, dX, X |, (iE Dae cdZ, dZ, Z which is the differential equation of the curve of curvature. This equation must therefore be satisfied by taking for Y¥+dX, Y+dY, Z74+dZ, the co-ordinates of the consecutive point along either of the curves of curvature,—say along that which is the intersection with the surface, xX? xe a Zia Rae eae 9. To verify this, observe that we then have XdX YdY | Z4Z_ 0 Veru | a bv c XdX . Vay Ge. @+n B+ C+n ? or, what is the same thing, XdX : YdY : ZaZ=e (Witla: P(P+n)B: &(?+n)¥. But from the equations —SyX?=a’ (a+ €) (a’+7) &c., these become Xe v5 Z XdxX : YdyY : ici re See e+e? or, what is the same thing, 2 G4 MR Gipgege § | @+E V+E o+E’ and substituting these values in the determinant equation, it becomes dX :dY :dZ= a XYZ iA ee Ie @+E) G+) (+E) | 1 wre | e,1, o+€ | which is identically true, since evidently the determinant vanishes. 10. Proceeding with the solution, we have from the three equations XdX + YdY + Zd4Z+% (A+ a2 <3 a) +dn (= + = 45 =) = 0, and observing that from the equation X°4+-V?4+27=04+h4+C+E+7, considering therein 7 as constant, we have XdX+ YAY + ZdZ=} dé, OF AN ELLIPSOID. 325 the equation becomes 4d&E+drx=0; and the three equations then are 0=aX(1 +%) PE Ge! a a or say 0 = dX (a+r) —4 XdE, &e, But from the equation —@yX*= a’ (a? + £) (a*+%), considering therein » as a constant, we have and the equations thus become viz. these are all satisfied if only Y= &. 11. The co-ordinates of the poimt of intersection of the two normals thus are o=-X(1+8), y=¥(1+8), -=2(1+8), or squaring, and substituting for X*, &c., their values as given by — ByX* =a (av + &) (V+), &e., the equations become — By aa? = (a? + £) (2+), — ya biy* = (BF + £)° (b +), — a8 ce! = (+ £) (c+ 2), viz. these equations give (x, y, 2) the co-ordinates of a point on the centro-surface, the intersection of the normal at the point (X, Y, 7) of the ellipsoid, (determined by the parameters & 7) by the normal at the consecutive point along the curve of curvature Saal ae 8 z8 e+n' +n C+yn or say 7 is the sequential parameter*. Of course by interchanging & and 7 we should obtain the co-ordinates of the point of intersection of the normal at the same point (X, Y, Z) by the normal at the consecutive point along the other curve of curvature: £& being in this case the sequential parameter. * The expressicns are given in effect, but not explicitly, Salmon, p. 143. 326 Mr CAYLEY, ON THE CENTRO-SURFACE 12. I stop for a moment to consider the foregoing two equations which at first sight appear inconsistent. But observe that in the foregoing solution 2 is the parameter of the point (a, y, z) of the centro-surface considered as a point on the normal at (X, Y, Z); %+da is the parameter of the same point considered as a point on the normal at the consecutive point (X+dX, Y+dY, Z+dZ): the value ¥+dvA=&+dé would belong to a different point, viz. the consecutive point of the centro-surface considered as a point on the consecutive normal—wherefore the dd of the solution ought not to be =dé. In further explanation, observe that the equations z=X(1 +%), &ec. where \=&, if we pass from (a, y, z) to the consecutive point on the centro-surface, give MV ae de =dX(1 +r) +5 de; but since by what precedes, 0=aX(1 +%)- 1A dg a this is X da = 32 dé. Or since a’x = X (a’ + &), this is de, a 2s? a’ + &’ and similarly dy _, 4 y 2B +e’ ds _, dt ae i e+é&’ which are the correct values of dx, dy, dz as derived from the equations — Bya’x® = (a* + £)° (a? +7), &e. 13. The equations — Pya’a* =(a’ + £)* (a? +), &e. give expressions for the co-ordinates (x, y, 2) of a point on the centro-surface’in terms of the two parameters (§ 7): the elimination of (&, 7) from these equations will therefore lead to the equation of the surface ; but the discussion of the surface may also be effected by means of these expressions for the co-ordinates in terms of the two parameters. Discussion by means of the equations — Bya'x* = (a + £)° (a’ +m), &c.; Principal Sections, &c. Art. Nos. 14 to 24 (several subheadings). 14. To fix the ideas consider the section of the surface by the plane z=0; we have in the surface z=0, that is £=—c’, or else »=—c’, values which give respectively iS) bo bss f OF AN ELLIPSOID. — Bya'x* = — 8° (a +7), | — Bya’a* = — B (a? + &)°, —yab'yr = a? (+7). — yab'y? = a (b+ €)%. Or, what is the same thing, 1 gow = a+, | 2 By =- 8-9, you? = (a =i &), py =— (0+ 8. The first set of equations gives which is the eauation of an ellipse. The second set gives (aar)* + Qy)§ = 97, or in a rationalised form (a*a* + by? re, yy)? + 27°? yx" y? = 0, which is the equation of an evolute of an ellipse. é aa By? ; 5 15. The ellipse (aoe a is a cuspidal curve on the surface, and the section by a the plane z=0 is consequently made up of this ellipse counting three times, and of the evolute; it is therefore of the twelfth order; and the order of the surface is in fact =12. 72 2 It is clear that the section of the centro-surface arises from the section at a 5 viz. the normal at any point of this ellipse lies in the plane Z7=0, and its intersection by a normal at the consecutive point of the ellipse gives a point of the evolute; the evolute being thus the sequential centro-curve of this section: the intersection by the normal at the consecutive point on the other curve of curvature gives a point on the ex bey? ellipse at we 1, which ellipse is therefore the concomitant centro-curve. Observe that 7 2 Dae this other curve of curvature cuts the ellipse Aan aE at right angles, and that the normals at the consecutive points above and below the point on the ellipse will meet each other and also the normal at the point of the same ellipse at the point on the ellipse aa by? ea surface. =1: this shows that the last-mentioned ellipse is a cuspidal curve on the centro- 16. The three principal sections ef the centro-surface are consequently 2,2 22 i % Oa is =1, and (by) + (cz) =ai; =0 = xv OR Be 2,2 2,2 . y=0, cage ee and (cz)® + (ax)* =A?; 22 2,2 2=0, Seat, and (ax) + Qyjt=a'; 328 Mr CAYLEY, ON THE CENTRO-SURFACE viz. each section is made up of an ellipse counting three times and of an evolute (of an ellipse). I have for shortness represented the three evolutes by their irrational equations. It will presently appear that the section (imaginary) by the plane infinity is of the like character. 17. Considering only the positive directions of the axes, we have on each axis two points, viz. axis of a, r= LN pe a a axis of y, y= = y= = axis of 2, peels L= 2. c c through each of which, in the two different planes through the axis respectively, there passes an ellipse and an evolute. In the assumed case a*+c¢’> 26%, the disposition of the points is as shown in the figure. Plane of xz, evolute is outside ellipse, YZ, a inside $3 zy, ” cuts ” but in the contrary case a* + c* < 2°, the disposition is Plane of xz, evolute is outside ellipse, ¥Y%, a cuts xy, » 1s inside 55 4 [SS there is no real difference, and to fix the ideas I attend exclusively to the first-mentioned case V+e> 2h. 18. In each of the principal planes, the evolute and ellipse, qua curves of the orders 6 and 2, respectively, intersect in 12 points, 3 in each quadrant; viz. of the 3 points two unite together into a twofold point or point of contact, and the third is a point of simple intersection; assuming for the moment that this is so, the figure at once shows that in the plane of zz or umbilicar plane the contact is real, the intersection imaginary; in the plane of zy, or major-mean plane, the contact is imaginary, the intersection real; but in the plane of yz or minor-mean plane the contact and intersection are each imaginary. The contacts arise, as will appear, from the umbilici of the ellipsoid, and may be termed ‘“‘umbilicar centres,’ or “omphaloi;” the simple intersections “points of outcrop,” or simply “outcrops.” By what precedes there are in the umbilicar plane, four real umbilicar centres (in each quadrant one); and in the major-mean plane four real outcrops (in each quadrant one); the other umbilicar centres and outcrops are respectively imaginary. 19. The surface consists of two sheets intersecting in a nodal curve connecting the outerop with the umbilicar centre. As to the form of this curve there is a cusp at the OF AN ELLIPSOID. 329 outcrop, and the curve does not terminate at the umbilicar centre, but on passing it, from crunodal becomes acnodal (viz. there is no longer through the curve any real sheet of the surface): moreover the curve is not at the umbilicar centre perpendicular to the plane of Nei xz, and there is consequently on the opposite side of the vA a plane a symmetrically situate branch of the curve, viz. the re ee umbilicar centre is a node on the nodal curve. Completing On a the curve, the nodal curve consists of two distinct portions, > 8 a one on the positive side of the plane of yz or minor-mean a plane consisting of two cuspidal branches as shown in the Va figure ; the other a symmetrically situate portion on the nega- tive side of the minor-mean plane. Intersections of Evolute and Ellipse. 20. Consider in the plane of zy the ellipse and evolute, 2,2 2 ettta, ( (a*a? + By? — 9)? + 27 ya"b?x*y? = 0. First, these are satisfied by ce=——, | % $ Co-ordinates of Umbilicar centres in plane of ay (imaginary), By? = — oa ys viz. the equations respectively become ~f-2o1, (- ete — 9) + 27a°8 = 0, the first of which is a+8+y=0, and the second is (a°+/°+9°)’—27a'6'y°=0. But the equation a+8+y=0 gives a +/*+7'=3a8y, and the two equations are thus identically Atisfied. Moreover the condition for a contact is at once found to be B[(atat + Diy? — of + 9 Vy] = at [(ata? + iy? — ) + 9a, or, what is the same thing, (a? — 6%) (a?a? +34? — 97)? + 99? (a7a*a* — Bb y’) = 0; and substituting the foregoing values, this is (e248 _ay +07 AE ESE that is, £OE (at Bt) + 910° (a 8)=0, Vor. XII. Part I. 42 330 Mr CAYLEY, ON THE CENTRO-SURFACE which, putting therem a+8=—y, and a°+6°+9°=3afy, is also satisfied; that is, the points in question are points of contact of the ellipse and evolute. 21. Secondly, consider the values ae pear Co-ordinates of outcrops in plane of ay (real). 059) Substituting in the equation of the ellipse, we have a (8 —y)+B(y—2)' +4 (2-8) =0, (8-7) (y- 4) @—-8)(@+B+y)=0, or the equation is satisfied identically: and substituting in the equation of the evolute, we have first which is ata 4 By op = SE FE ea eB which in virtue of a(8—y)+B8 (y—2)+¥7(«—8) =0 becomes _ 3a8y (8-7) (y¥—2) Cs) a Soa ¥ (a—B) _ _ 348 (8 — 9) (y—2) @=5) 7 and then, completing the substitution, it is seen that the equation of the evolute is also satisfied. The points last considered are simple intersections, and we have thus the com- plete number 8+4=12 of the intersections of the evolute and ellipse. 22. We have a, y positive, 8 negative; whence a—8 is positive, B—y negative; y—a(=a°+c*— 26°) is positive, and hence, the outcrops in the plane of ay are real; the umbilicar centres are imaginary for this plane, but real for the plane of za, the co-ordinates being a? | 4 Co-ordinates of Umbilicar centres in plane of xz (real). 2,3 Y | Nodes of the Evolute. 23. The Evolute is a curve with four nodes, all of them imaginary; viz. for the evolute in the plane of xy, the equation of which is (a@ax* + By? — 9°)? + 27 yab'a*y* = 0, these are ax? = — 4%, ay? - Co-ordinates of Nodes of evolute in plane of ay (imaginary). OF AN ELLIPSOID. 331 in fact these values satisfy as well the equation of the evolute, as the two derived equations 6a°ax [(a*x* + b’y? — x")? + 9ry*by"] = 0, 6b?y [(a°ax” + b’y? — yy")? + Ina7x"] = 0, or the points in question are nodes of the evolute. The evolute has the four cusps on the axes and two cusps at infinity, in all 6 cusps as just mentioned; it has 4 nodes: and the order being 6, the class is 30 —2.4—3.6,=4 Section by the plane infinity. 24. The surface itself is finite, and the section by the plane infinity is therefore imaginary, but by what precedes the nodal curve must have real points at infinity, viz., there must be real acnodal points on this imaginary section. The section by the plane infinity resembles in fact the principal sections; viz., writing successively = 0, and n= ©, we have — Byala? : —yab*y? :— oB@Z2=AO+n:P+yn:C +n or H=@+ EP (P+ EP: (8 + £)%, giving respectively Z a +by+ce=0, and (aax)' + (bBy)? + (cyz)?=0, where the first equation represents an imaginary conic which counts three times; and the second equation, the rationalised form of which is (arava? + bB?y? + ee 27 Be By'x*yPa? = 0, an imaginary evolute. The conic and evolute have four contacts and four simple intersections (in all 4.244=12 intersections) which are all of them imaginary. But the evolute has four real nodes (acnodes) a’a’a* = 6°8’y" = c’y’2"; or, what is the same thing, there are four real lines a’a°a*=6°y*? =c’y’z*, which are respectively asymptotes of the nodal curve: viz., inasmuch as the equation of the surface contains only the squares 2”, ¥,2°*, the lines in question will be not merely parallel to, but will be, the asymptotes of the nodal curve. The plane infinity may be reckoned as a principal plane, and we may say that in each of the four principal planes there are four umbilicar centres, four outcrops, and four evolute-nodes. The generation of the surface considered geometrically. Arts. Nos. 25 to 28. 25. I have deferred until this point the discussion of the generation of the centro- surface by means of the centro-curves, for the reason that it can be carried on more precisely 42—2 332 Mr CAYLEY, ON THE CENTRO-SURFACE now that we know the forms of the principal sections and of the nodal curve. The two figures exhibit (as regards one octant of the surface) the portions already distinguished as (A), and (B): they intersect each other in the nodal curve, shown in each of the figures, 26. Consider first the generation of the portion (A) by means of the major-mean sequential centro-curves. The major-mean curves of curvature (attending to those below z (@) the plane of wy) commence with a portion (extending from umbilicus to umbilicus) of the . te fi . . ellipse 7s fen this may be termed the vertical. curve, and they end with the whole Ve ; : a= 1, which may be termed the horizontal curve. The normals at the several ellipse =a + a h OF AN ELLIPSOID. 333 points of the vertical curve successively intersect along a portion (terminated each way at an umbilicar centre) of the evolute in the plane of xz or umbilicar plane; viz. this portion of the evolute, shown fig. (a), is the sequential centro-curve belonging to the vertical curve of curvature. The curve of curvature is at first a narrow oval surrounding the vertical curve; the corresponding form of the sequential centro-curve is at once seen to be a four- cusped curve as in fig. (b), and which we may imagine as derived from the curve (a) by first doubling this curve and then opening out the two component parts thereof: the two upper cusps of the curve (2) are situate on the yz-ellipse of the centro-surface, and the two lower cusps upon two detached portions respectively of the «z-ellipse of the centro- surface. And as the curve of curvature gradually broadens out and ultimately coincides with the XY-section of the ellipsoid, the four-cusped curve continues to open itself out, and ultimately coincides as shown figure (c) with the wy-evolute of the centro-surface, viz. this evolute is the sequential centro-curve belonging to the horizontal curve of curvature or XY-section of the ellipsoid. The successive sequential curves are also shown (so far as regards an octant of the surface) in the figure (A). 27. We consider next the generation of the portion (B) by means of the major-mean concomitant centro-curves. Starting as before with the vertical curve of curvature, the con- comitant centro-curve is a finite portion (terminated each way at an umbilicar centre) of the wz-ellipse of the centro-surface. As the curve of curvature opens itself out into an oval, the concomitant centro-curve in like manner opens itself out into an oval, the two further vertices thereof situate on two detached portions of the xz-evolute of the centro-surface, and the two nearer vertices on the yz-evolute of the central surface. And as the curve of curvature continues to open itself out, and ultimately coincides with the horizontal curve or XY-section of the ellipsoid, so the concomitant centro-curve continues to open itself out and ultimately coincides with the ay-ellipse of the centro-surface. The successive forms (so far as relates to an octant of the surface) are shown in the figure (B). We have in each case attended only to the curves of curvature below the plane of zy, and the corresponding centro- curves above the plane of «xy, but of course every thing is symmetrical as regards the two sides of the plane. 28. There is a precisely ‘similar generation of the portion (A) by the minor-mean concomitant centro-curves, and of the portion (B) by means of the major-mean sequential centro-curves. The Nodal Curve. Art. Nos. 29 to 60. 29. If two different points on the ellipsoid correspond to the same point on the centro- surface, this will be a point on the Nodal Curve: the conditions for this if (& m), (&, 7,) are the parameters for the two points on the ellipsoid, are obviously (a+ &) +n) =(@+ &) (a +0,), G+ £)° F+m) = (H+ &) +n), +e C+n = +E E+); 334 Mr CAYLEY, ON THE CENTRO-SURFACE these equations in effect determine 7 as a function of & so that the equations — Byata? = (a + £)° (@ +7), &e. then determine the co-ordinates (a, y, z) of a point on the Nodal Curve in terms of the single parameter &. The relation between € and 7 would be obtained by eliminating &, 7, from the fore- going equation: but it is easier to eliminate 7 and 7,, thus obtaining between &, and & a relation in virtue of which £ may be regarded as a known function of &; 7 and y, can then be expressed in terms of & &, so that each of these quantities will be in effect a known function of &*. 30. The relation between & & is in the first instance given in the form a*[(a* + £)°—(a°+&)"], (a +8), (+E) | =0. e[e+ey-E+E)] C+, H+) ee +é—-C+8) 1, +8), +8) Throwing out a factor (&—&,)*, this becomes & [a? (Ba* + Ba” (E+ &) + B+ 6, + x@-e).LLU(E+h (+8) C +8) E+H)1=0, where the left-hand side is a symmetrical function of & & vanishing for £=€, and there- fore divisible by (—&)?; it is also divisible by A, = (8°—c*) (@—a’) (a’—20°) (=aBy). To work this out, write + &=p, &€,=g, the equation may be written T{(P-—ce)a’ | 3a" 3b*ct }=0, + 3a*p + 307c? (B? + c*) p +p'—q || + (+e (p*—9) + bic’ (p* + 89) +3 (0+ c’) pg +37 where the left-hand side divides by A (p* — 4g). 31. Developing and reducing, and omitting this factor, the final result is 6R+3Qp+ P(p*+ 29) + 3pq = 9, where as before P, Q, R denote a’? +b°+c’, Bc’ +c'a’+a’b’, a’b’c’, respectively ; that is 6R+3Q (E+E) + PE +t 486, + &') +3 (E+€) £F, = 0. or as this may be written 6R+3QE+ PE + & (3Q + 4PE + 3&) a a a A ed viz. the parameters £, £, have a symmetrical (2, 2) correspondence. nee eed * This was my first method of solution; and I have | —but it will appear in the sequel that I have succeeded in thought the results quite interesting enough to retain them | expressing £, 7, &, 7, in terms of a single parameter c. OF AN ELLIPSOID. 335 32. From the equations (a*+ &)° (a+) =(a@+ &)* (@+7,), &e, we have = (B —c*) {(a' + €) (a +) -@+ &) (a + 9)} =9, Soc’ (b’ —c’) {(a* + £)’ (a +m) — (a +E)" (@+n)} =0; and observing that the term in { } is a (3E + 9 — 3&, — m,) + a! (3& + 3& — 3E°— 3E,n,) + a* (E+ 36% — E° — 3E,n,) + (oy = 'm,)> these are readily reduced. to (3& +9 —3&, —,) P+ B& + 38 — 38, — 3&1.) = 0, (BE +79 —3&, —9,) B+ En — En, =0, or what is the same thing 3(E-&) (P+E+&) +0 (P+ 3&) — 0, (P+ 38) = 0, 3 (E—§) k +0 (B+ &)—1, (B+ &) =0, and if we hence determine the ratios 3(€—&): 7: m, the first of the resulting terms divides by €—&, and we have 3:9: 9=—P(E+ E64 &') + 3h — 386, (€+ &) : B(2E,—£) - &°(P+&+&) > RQE-£)-H (P+ E+ &). Hence observing that by the relation between &, & the first term is = 3 (PEE, + QO(E+ &,) + 3R}, the equations become 1:n: 9, =P&E,+Q(E+E)+32 2 HiGE = 6) — Sr (B+ E+ 6) eRe) ie (bP EAE); and we thus have _RQE-9-E(P+E+E) 1 Pe +QEFE) +3R ’ which, considering £, as a given function of &, gives as a function of & 33. I write €+&,=2x, €-&=2y, so that p=2x, g=x’—y’, the relation between &, & takes the form ? 6 (R+ Qx + Px’ — x°) — (6x + 2P) y?=0, or, what is the same thing, _ (x +a) (x +0) (x+e), y= x+i(¢?+0'+¢) ? so that taking x at pleasure and considering y as denoting this function of x, the values 336 Mr CAYLEY, ON THE CENTRO-SURFACE of & & belonging to a point on the nodal curve are €=(x+y), &=(x—y); and the value of 7 is then given as before. 34. The form just given is analytically the most convenient, but there is some ad- vantage in writin a as , in the place of x, y respectively; viz. we then have s s Va” Ve yi: P y Tesp' y ie (x +a?V2) (x + 0°V2) (x +e V2) x+4$4V2(74+0+¢) where = 55 (x+y), &= 4 (x—y), so that if (&, &) be taken as rectangular co-ordinates of a point in a plane, (x, y) will be the rectangular co-ordinates of the same point referred to axes inclined at angles of 45° to the first-mentioned axes respectively. 35. The curve is a cubic curve symmetrical in regard to the axis of x, and having the three asymptotes, x=—3 (748 40)V2, y=t{xth@4+h4e) V3}, “I OF AN ELLIPSOID. 33 viz. these all meet in the point P the co-ordinates of which are x=—4(74+8+0)V2, y=0: moreover we have y=0 for the values x=—a?V2, —0V2, -—¢ V2, that is, the curve meets the axis of x in the points A, B, C; the order in the direction of —x being C, B, P, A as shown in the figure: and with these data it is easy to draw the curve: the portion which gives the crunodal part of the nodal curve is that extending from B to the points ©; viz. at B we have €=£ =—J* corresponding to the umbilicar centre; and at ©, QO we have & or & =-c’, & or fact 2 2 corresponding to the outcrop. 36. The nodal curve passes through (1) the umbilicar centres, (IT) the outcrops, (III) the nodes of the evolute. The geometrical construction led to the conclusion that the real umbilicar centre was a node on the nodal curve, and that the real outcrop was a cusp (the tangent lying in the principal plane). It will presently appear generally, as regards the several points real or imaginary, that the umbilicar centre is a node on the nodal curve, and the out- crop a cusp—the tangent at the outcrop being in the principal plane: as regards the node on the evolute this is a simple point on the nodal curve, and by reason of the-symmetry in regard to the principal plane, the nodal curve will at this (imaginary) point cut the principal plane at right angles. Hence considering the intersections of the nodal curve by a principal plane, the umbilicar centre, outcrop and node of the evolute count respectively as 2 points, 3 points and 1 point, and as for each kind the number is 4, the whole number of intersections is 4(2+5+41),=24. It may be shown that these are the only intersections of the nodal curve with the principal plane ; and this being so, it follows that the order of the nodal curve is = 24; which agrees with the result of a subsequent analytical investigation. 37. The umbilicar centres or points (I) belong to values such as = £ =—a? which are the united values in the equation between (€, &), viz. writing herein £=£€ the equation becomes (E+ a’) (E+ 8%) (E+!) =0, so that the united values are §=£ =—a’, —0° or —c*. (It may be remarked, that treating this cubic as a degenerate quartic, a united value would be €=£ =, corresponding to the umbilicar centres at infinity.) To a value such as €=—a* there corresponds (not only the value £,=—a’*, but also) a value & =— a+ a , as it is easy to verify. And the outcrops or points (II) belong to such 3By THe; ZSSon (aS — GPa F=-0', & a, And the nodes of the evolute or points (III) belong to values such as £= ab’ + oc’, —,='b* + wc’ (» an imaginary cube root of unity) which, as it is easy to see, satisfy the relation between (&, &). But to complete the theory we require to have the values of 7, , and also the co-ordinates of the points on the centro-surface, and of the two points on the ellipsoid. Vou. XII. Parr I. 43 338 Mr CAYLEY, ON THE CENTRO-SURFACE 38. Iexhibit the results first for the umbilicar centres (imaginary), outcrops (imaginary), and nodes of the evolute (imaginary), in the plane 2=0; secondly for the real umbilicar centres in the plane y=0 and for the real outcrops in the plane z=0. The formule contain an expression © which is a symmetrical function of «, 8, y (or a, 4, ¢), viz. it is =e — By=—?—ya2=7' —48 =% (+8 +7) =— (By + 4+ a8). We have L E=-a', n=—- 0; &£=—a', n=O. 0 ese 5 Peay Fawn TY De Pe a” ” a a’ * (Umbilicus). Bee TF meee a aj a Oe 3 CC ee Ls a” | (Umbilicar centre). eZ =— Bi | a} 9ByQ II =-—¢ yi el a, n a+ (@—9) oS eee)" 2_ B(y—a)" (orn +0 Yay 7 t8 Bay)” 2, 8PY 2 &= a 7 Ba Ui =a Xx =U, X,= 0, ya_ py @-8) A ical a (@—¥)”’ : nan oS B (y—4)° 2 2B y-a See Bae ae BE Baa)" C—O Bay” | Ellipse, concomitant. Ellipse, sequential. 2= 0; 3 (a — p) by? = ell tl Se bt 7 ~a(B=9)! 5 : (Outcrop.) Giger & (y- 4) a(e=y)" 72 2 (Observe that at point Y,, Z of ellipse ws =1, the co-ordinates of the centre of 73 ae ; curvature are y=, z=— 4, and it thence appears that this is the point in regard to which the ellipse is sequential.) III. E=o)'+o'c’, n=—-a’; E=ab' +c’, ,=—a’, OF AN ELLIPSOID. 309 X=0 X,=0, = —/jf Y~=-Fo, Z=— Co, Z=-Co, =((). } bey? = — a’, (Node of evolute.) C2 =a 39. Observe that these are the only ways in which it is possible to satisfy the equations = (a + &) (+n) =(@ +E) (a +9,), viz., starting from this equation we have if a+&=0, @ +E =0, whence in the equations for 7, 7,, substituting the values £=£=—a*, we have 1:9: 7,= Pa'—2Qa +3R, :—a'R +a° (P — 2a’), :-a@R + a’ (P—2a’), that is 1:7: ,=—a’By : a’By : a’By, or, n=, =- a. 40. Il, a+£&=0 without a’? +£,=0, consequently a?+7,=0; writing €=—a’, in the relation between (& &) this is 6R+3Q(E,—-a*) + P(E’ —40°E, + a’) — 30°E, (E, + a’) = 0, yiz., this is E? (P +c — 2a’) + &, (—a* — a’b* — a’ + 30°C*) + a (a* — 20°b* — 2a’c* + 30°c*) = 0, where the left-hand side should divide by £,+.a*; the equation in fact is (€, + a”) {&, (+e —2a*) + a* — 20°’ — 2a’? + 3Bc*} = 0; or, what is the same thing, (E, +a") {(E, +a") (@—) — 38y} = 41. Considering these values of &, & as given, the verification of the value 7,=—a’, ee 9 and determination of »=- a@+ ( ee 7 is somewhat complex. 3By Writing for a moment A eh we have 1:n:7,=P(a'+@A)—- Q2e+A)+3R :— BR (a + 2A) — (a? + A)® (2a*— P+ A) : — R (a — A) — a’ (2a? — P+ A), 43—2 340 Mr CAYLEY, ON THE CENTRO-SURFACE The first term is a'P—2¢Q+3R+A (a’P- Q), which is =— a By tA (a'— bc’), and for the value of », proceeding to the third term, this is =-—@R-a' (2a —P)+A(R-a’), which is = *By-—a@A (at — ¢°), so that without any further reduction 7,=— a’. 42. We have then — a'By + A (a* — b%c*) 3 and I assume Q=—a + Go QO, and investigate the value of 0. We have — R (a+ 2A) — (a? + A) (@?- B—c + A), =a’By + AO, suppose. The equation therefore is a'By + A© sn Uy t AO. scopes pay, BY. —apy+A@—oe) (ae that is AO=-— 2A (a‘ — B°c*) + gino {— By +A (at — b°c*)} =( or writing 7 Fo =; a omitting the factor A, and multiplying by (@—y)*, this is (8-9)? {© +a’ (a' — Bc*)} + 80 { — a?By + A (a*b%c’)} = 0, in which equation © =— 2R—-a’— (8a*+ 8a°A + A’) (?-B—c' + A), and thence © +a’ (a‘ — bc?) = same + a’ (a* — b'c’), =—3a'+ 8a‘ (?+.c*) — 3a°B'e? + A {-—6a*+ 3a’ (0° + c*)} +A?(-4+ +e) -—A? = 3a*By + 8a°A (8 —y) + A’ (B—¥ — 2a’) — A’. 43. Hence, substituting for A its value and multiplying by (8 —¥)*, we have (8 —4)° {O+a' (at —B'e’)} = 8a°By (8 — 7) — 9a’By (8 — 7) + 9B*y* (8B — 7 — 20°) (8 —y) + 27By', OF AN ELLIPSOID. 341 which is = —6a"By (8 —y) + 9B’ (8 — 1)? — 182°B 7 (8 — ¥) + 278 %y'; viz. this is = {- 6a" (8 —y) + 984} {(8 — 7)" + 38%} By, = {—Ga? (8 — y) + 98y} (8 + By +7) By, and the equation thus is {— 24° (8-7) +884} (8° + By +11) By +0 {— aay — FO" (at Ve} (8-7) =0, or finally 0 {a? (By) +3 (a* —B'c’)} = (— 2a” (8 —y) + 38y) (B+ By +7’). But ?=a?+8, P=a'—y, and hence a‘ —b’c? =—a& (8 —y) + By, and therefore a’ (8—y) +3 (a* —b%c') = — 2a* (8-4) + 8By; the equation thus divides by —2a* (8—y)+38y and we have 2= B+ By +7, or as this may also be written Q=a°—fy, =6’—ya, =y°—a8. So that O has the value originally so denoted, and we have then 9By (a 44, III, Lastly the equation 0=(a7+&)* @+n)=(+£&)* (@’+7,) is satisfied if a+n=0, a+7,=0: the equations (+E F+n= C+) (+n), (P+ &)° (+n) = (6+ &)*(e+m,), (+ EP=O+E), (+ €= (C+ &)’, which can be satisfied by €=&,, leading to £=£,=—a’, which is the case I, or else by e+E=a (0+ &), + E=a" (b+), q=—a+ then give that is E=ol’ +, &=a'b’ + we’. To show that these values satisfy the relation between & &,, observe that they give E+£=-B—c, &, =e te, , whence also 244664 62=3(' +c’), 342 Mr CAYLEY, ON THE CENTRO-SURFACE and the relation becomes 60°F'e — 3 [a* (P +e) +7] +c) +[a*+ (BP +0)].3 G+c)-3 0 +e) (&-Be? +c’) =0, which is an identity. 45. I will show that these values of & & give the foregoing values n=7,=—a’. We have 1:n—7,:97+7,=Pé£,+ O(E+&)+3R : (&— &) (BR — (E+ €6, + &') (P+E+&)} > (6+ {(R-(@— 6.48) (P+E+&)}, this is Linn intn=a (+e) : OU, — 8) :— 2a B42), oF 7—-7,=9, 7 +7,=—2a°; that is y=y,=-a’. 46. For the real umbilicar centres and outcrops we have 1 g=-U, n=-0, £=-0, =—8. BV=—gt X=- aS, Y=0, Ye= 0) Pa 2& a aa? Ie eae? scapes CcZz= Bp’ 2 9a8 II —— lias age ae aa ~_ @l(¥—-%) 2_a(8—y)* (= 1ef Sapa) g=-0+ 7}, == 72 2B (y —a)* fat ee By-a@ ee 8” ey EB & ys___ 722 (8—4)° 2__ 2 aB—-y¥ ‘4 y (a—8)°’ a em Z=0, ZU) ellipse concomitant. ellipse sequential. OF AN ELLIPSOID. 3438 aa? Boa Sh (y = a)? | y («—B) 5 . (real outcrop). meee (B=) y («—8) 2=0. 3 3 Nodal curve in vicinity of umbilicar centre, a’a?=—%, Y=0) Cz: = =s . Art. Nos. 47 to 49. 47. Write —E=-B +4, n =—-U +r, £=-0 +49, 7, = — 0 +15 we have to find the relation between gq, g,, 7, 7,3 first for g, gq, the equation of correspondence gives 6R +8Q(—20+9¢+4%m) + P (6b' — 60° ¢+q,4+9 + 49m + 7) +3 (—20°+30'¢+q—-2P (¢+4¢n4+H) +949+4H) =9, that 1s 3 (¢+4,) (80*— 2B-P+ Q) + (9° +99,+9:) (— 30° + P) + 399, (¢+4,) =9, viz. this is —3 (q+) 4 + (¢+499,+%) Y- 4) +3499, (¢+4,) =9, whence approximately g+q,=0; but it will appear that the value is required to the second order; we have therefore — a iD g+q=t (+ 4qn+%) 48, Now the equations (+8 (a+) =(@+E) @+m,), and (P+ €) (+n) = (P+ &) (+), putting therein for &, », &,, 7,, their values, give the first of them log (1 +7) 43 log (1 +2) =t0g(1 +7) 43 log (1 +4), ny Y i oF that is 1 37 1 jh a, on r+3q- By (7? + 3q°) +a (7° + 8q°) =7,+39,- oF (77+ 39°) + By? (7° + 39,), 344 Mr CAYLEY, ON THE CENTRO-SURFACE and similarly the second equation if 1 1 1 r+3q+5, (PF +3¢/) + 55+ 39/) = 7,43, + 5 (OY + 8a) +35 + 84) 5 whence multiplying by y, a, and adding, (y +4) {r+ 3q+ iy (P+ 30) =(y+ 4) {r, +39,+ Sag (r+ 3q.)} ; which, neglecting terms of the third order is r+3g=r,+ 3q, Subtracting the two equations we have A (| a ‘) (r? + 3°) +4 (3 = 7) (7° + 39°) =4 é SF =) (r? +39.) +4 G > =) Sg), viz. this is —a r+ 3q?+ , q tere (r + 3q°) = i a5 3q, a3 3 i - Ge + 3); a a or, what is the same thing, r—r?+3 (¢—g)) +3 = {FP —7$+3(¢—- 7} =0, which, putting therein »—7,=—3(q—q), 1s ae! —r—7, 4949.43 (—?—9r,— 7 + 9° +99, + 7°) =0, say this is Bos F —r—r+9t+9,+24=0; combining herewith r—7r,+3¢—-3q, =0, we have r+q—2qm—A=0, and 7, —2¢+9,-A=0, where A= Vii (ee ei 0 ee 2 =} a ( i LT +¢+49%+ 9%); but substituting herein the values r=—g+29,, 7 =2q¢—q, this becomes er he ee BS ACEC 3 Se (—2¢° + 499, — 24’), = —§ ya “e) and then _— —g+2n, +A, that is Alte r43q=2 (+g) +d, =-— 0-9 ®) 49. We have then OF AN ELLIPSOID. 345 Heth) ~—Hfie(—9- and in the same way from ez? =— B ¢ = 4) (1 = ), we have CZ= -S{1- g 2 aE Te, LE ya ya moreover we haye at once Hence, writing x+ 6a, 0+ 8y, 2+62 for a, y, 2, we find 2(B— da=4a. =o”) ” aC ie ee (Ses dy=+ BN ya? » —2(a—-8) , , dz 3 ye -q - or, what is the same thing, a eee eee) tate See ae 9) 8 9 dy Oe ea eae ya’ qt? where x, z denote the values at the umbilicar centre. Nodal curve in vicinity of real outcrop, viz. ata? = — & = » Uae a 5 a , z=0. Art. Nos. 50 to 52. 50. Write 8 aBQ é=-c+4q, n=— 6 + Ea +8 3 g=-¢ + +4, n=—e +05 and first for the relation between g and g,, writing for a moment = a8 say a Q,, and therefore &,=—c*+ Q,, the equation of correspondence gives —8a8 (q+ Q,) + (¢ +499,+ Q,’) (2-8) +3qQ, qt Q) = 0, Vou. XII. Parr I, 346 Mr CAYLEY, ON THE CENTRO-SURFACE which putting for Q, its value is - 328 (g++ 75) 3 9a? ‘ +(2-8) (7 + 499, + 9° + (49 + 24,) = 22+ +a 3 9 2Q2 + 3q {o@+a) + (+20) 24+ ea " 0; that is — 3a8 (¢+ q,) + 348 (4q + 29,) + (2-8) (7° +499, + 2) Pigs ataao or, what is the same thing, 27a°B" 3 (908 + Ge em) et a8q, 2 16 ; - 1g EERE oy, ETE 4 gra) + 399, (7+ 9,) =9, or for small values 9a8 302 (8+ gage) atn= 0, that i STP — Ape ea 0. 51. Moreover, from equation (¢’+ &)* (*+m)=(C+ &)° (e+ = we have ¢ ea > ea or, since q and gq, are of the same order, @, is of the order g,°. Hence, starting from the equations —Pya’s* = (a°+&)*(a*+7,) &c., the terms of a, y arising from the varia- tion of », are indefinitely small in regard to those arising from the variation of £; and we have 6,, that is, 0, =4.9 age’ qs ae Le. Sega 30 (a—8) ot) ty 2. a 1 B(y—a)’ a= 20y' | ees a-B8 y a 4. 38 oe 3% a(B—y) Poe 2(8)2 = § _ 92 _(a-B) c’(8z) =~ 33 (—a) a (a— (a—B)’ > 32°, Coe OF AN ELLIPSOID, r 347 so that writing for greater simplicity, («—8)¢,=—-a8w, the formule become 280 zyma”’ 267 __ 38 7) Bas 8 céz ieabey OV3 52. This shows that there is at the outcrop a cusp, the cuspidal tangent being in the plane of ay. It appears moreover that this tangent coincides with the tangent of the evolute. In fact from the equation (ax) + (by)i—9=0 of the evolute we have 3 3 di (ax)*. dx 8 (by). dy 50. x or substituting for (x, y) their values at the outcrop, Biel ge palB ok de egy. y(a—B) = (a—f) y ? that is, ee ee et Bly-a) +089) 2 =0, which is satisfied by the foregoing values of = and °Y, and the two tangents therefore coincide. We have 4 {(82)? + (8y)"} = a Gee) ae a a which in virtue of a’a (8B —y) + UB (y— a) + c’y (4-8) = 3287, =~ 9a70'R" is 4 { (Sax)? 45 (8y)’ j= wb? (Ga— ab? (a — 8) {348 =e (a iz. B)} (observe 328 —c*(a— 8), =—c" (y—@) — 3a’a, is negative) — 9a? * ab (a— By E if &, be the value at the outcrop. Writing 6s for the element of the arc we have ne OB. os =— CTS) mao E,.@; ON3 ’ which exhibit the form at the outcrop. Ad? 348 Mr CAYLEY, ON THE CENTRO-SURFACE The Nodal Curve; expressions for the co-ordinates in terms of a single parameter o Art. Nos. 53 to 60. 53. After the foregoing investigation of the nodal curve, I was led to perceive that it is possible to express & », &, 7», in terms of a single variable o, and thus to obtain expressions for the co-ordinates of a point of the nodal curve in terms of the single vari- able ¢. The result was obtained by the consideration that the acnodal portion of the nodal curve could only arise from imaginary values of £, 7; the question thus was, what imaginary values of these quantities give real values for the co-ordinates a, y, z. To make y real we may assume : : ae — p (8 — gz), 7=—b+p (0+ di), (¢=V—1 as usual): this being so, if A denote one or other of the quantities Y -—a(= eS B*, eS 2), the expressions for — Bya°x”, — yab*y’ will be = {A —p (0-41) {A + p (0 + 41)4, and we have therefore the condition that this shall be real (for the two values A=y, A=—a): being real, it will in certain cases be positive, and we shall then have real values for the remaining co-ordinates 2, z. ; 54, The condition of reality is easily found to be A’ (360° — ¢°+ 3) — 6OpA (# + ¢? +1) +p" {3 (# + 6°)? + 36 — g} =0, viz. this equation in A must have the roots y, —a, or the expression on the left hand must be = (38 — $*+ 3) {A*— (y— a) A— ay}: we have therefore (y—a)? _ 36 (@ + ¢* +1) —2 ~ (86 — ¢* + 3) {3 (@ + $*)? + 36 — ¢*}’ 6Op (F +g? +1). Yo a BFP 43? and writing @+¢*=X, 3—¢°=Y, the first of these is (y—a)? _ 9(X+Y)(X+1)? —ya © (Y¥+38) {3(X*-1)4+ ¥+3}’ which regarding X, Y as the co-ordinates of a point in a plane is a cubic curve having the point X+1=0, Y+3=0 as a node: hence writing Y+3=3e(X+1), X and Y will be each of them a rational function of ¢. The second equation is 6Op (X +1) _ wee Oe G4 e)a, “Yee a Se aie oy eas OF AN ELLIPSOID. ; 349 and we have also 20=VX+Y, 2=V3X-Y; the equations thus become 2, ¥—a)o(X+ VY) - (/3xX—Y)* + 8 {1+é ray: which are better written in the form g=-0-}(y-a)o{1- 7) where X, Y are given functions of o. We in fact thus obtain an analytical expression of the nodal curve, quite independent of the considerations as to real and imaginary which suggested the process: the foregoing values substituted for £, » will give — Bya’a*, &c. equal to rational functions of ¢, so that taking for £, », the same expressions, only changing therein the sign of the radical of ee = values of — Bya’x’, &c., or the values of & , &,, , satisfy the conditions (a+ &P (@ +n) = (+E) (+n), &e, these values of &, 7, give the very same for a point on the nodal curve. 55. To complete the investigation, writing as above Y+3=30(X+1), we obtain (y—9)*_ (Ba +1) X+30-3, — ya a(X+oa-1) ’ or putting for a moment Kets.) Os oes we have _ (K-83) (¢-1) >, 4 _K(o—2)4+4. eas RII ee. aE Lae 3Ke(o-2) +12 _ 3 (e-1) {K(e-1) +1] Br a Ore EN a ’ 3a B= K ; 3 (o-1){K(o-2) +4}. (¢ —1) (80-2) K *3xeys ceep ls See ss pea glial oa Rew ; 300 Mr CAYLEY, ON THE CENTRO-SURFACE or substituting for K its value we have = (y—«)? {_. tb 4ya K(e-2)+4=— 92" [or 90 Go nS) EN A, nh =— ya (o+ = )(¢ se 3041-K= {Bo +1) ya+ (y—a@)*o}, if as before Q=8*—¢yx; and the result is 2a 2 WR 0 S +24 ) Qy ) 2 oc Oi s Se RS OS (¢ —1) (30 — 2) af ( Spee! ( sa B 8 (y a) Qo + ya 1 aT ae ee and changing the sign of the radical we have the values of &, 7,. 56. Write for moment oe 2a ee iy" [5-4 creo eer erae ia] =(A-a+aVS)'=A+BVS, 7 RE eee then in the product of these two expressions the rational part is =.4A'+ BBS; but from the manner in which they were arrived at we have 0 =AB'+ A’B, and the rational part is thus B 2 2 == (4 — Bs). We have A?— BS = {(A—a)’—a'S}’, .o(¢—1) (30 — 2) B=} (ya) {8 (A—a)' +2°S} ; henée the rational part in question is (y—a)" o(c—1) (30 — 2) (8 + 9) ee cme. Fnaeas — (A-a)'+a's)" which putting therein A=0 gives the value of ~yab’y’; and putting A=y, or A=—a, gives the value of —Bya*a* or — aBe's’. OF AN ELLIPSOID. 351 57. We have toe gag [oof | ee ito Go. Ban 4 9= gray [Bee + [ote Se) Geren eee Hence we have at once the value of 22 1 (Y-% s(c—1)(3c-2 n'y, = 4 PaO. =D GED) gy where a=}(8—-y)c. 58. Moreover (A —a)* — a? S = A?— 2aA + a? (1— 8) = = 3 [(8a— 2) {—(y—@) Ac + A} + (y— a)? o* + 3a], where the term in [ ] is a” (y —a) (y —a — 8A) +o {8A* + 2 (y—a) A + Say} — 2A%, and sincee A=y or —a, that is A’—(y—a)A—ay=0, the coefficient of o is = A [6A—(y—a)}, or the term is the product of two linear functions of ¢; and we have (A — a)? a $= fo (y—a) — 2A) fo (y—a— 3A) +A}. Similarly 3 (A —a)?+a°S= 3 (A?— 2a) + a? (348) = go -2) + y- 2) oA +84 +0 [(y—a) 044} (ya) oy} where the term in [ ] is (y— a)? o° — (y— 2) (y—at 3A) o* + {3A* +2 (y— 2) A — ay} ao — 2A in which the coefficient of ¢ is =A{2A+3(y—a)}, and the term is a product of three linear functions: hence 3(A—a)* +08 S=92— 2) yao A) ae — 2A). 352 Mr CAYLEY, ON THE CENTRO-SURFACE 59. Substituting these values we have the expression 1 {(y—a)o tah {(y—a) o— 9} f(y —@) o — 2A} f(y —a— 3A) o + AP Qe + ya {(y¥ — a) o — A} (30 — 2)? ; which writing therein A=y¥ gives — Bya'a’, and writing A=—a gives —a@c*z*; we have above an expression for —-yab*y* requiring only a simple reduction, and the final results are 22 _ iC —a)o+a} {(y—a) o— 2} (8-9) o +4} Botere (Qe+ ya) Br 2)" (6-1) 0? {(y a)" + Say)? (Qe + ya) (8a — 2)? ans yab’y® = een Sale Si 2a ee (Qe + ya) (80 — 2)? ; where it is to be observed that, equating the denominator to 0, we have a triple root «=o; to indicate this, we may insert in the denominator the factor (1 — 0c)’, 60. We see here the meaning of all the factors, viz. Planes. Evolute nodes Umbilicar centres Outcrops For the real curve o extends from —3 through 0 to aS viz, B Q c= — gives outcrop in plane z=0, o= 0 .,, umbilicar centre in plane y=0, a ‘ o=—— ,,_ evolute-node in plane ©, OF AN ELLIPSOID, 353 It is to be noticed that the order of magnitude of the terms in the table is -¥ a —ya -a —-24% —3ya (a) sh ga ok ile 0 5) 5) B-y’ a—p’ ? Om y—a@’ y—a’ (y—a)*’ Y=a ya, 2, — 0. —ye 4. ‘ ‘ =~ which belong to the real curve are contiguous; this is as it should be. Several of the preceding investigations conducted by means of the quantities & », &,, 7, might have been conducted more simply by means of the formule involving o. so that the values oe 0, a—B The Hight Cuspidal Conics. Art. Nos. 61 to 71. 61. The centro-surface is the envelope of the quadric ax by? Cz 2 a+ 7 Fate 7a — (CO GRE) C8) Hence it has a cuspidal curve given by means of this equation and the first and second derived equations iL (0). aa? By? ce? 2 va ith (pa eNe ies 3: 0, (GEE) GPE (C48) a2x? By? ee? 72 4 +7 ASR Er 7 es 0, (Hey Gi Seu Fe) which equations determine a’a’, b*y’, cz* in terms of &, viz. we have — By ata" = (a +) —yaby? =O +2), — Bes = (6 +8) so that, comparing with the equations — Py a’a*= (a’+ &)° (a?+%) &c. which give the centro- surface, we see that for the cuspidal curve €=7; or the cuspidal curve now in question arises from the eight lines on the ellipsoid, which lines are the envelope of the curves of curvature: it is clear that the curve is imaginary. 62. From the foregoing equations we have: Vaaxr+NnB by + Vy cz =" — aby, at Vax + BI Vby +9' Vez =0, the second of which is best written in the rationalised form (1, 1, 1,-1,-1,—1) @Vaaa, BVB by, yVycz)?=0, and combining herewith the equation Vaac+NVB by +Vycz=V —aBy, then for any given signs of Va, VB, Vy and V—afy the first of these equations represents a quadrie surface, the second a plane, or the two equations together represent a conic. Vou. XII. Parr I. 45 354 Mr CAYLEY, ON THE CENTRO-SURFACE By changing the signs of the radicals (observing that when all the signs are changed simultaneously the curve is unaltered) we obtain in all 8 conics, but only four quadric surfaces ; viz. the two conics Vaax+NVB by Vyez=4tV —apy lie on the same quadric surface. 63. The conics form two sets of four, corresponding to the two sets of four lines on the ellipsoid. The analysis seems to establish a correspondence of each conic of the one set to a single conic of the other set; viz. the conics have been obtaimed in pairs as the intersections of the same quadric surface by a pair of planes: there is a like correspondence of each line of the one set to a single line of the other set, viz. the lines meet in pairs on the umbilici at infinity, but this correspondence is included in a more general property: in fact each line of the one set meets each line of the other set in an umbilicus; and the corresponding conics (not only meet but) touch at the corresponding umbilicar centre; and qud touching conics they have two points and intersection, and consequently lie on the same quadric surface. It is to be added that the two conics touch also at the umbilicar centre the cuspidal conie of the principal section. 64 The 8 conics form two tetrads, and the principal conics (reckoning as one of them ‘the conic at infinity) another tetrad: the complete cuspidal curve consists therefore of three tetrads of conics: with these we may form (one conic out of each tetrad) 16 triads; viz. each conic of one tetrad is combined with each conic of either of the other tetrads, and with a determinate conic of the third tetrad, to form a triad. And the conics of each triad, not only meet but touch at an umbilicar centre, the common tangent being also by ‘what precedes, the tangent of the evolute at that point, which point is also a node of the nodal curve. 65. In fact consider the two conics Vaar+NB by +Vycz="V —aBy, (1, 1, 1,-1,-1,-1) (aVaax, +BVBby, y Vycz)*=0; for the intersections with the plane y=0 we have Va ax + Vy cz = — ay, (aVa ax —yNycz)?=0; so that the two conics each meet the plane in question in the same two coincident points, that is they each touch the plane y=0 at the same point, viz. the point given by the equations Vaax +Ny cz =V—aBy, aNvaax—yNycz=0; bots oa f yi v ai 2 a viz. this is the point, ax “Ja4 ce rast which is one of the umbilicar centres (ata*= = * ; cz? = — *) : OF AN ELLIPSOID. 355 and the common tangent at this point is Vaan+Wycz=" — aBy, which is also the common tangent of the ellipse and evolute in the plane y=0. 66. It has been seen that the nodal curve meets each principal conic at four outcrops, which points are cusps of the nodal curve: it is to be further shown that the nodal curve meets each of the 8 cuspidal conics in four points (giving 32 new points, which may be called ‘outcrops, the 16 pomts heretofore so called being distinguished as the principal outcrops or 16 outcrops, and the points now in question as the 32 outcrops), which points are cusps of the nodal curve. In fact to obtain the intersections of the nodal curve with the 8 cuspidal conics, we must in the equation of the nodal curve, or (what is the same thing) in the expressions of &, 7 in terms of o, write n=&. 67. Putting for shortness, 2g Cipieie les: Sie as ee: Qa + ya 2+ 2)“ 24) is ya ni (ed o (80 — 2) ; and as before we have thus ne @ (14+ VS) =1_-VS, or, what is the same thing, © (1+38) -1+¥VS{0 (34+ 8)+4+1}=0: we have without difficulty (y — 2)" fois _ o Arya : OB +S)+1= yo" |30"—6 +ho+ ese: (Bc — 2) (+) («- Qy ) U feats © (1+3VS)-1 Oot ya so that the resulting equation contains the factor omitting it, the equation becomes Pa ae 7 VA (30-2)! Wa + V5 {30° — bo" ed x eae 0, aie = = WM, and rationalising, this is S52): — (oc? — 26 — M) (3c — 2)? o + 38 (80° —60* + 40+ W)*= and, working this out, the terms in o°, o° disappear, and the result is (36 + 27M) ot — (64+ 361) o + 3207+16Mo+3M*=0, or, as this may also be written, or putting for shortness 356 Mr CAYLEY, ON THE CENTRO-SURFACE 3M? + M (270 — 360° + 16c) +4 (90*— 160° + 80%) = 0, a quartic equation in o: to each of the 4 roots there correspond 8 intersections, viz. there 5 will be in all 32 intersections, lying in 4’s upon the 8 cuspidal conics. 68. To show that these points are cusps, or stationary points on the nodal curve, starting from the expressions of — Sya‘* &c. in terms of o we have, first for dy, Qdy _ i 3 (y— a)? 9) 6 =ds\— +s + G=a)e+8ay. Qe+ya 30-2)’ or, as this may be written, Lyne 2 12 443M 6 da | + tae SM aM Fit Seca _ ag § 82 ~ bot 4 (32 + 2401) o—9M?* ) = 40 135° — Bat + 20 (16+ 12M)o* + (16M + 9M) o +3)’ _ do. 4{(36 + 27M) of — (644+ 36M) o° + 320° + 16Mo +37} o (¢—1) (30 —2) (40 + 3M) [(44+3M) c+} ; viz. the numerator vanishes when ¢ is a root of the quartic equation. 69. We have next 2 dx (aie 2(y —@) 3 Gai) er Oe 7 ru: “\qeaota’ (y—a)o—2y Ba ot7 Qe + y2 ae which, putting ~ 1 = B, and therefore Fa =B-1, adz=G is ¥ B-y q 2 3 443M 6 — + aE fn 2o8 o+C (44+3M)c+M 3c—2)’ , the numerator is o ee + 36B—4) 40 (54 (B?— B) M+72B" — 80B +8} + 4M—16B°+16B, which observing that B’- B=} M is = o° (27MB + 36B- 4) +0(5 27 y+ 18M —8B+ 8), and, substituting for M and B their values, this is found to be _ 4 (2y+a)? ot 8 (2y+4)*a (y—a)® (y—a)* ae eee (0 e+), (y=4) y=a ? OF AN ELLIPSOID. 70. Hence observing that OF sor the whole coefficient of do is 4 (Qry +a)? 2 pet eat) y — a) ya 3 ~ Go—2)(¢+ B-1) @—-28) [GM Dot | a+? and the numerator of this is ¥ 4 (Qy+a)? - (y—ay G Ff = [(2y +4) « —y], +8 (80-2) (°-c-4M-— Bo) {(3M + 4) 0+ M}, which is =3 (3c — 2) (?—-c-3M> {81+ 4)0+M}, + o| - 3B (80 — 2) {(8M +4) 0+}, at Mera] the term in { } is (2y-44)° . o |- (27M +36) Bye | (y —a)° 4 (2y +a)? Qa (Qy + a) | , zelehi as he fine, ea Net) +o [s2(ear-+8) + =a eee ) d [oo Crt™ |, (y— 4) which is found to be =—4¢+6(84+15M+ 2 M’) -2M-3 NM’, and the whole numerator is thus 3 (80 — 2) (oc? -c—4 M) [(38M@+4)o+M] — 4c° +o” (8+15M+2 M*)+o(-2M -—?M"), which is = (36 + 27M) ot — (644 386M) o° + 820° +16 Mo + 3M". 71. We have thus 9 2da _ ae (36 + 27M) o' — (64 + 36M) o° + 320? + 16Mo + 3M? 2 o (o —1) (80 — 2) (40 + 3M) {(4+4+3M) 0+ UY} («+ 5%) and thence also _ Adz _ Ae (36 +27M) of — (644361) o° + 320°4+16 Mo + 3M* # “a (e=1) (Bo—2) (So + BM) (4+ 8M) o + M} ( --*,) C3 | 358 Mr CAYLEY, ON THE CENTRO-SURFACE so that dx and dz also vanish when o is a root of the quartic equation: the points in question are therefore cusps of the nodal curve. Centro-surface as the envelope of the quadric Ya’x* (a’°+£)*=1. Art. Nos. 72 to 76. 72. The equations — Pya*a*= (a? + £)° (a? +), &e. considering therein & 7 as variable eive the centro-surface: considering 7 as a given constant but & as variable they give the sequential centro-curve; and considering £ as a given constant but 7 as variable they give the concomitant centro-curve. 73. Suppose first that » is a given constant; to eliminate € we may write the equations in the form — (8y7)* (az) @ +0) =(@+8), &e., and then multiplying first by a(a*+ 7), &e. and adding, and secondly by a, &c., and adding (observing that Sa (a? + &) (a*+7)=—aBy, =a(a’+&)=0); we have = (aaz)’ (a? + »)* = (aBy)%, = (aax) (a? +1) =0, which equations, considering therein 7 as a given constant, are the equations of a sequential centro-curve. If from the two equations we eliminate 4 we should have the equation of the centro- surface; the second equation is the derivative of the first in regard to ; and it thus appears that the equation of the centro-surface might be obtained by equating to zero the discriminant of the rationalised function norm. [{= (aaa)3 (a? + n)>} — (a8y)*]; but the form is too inconvenient to be of any use. 74. Taking next & as a given constant; and writing the equations in the form — Bya’a? (a? + &)* = (a +9), &e., then multiplying by a(a°+&), &e. and adding; and again multiplying by a, Ke. and adding, we have La'n" (a? + &)?=1, daa? (a? + E)° =0; or writing these equations at full length, a2? By? cz? 2 \2 3h Py 2 2 a t= 0, @+h' @+ +h aa by? ez 7.2 \3 Tar 72). £\3 = 0, @+e* @+e* +8 which equations, considering therein € as a constant, are the equations of any concomitant centro-curve: since the equations are each of the second order it thus appears that the concomitant centro-curves are quadri-quadries. OF AN ELLIPSOID. 309 75. If from the two equations we eliminate & we have the equation of the centro- surface; the second equation is the derivative of the first in regard to &; and it thus appears that the equation of the centro-surface is obtained by equating to zero the dis- criminant in regard to & of the integralised function (a? +£)? (B+ £2 (e+ £)*{ Saiz? Geet or, what is the same thing, the discriminant of the sextie function (a* + £)? (b° + £)? (c* + £)? — Sa’x* (b? + E)* (c+ €)*. 76. If instead hereof we consider the homogeneous function w* (a* + &)* (0 +E) (c? + EY — Sava® (6° + £)* (c* + E)’, then the coefficients are of the second order in (a, y, z, w), and the discriminant, being of the tenth order in the coefficients, is of the order 20 in (a, y, z, w). But the sextic function has a twofold factor @ + =) if w'=0, and it has evidently a twofold factor if a =0 or y’=0 or 2=0, that is, the discriminant contains the factor 2*y*z*w*; or, omitting this factor, it will be of the order 12 in (@, y, z, w); whence writing w=1, the centro-surface is of the order 12. I have in this manner actually obtained the equation of the centro-surface: see the memoir “On a certain Sextic Torse,” Camb. Phil. Trans. t. Xt. (1871), pp. 507—523. Another generation of the Centro-surface. Art. Nos. 77 to 83. 77. By what precedes the equation of the centro-surface is ebtained as the condition in order that the equation e: {a’a* (a? + E)*} -1=0 may have two equal roots. But taking m an arbitrary constant, this is the derived equation of {Sa?a? (a? + £)7} + E+ m=0, and as such it will have two equal roots ‘if the last-mentioned equation has three equal roots; and conversely, we have thus the equation of the centro-surface by expressing that the last-mentioned equation, or, what is the same thing, the quartic equation (E+m) (Eta?) (E+ 0*) (E+ e) — Sa’? (E+ 0) E+’) =0 has three equal roots. The condition for this is that the quadrinvariant and the cubin- variant shall each of them vanish; the two invariants are respectively a quadric and a cubic function of m; viz. the equations are (a, 6, c) (m, 1)/°=0, (@, B, c, ad’) (m, 1° =0; where the degrees in (a, y, z) of a, 6, ¢ are 0, 2, 4 and those of a’, b', c', d' are 0, 2, 4 6 respectively: the equation of the centro-surface then is a, 65 ¢) | 08 (i, (Ee a, 6, c Cipwile (os GE iw Ong ae 360 Mr CAYLEY, ON THE CENTRO-SURFACE which is of the right order 12; but it would be difficult to obtain thereby the developed equation, 78. For the nodal curve the cubic equation must be satisfied by each root of the quadric equation, or, what is the same thing, the quadric function must completely divide the cubie function; the conditions are an iy G ||) Soy a, b, ©, CNR IA: & where the degrees may be taken to be > > 4, and the order of the nodal curve is thus =24: two of the equations in fact are a, b ==)! b, c = 0, a, b, c a, C, a, b,c aie Mids which are surfaces of the orders 4, 6; or the nodal curve is a complete intersection 4 x 6, By the results above obtained as to the nodal curve, it appears that the two surfaces must have an ordinary contact at each of the 16 umbilicar centres, and a stationary or singular contact at each of the 48 outcrops. 79. The derivation of the centro-surface from the surface requires to be further explained. The surface, say V=0, is a quadric surface depending on the two parameters £ m; the axes coincide in direction with those of the ellipsoid, and their relative magnitudes are as INGE : VB+E: ETE, viz. these are as the axes of the confocal surface 2 2 2 a +E +E +E divided by a, 0, ¢ respectively; to fix the absolute magnitudes observe that the equation may be written 1=0 a+ yf +e8—m— E( 2 + zl tps -1)=0 @+E F+E S+E ; viz. the surface V=0 is a surface through the spheroconie which is the intersection of the confocal surface by the arbitrary sphere a*+y°+2*—m=0; but, while the surface is hereby and by the preceding condition as to the axes completely determined, the geo- metrical significance is anything but clear, OF AN ELLIPSOID. 361 80. Considering then the quadric surface V=0, depending on the parameters £, m; suppose that m remains constant while £ alone varies; we have thus three consecutive surfaces V=0, V’=0, V"=0; and these I say intersect in a point of the centro-surface ; the point in question will depend on the two parameters (£, m), and if these vary simul- taneously we have the whole system of points on the centro-surface; but if only ene of them varies, the other being constant, we have a curve on the centro-surface. The three equations may be replaced by V=0, 6:V=0, 6°V=0; of which the first alone contains m; and it thus appears that if m be the variable parameter, the equations of the curve are 6;V=0, §°V=0, viz. the curve is then the quadriquadric curve which is the concomitant centro-curve of the curve of curvature for the parameter & But if the variable parameter be &, then this is a curve on the 12-thic surface Q=0 obtained by the elimination of & from the equations V=0, 6:V=0; viz. we have Q=S*—7T*=0, where S=(a, b, c)(m, 1)’, T=(a’, b', ¢, d')(m, 1)’, and the curve in question is the curve S=0, T=0, which is the cuspidal curve on the surface Q=0; the elimination of m from the two equations S=0, 7=0 gives as above the equation of the centro-surface. 81. The surface 2 =S*—T°?=0 obtained as above by the elimination of & from the equations V=0, 6:V=0, (or, what is the same thing, by equating to zero the discriminant of V in regard to &) may be termed the sociate-surface: we have then the quartic and sextic surfaces S=0, Z7’=0 intersecting in the before-mentioned curve, which may be called the sociate-edge; and the locus of these sociate-edges is the centro-surface. 82. We may if we please, changing the parameter in one of the functions, consider the two series of surfaces S=0, 7’=0 depending on the parameters m, m’ respectively ; a surface of the first series will correspond to one of the second series when the para- meters are equal, m=m', and we have then a sociate-edge. Taking a point anywhere in space, through this point there pass two surfaces S=0, and three surfaces 7=0; but there is no pair of corresponding surfaces, or sociate-edge. If however the point be taken anywhere on the centro-surface, then there is a pair of corresponding surfaces S=0, 7=0, that is, through each point of the centro-surface there passes a single sociate-edge; and if the point be taken anywhere on the nodal curve of the centro-surface, then there are two pairs of corresponding surfaces; that is, through each point of the nodal curve there are two sociate-edges: this explains the method above made use of for finding the equations of the nodal curve, by giving to the equations S=0, 7'=0, considered as equations in m, two equal roots. 83. The d-posteriori verification that the surfaces V=0, V’=0, V”=O intersect in a point of the centro-surface, is not without interest; the parameters £&,, 7, of the point of intersection are found to be & =&, »,=m—a*—b’—c’—3&; whence in the equation V=0, writing — By’ =(a°+ &)° (a +,) and m=a?+'+04+3,+7,, the resulting equation con- sidered as an equation in & should have three roots €=£€: the fourth root is at once seen to be £=7,, and we ought therefore to have identically Vou. XII. Part L 46 362 Mr CAYLEY, ON THE CENTRO-SURFACE —a (a’ + &)* (+m) a+é — &e. + aby (F-—3&,-— 9, -@ —U-c’) at apy (E = Ee (E a ”,) < (a° — &) (b+ &) (+ &)’ and by decomposing the right-hand side into its component fractions this is at once seen to be true. Third generation of the Centro-surface. Art. Nos. 84 and 85. 84. Instead of the foregoing equation V=0, consider the equation a a B yf 2 2 ) fae ies b? 2 ° ez? AF : = _ — : e (erete+E eset e (erg hag tees The equations d;V=0, d?W=0 contain only & and are in fact identically the same as the equations d V=0, d'V=0; the elimination of & from the equations d;V=0, d?W=0 would therefore lead to the equation of the centro-surface: and the centro-surface is connected with the surfaces W=0, 6:W=0, d?W=0 and the parameters £, 7 in the same way as it is with the surfaces V=0, d:V=0, dV=0 and the parameters & m. That is, if from the equations W=0, d:/7=0 we elimimate € we have a surface Q2=0 (depending upon ») and having a cuspidal curve; and the locus of the cuspidal curve (as m varies) is the centro-surface. But the equation W=0 divides by &—y, and throwing out this factor it becomes a: 2 Be 2 22 ceca t aa et —— —1 (+&)(@t+n) (+E) G+) +8) C+) so that the surface Q=0 is obtained by eliminating € from this equation and the derived equation in regard to £; or, what is the same thing, by equating to zero the discriminant in regard to & of the cubic function (@?+£&)(@+£&)(P+£)- > +)=0. —( ox wae CHOCO: this surface is in fact the torse generated by the normals at the several points of the, curve of curvature belonging to the parameter 7; the cuspidal curve is the edge of regression of this torse, that is, it is the sequential centro-curve of the curve of curvature; and we thus fall back upon the original investigation for the centro-surface. 85. In verification I remark that if X, Y, Z be the coordinates of a point on the curve of curvature in question, and (a, y, 2) current coordinates, then the tangent plane of the torse, or plane through the normal and the tangent of the curve of curvature, has for its equation 3 ie ae ae a+n +n e+ and if in this equation we consider the point (X, Y, Z) to be the point belonging to the parameters (n, &), viz. if we have — Py X*=a*(a’+ &) (a’+”), &ec, then this plane will be always touched by the before-mentioned ellipsoid, 1=0, ax B y 2,2 Cz @+8 @ia * @+h OF CHOC)? 7 OF AN ELLIPSOID. 363 the condition for the contact in fact is 5X? @+H (+n) _, “(a +n)? a = 4} viz. substituting for (X, Y, Z) their values, this is 1 iS tS 2 reas ag ee + &) il which is true. And this being so the ellipsoid and the plane have each the same envelope, viz. this is the torse in question. Reciprocal Surface. Art. No. 86. 86. The centro-surface is the envelope of wae is by? ig Cz CaS rta NSP) (SES) hence the reciprocal surface in regard to the sphere e+yt+e2—h?=0, is the enve- lope of 20; (a’ + &)? ey tee pe Z’—k=0, CPS that is 22 272 272 2 72, 72 2 2 LGD Pee G CX?+ BY? +02? —h + 26 (X?4+ V°+ ZA) +E atate =), viz. the envelope is 7 2 72 2 (xe yy? ave eZ? — k*) c or, expanding and multiplying by a’b’c’, this is a (P—c’)? WOW a. b? (e— a’) axe + om (v— by oye ys nee) ki Ce De + Ca yea +0" Z’) = 0, or, what is the same thing, CH age a igen PAC + cy DAVES =i (b'e? Xe +cea* we de ab? Lig =((), which may be written YZ +X eX +X +e VY? +h'Z*=0, where (a, b, c, f, g, h)=(aa, BB, cy, 2k*be, 2k*ca, 2k*ab), and consequently, af + bg + ch = 2h*abe (a2 +B +4) =0. It would doubtless be interesting to discuss this surface as it here presents itself, and with reference to its geometrical signification as the locus of the pole, in regard to the sphere, of the plane through two intersecting consecutive normals of the ellipsoid : but I abstain from any consideration of the question. 46—2 364 Mr CAYLEY, ON THE CENTRO-SURFACE Delineation of the centro-surface for given numerical values of the semiaxes. Art. Nos. 87 and 88. 87. I eonstrueted on a large scale a drawing of the centro-surface for the values CS) FSU, Fas. (These were chosen so that a, b, ¢ should have approximately the integer values 7, 5, 4, and that a*+c® should be well greater than 26°; they give a good form of surface, though perhaps a better selection might have been made; there is a slight objection to the existence of the relation a’ = 26°, as in the ay-section it brings a cusp of the evolute on the ellipse). We have therefore a=10, 8B=—35, y= 25; the ellipses in the prineipal planes of the centro-surface are 2 2 (* Guara Z 02 (@ssay * 585)» ore 2 =j. (F930" 7 (2) and these determine on each axis the two points which are the cusps of the evolutes, We have moreover for the umbilicar centre «=2°988, y=0, z=1'380, and for the outcrop x=1127, y=1-947, 2=0. 88. For the delineation of the nodal curve (crunodal portion) we have first to find the values of & &; these are given in terms of x, y ante No. 33, where y is a given function of x, and x extends between the values {—6? and —} (a*+6?+c’)} —25 and —263. It was thought sufficient to divide the interval into 6 equal parts, that is, the values of z were taken to be —25, —25°3, ...—26°6. The values of & & being found, those of , 7, were obtained from them by means of the origmal equations (a?+£)*(a’+n)=(a°+€,) (@+7) &e. viz. we have thus for the determination of 7, 7, three simple SCUaUIOnE affording a verification of each other. For the performance of these calculations (viz. of the values of y, & &, 9, »,) I am indebted to the kindness of Mr J. W. L. Glaisher, of ‘Trinity College. The results being obtained it is then easy to calculate as well the co-ordinates (2, y, 2) of the point on the nodal curve as also the co-ordinates (X, Y, Z) and (X,, Y,, Z) of the corresponding two points on the ellipsoid (these last are of eourse not required for the delineation of the nodal curve, but it was interesting to obtain them). The whole series of the results is given in the annexed Table, and from them the drawing was constructed, OF AN ELLIPSOID. 365 I find also in the neighbourhood of the umbilicar centre (if &=— 25 +4), dz= 02868 9’, Sy = + 02484 g°, Sz= 02191 ¢. and in the neighbourhood of the outcrop if £,=—38°333+% a, be@= 1127a, 6y =— 1704 a, 8z = + 4582 oF, x y & é n "; x fam Zee aT WZ |X" Ya we n j 7 T —25 0° — 25° —25- — 25° — 25° 2-988 | 0: 1-380 || 5°326 | 0- 2-070 || 5°326 | 0- 2-070 | ~25-3| 4:2669| — 21-0664 | — 29-6002 | — 38-3193 | — 16-6728 | 2-543 0-360 | 0-996 || 4-394 | 2-289 | 2-491 || 6-233 | 1-957 | 1-023 —25-6| 6°3191 — 19-3475 | - 31-9858 | - 42-9911 ~ 15-4693 | 2-148 | 0-721 | 0-662 || 3-504 | 3-189 | 2-283 | 5-956 | 2-580 | 0-584 | | | \| —26: | 8:1240 | ~ 17°8760| ~ 34-1240 — 45-7879 | — 15-1047 | 1-786 | 1:175 | 0-373 | 2-780 | 3-847 | 1-948 || 5-626 | 3-005 | 0-293 —26-3| 9-8760 |- 16-4573 Aas — 15-0106 | 1-448 | 1-500 | 0-139 | 2-159 | 4-391 | 1-426 || 5-251] 3.346 | 0-098 | — 26-6 | 11-6667 Bos -38:3938] 48-7037 — 15-0000 | 1-127 a '1-610|4-869|0- || 4-829 | 3-651) 0- The calculations were performed before I had obtained the formule in c, which would have given the results more easily. V. On Dr. Wrener’s Model of a cubic surface with 27 real lines; and on the construction of a double-sier. By Prop. Cay.ey. [Read May 15, 1871.] Te I cart to mind that a cubic surface has upon it in general 27 lines which may be all of them real. We may out of the 27 lines (and that in 36 different ways) select 12 lines forming a “ double-sixer,” viz. denoting such a system of lines by Qj5 Ary Az, Ay, Asp ey Bis by, bs, by, Bs, bs 5 then no two lines @ meet each other, nor any two lines 6, but each line a meets each line 8, except that the two lines of a pair (a,, b,), (az, bz), ... (ds, 6s) do not meet each other. And such a system of twelve lines leads at once to the remaining fifteen lines; viz. we have a line €,, the intersection of the planes which contain the pairs of lines (a, 6,) and (a,, 6,) respec- tively. The model is formed of plaster, and is contained within a cube, the edge of which is = 18-2 inches: the lines a, 6, ¢ are colored blue, yellow, and red respectively; the lines a, 6,, b; being at right angles to each other, in such wise that taking the origin at the centre of the cube, the axes parallel to the edges, and the unit of length = 1°6 inches, the equations of these three lines are a, vz=0,y= 0, Des w=0, z= 1, bs, y=0, s=-1 The model is a solid figure bounded by portions of the faces of the cube, and by a portion of the cubic surface, being a surface with three apertures, the collocation of which is not easily explained. To determine the construction I measured on the faces of the cube, the coordinates of the two extremities of each of the twelve lines; these were measured in tenths of an inch ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, &e. 367 (taking account of the half division, or twentieth of an inch), and the resulting numbers divided by 16 to reduce them to the before-mentioned unit of 1°6 inches. These reduced values are shewn in the table: knowing then the coordinates of two points on each line, the equations of the several lines became calculable; the true theoretical form of these results—(viz. the form which, but for errors of the model, or of the measurement, they would have assumed)—is b, v©=Bx+D, y=B/z2+D, be, a2=0; aS i b,, w= B(s+P;), y=B; @+8,), bis c= B, (s+), y=B, (e+ Bp), bs, i=10; s=-1, Des 2= B(z+:), y= By (zs + B). Qh; a2=0, y = 0; As, e=4As4+0, y=A,'(z—-1), Qs, w= A,(x¥+1), y=A,; (e—-1), a, @=A,(e +1), y=A, (z—-1), as x= A, (# + 1), Net Oe ss x=A,(x+1), y=, (s—-1); but in consequence of such errors, the results are not accurately of the form in question The faces of the cube being as in the diagram A the Table is 368 PROF. CAYLEY, Equations enleniated aa ABCD EFGH AEBF BCFG CDGH roe nea | | l= — 5688) x = +5°688 y = +5°688 jw = — 5°688| y = — 5-688 a2=0 “r=0 2=0 y=0 a, |\y=0 y=0 w= — "7808 — 187 | | |w=4'250 v= — 4625 y = — ‘4232 + °406 2 )y=2°812 |y=—2°000 w= —654x— 656 | | jw= 3-062 |w=— 4375 = — *588z + °531 Sly=3'875 |y=—2°812 @ = ~2°912% — 2°959 | y = 0625 ay Spee 4p) o7bal| Mallee eeeeeel| eesoseaes Wiese Me eoare te || nets v= 1°024% 4+ 1°014 v= — 4812 = —1°049x — °277| % a OpGanl eeteg| eres ieee een ere see = ‘2642 + ‘187 v@=—1°313\e= 1:687 = — 1042 4°219 | “6 jy="8125 |y=—-375 w@=—1'611lz 4°151 b y= 5:°500 y = — 1'438x + °288 L | teeeeeeee sev onieiai) | eae var? 9 Mss on Gag aseeseanas ——— oe v2=0 b vx=0 v=0 zZ=-1 2 eocceetee eeecescee Sel eoerescee ——_ w= — 1°352x — 685 | , @= $°750 ve — 3812 y= = 2°034% — ‘984 3 wee eeeeee Bee eet ene z= —3°281 weer eee eS 2313 w= — "753% — “0315 | » |w=4250 |w= — 4-318 y = —'500x —°0315 | * |y=2°812 |y=—2°875 v=4+1 5 . | wee eee lg=1 eee eee w= 1'°123% —*702 b v= —5'688 y = — 11282 + "702 | Fa] certeters | ceeeeeees [2 qeggg| coetettee | ceeeseees 1 AEDH 2 = + 5.688 Y= 2937 s= — 2-969 y=—5-063 Z= 4562 = 3$°437 y=0 S=1 y=— 5688 z= 5°688 I hence calculate the intersections: considering any two lines which ought to intersect, then projecting on the horizontal plane and calculating «#, y the coordinates of the point of intersection of the two projections, these values of xv, y substituted in the equations should give the same value of z; but if the lines do not accurately intersect, then the values of z will be different, a as ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, 369 b, b, bs b bs b, ) - 0 Sil —*495 £°011 + *625 —‘077 +°381 —°967 — °398 +°455 ‘i +°771 —008+="008 | +°227 —°129 +013 —°796 +*067 +1 +346 £°076 — 423 —*001 = '001 —1°310 — ‘673 +699 +1:119 * —028 "028 | +°344 —'264+'021 | -1 +6°350+:042 | +1 +°162 £146 | —-957 — 023 ='023 | +1-286 —5°871 —1°330 + 1°238 +1°488 +1°736 +°008+:008 | +°847 —°674£°014 | -1 — 1°398 £:060 +1 —°3444°215 +2519 —"005+'005 | +'282 +°412 +18°764 —1°801 +'772 +°476 +7194 7 — 14155 +1°462 £:008 | -1 —°717£'001 | —:518+*070 +15'282 £4°052 | +194 —"038=:038 | +°045 —"131 +451 +214 + °323 +283 +284 | +°057 +057 * +°040 £:022 | —1 —"582£°042 | — +423 4-208 oil Starting from the assumed equations of b,, b;, b,, 6; b;, a, and calculating by the theory the remaining lines, the equations of the 6 lines (those of 6, being calculated) are f= 15321 = — 310, y = — 1295 = +581, v=0, s=-1, @ = — 1°352 (x + °510), y = — 2034 (x + °510), v= — "753 (z + °052), y = — *500 (% + *052), y=0, s=+1, @= 1123 (x — °624), y = — 11123 (x — 624); and the equations of the a-lines (those of all but a, being calculated) are Ay Wow, SOUL ie dh v=0, y=0, 2= —‘7532 — "091, y = — ‘498 (x — 1), a = — ‘609 (z+ 1), y = — ‘677 (= - 1), 370 PROF. CAYLEY, a, = — 2506 (z + 1), =— ‘841(%—-1), As5 z= "874 (s +1), Yy = — ‘967 = — °288, Uss @= °170(z% +1), y = — ‘071 (= —1); and thence for the points of intersection the coordinates are b, b, bs b, bs bs | 0 0 0 0 0 a, - ny) ty) 0 ty) 0) =i — +510 — 052 il + *624 —"170 + 662 +197). | — "844 — °336 § x ‘906 , (ae. lines ay, bs 2 as + 446 2 +131 nearly parallel. 0 + °336 + °105 —1 — 262° +1 + °325 = "515 0 — 1°805 ; — 1:218 — ‘641 a3{ +4 °782 + 1°354 = — 2:007 0 — 641 —*155 -1 + 3°964 +1 + '053 — 1°071 0 + 1°438 — 5°012 — 1'259 Qs} + 1:323 + 1°682 + 2164 ey 0 + 1:259 = Ge -1 — 1°574 +1 — °497 + 3°189 0 + °259 + °383 + 6410 a,} —2°849 + ‘679 + 291 + °255 4 — 6°410 + 2'649 = — ‘702 — 561 + 6383 + 241 0 — ‘074 +131 + °340 Gs) + 041 + 142 +112 +- °087 0 22 + °417 =i — °565 — '226 261 II. I have in a paper “On the double-sixers of a cubic surface,” Quart. Math. Journal, t. X. (1868), pp. 58-71, obtained analytical expressions for the twelve lines of a double-sixer, and also calculated numerical values, which however (as there remarked) did not come out convenient ones for the construction of a figure. A different mode of treatment since occurred to me, by means of the following equation of the cubic surface @ Y & WwW) (ax y *) 2 yY S$ Wy fas yw —~-st—-s)(—-® ~b (24 54" -5) ( - Fe) =o (; Y y) Ss Be (. B y 9o/ \a'y ps ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, &c. 371 which as will appear is a very convenient one for the purpose; we in fact obtain at once eight lines of the double-sixer ; viz. these are 1. w=0, w=0, 2. #2=0, y=0, 2 97SQ; P=: 4, #=0, w=0, fn Ss ow eS se) yY 8 5 =--—-==0, --—-~=0 5. 3z-yvy=90, =-— =0 a B "oy a 3, a ay Bs y ’ x ze w 5 ty OD Pa 6. ay ENO: haa) = 0, 6. - =——~=0, Ut =o) mi 6) y 38 a o Bo¥ and also five lines not belonging to the double-sixer, viz. eee (BS Da (fed Ngee a ft y / ay a ay oye sew ED) 56. >= SS —--=+=0 —_- — w= Oh a Bryne a Boy 8 The remaining lines of the double-sixer are then easily determined ; viz. the lines 3, 5, 6, and 12 are met by the line 2’, and by a second line 1’; this, as a line meeting 3, 5, 6, will be given by equations of the form a ee ({2-«), 2-5 y= 6(Se-~), a and observing that these equations, writing therein x = 0, give , w a xs w z Mos me: = a DAC oe ae aaa the condition of intersection with the line 12 gives which is the value of @ in the foregoing equations: and to these we may join the resulting equation yyy (aB' - a’B) = xB’ (yo! - 79). 372 PROF. CAYLEY, Proceeding in like manner for the lines 3', 2, 4, the equations for the remaining four lines of the double-sixer are a — ka : a —ka O7 SER = Po 8 = Bae @-We= (y--"), v-95= (2 -w), e-ws=o(y 2). t-vg=6(=5-~), wry’ (ad — ad) = zed’ (By' — B'y). yyy (a8 — aB) = 2@BB (yd - 79). n yeas a. o-- 33, o(e5-w) 9% -» o (eo -y)=2-w%, o(ct-w) 13» #(eB-a) end 2QBB' (ad —a’d) = yaa’ (By' — B’y). apo (aB’ - a’) = waa’ (yo' — y'8). It may be added that In plane wv = 0, intersection of 1' lies on line z : w = (af —a'B) yy’: ayBs - a’y’ Be, 2 35 y:% =ayBd -ay Bo: (ad — a’) YY> and that line joining these intersections is line 12. > In plane y = 0, : intersection of 2 lies on line w : w = ayBd — a'y'Bd = - (By' — By) 80, ‘* 3 b z:w=ayBd —ay'Bo: (a8 — a’) ds, and that line joining these intersections is line 23. In plane z = 0, intersection of 3’ lies on line a: y = (yo — y'0) aa’: ay3'S - a’y’ Bd, my $y w= — (Br - Bly) aa’: ary B'S’ - a'y'Bd, and that line joining these intersections is line 34, And in plane w =0, intersection of 4 is on line y: x = (a0 - a’) BB’: ays’ - a’y/Bd, fe Voy wry — arid) - a’y'Bd: (5 — y'd) BB’ and that line joining these intersections is line 14, ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, &c. 3/0 The equations of the remaining ten lines of the surface may be obtained without difficulty, and also the forty-five triple planes, but I do not stop to effect one the Baur v=0, y=0, 2=0, w=0, are, it is clear, triple planes, containing the lines 1, 2’, 12; 2", 3, 23; 3, 4’, 34; and 4’, 1, 41 respectively. If, to fix the ideas, the planes # = 0, y = 0, z =0, w= 0 are taken to be those of the tetrahe- dron ABCD (w = BCD &c., as usual), then the edges AB, BC, CD, DA (but not the remain- ing opposite edges AC, BD) will be lines on the surface. Each plane of the tetrahedron, for instance ABC (w=0), is met by the ten lines not contained therein in two vertices “As Gs three points on the edge BA, three points on the edge BC, and two other points, viz., these are the intersections of the plane ABC by the lines 4 and 1’. For the construction of a model it is sufficient to determine the three points on each edge, and the two points say in the plane ABC, and in the plane DRC («x = 0) respectively, for then each of the remaining eight lines will be determined as a line joining two points in these two planes respectively. If in the first instance / is considered as a variable parameter, then the two points in the plane w = 0 are given as the intersections of two fixed lines by a variable line (14) rotating round the fixed point me ee: 0, Bee + =0; and the like as regards the two points in the plane « = 0. a cu ay a B By making with assumed ane of the other parameters the proper drawings for the two planes w= 0, #=0, it is easy to fix upon a convenient value of the parameter k; and I have in this manner succeeded in making a string model of the double-sixer; viz., the coordinates w, y, x, w were taken to be as the perpendicular distances of the current point from the faces of a regular tetrahedron (the coordinates being positive for an interior point); the values of a, B, y, 6 were put = 3, 4, 5, 6 and those of a’, B, ry’, & =1, 1, 1,1; the value of & fixed upon as above was k = —1; this however brings the lines 2 and 4 too close together (viz., the shortest distance between tent is not great enough), and also their apparent inter- section too close to their intersections with the line 6’; and it is probable that a slightly different value of k would be better. 374 PROF. CAYLEY, The results just obtained may be exhibited in a compendious form as follows: l’ and 2 meet BCD on line ( = Zand 3 4, CDA 3’ and 4 ,, DAB 4 and 1’ ABC ” ” + or calculating the numerical values from the foregoing assumed data, | xz :y ae 10 | 1 is line BC 0 | 0 Fly tae br 9 0 0 | | ea I 0 Ate ig. A 0 0 5 meets CD if | 8 5 Aas, a B Barght SOD 7 | FY 5) ALB: a’ B iGiaees BC B Y ” AD a 8 (ist ke B’ 7 | cm At) a’ a [et gy Pp 8-8 » BCD —(a’— ka) (y8'— '8) BB’ | (© =) (a 8" — a’B) yy’ | (8-48) (ayB'S'—a'y B8) » ABC | (a-ha) (ayB’S'—a'yB8)| (a — ka) (y8’ — 8) BB’ | — (8'— k8) (a SAR “ky eD TB | ep tyyteye’8-a/68)| (y/— ty) (of —0'B) 88 | 4) ACD | —(B'—kB) (y8'—y'8)aa’ ¥—ky)(ayB'S— a’y’B8)| (y' — Fy) (aB’ — a! » ABD|(p'-kp) Gi — 8) aa’ | (B- — a’y'B8) — (7 — Ay) (@B’— a’) 85 Sch tA rae kB » BCD (B'- ipo d—a’yB8) Ce at Ve '8) 77’, | (a — ka) (By — B’y) 88" | 4 ACD | (a'—ka) (ayf'8'-a'y 8) KB) (a®’ —a'8) yy/ | —(a’ — ka)(fy'— By) 38 A. « OD | y¥—k y_kS | » ABD|—(8'—%8) (By — B’y)aa’ | — (y/—ky)(a8'—a'3) BB") fey ager | O48) (@yB'8'—a'y’ BB) » ABC | (8 —k8) (By — B’y) aa’ | (7 — ky) (a8! — 8) BB’ | (y'—hy)(a7B'S—a'y'B8) 1 ij ace Nl m—*(- 442 _2) 75 =0, Bs Boy 8/p% = -%(2 .+2_5) ert ay a oy 8 ay LF /e DV au a-*(G-B --3)ea7° a -k(e- 842 ) =O, ay a B Y ay ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, ke. 375 Le - : : x :y 28 :w say 1 is line BC (0) 0 Beer OD 0 ty) She's cay. ad 0 0 4h 8.) AB 0 0 | 5 meets CD 5 6 2=45°5, w= 545, ae ae 82} 3 4 = 42°90, y=57'1. 6 meets CD 1 1 2=50, w= 50. nae 8} 1 1 e=50, y= 50. 6 meets BC 4 5 y = 444, 2 = 55°6. were (a AOD 3 6 @ = 33°3, w = 66-7. 5’ meets BC 1 1 y=50, 2=50. a AD 1 1 ®=50, w=50., 1’ meets AD 11 14. w=44, 2= 56, ana Jat d) — 44 70 126 y =—25'1, 2 = 39'9, w = 71°'8. Son WEG: 99 44 — 70 Not required. 3’ meets BC 12 13 y=48, 2 = 52, Bee eA CD 80 117 78 Not required. nee ABD 36 108 — 78 @=47°2, y=141°7, w= — 102'S. | 2 meets AB 11 12 V=47'8, y = 52:2. eo BCD 100 180 66 y = 264, 7 = 44, w = 162, Bee PACD — 199 189 66 Not required. 4 meets CD 13 14 z= 481, w= 51°9. eB 42 156 —126 | #=50°5, y=187°6, w=—151'5. aso ABC 42 156 117 Not required. 1’ and 2 meet BCD on line 35y — 32z + 30w = 0, 2and3 ... CDA ...... 26” + 222 -21w =0, and4 ... DAB ...... 8v— Ty— 6w=0, andl... ABC ...... 520 — 47y — 44% tl S which last four equations serve as a verification. The outside numerical values are given in the manner most convenient for the con- struction of a drawing; viz. when the coordinates refer to a point on an edge of the tetrahedron, or say on the side of an equilateral triangle, then taking the length of this edge (or side) to be = 100, the numerical values are fixed so that the sum of the two coordinates may be = 100, and the two co-ordinates thus denote the distances from the extremities of the edge or side: but when the three co-ordinates belong to a point in 376 PROF. CAYLEY, the face of the tetrahedron, or say in the plane of an equilateral triangle, then the sum of the coordinates is made = 86°6, and the three coordinates thus denote the perpendi- cular distances from the sides of the triangle. if) fe It is possible to find on a cubic curve a double-sixer of points 1, 2, 3, 4, 5, 6 and 1, 2’, 3’, 4, 5, 6 such that any six points such as 1, 2, 3, 4’, 5’, 6’ lie in a conic. In fact con- sidering a cubic surface having upon it the double-sixer of lines 1,2,3,4,5,6 and 1’, 2’, 3’, 4’, 5’, 6, the section by any plane is a cubic curve meeting the lines, say in the points 1, 2, 3, 4, 5, 6, 1’, 2’, 3, 4, 5’, 6: each of the lines 1, 2, 3 meets each of the lines 4’, 5’, 6’, and consequently the six lines lie in a quadric surface: therefore the points 1, 2, 3, 4’, 5’, 6 lie in a conic: and so in the other cases; the number of the conics is of course = 60. The cubic curve may be a given curve, and six of the points upon it (not being points on a conic) may also be taken to be given; for instance the points 1, 2, 3, 1° 4’, 5’. For take through the points 2, 3 respectively any two lines 1, 2; through 1’, 4’, 5’ respectively the lines 1’, 4, 5’ each meeting each of the lines 2, 3: and through 1 a line meeting each of the lines 4’, 5°. It is easy to see that a cubic surface may be drawn through the cubic curve and the lines 1, 2, 3, 1’, 4, 5: for the passage through the cubic curve is 9 conditions; the surface then passes through the point 2 and to make it pass through the line 2 is 3 conditions; simi- larly the surface passes through the point 3, and to make it pass through the line 3 is 8 conditions. The surface now passes through 1’ and through the points of intersection of the line 1’ with the lines 2, 3: to make it pass through the line 1’ is 1 condition; similarly to make it pass through the lines 4’, 5’, 1 is in each case 1 condition; or there are in all 19 conditions, so that the cubic surface is completely determined. Take now through the points 1, 2, 3, 4’, 5’, a conic meeting the cubic in the point 6’: then through the lines 1, 2, 3, 4, 5’ we have a quadric surface passing through this conic, and therefore through 6: hence through 6’ we may draw a line 6 meeting each of the lines 1, 2, 3; and since the cubic surface passes through the point 6’ and also through the intersections of the line 6’ with the lines 1, 2, 3, it passes through the line 6’. We complete in this manner by constructions in the plane of the cubic the system of the twelve points, viz., each new point is given as the intersection of the cubic curve by a conic drawn through five points of the cubic curve; and it is then shown as for the point 6’ and the line 6’ through it, that through each new point there can be drawn a line denoted by the same number and meeting each of the lines which it ought to meet, and hence lying on the cubic surface: the twelve points are thus the intersections of the plane of the cubic curve by the twelve lines of the double-sixer; and it follows that the six points which ought to lie in a conic (in every case where such conic has not been used in the plane construction) do actually lie in a conic, I was anxious to construct such a double-sixer of pounds on a cubic curve; for this 2 r av él wv v purpose I take the equation of the curve to be y° = (1 - “) (2 - *) (: - “) » or say for a c shortness y” = 3 where, to fix the ideas, a, b are supposed to be positive, @ greater than 06; and ¢ to be negative. ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, &c. 377 The cubic curve is thus a parabola symmetrical in regard to the axis of #, and consisting of a loop and infinite branch; and I take upon it the points 1, 2, 3, 1’, 4, 5’ as shown in the Y figure, viz., the coordinates of these points are as stated in the Table, where m is the « coordi- == m = nate, and /M= J (1 - ") (1 - a) (: ") and so in other cases, \/14 = 3°74165. @ 7] x y 1 m / 6 V/ 14 = 3742 Q 0 1 0 1 8 0 —1 0 f =i ws 75 2280 ,— 6 Qa = 1945 SS T/A atl Q Vv 1369 ane e = 7 OM : —1+4+ — = — 0°792 ee a 5 p Vo i 79 ag” a = “606 a 13 QS 6 m, —/M, ane 4°333 ~ ov it = — 1°641 7 ona 2 JM 6 - / ihe — 3°742 a 1560 (14 — 4/14) 9! 3 a ee ee 1°299 => ‘676 x G Vv (814/14 — 5)? ca = 1560 (14 + 4/14 ava 3 T JT ee Z ue 1887 ena (814/14 + 5)? 4 c (0) | ° 5 b 0 2 0 = 18 A ar 6 om, VM, 3 = £933 5 V1 = 1641. - wv a The numerical values belong to the curve y* = (: > | (: - a (: + x) and to m= 6, ~ Won, S006 Teme Ie 48 378 PROF. CAYLEY, Starting with the points 1, 2, 3, 1’, 4’, 5’ we have to find the remaining points 6, 6, 4, 5, 24 8% Point 6’ by means of the conic 1234’5'6. The equation of the conic is (a — 6) (w—c) —be yw +k ay =0, (2, 3, 4, 5), and making this pass through the point 1 (w =m, y = VM) we find (m—b) (m-c)+ka/M=0. (1). Hence taking the coordinates of 6 to be m, / M, we have (m,—6) (m,—c)+ka \/M,=0, (6), and thence VATE fx (m, — b) (m,—ce) pel M, (m—a) /M (m —b) (m —c) M (m,— @)” that is VM, _ (m—b) (m—0) _ m4 /M (m—b)(m—c) m—a’ we have thus for m, a quadric equation satisfied by m=m,, so that throwing out the factor m —m,, the equation is a linear one, viz., we find ma — ab—ac + be Mm, = > m—a or what is the same thing m—a and thence also (a—b) (a—c) (m —a)* VM,= VAUE viz., \/ M, is determined rationally in terms of m, / M;; this is of course as it should be, since the point 6’ is uniquely determinate. Point 6 by means of the conic 2361'4'5. In precisely the same manner the coordinates are m,—V/ M,, where m,, VM, denote as before. Point 4 by means of the conic 2341'5'6'. The equation of the conic is 1-y Fo+Gy+H= y : 2, 3), @ ON Dr. WIENER’S MODEL OF A CUBIC SURFACE, &c. where 1 Fb +H= p (5’) she 1—M. Fm,+ G/M, +'H = = 1 (6’) 1 —. 1— Y Fm-G/ M+ =1— "| 1’) which give without difficulty where P = 2a—c— 2 (a—c) (6-0) abe F=-a—-c+P, /M abe G = (m — b) (-m +P), abe H = ab+ac+be — bP, m+ mM, — 2c of m only. And then or say that is feGs/64 He RiGee Gk 6 Se OL ab 1 1 Ly ay ee ac) ( ) (St abe }’ (@ — 6) (abe F+tatec-6)+ Gabe /0 =0, viz., that is (9 —b) (P= 6) + (m= 1) 2 (P= m) =0 or, rationalising and throwing out the factor @ — 5, this is (9 — b) (0 - Py’ — (m - b) (m- PY? (24) Cle, “(m—a)(m—b) ” 379 , a quantity which will presently be expressed in terms which is a cubic equation satisfied by 9=m and 9=m,; so that throwing out the factors @ — m, @—™m, we have for @ a linear equation. Putting for shortness {4 =(m— a) - (a - b) (a-e), 4 B =(m —b)? - (6 -c) (b - a), | = (m-e)* — (¢ -a) (c — b), 48—e2 380 PROF, CAYLEY, the value of @ may be expressed in the forms 3 2 2 = =— — = =_ g= a= a(e-0), @-b=A(¢-0),0-0n VO IDE 9 0-9-9 We have moreover 9 — — — — es ORI 7 Ae Ci equations which express P in terms of m only; also —2(a-—c)(m—b) (m-c)B C > §-P= and then ~ —@-—b P-—@ Ue ee ia whence M0 =2 / M (b—c) (e—a) a: so that 6, »/@ are now determined. Point 5 by means of the conic 2351'4'6’, The conic is 12 Fo+Gy+H="—", (2, 3) where 1 Fe tiles, (4) ib Fm,+G/M,+H = oan (6) 1 Fm —G /i+H=———, (/). Every thing is the same as for the point 4 except that 6, c are interchanged: hence writing Q instead of P, and using A, B, C to denote as before, we have abe F = —a—b+ Q, VM abe G = (m—c) (—m+ Q), abe H = ab + ac + be—cQ, and er dies (m—b) (m—c) (e—b) (a—54) B ‘ guee tn) ON Dr. WIENER’'S MODEL OF A CUBIC SURFACE, &c. _2 (m—b) (m—c) (a—b) Q—b B = m= os , _ 2 (m—b) (m—c) C (a—b) @-Q=—- B: > and AC /db= 2/ M (c—d) Cae) se which determine @, Jo. Point 3’ by means of conic 1263'4'5’. Equation of conic is v—b) («— Fo+Gy+H= aa’ =) 5 (4, 5’) and we have G+ H=be, (2) can (m—b) (m—c) F i = m+ G/M + H /M ’ a2 (m, — b) (m,— ce) Fm,— G/M, + H= Maree (6). Eliminating F’, we have m, (m—b) (m—c) m (m, — 6) (m, —e) G (m, /M+m/M,) +H (m—m,) = JM + Wap which is easily reduced first to (m — a) (m — b) VM 2mm, —a(m +m) iy (m — a) \/M ee em) (m—«) A and then to § - = = 2) +abe§-A+2 - = G (aA + 2m (a—b) (a c)} H VAT a abe 4 2m (m a)} = (0) and combining herewith G + H = be, we have = abe m[a (m—a) + (a—b) (a-c)] - i (m-a) A’ A +2m (a — b) (a—c) + ———— ad 42m (a~0) (0-0) + G=be-H; 382 PROF. CAYLEY, and we have then F (m +m) + G/M —/M) + 2H=0, that is A\/M F {2m (m-a) — A} + GAM 2 H(m—a)=0, or what is the same thing Tal Oe We then have Fre +Heay (- ge (w@- = 9) ~ (aGite we). 2) y\ ri—a eG—a ; that is (Fa + H)’? = - ee (Ha + Ga)’, or abe (a@ — abe (w — a) (Fx + H)' + (a — 5b) (w —c) (Gx + Hay = 0, or, developing and throwing out the factor x, this is G x + + {2a GH—(b +) G? + abe F°} a + fa H* - 2a (b +c) GH +be G + abe @FH—aF")} @ +§—(b+c) a H* + 2abe GH + abe (H*—2aFH)} = 0. This must be satisfied by # =m, #=m,; hence the left hand must be = G?(w — m)(w — m,)(w—o), or equating the constant terms we have G? mm, o = aH { —2abe F + 2be G + (be—ab—ac) H}, which gives ¢; and we then have /i=- o—a Go + Ha but I have not attempted the further reduction of these expressions. (Fo ot ff), The numerical values for the example are froaal cia 2 = gp Pa Tit 2 V4 oes 10 + 62/14 Pla 104 14 5 +21 4/14’ 5+ 21 4/14” 5+ 14 whence o as in the Table, ON Dr. WIENER’'S MODEL OF A CUBIC SURFACE, &c. 383 Point 2’ by means of conic 1362'4'5’, The equation of the conic is w—b) («#— Fu+ Gy+H= aos (4°, 5') where —-G+H= —be, (3) = (m—b) (m — e) i =; 1 Fm+G/M +H ar (1) as (m, — b) (m, — ©) Fm, —G /m,+H=—~—— 7? (6) which are the same as for point 3’, if only we reverse the signs of F, H and WM, / M,- Hence the formule are wa — 20cm [a (m — a) + (a — 6) (a -c)| A 42m (a - b) (@-c) (ar aA +2m (a- -c) - —=— VM G= be + H, A\/M M F {2m (m - a) - At = —be ee - {2 (m— a) 4 4V MN, m—a m—a G mm, 7 = aH § - 2abe F — 2be G + (be — ab - ac) Ht, which, gives 7, and then JF a Qonan (Pr +B, which are also unreduced. The numerical values are gp - 0 + eV 4 _ = 10 - 62/14 — 104\/14 BOSS a1 / 1h oe Ws eh / 14 4 5S aa whence + as in the Table. VI. Tables of the first 250 Bernoulli’s Numbers (to nine figures) and their logarithms (to ten figures). By J. W. L. Guatsuzr, BA. F.RAS, Fellow of Trinity College, Cambridge. [Read May 29, 1871.] Tue only table of the logarithms of Bernoulli's Numbers that has hitherto been calcu- lated with which the author is acquainted, is that given in Grunert’s supplement to Kliigel’s Worterbuch (Article, Bernoullische Zahlen) ; it contains the logarithms of the first eighteen numbers, This table, which is stated to be copied from Eytelwein’s Higher Analysis, is also reprinted in the Penny (and English) Cyclopedia. The present table which contains the logarithms of the first 250 Bernoulli’s Numbers to ten decimal places was constructed as follows : The first seven logarithms were found by merely taking out the logarithms of the corre- sponding numbers; the rest were calculated by means of the formula* Die. 27) 1 1 ; B, = (Qa)™ (1+ 545+), \ “x from which we obtain u log B, = log 2 + log 1 + log 2... + log (2m) — 2n log (27) +m (= en ) ; (where » is the modulus *43429448...), the square and higher powers of being insensible to 10 places, when n is greater than 7. Several of the logarithms were calculated by taking out the logarithms of the number as well as by this method. A table of the values of log (1.2...7) as far as v = 1200 to 18 decimal places, was published by C, F’. Degen, at Copenhagen, in 1824; this has been used in the construction There is some confusion in the notation for Bernoulli's | sponds to B,,_, in the other. The former system is here Numbers; they are sometimes written B,, B,, By, &c., and | adopted as being obviously the better. sometimes B,, B;, L;, &c. B, in the one system corre- MR GLAISHER’S TABLES OF THE FIRST 250 BERNOULLI’S NUMBERS, &. 385 of the accompanying table, but every number quoted has been verified, to guard against any misprint of Degen’s. The whole work has been performed in duplicate, and as the calculation was not of a difficult nature, there is a high probability that the table is free from error. It would have been easy to give more decimal places, but as there exists no printed table of logarithms of natural numbers to more than ten figures, it seemed useless to calculate the present logarithms farther; the work was however extended to fifteen places, so that the last figure (which has been corrected) is in all cases accurate. The table in the Wérterbuch was found to be very inaccurate; the following is the list of errors*. In B, 1.4033154004 should be 1.4033154003, B »» Bz 0.8507783387 u 0.8507783327, »> By 4.9574188514 _ 4,9374188511, »» By 7.4361345055 a 7.4361345056, »» Bis 8.7792940212 FS 8.7792940203, B,, 11.6330790754 ee 11.6330790755, »» Big 13.1370898829 a 13.1370898839. For values of n greater than 250, we can replace 1 1 1.2...2n by 2°+1, /(an) n™ e-™ (2 te eee sent) y (a7) 24n 1152n?) ’ so that 1 1 log B, = log 4 + (20 +) log n — (2n -;) log 7 — 2un + = the term F s being cancelled by an equal term of opposite sign in the expansion. 152n° Bernoulli's Numbers have been calculated as far as B,,, the first 15 by Euler and the rest by Rothe; they are all given in Crelle’s Journal, t. xx. p. 11. These as far as B,;, have been verified by two calculations of Euler’s constant by means of them}. An error should be mentioned in Euler’s original calculation of B,,, which is given (Acta Petropol. pt. u. for 8553103 , 8553103 < : : 1781, p. 46) as rae instead of ——. This error is reproduced in the Penny Cyclo- pedia (Article, Numbers of Bernoulli) and probably elsewhere. * All these errors occur also in Eytelwein’s Grundlehren | Euler’s error in B,,, alluded to further on, is not reproduced der hihern Analysis, Berlin, 1824 (t. 1. p. 488); so that they | by Eytelwein. were not introduced by inaccurate copying on Grunert’s part- | + Proceedings of the Royal Society, No.129, 1871. Vout. XII. Parr I. 49 386 MR GLAISHER’S TABLES OF THE TABLE OF THE LOGARITHMS OF BERNOULLI’S NUMBERS. 1 1:22184 87496 2 252287 87453 3 237675 07096 | 4 2-52287 87453 5 2:87942 60688 6 1-40331 54003 7 0:06694 67896 8 0°85077 83327 9 1:74013 50433 10 2-72355 76597 1l 3:79183 95878 12 493741 88511 13 615397 24516 14 743613 45056 15 877929 40203 16 | 10:17944 59554 17 | 11-63307 90755 18 | 13:13708 98839 19 | 14-68871 54679 20 | 16:28548 03295 21 | 17-92515 37399 22 | 1960571 51352 23 | 21-32532 57440 24 | 23-08230 51026 25 | 24-87511 14502 26 | 26-70232 52332 27 | 28-56263 51260 28 | 30°45482 61057 29 | 32°37776 92183 30 | 34:33041 27436 31 | 36°31177 45314 32 | 38-32093 53181 33 | 40-35703 28735 34 | 42-41995 68522 35 | 4450684 42463 36 | 4661907 53547 37 | 48°75527 01978 38 | 50:91478 53168 39 | 53-09701 09079 40 | 5530136 82495 41 | 5752730 73841 42 | 5977430 50258 43 | 6204186 26660 44 | 6432950 48541 66:63677 76334 log B, 68-96324 71:30849 73°67213 76:05377 7845304 80-86960 83°30311 85°75325 88°21970 90-70216 93°20034 95°71396 98-24276 100°78647 103°34483 10591762 108°50459 11110551 113-72016 11634834 118-98982 121-64442 124:31193 126:99217 129-68495 132°39010 135°10744 137°83680 140:57802 143°33094 146:09540 148°87125 151°65835 154°45654 157°26570 160:08568 162:91636 165°75759 168-60925 171°47123 174°34339 177°22563 180711782 183°01985 185°93162 71164 |, 81818 32834 13567 68146 86234 94507 48783 26748 21211 33859 69440 30368 11692 || 96399 || 51042 21641 29855 69394 02653 57554 24580 53982 53150 84142 61345 49274 60487 53621 31529 39514 63663 29261 99288 72990 84529 01686 24642 84803 43696 91902 48049 57847 93166 51160 log B, 18885301 191-78392 194°72424 197-67388 200-63274 203°60071 206-57771 209°56364 212°55842 215°56194 218-57413 221:59489 224°62416 227°66183 230°70784 233°76209 236°82453 239°89506 242°97362 246-06012 249715451 252°25670 255-36663 25848423 261:60944 264:74218 267°88239 271:03001 27418498 277 °34723 280-51670 28369333 286°87708 290-06786 293°26564 29647036 299-68195 302:90037 306712557 309°35748 312°59606 315°84126 319°09303 322°35132 325°61607 53421 45182 94541 91731 48416 97008 90030 99490 16287 49641 26542 91229 04676 44113 02554 88349 24750 49493 14401 84990 40104 71551 83756 93431 29248 31528 51945 53231 08894 02954 29672 93304 07852 96825 93016 38271 83282 87373 18298 52052 72671 72058 49796 12983 FIRST 250 BERNOULLIS NUMBERS AND THEIR LOGARITHMS. n log B f—] 136 | 328:88725 137 | 33216480 138 | 33544869 139 | 338-73885 140 | 342:03525 141 | 345-33785 142 | 348-64659 143 | 351-96145 144 | 355-28236 145 | 358-60930 146 | 361-94221 147 | 365-28107 148 | 36862582 149 | 371-97643 150 | 375-33287 151 | 37869508 152 | 382-06304 153 | 385-43670 154 | 38881603 155 | 392-20099 156 | 395-59154 157 | 39898766 158 | 402-38930 159 | 405-79642 160 | 409:20900 161 | 412-62701 162 | 416-05039 163 | 419-47913 164 | 422-91320 165 | 426-35255 166 | 429-79715 167 | 433-24698 168 | 436-70201 169 | 440-16219 170 | 443-62751 171 | 447-:09793 172 | 45057343 173 | 454:05396 174 | 457°53951 175 | 461-03005 log B 5 1464-52554 468-02595 471-53127 47504146 478-55650 482-07636 485-60101 489-13042 492-66457 496-20344 499-74700 503-29523 506-84808 510-40555 513-96762 517-53424 521-10541 524-68110 528-26129 53184594 535-43504 539-02857 542-62650 546-22882 549°83549 553-44650 557-06183 560-68145 56430535 567-93349 57156587 575-20246 578-84324 582-48820 586-13730 589-79054 593-44788 597-10932 600-7483 604-44440 n n log B. 03298 | 216 | 608:11800 48903 85605 || 217 | 611-79562 27795 71748 || 218 | 615:-47723 87890 86808 || 219 | 619-16283 45997 58935 || 220 | 622°85239 20597 19292 || 221 | 626:54589 31818 02012 || 222 | 630°24332 01415 44143 || 223 | 633°94465 52744 85605 || 224 | 637:-64988 10748 69140 || 225 | 641°35898 01929 40268 || 226 | 645:07193 54329 47241 || 227 | 648-78872 97510 41000 || 228 | 652:°50934 62536 75133 || 229 | 656:23376 81950 05832 || 230 | 659:96197 89755 91851 |} 231 | 663-69396 21397 94467 || 232 | 667:-42970 13746 77442 || 233 | 671-16918 05075 06981 || 234 | 674:91238 35044 51697 || 235 | 678:65929 44683 82574 || 236 | 682:40989 76374 72929 || 237 | 686:°56417 73831 98377 || 238 | 689-92211 82087 36798 || 239 | 693°68370 47476 68301 || 240 | 697:-44892 17617 75191 || 241 | 701:21775 44396 41937 || 242 | 704:99018 68953 55137 || 243 | 708:76620 51664 03493 || 244 | 712°54579 42129 T7774 || 245 | 716°32893 94154 70785 || 246 | 720711562 62735 77346 || 247 | 723-90584 04050 94252 || 248 | 727°69956 75437 20253 || 249 | 7381-49679 35385 56019 || 250 | 735°29750 43517 04120 68992 56917 75990 36100 | i I 49—2 387 [Tables of the first 250 Bernoulli's Numbers (to nine figures) and their logarithms (to ten figures)]. SUPPLEMENT. Added February 29, 1872. [Reap March 11, 1872.] Arter the reading of this paper, which originally contained only the logarithms of the first 250 Bernoulli’s Numbers, it occurred to me that it would greatly increase the value of the Table if the first nine or ten figures of the numbers themselves were also tabulated. If only seven figures are required, the number is taken out from the logarithm so easily that it is of no great consequence whether the numbers are or are not tabulated; but if more than seven figures are required, the case is very different. In the first place the operation of taking out the number is far more laborious, and secondly, ten-figure logarithmic tables are very difficult to procure. The only complete table is Vlacq’s (often erroneously called Briggs’s), published at Gouda, 1628, and at London, with an English introduction, 1631. This was reprinted by Vega in his Thesaurus Logarithmorum Completus, Leipsic, 1794, but the form is not quite so convenient as in the original. Both Vlacq and Vega are now very scarce. The former, further, contains over 400 errors which were found by M. Lefort, by comparison with the great French Manuscript Tables, and published by him in tome tv. of the Annales de V Observatoire Impérial de Paris (1858), pp. [148 ]...[ 150], and very many of these occur also in Vega. The accompanying table of Bernoulli’ss Numbers was calculated from By to B,. by taking out the numbers answering to the logarithms by Vlacq’s ‘table, the logarithms corresponding to these numbers were then taken out from the same table and compared with the originals, so that the verification was perfect. The first five figures of the numbers were then read with Lefort’s table of errata pre- viously referred to and one error* found thereby was corrected. * It corresponded to By, ; log 44656 should be 64987 98191 | Lefort: see Monthly Notices of the Royal Astronomical not 64987 48191. Since this paper was read I have formed | Society, for May and June, 1872. a list of errata in Vlacq, supplementary to that given by MR GLAISHER’S TABLES OF THE FIRST 250 BERNOULLI’S NUMBERS, &. 389 As the tenth figure, obtained by ten-figure logarithms, is not to be depended on as accurate, it seemed best to reject it and only tabulate nine figures of the numbers. The first eighteen Bernoulli's Numbers in the table were formed by division from the exact numbers (as vulgar fractions) given by Ohm in Crelle’s Jowrnal, Vol. xx. p. 11. It should be mentioned that Ohm has given the first thirty-one numbers, the values of the first twenty-five of which have been verified rigorously, and the rest partially by a calculation made by means of them for the determination of Euler’s constant (Proc. Roy. Soc. 1871, p. 514)- The numerals in square brackets denote the number of decimal places that follow the figures tabulated before the decimal point; for example Big) is 161811355 followed by 401 figures before the decimal point is reached, so that the integral portion of B,,. consists of 410 figures. If a, be the characteristic of log B,, then the quantity in square brackets corresponding to B, is a,— 8; as there are altogether a, +1 figures before the decimal point, of which the first nine are tabulated. It may be remarked that the value of a table often consists as much in its insuring accuracy, as in its saving the user the trouble of calculating any of the results tabulated. Even if the formula from which a table is calculated is very simple and admits of ready computation, it by no means follows that on that account the table is not worth constructing, as although it may not savea great deal of labour, it gives an amount of confidence to the consulter that he might not feel in his own calculations. This remark does not apply in full force to the present case, as the calculation of the results was quite as laborious as in the average of tables, but it affords one of the chief reasons that seemed to render it desirable to supplement the logarithms by the numbers, J. W. L. GLAISHER. 390 MR GLAISHER’S TABLES OF THE TABLE OF BERNOULLIS NUMBERS. B B B 3 | 03333 3333 60802 02380 9524 52996 03333 ©3333 47194 07575 =67576 | 28382 2496/70 42928 °25311 3553 7 2490 ce 39876 11666 6667 } O92) 2 4548/75] | 37819 70921 5686 7 1701{77] | 36614 54-971 1779 5112/80] | 36176 529-12 4242 8 8599 2 | 36470 6192-1 2319 D 4682/8 37508 86580: 2531 7 7436 oF 39345 14255 17:17 7 9218/90 42088 27298 231-1 5200 a 45902 60158 0874[0] 22122 7769[9 51031 15116 3158/2 2722 7768 97] 57822 42961 ‘ : 9251] 100 66762 13711 6552 27 0822/103 78535 48833 : 2 9231) 105 94106 19296 y 8179[108 11484 84169 7 ce 14272 16666 6667 52 igs 71287 8213/180 iT COMMA OF WWe 40338 0719 3620) 113 18059 23261 30495 75008 6675 2 6008/121 40685 50387 7810 2 0888 d 55231 21150 7486 2 5709/ 116 12086 6265 82 4333/118 36528 7765/20 : 2 9268] 127 76277 28498 7693) 2: é 76 «1671/1129 10715 23865 4275/2 7 4686] 132 15310 21399 9493]: 21426 1013/135 22244 20500 9757{: 2456 7271[138 32862 20938 0059} ¢ 74345 7875|140 49355 22752 6965/< 5E 7953] 143 75349 26257 7103} ¢ 28612 vane 146 11691 32125 0821/36 37 6{149 18435 41598 2782[¢ 2 et 152 29536 56920 6955): 2 2482 1872/154 48079 82183 225 8779157 79502 12502 68 5305/160 13352 20015 2 rt 295 0921) 163} || 22776 33674 9829/49 y 5226[166] || 39451 8404/304 59470 9705/51] | 6812 5971|169 2| 69385 2577/307 11011 9103/54] | 7362} 172} | 12388 9637/311 21355 2595/56 8940] 175 22455 4260/314 43328 8970/58 5432 8936177 ‘ 41312 1318317 ‘The numerals in square brackets denote the number of additional figures before the decimal point, thus B,,, to nine figures, is 750086675 followed by 16 ciphers before the decimal point. FIRST 250 BERNOULLI’S NUMBERS AND THEIR LOGARITHMS, Be n B n 8078 337] | 9879[340] | 2566 a 6735/3 go30 0 3869] 390 6362/394 9210] 397 5480 387 1355|401 2880/ 404 4707|408 7179/4111 3757{414 1059418 1129/421 9085] 425 54441428 8103) 432 9089} 435 6600[439 3005] 442 1581/446 1085] 449 4338] 453 al || 33538 10615 33984 11001 36016 11922 39903 13502 46193 15975 55847 19754 70482 25442 92815 34217 12747 47985 18251 70136 27230 10680 42316 16936 68469 27958 11530 48023 20199 85802 36802 15939 69702 30775 13718 61736 28047 12862 59543 27822 2097) 463 9250) 467 8638[470| ot 2448/456 9426/460 4275/47] 8180/48] 2544, 484 2224[488] aia 5373 491] 4362 ra 9544 498) 3670/502| 5160 te z : 3364 2481 1695 3201 6744 5931 0386 531 1485/5311 5095/534| 5005 ae 4485] 541 aS 513] 516] 1297/549 6885/552 9526/556 0724/559 4794(563 2446/567 6718|570 2809/574 4676[578 2736 0313 5090 9442 9779 581 585 589 592 596 B n 9915[603 1201/607 0488/611 5852[614 3982/618 3707[622 9809] 625 1291[629 9457[633 oe ae 4185[640} 6916[644 4273 648 6136[651] 7597[655 8471[659 1899/663 3774[666 6231[670 9002[674] 1022 678| 0488 681] 0509 685] 9431 689] 2686[693] 7858[696 0796{700 3896[704 4737[708 4736[712 2534(715 8416/719 1607[723 2954[727 391 The numerals in square brackets denote the number of additional figures before the decimal point, thus B,,,, to nine figures, is 107164338 followed by 453 ciphers before the decimal point. VIL Further Observations on the state of an Eye affected with a peculiar malformation. By Grorce Bippert Airy, M.A., LL.D., D.C.L., Honorary Fellow of Trinity College; formerly Lucasian Professor, late Plumian Pro- fessor, in the University of Cambridge; Astronomer Royal. [Read Feb. 12, 1872.] For the method which I have employed now for the fourth time in examining the state of the eye, I refer generally to my communication to the Cambridge Philosophical Society dated 1825, February 5. A very minute hole is made, by the point of a fine needle, in a blackened card, which is so pierced in another part that it can be slid upon a graduated scale, of which one end abuts against the orbital bone of the eye; the graduated scale thus giving a very approximate measure of the distance of the minute hole from the cornea of the eye in every experiment. With a properly-formed eye, and with the card raised between the eye and the bright sky, the minute hole is seen, at the distance of distinct vision, as a brilliant point. With the anomalous eye, the hole is seen at one distance as a nearly horizontal line pretty sharply defined, and at a greater distance as a line at right angles to the former line (and thus approaching to a vertical direction) pretty sharply defined. At no distance is it seen as a point. To this form of refraction Dr Whewell gave the name of astigmatism, which it has since retained. Without further explanation, I will give the results of a late examination, in combination with those of previous examinations, in the same form as in my paper of 1866, November 19 (Proceedings of the Camb. Phil. Soc. Pt. Iv.). I. Distance from the cornea of the left eye at which the luminous point presents the appearance of a nearly horizontal line. In 1825, 3°5 inches; Reciprocal ‘286 In 1846, 4°7 esses Hibas.c.2- iby ee ee a In 1866, F4 ch Sete... ASK shake = ve [isl -b:Ge es 179 Selb Il. Distance from the cornea of the left eye at which the luminous point presents the appearance of a nearly vertical line. In 1825, 6°0 inches; Reciprocal = ‘166 In 1846, 189. ...... Seumateeens 112 Inpl866,e10'6e . 2... ist sAconce 0940” =e a In 1871, 100 ...... Oe abet: “i= at Difference — ‘054. — ‘018. MR AIRY, ON AN EYE WITH A MALFORMATION. 393 TMH Measure of the astigmatic power of the left eye at different epochs; estimated in each case by the differences of the reciprocals for the same date in the two pre- ceding tables. In 1825, astigmatism = ‘120 TnVlIS4iGse ae sececsss 101 Tha TUSKTG, — edodsogue ‘091 Tint TSY/lly Gaenatoae ‘079 Difference = ‘019. 010. eeeeee ee eeee IV. Distance from the cornea of the right eye at which the luminous point is seen distinctly. In 1846, 47 inches; Reciprocal = linwl| SG 6s p:ommseses c In 1871, 5-4 28 hie 031 182 imerence = — ae Fe See + 003. The changes in the last period of five years are small. The element which appears to have undergone the greatest change is the astigmatism; this result of observation is opposed to that of earlier years. I am inclined to think that the self-adjusting power of the eyes for different distances is sensibly less than it was in 1866; and that the stigmatic refraction of the right eye is less perfect than it was formerly. RoyaL OBSERVATORY, GREENWICH, 1871, December 27. Vou. XII. Parr I. G. B. AIRY. 50 Convolvulacex. (Eryetbex ) Phil. Sow. Trans. Vol XI. Pel. Le YP 0 / ii Kricacee. f, ayy. L = a Ba, on =) Olacineee. Te f, iS hoveng s Si pie - “Ge S S ; Loy Ss Sy LbCNACCE: S Byes y S £ a (e oO I \ s \ DS 7, Dra. sud Al \ (i \ Pi i" vy \ ip U ea GQAsgyan iy \ yy) ! \ ee Camo. fhrae. frars. vol All. Plate.Il. A.T Halhick del et hth ae Es Plate TIL. Camb. Phil. Trams. Vou.XIl. ft UME LY ° r ie, U] 8 ( ; a Plate. VI Camb. Phil. Trams. Vou.XIL yh Camb Phil. Troms Vol _XIC ACT. Holtick del et kth Wirt.ecn Bros imn | = bya Plate VIL Camb Phil. Trans Vol XIL aay 3 Plate IX, Cam.Phil. Trans. Vol XII. MinternBros mp Let lth orle A THabck a Kas Comb. Phal. Treas. Vou XIL Plate.X Plate.XT. Cornb Ful Trans Vol Xi I. On the geometrical representation of Cauchy's theorems of Root-limitation. By Professor Cay.ey. [Read Feb. 16, 1874.] THERE is contained in Cauchy's Memoir ‘Calcul des Indices des Fonctions,’ Journ. de [Ecole Polytech. t. Xv. (1837) a general theorem, which, though including a well-known theorem in regard to the imaginary roots of a numerical equation, seems itself to have been almost lost sight of. In the general theorem (say Cauchy's two-curve theorem) we have in a plane two curves P=0, Q=0, and the real intersections of these two curves, or say the “roots,” are divided into two sets according as the Jacobian d,P.d,Q—d,Q.4,P is positive or negative, say these are the Jacobian-positive and the Jacobian-negative roots: and the question is to determine for the roots within a given contour or circuit, the difference of the numbers of the roots belonging to the two sets respectively. In the particular theorem (say Cauchy’s rhizic theorem) P and @ are the real part and the coefficient of 7 in the imaginary part of a function of «+ dy with, in general, imaginary coefficients (or, what is the same thing, we have P+iQ=f(a + cy) + 7p (+7), where f, @ are real functions of «+7y): the roots of necessity are of the same set: and the question is to determine the number of roots within a given circuit. In each case the required number is theoretically given by the same rule, viz., con- sidering the fraction a it is the excess of the number of times that the fraction changes from + to — over the number of times that it changes from — to +, as the point (2, y) travels round the circuit, attending only to the changes which take place on a passage through a point for which P is = 0. In the case where the circuit is a polygon, and most easily when it is a rectangle the sides of which are parallel to the two axes respectively, the excess in question can be actually determined by means of an application of Sturm’s theorem successively to each side of the polygon, or rectangle. Weir, 220 eae oe 51 396 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF In the present memoir I reproduce the whole theory, presenting it under a completely geometrical form, viz. I establish between the two sets of roots the distinction of right- and left-handed : and (availing myself of a notion due to Prof. Sylvester*) I give a geometrical form to the theoretic rule, making it depend on the “intercalation” of the intersections of the two curves with the circuit: I also complete the Sturmian process in regard to the sides of the rectangle: the memoir contains further researches in regard to the curves in the case of the particular theorem, or say as to the rhiziec curves P=0, Q=0. The General Theory. Articles Nos. 1 to 19. 1. Consider in a plane two curves P=0, Q=0 (P and Q each a rational and integral function of a, y), which to fix the ideas I call the red curve and the blue curve re- spectively+: the curve P=0 divides the plane into two sets of regions, say a positive set for each of which P is positive, and a negative set for each of which P is negative: it is of course immaterial which set is positive and which negative, since writing —P for P the two sets would be interchanged: but taking P to be given, the two sets are distinguished as above. And we may imagine the negative regions to be coloured red, the positive ones being left uncoloured, or say they are white. Similarly the curve Q=0 divides the plane into two sets of regions, the negative regions being coloured blue, and the positive ones being left uncoloured, or say they are white. Taking account of the twofold division, and considering the coincidence of red and blue as producing black, there will be four sets of regions, which for convenience may be spoken of as sable, gules, argent, azure: viz. in the figures we have Bei = = — — sable, shown by cross lines, — + gules, ,, » vertical lines, + + argent, left iwhite, + — azure, shown by horizontal lines, sable and argent (—— and + +) being thus positive colours, and gules and azure (— + and + —) negative colours. See figures towards end of Memoir. 2. Consider any point of intersection of the two curves. There will be about this point four regions, sable and argent being opposite to each other, as also gules and azure; whence selecting an order sable, gules, argent, if to have the colours in this order we have to go about the point, or root, right-handedly, the root is right-handed: but if left-handedly, then the root is left-handed: or, what is more azure ; “ See his memoir, A theory of the Syzygetic relations | fact the intercalation of these roots: but, not being concerned &c. Phil. Trans. 1853. The Sturmian process is by Sturm and Cauchy applied to two independent functions gx, fx of a variable x; but the notion of an intercalation as applied to the order of succession of the roots of the equations (x)=0, f(x)=0 is due to Sylvester, and it was he who showed that what the Sturmian process determined was in with circuits, he was not led to consider the intercalation of a circuit. + It is assumed throughout that the two curves have no points (or at least no real points) of multiple intersection ; i.e. they nowhere touch each other, and neither curve passes through a multiple point of the other curve. CAUCHY’S THEOREMS OF ROOT-LIMITATION. 397 convenient, going always right-handedly, then, if the order of the colours is sable, gules, argent, azure, the root is right-handed: but if the order is sable, azure, argent, gules, the root is left-handed. 3. The distinction of right- and left-handed corresponds to the sign of the Jacobian d(P, Q) 4 =d,P.d,Q—d,Q.d Dae) (=d,P.d,Q—d,Q.d,P), and we may (reversing if necessary the original sign of one of the functions) assume that for a right-handed root the Jacobian is positive, for a left-handed one, negative. 4, I consider a trajectory which may be either an unclosed curve not cutting itself, or else a circuit, viz. this is a closed curve not cutting itself. A circuit is considered as described right-handedly: an unclosed trajectory is considered as described according to a currency always determinate pro hdc vice: yiz. one extremity is selected as the beginning and the other as the end of the trajectory: but the currency may if necessary or con- venient be reversed: thus if an unclosed trajectory forms part of a circuit the currency is thereby determined: but the same unclosed trajectory may form part of two opposite circuits, and as such may have to be taken with opposite currencies. It is assumed that a trajectory does not pass through any intersection of the P and Q curves. 5. A trajectory has its P- and Q-sequence, viz. considering in order its intersections with the two curves, we write down a P for each intersection with the red curve and a @ for each intersection with the blue curve, thus obtaining an intermingled series of P’s and Q’s, which is the sequence in question. In the case of a circuit, the sequence is considered as a circuit, viz. the first and last terms are considered as contiguous, and it is immaterial at what point the sequence commences. The sequence will of course vanish if the trajectory does not meet either of the curves. > 6. A P-and Q- sequence gives rise to an “intercalation,” viz. if in the sequence there occur together any even number of the same letter these are omitted (whence also any odd number of the same letter is reduced to the letter taken once): and if by reason of an omission there again occur an even number of the same letter these are omitted: and so on. ‘The intercalation contains therefore only the letters P and @ alternately: viz. in the case of an unclosed trajectory the intercalation may contain an even number of letters, beginning with the one and ending with the other letter, and so containing the same number of each letter—or it may contain an odd number of letters, beginning and ending with the same letter, and so containing one more of this than of the other letter; say the intercalation is PQ or QP, or else PQP or QPQ. The intercalation may vanish altogether, thus if the sequence were QPPQ this would be the case. 7. In the case of a circuit the intercalation cannot begin and end with the same letter, for these, as contiguous letters, would be omitted; and since any letter thereof may 51—2 398 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF be regarded as the commencement it is PQ or QP indifferently. A little consideration will show that the whole number of letters must be evenly even, or, what is the same thing, the number of each letter must be even. Thus imagine the circuit beginning in sable, and let the intercalation begin with P@; viz. P we pass from sable to azure, and Q we pass from azure to argent: in order to get back into sable we must either return the same way (Q argent to azure, P azure to sable), but then the sequence is PQQP, and the intercalation vanishes: here the number of letters is 0, an evenly even’ number: or else we must complete the cyele of colours P argent to gules, @ gules to sable: and the sequence and therefore also the intercalation then is PQPQ, where the number of letters is 4, an evenly even number. 8. In the ease of any trajectory whatever, the half number of letters in the inter- calation is termed the “index,” viz. this is either an integer or an integer + 3. But in the case of a circuit the index is an even integer, and the half-index is therefore an integer. The indéx may of course be = 0. 9. But we require a further distinction: instead of a / and @- sequence we have to consider a + P- and Q- sequence. To explain this observe that a passage over the red curve may be from a negative to a positive colour (azure to sable or gules to argent), this is + P, or from a positive to a negative colour (sable to azure or argent to gules), this is —P. And so the passage over the blue curve may be from a negative to a positive colour (gules to sable or azure to argent), this is +Q, or else from a positive to a negative colour (sable to gules or argent to azure), this is —Q. The sequence will contain the P and @Q intermingled in any manner, but the signs will always be + — alternately; for +(P or Q), denoting the passage into a positive eolour, must always be immediately suc- ceeded by —(P or Q), denoting the passage’ into a negative colour. Whence, knowing the sequence independently of the signs, we have only to prefix to the first letter the sign + or — as the case may be, and the sequence is then completely determined. 10. Passing to a + intercalation, observe that in omitting any even number of P's or Q’s, the omitted signs are always +—-+— &e. or else —+—+ ce. viz. the omitted signs begin with one sign and end with the opposite sign. Hence the signs being in the first instance alternate, they will after any omission remain alternate: and the letters being also alternate, the intercalation can contain only + P and — Q or else —P and +Q. Hence in the, case of a cireuit the intercalation is either (+ P—Q), say this is a positive circuit, or else (-P+Q), say this is a negative circuit. There is of course the neutral circuit (PQ), for which the intercalation vanishes. 11. Consider a cireuit not containing. within it any root; as a simple example let the cireuit lie wholly in one colour, or wholly in two adjacent colours, say sable and gules: in the former case the sequence, and therefore also the intercalation, vanishes: in the latter ease the sequence is + Q— Q, and therefore the intercalation vanishes: viz. in either case the intercalation is (PQ),. CAUCHY’S THEOREMS OF ROOT-LIMITATION. 399 12. Consider next a circuit containing within it one right-handed root; for instance let the circuit lie wholly in the four regions adjacent to this root, cutting the two curves each twice; the sequence and therefore also the intercalation is +P—Q+P—Q; viz. this is a positive circuit (+ P—Q),, where the subscript number is the half-index, or half of the number of P’s or of Q's. Similarly if a ciretit’ contains within it one left-handed root, for instance if the circuit lies wholly in the four regions adjacent to this root, cutting the two’ curves each twice, the sequence and therefore also the intercalation is -P+@Q—P+ Q, viz. this is a negative circuit (-P+Q),: and the consideration of a few more particular cases leads easily to the general and fundamental theorem : 13. A circuit is positive (+P—Q)s or negative (-P+Q)s according as it contains within it more right-handed or more left-handed roots; and in either case. the half-index 6 is equal to the excess of the number of one over that of the other set of roots. If the circuit is neutral (PQ),, then there are within it as many left-handed as right-handed roots. 14. The proof depends on a composition of circuits, but for this some preliminary considerations are necessary. Imagine two unclosed trajectories furtning a circuit, and write down in order the inter- calation of each. The whole number of letters must be even: viz. the numbers for the two intercalations respectively must be both even or both odd. I say that if the terminal letter of the first intercalation and the initial letter of the second intercalation are different, then also the initial letter’ of the first intercalation and the terminal letter of the second intercalation will be different: if the same, then the same. Im fact the intercalations may be each PQ or each QP, or one PQ and the other @P: or each PQP, or each QPQ, or one PQP and the other QPQ. Supposing the letters in question are different, then the intercalations may be termed similar; but if the same, then the imtercalations may be termed contrary. 15. In the first case, that is when the intercalations are similar, the two together form the intercalation of the circuit; the sum of their numbers of letters (that is twice the sum of their indices) will be evenly even, and the half of this, or sum of the indices, will be the index of the ciretit; each intercalation will be (+P— Q) or else each will be (—P+ Q); and the circuit will be (+P— @Q) or (- P+ Q) accordingly. In the second case, that is when the intercalations are contrary, they counteract each other in forming the intercalation of thé circuit: it is the difference of the numbers of letters, or twice the difference of the indices, which is evenly even, and the half of this, or difference of the indices, which. is the index of the circuit: one intercalation is (+ P— Q), and the other is (— P+ Q): and the circuit will agree with that which has the larger index. In particular if the circuit consist. of a single unclosed trajectory, taken forwards and backwards; then the trajectory taken one way is (+P—Q), taken the other way it is (— P+ Q); the number of terms is of course equal, and the circuit is (PQ),. 400 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF 16. Consider now two circuits ABCA and ACDA, having a common portion CA, or, more accurately, the common portions AC and CA: write down in order the intercalations of ABC, CA, AC, CDA: the two mean terms destroy each other, and we can hence deduce the intercalation of the entire circuit ABCDA. Suppose jist, that ABC and CDA are similar; then if CA is similar to ABC it is also similar to CDA, that is AC is contrary to CDA: and so if CA is contrary to ABC, then AC is similar to CDA. To fix the ideas suppose CA similar to ABC, but AC contrary to CDA, then ABCA is similar to CA; but ACDA will be similar or contrary to AC, i.e. contrary or similar to CA, that is to ABCA, according as index of AC > or < index of CDA. Suppose Ind. AC < Ind, CDA, then ACDA is similar to ABCA. Ind. ABCDA =Ind. ABC+ Ind. CDA, Ind. ABCA =Ind. ABC+ Ind. AGC, Ind. ACDA =Ind. CDA-—Ind. AC, and thence Ind. ABCDA =Ind. ABCA + Ind. ACDA, the whole circuit being in this case similar to each of the component ones. But if Ind. AC > Ind. CDA, then ACDA is contrary to ABCA. Ind. ABCDA =Ind. ABC +Ind. CDA, Ind. ABCA =Ind. ABC+Ind. C4, Ind. ACDA =~—Ind. CDA +Ind. AC, and thence Ind. ABCDA =Ind. ABCA —Ind. ACDA, and the investigation is like hereto if CA is contrary to ABC but AC similar to CDA. 17. Secondly, if ABC and CDA are contrary, then if CA is similar to ABC it is contrary to CDA, that is AC is similar to CDA; and so if CA is contrary to ABC it is similar to CDA, that is AC is contrary to CDA. Suppose CA similar to ABC, and AC similar to CDA; then ABCA is also similar to ABC, and ACDA similar to CDA; viz. ABC, CA and ABCA are similar to each other, and contrary to 40, CDA, ACDA which are also similar to each other. Ind. ABCDA =Ind. ABC ~ Ind. CDA, Ind. ABCA =Ind. ABC+Ind. CA, Ind. ACDA =Ind. CDA +Ind. AG, and thence Ind. ABCDA =Ind. ABCA ~ Ind. ACDA, and the investigation is like hereto if CA is contrary to ABC and AC contrary to CDA. CAUCHY’S THEOREMS OF ROOT-LIMITATION. 401 18. It thus appears that in every case Ind. ABCDA = Ind. ABCA + Ind. A CDA, or = Ind. ABCA ~ Ind. ACDA, according as the component circuits are similar or contrary, and in the latter case the entire circuit is similar to that which has the largest index. Moreover, any circuit whatever can be broken up into two smaller circuits, and these again continually into smaller circuits until we arrive at the before-mentioned elementary circuits, and the theorem as to the number of roots within a circuit is true as regards these elementary circuits ; wherefore the theorem is true as regards any circuit whatever. 19. In the case where a trajectory is a finite right line, y is a given linear function of w, or the coordinates z, y can if we please be expressed as linear functions of a parameter u, so that as the describing point passes along the line, w varies between given limits, say from w=0 to w=1. The functions P, Q thus become given rational and integral functions of a single variable w (or it may be # or y), and the question of the P- and @- sequence and intercalation relates merely to the order of succession of the roots of the equations P=0, Q=0, where P and Q denote functions of a single variable as above. To fix the ideas let the trajectory be a line parallel to the axis of #; and in this case taking a as the parameter, and supposing that yp is the given value of y, P and Q are the functions of 2 obtained by writing y, for y in the original expressions of these functions. Of course the theory will be precisely the same for a line parallel to the axis of y: and by combining two lines parallel to each axis we have the case of a rectangular circuit. We require, for each side of the rectangle considered according to its proper cur- rency, the intercalation PQ, QP, PQP or QPQ as the case may be, and also the sign + or — of the initial letter of the first intercalation; for then writing down the intercalations in order, with the signs for the several letters, + and — alternately (the first sign being + or — as the case may be), we have or deduce the intercalation of the circuit, and thus obtain the value of the difference of the numbers of the included right- and left-handed roots. We thus see how the whole theory depends on the case where the trajectory is a right line. Intercalation-theory for a right line. Articles Nos. 20 to 31. 20. Considering then the case where the trajectory is a line parallel to the axis of 2, P and @ will denote given rational functions of «; the curves P=0, Q=0 being of course each of them a set of right lines parallel to the axis of y: the regions will be bands each of them included between two such lines; and colouring them as explained in the general case, the colours will be as before, sable, gules, argent, azure, each region having in the neighbourhood of the trajectory (what we are alone concerned with) the same colour that it had in the original case where P and @Q were functions of (x, y). We may regard the trajectory as described according to the currency z=—o to w=+o: we have in regard to the trajectory a P- and Q- sequence, and intercalation, a + P- and Q- sequence, &c., as in 402 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF the original case. The intercalation may be as before PQ, QP, PQP or QPQ, and in each of these cases it may be positive, that is (+P —Q), or else negative, that is (-P+ Q). 21. The question of sign may in the present case be disposed of without difficulty. For the initial point of the trajectory, we know the signs of P, Q, that is the colour of the region: suppose for example that we have P=—, Q=+, or that the region is gules: then if the intercalation begin with P, this means that we either first pass a red line, or before doing so we pass an even number of blue lines: but in the last case the colours are sable gules sable gules,... always ending in gules; and the passage over the red line is gules to argent, viz. this is +P; and so in general the initial P or Q of the intercalation has the sign opposite to that of the P or Q belonging to the commencement of the trajectory. 22. For the solution of the problem we connect with P, Q a set of functions R, S, T, &c.: the intercalation is in fact given by means of the gain or loss of changes of sign in these functions on substituting therein the initial and final values of the variable z. It is convenient to consider the functions as arranged in a column ~~ RAS ty say this is the column PQRS..., and to connect therewith a signaletic bicolumn: viz. the left-hand column is here the series of signs of these functions for the initial value of z, and the right-hand column is the series of signs for the terminal value of «: the bicolumn thus consisting of as many rows each of two signs, as there are functions. But such a bicolumn may be considered apart from any series of functions, as a set of rows each of two signs taken at pleasure. We say that the “gain” of a bicolumn is = —(No. of changes of sign in left-hand column) + (No, in right-hand ditto), the gain being of course positive or negative; and a negative gain being regarded as a loss. Also if a positive gain be converted into an equal negative gain or vice versd, we may speak of the gain as reversed. 23. A bicolumn may be divided in any manner into parts, taking always the last row of any part as being also the first row of the next succeeding part. This being so, the gain of the whole bicolumn is equal to the sum of the gains of its parts. In a bicolumn of two rows, if we reverse either row (that is write therein — for + and + for —), we reverse the gain: and hence dividing a bicolumn into bicolumns each of two rows, viz. first and second rows, second and third rows, and so on, it at once appears that if we reverse alternate rows (viz. either the first, third, fifth, &e., rows, or the second, fourth, sixth, &e., rows) we reverse the gain. It of course follows that reversing all the rows, we leaye the gain unaltered, CAUCHY’S THEOREMS OF ROOT-LIMITATION. 403 24. If to any bicolumn we prefix at the top thereof the second row reversed, we. either leave the gain unaltered or we alter it by +1. Im fact, as regards either column, if this originally begin with a change, the process introduces no change therein; but if it begins with a continuation, then the process introduces a change. Hence if the columns begin each with a change or each with a continuation, the gain is unaltered: but if one begins with a change, and the other with a continuation, then the gain is altered by +1; viz. the left- hand column beginning with a continuation the gain is altered by —1, and the right-hand column beginning with a continuation the gain is altered by +1. The column PQRST... is taken to satisfy the following conditions: two consecutive terms never vanish together (that is, for the same value of the variable): if for a given value of the variable, any term vanishes, the preceding and succeeding terms have then opposite signs; the last term, say V, is of constant sign. 25. Considering P, Q as given functions without a common measure, such a column of functions is obtained by the well-known process of seeking for the greatest common measure, reversing at each step the sign of the remainder: viz. we thus derive a set of functions Rk, S, 7.,. where P=rQ-R, R=v78—-T, S=pT-U, the degrees of the successive functions R, S, 7, ..., being successively less and less, so that the last of them, say V, is an absolute constant: or we may stop the process as soon as we arrive at a function V, the sign of which remains unaltered for all values between the initial and final values of the variable. It may be observed that the process may be regarded as applicable in the case where the degree of @ exceeds that of P: viz. we then have X=0, R=-—P, and the column begins (P, Q,—P, S,...), the subsequent terms being, except as to sign, the same as if P, Q had been interchanged. Reversing the sign of P or Q, we reverse “in the bicolumn a set of alternate rows, and thus reverse the gain: and reversing both signs we reverse all the rows, and leaye the gain unaltered—of course the intercalation (considered irrespectively of sign) is in each case un- altered. It is convenient to take the signs in such manner that for the initial value of z, the signs of P, Q shall be each positive: or, what is the same thing, taking PB, Q with their proper signs, we may in the bicclumn, by reversing if necessary each or either set of alternate rows, make the left-hand column to begin with the signs + +. 26. The complete rule now is—for a given trajectory form the bicolumn for PQRS..., and if necessary, by reversing each or either set of alternate rows, make the left-hand column to begin with ++: then if there is a gain the intercalation begins with P, if a loss with Q, the gain or loss showing the number of P’s. To find the number of Q’s prefix at the top of the bicolumn the second row reversed—then the gain or loss (equal to or differing by unity from the original value) shows the number of Q’s. It may-happen that for P the gain is =0; then for Q the gain is 0 or +1, and the intercalation vanishes or is Q. Ws SGU, Levant UE 52 404 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF 27. I give some simple examples. 0 2 4 IDEM G) |) =F Q=ax-1|]— + + ES Ree oe ial’ Sune adie © Maa OO Som pe ene In the left-hand example taking the intervals to be successively 0—2, 0—4, 2-4, the bicolumns modified as above are 0-2 0-4 2-4 - -|- +|- + + —-|+ -|+ + + +}/+ -|+ - + +]+ 4/- - viz. Interval 0O—2; for P gain =1, P first; for Q gain =0; Intercalation is P; » O—4 ” ” =1, ” ” ” et ” ” PQ; 1G R= —-1/+{- 4 | —| wherefore intercalation is QP, Shea | or since at origin P=-, R=-, "or region is sable, it is —Q+P. 0 8 Onos (2) ee ab ee thet is) | + |, for P gain = 0, ee as aso — pe hg 8 ee iar =-] a 4 Intercalation is —Q. CAUCHY’S THEOREMS OF ROOT-LIMITATION. 407 -3 0 -3 0 (8) P= 97$5 | +) +], that is | SS ; for P gain =0, Q=-y +4 =) = + + ” Q ee =(} i= —1]|/-—]|—- fol lt Intercalation vanishes. —3 0 —-3 0 (4) P= w?-—4/ + | — |], that is | —| — |, for P, gain =+1, P first, Q = av —-1l/—|—- can =a ” Q p= 0. R= +1} +)+ Pa [are Intercalation is + P. +|+ Hence for the four sides, combining the intercalations, we have —-Q+P-—Q+P, and since there are no terms to be omitted, this is the intercalation of the N.E. square: which is right. The Rhizic Theory. Articles, Nos. 32 to 38. 32. Consider now /'(z) =(%)(z,1)” a rational and integral function of z, of the order m with in general imaginary (complex) coefficients, or, what is the same thing, let. F(z) =f (z) +t (2), where the functions f, ¢ are real*. Writing herein z= «+ ty, let P, @ be the real part and the coefficient of the imaginary part in the function F (x+ty): or, what is the same thing, assume P+iQ=f (a+ ty) +id (w+ ty), then it is clear that to any root a+7@ (real or imaginary) of the equation F(z) =0, there corresponds a real intersection, or root, c=a, y=£, of the curves P=0, Q=0. The functions P, Q as thus serving for the determination of the roots of the equation F(z)=0, are termed “rhizic functions,” and similarly the curves P=0, Q=0 are “rhizic curves.” The assumed equation shows at once that we have d, (P +7Q) =td,(P+7Q), or, what is the same thing, d,P =— 12» d,P = d,Q. And we hence see that d (P , Q) fee 2 2 2 2 Teg? 7 GEN + (GP) oF (2.0) + 0" is positive: viz. that the roots P=0, Q=0 are all of them right-handed (the essential thing is that they are same-handed; for by reversing the signs of P and Q they might be made left-handed: but it is convenient to take them as right-handed): hence the theorem—which in the general case, P and Q arbitrary functions, serves to determine * It is assumed that the equation F(z)—0 has no equal roots: this being so, the curves P = 0, Q = 0, will have no point of multiple intersection ; which accords with the assumption made in the general case of two arbitrary curves. . 408 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF the difference of the numbers of the right and left-handed roots—in the particular case where P and @Q are rhizic functions serves to determine the number of intersections of the curves P=0, Q=0: or, what is the same thing, the number of the (real or imaginary) roots of the equation #’'(z)=0: viz. we thus determine the number of roots within a given circuit. 33. The rhizic curves P=0, Q=0 have various properties. 1°. Each curve has n real points at infinity, or, what is the same thing, n real asymptotes: and the P and Q points at infinity succeed each other, a P-point and then a Q-point, and so on alternately. In fact from the equation P+iQ= (a +%a") (a+ ty)*...+ (K+ K%, writing herein a’ + ia” =a (cosa+isina), and x +7iy=p (cos@+isin @), we have P+ iQ =ap" [cos (nf +a) + 7sin (nO +a)]...+h +k; and it thus appears that for the curve P=0, the points at infinity are given by the equation cos (n®+a)=0, while for the curve @=0 they are given by the equation sin (n8+a)=0: which proves the theorem. Representing infinity as a closed curve or circuit, each point at infinity must be represented by two opposite points on the circuit; so that writing down P for each P-point and Q for each Q-point we have 2n P’s and 2n @’s succeeding each other, a P-point and then a Q-point, and so on alternately. It may be assumed that taking the circuit right-handedly, the P’s are + and the Q’s—, (this depends only on the colouring, but it corresponds with the foregoing assumption that the roots P=0, Q=0 are right-handed): the theorem just obtained then really is that for the circuit infinity, the intercalation is (+P—Q),: and we have herein a proof of the theorem that a numerical equation of the order x with real or imaginary coefficients has precisely » real or imaginary roots. But the force of this will more distinctly appear presently. 34. 2°. Neither of the curves P=0, Q=0 can include as part of itself a closed curve or circuit. The foregoing relations between the differential coefficients give d:P+d2P=0, d2Q+4,;Q=0, which equations for the two curves respectively lead to the theorem in question. For as regards the curve P=0, take z a co-ordinate perpendicular to the plane of ay, and consider the surface z=P: if the curve P=0 included as part of itself a closed curve, then corresponding to some point (x, y) within the curve we should have z a proper maximum or minimum, viz. there would be a summit or an imit; at the point im question we should have d,P =0, d,Q=0; and also (as the condition of a summit or imit) d?P.d/P —(d,d,P)*=+, implying that d,°~P and d,’P have at this point the same sign: but this is inconsistent with the foregoing relation d,’P + d,’P =0. CAUCHY’S THEOREMS OF ROOT-LIMITATION. 409 35. 3°. The curves P=0, Q=0 have not in general any double (or higher multiple) points, A point which is a double (or higher multiple) point on one of these curves is not of necessity a point on the other curve: but being a point on the other curve it is on that curve a point of the same multiplicity. For changing if necessary the co-ordinates, the point in question may be taken to be at the origin: forming the equation P+iQ=(¢+0'%) (wt+iy)”...+ (WM +R% (w+ ty) t+ (U4 U%) (e+ ty) 4+m'+ mi =0, the point «=0, y=0 will not be a double poimt on the curve P=0, unless we have m =0, U=0, l'=0; these conditions being satisfied, it will not be a point on the curve @=0 unless also m”=0; but this being so, it will be a double point on the curve Q=0: and the like for points of higher multiplicity. But a point which is a multiple point on each curve, represents four or more coincident intersections of the curves P=0, Q=0, that is four or more equal roots of the equation F(z)=0; so that assuming that the equation has no equal roots, the case does not arise: and we in fact exclude it from consideration, To fix the ideas assume that the curves P=0, Q=0 are each of them without double points. As already seen, neither of them includes as part of itself a closed curve. Hence in the figure the curve P=0 must consist of branches each drawn from a point P in the circuit (viz. the circuit infinity) to another pomt P in the circuit; and in such manner that no two branches intersect each other: this implies that the two points P of the same branch must include between them an even number (which may of course be =0) of points P. And the like as regards the curve Q=0, 36. 4°. No branch of the P-curve can meet a branch of the Q-curve more than once, In fact drawing the two branches to meet twice, the colouring would at once show that of the two intersections or roots, one must be right, the other left-handed: whence, the roots being all right-handed, the branches do not meet twice. And in exactly the same way it appears that no P-branch can meet two Q-branches, or any (-branch meet two P-branches. And under these restrictions it requires only a consideration of a few successive cases to show that the nm P-branches, and the n Q-branches can only be drawn on the condition that each P-branch shall intersect once and only once a single Q-branch; which of course implies that each @-branch intersects once and once only a single P-branch: and further, that there shall be precisely x intersections: viz. the n P-branches and the n Q-branches must satisfy the conditions just stated. And the theorem of the m roots is thus obtained as a consequence of the impossibility (except under the same conditions) of drawing the n P-branches and the @-branches, so as to give rise to right-handed roots only. But the case of double or higher multiple points would need to be specially considered. 37. It is interesting for a given value of » to consider ¢(n) the number of different ways in which the P-branches and the Q-branches can be drawn. We have 2n points P and 2n points Q, in all 4n points: starting from any point P, these may be numbered in order 1, 2, 3,...4m, the points P bearing odd numbers and the points Q even numbers. 410 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF We may consider the P-branch which joins 1 with some P-point 8, and (intersecting this) the Q-branch which joins some two Q-points a and y: the numbers lay are then in order of increasing magnitude: and excluding these four points there remain the points corre- sponding to numbers between 1 and a, between a and £, between 8 and y, and between y and 1. Now since the P-branch 18 meets the Q-branch ay, no branch from a point between 1 and g can meet either of these curves; hence these points form a system by themselves, capable of being connected together by P-branches and Q-branches: the number ° of them must therefore be a multiple of 4: and the like as to the points between a and 8, between 8 and y, and between y and 1. Taking the number of the points in the four systems to be 4a, 4y, 42, and 4 respectively, we have «+y+z+w=n- 1, and the first mentioned four points bear the numbers 1 a=4r+2, B=4r+4y +3, y = 4a + 4y + 42 + 4. For the four systems the number of ways of drawing the P- and Q-branches are dx, by, $2, dw respectively: that is a, y, 2, w being any partition whatever of n—1 (order attended to), and $ (0) being =1, we have E $ (x) = =4 (2) 6(y) $(2) $(w), which is the condition for the determination of gn. Taking then 6 for the value of the generating function 1+ td (1) + Pg (2)...+ eG (n) +... it hereby appears that we have 6=1+ 6; or writing this for a moment 6=u+t6*, and expanding by Lagrange’s theorem, but putting finally «=1, we have the value of @, that is of the generating function, t Ga ae: pe auee dies =1+[4P5+ [8] 1.97 12 pase + [4n] "Tount =1+ t+ 4¢ + 224 + 140¢ +..., that is $(1)=1, $(2)=4, $(8)=22, (4) =140,... es _ [4nJ* An. dn — 1...3n + 2 and generally o (n) = eee ae ee ee The results are easily verified for the successive particular cases ; thus n=1, the points are 1, 2, 3, 4, and the P- and @Q-branches respectively are 13, 24: ¢(1)=1. Again n=2, the points are 1, 2, 3, 4, 5, 6, 7, 8: we may join 13, 24 or 13, 28 or 17, 28 or 17, 68, leaving in each case four contiguous numbers which may be joined in a single manner: that is #(2)=4. Or, what is the same thing, the partitions of 1 are 0001, 0010, 0100, 1000, whence ¢ (2) =4 { (0)}°(1)=4 Again n=3, the partitions of 2 are 0002, &e, CAUCHY’S THEOREMS OF ROOT-LIMITATION, 411 (4 of this form) and 1100 (6 of this form): that is $(3)=4 {p(0)}’ 6(2) + 6 {[6(O)} [H(1) =4.4+6.1=22, and so on. 38. Starting from the 4n points P and Q, and joining them in any manner subject to the foregoing conditions, we have a diagram representing two rhizic curves; and colour- a | i) AMI Hi ing the regions we verify that the n roots are all of them right-handed. We have for instance the annexed figure (n=3) Having drawn such a figure we may by a continuous variation of the several lines, in a variety of ways introduce a double point in the P-curve, or in the Q-curve: and by a continued repetition of the process introduce double points in each or either curve: thus for instance we may from the last figure derive a new figure in which the P-curve ‘has a Vot. XII. Part IT. 53 412 Pror. CAYLEY, ON THE GEOMETRICAL REPRESENTATION OF node at N. It will be observed that here it is no longer the case that each P-branch intersects one and only one @-branch: the P-branch 1—9 does not meet any Q-branch, but the P-branch 7—11 meets two Q-branches. But looking at the figure in a different manner, and considering the P-branches through N as being either 11-N—1 and 7—N—-9,. or 1—N—7 and 9— _N—11, then in either case each P-branches intersects one and only one Q-branch: and in this way, in a diagram in which the two curves have each or either of them double points, but neither curve passes through a double point of the other curve, the theorem may be regarded as remaining true—we in fact consider the diagram as the limit of a diagram wherein the curves have no double points. It will be recollected that the equation F(z) being without equal roots, we cannot have either curve passing through a multiple point of the other curve. And we thus see that the various figures drawn as above, without double points are, so to speak the types of all the different forms of a system of rhizic curves P=0, Q=0. In connexion with the present paper I give the following list of Memoirs: — Caucuy. Calcul des Indices des fonctions. Jour. de [Ecole Polyt. t. xv. (1887) pp. 176—229. First part seems to have been written in 1833: second part is dated 20 June, 1837. Refers to a memoir presented to the Academy of Turin the 17th Nov. 1831, wherein the principles of the “Calcul des Indices des fonctions” are deduced from the theory of definite integrals: I have not seen this. Sturm AND LiouVILLE. Demonstration d’un theortme de M. Cauchy relative aux racines imaginaires des equations. Jiowv. t. I. (1836) pp. 278—289. Sturm. Autres demonstrations du méme theortme. Do. pp. 290—308. These two papers contain proofs of the particular theorem relating to the roots of an equation F(z)=0, but do not refer to the general theorem relating to the intersection of the two curves P=0, Q=0: the special theorem of the existence of the n roots of the equation #(z)=0 is considered. SYLVESTER. A theory of the Syzygetic relations of two rational integral functions, com- prising an application to the theory of Sturm’s functions and that of the greatest algebraical common measure. Phil. Trans. t. CXLUI. 1853, pp. 407—548. Dre Morcan. A proof of the existence of a root in every algebraic equation, with an examination and extension of Cauchy's theorem on imaginary roots, and remarks on the proofs of the existence of roots given by Argand and Mourey. Camb. Phil. Trans. t. x. (1858). ; Contains the important remark that the two curves P=0, Q=0 are such that two branches, one of each curve, cannot inclose a space; also that the two curves always [i.e. at a simple intersection] intersect orthogonally. Airy, G. B. Suggestion of a proof of the theorem that every algebraic equation has a root. ° Camb. Phil. Trams. t. x. (1859). CAUCHY’S THEOREMS OF ROOT-LIMITATION. 413 Caytry, A. On a proof of the theorem that every algebraic equation has a root. Phil. Mag. t. xvitl. (1859), pp. 436—439. Watton, W. On a theorem in maxima and minima. Quart. Math. Jour. t. x. (1869) pp. 253—262. Caney, A. Addition thereto, pp. 262—263. (Relates to the curves P=0, Q=0.) Watton, W. Note on rhizic curves. Quart. Math. Jour. t. x1. (1870) pp. 91—98. First use of the term “rhizic curves:” relates chiefly to the configuration of each curve at a multiple point, and of the two at a common multiple point. Watton, W. On the spoke-asymptotes of rhizic curves. Q. VM. J. t. x1. (1871) pp. 200—202. Watton, W. On a property of the curvature of rhizic curves at multiple points. Do. pp. 274—281. Bsoruine. Sur la séparation des racines d’equations algebriques. Mem. d Upsal (1870) pp. 1—85. (Contains delineations of some rhizic curves.) 53—2 Il. On the Inequalities of the Earth’s Surface viewed in connection with the secular cooling. By Rev. Osmonn Fisuer, M.A., F.G.S., F.C.P.S., late Fellow and Tutor of Jesus College. [Read December 1, 1873.] Ix a paper which I read before this Society in 1868, I attributed the elevating force which has raised mountain-ranges to the contraction of the heated interior of the earth, and consequent wrinkling of the crust so as to accommodate itself to the diminished nucleus. This was an old hypothesis; but I believe the amount of horizontal pressure produced by that means had not been estimated before. I shewed that it is equal at the earth’s surface to the weight of a piece of rock of the same section as the stratum, and 2000 miles long ; enough to crumple up and distort any rocks*, and I also proved that a still greater horizontal pressure than this would be produced at any moderate depth. Towards the conclusion of the paper I made a rough estimate of the dimensions of the mountains which such a process might produce, upon certain hypotheses as to the amount of compression and thickness of the crust, which on a cursory view appeared to me probable. The object of the present paper is to attempt to arrive at a more definite conclusion upon this part of the subject. In order to render clear what follows I am obliged to recapitulate the substance of a small part of a paper upon the formation of mountains which I have already published in the Geological Magazine. Let ABCD be a layer of rock of unit of width, length 7, and depth &. And suppose the abutments at AC and BD to approach each other through the space le, where e is a * See Pratt’s Figure of the Earth, fourth edition, p. 203, note. + Geol. Mag. Vol. x. p. 248. Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE. 415 small fraction. Then the layer of rock in question would assume some new form, as one of those given in the figure, or any other whatsoever possible. Let us now seek for some simple laws which must govern the disturbed strata in spite of the confusion which appears to reign among them. Let a, a, &c., be the areas formed by the upper curved line above AB, and b, b, &c., the areas formed by the same line below AB, It is not necessary that the as should be equal to one another, nor yet the is. They are used simply to designate the areas in respect of their positions. We will call AB “The datum level.” In like manner let a, 8 be similar areas for the lower datum level CD. Then the space included between the curved lines must be equal to Ab Cd =kl (1 +e). It is also evidently equal to ABCD +a+a+&.+B8+8 + &e. —b—b—-—&c.-—a—a—e, or, denoting the sums of the quantities of the same sort by the symbol =, we get kl (1+e)=kl+% (a) — = (0) + (6) —¥ (a). “ kle= (a) —> (6) + & (8) —= (a). (1). Since the pressure is supposed to take place in a horizontal direction, it will not have any direct effect to raise the centre of gravity of the portion of the crust under considera- tion; so that, if the layer in question rest upon a liquid substratum, we may expect some portions of the disturbed crust to dip into the superheated rocks. But in that case a corresponding volume of such subjacent rock must rise into the anticlinals. Hence, > @=a (6). And the equation becomes 5 kle =% (a) —& (6). In order to render this reasoning applicable to the section of a surface of any form it is only necessary that the pressure, which causes the compression, should be everywhere tangential to the surface, and that gravity should be perpendicular to it. Hence it is applicable to the earth’s surface, although that surface is not strictly regular, and may contain local elevations and depressions affecting the mean figure (that is the figure as unaffected by corrugation), which, though of small amount as compared to the dimensions of the earth, may be large as compared with the quantities of which we have to take cognizance in this investigation. But the above suppositions cannot represent accurately what has occurred in nature. For they assume the upper and under parts of the crust of the earth to have been compressed horizontally by the same amount. In reality compression must have gone on gradually ever since a solid crust was first formed, and must be greatest at the upper datum level. We must therefore consider e to be the mean coefficient of compression. In the next place it will be necessary to extend our considerations from a section of unit of horizontal thickness to any proposed area of the earth’s surface, and eventually to that of the whole globe. 416 Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE A clear conception of what will then be the upper “datum level” is important. It will be an imaginary surface which occupies the position that the surface of the crust would occupy at the present time, had it been perfectly compressible in a horizontal direction ; so that no corrugations would have been formed in it. As soon as elevated tracts had once been formed by the corrugation of the cooled crust, all the material above the datum level, however distributed, must have been derived from matter originally beneath it, and subsequently raised above it. As often as further compression of the crust has taken place, every fresh addition to the sum total of the quantity of matter above the datum level must have accrued by the elevation of matter from below it. For the matter which was already above it, however freshly corrugated, or rearranged by water action, or otherwise, cannot have been thereby altered in quantity. ; Moréover each additional contraction will have acted upon a crust thicker than it was before on account of its having become in the meanwhile solid to a greater depth. We have hitherto confined our symbols to a vertical section of the earth’s crust of unit of width. We will now extend them to a portion of the crust whose length is 7 and width w, and the depth k; e and e’ being the mean coefficients of compression in the directions of length and width. Then, if we neglect the product ee’, our equation will become klw (e+e) == (A)—= (B), where A and B are now the volumes of the elevations above and depressions below the datum level. If in this equation we put w=1 and e’=0, it reduces to our former equation. If we put e=e' we get 2klwe => (A) — = (B)...... (A), which we may take as the general expression corresponding to any area of the surface. The tendency of the corrugations over a given area will be to form two systems at right angles to one another, thus relieving the whole compression, Where one corrugation intersects another, we shall have a part of the volume common to both. But physically the same space cannot be occupied by two distinct volumes of rock. Hence at every such intersection there must be an increase of altitude in the ridges sufficient to contain within the contour an additional volume equal to the common portion. This appears to occur in nature where a peak often occupies the place of intersection of two ranges, as if the VIEWED IN CONNECTION WITH THE SECULAR COOLING. 417 denudation which has shaped out the mountains has had a greater amount of matter to remove in such a situation. The result expressed by the equation may be extended to the whole globe by con- sidering its surface composed of elementary rectangular areas, and summing them; whence Sarke => (A) —% (B)...... (B). In which expression r is the radius of the earth measured to the present position of the datum level. k is the present thickness of the cooled crust. e is the mean total linear compression of the crust. = (A) is the volume of the total elevations above the datum level. = (B) is the volume of the total depressions below it. And 7=3-14159. It is worthy of remark, that the relation between lateral compression and elevation expressed by the above equation, (B), in no way depends upon the arrangement of the disturbed rocks, nor upon the time at which the successive movements have taken place, nor upon whether or not subsequent denudation, or any other mode of action, has redistributed the elevated matter. Nay, even if by human agency excavations have been made, their results will be included in our equation, which is perfectly general and true, so long as it is strictly interpreted. It requires also im particular to be noticed that it is not assumed that moun- tain-ranges need be anticlinals. All that is assumed is that they are carved out of tracts elevated as a whole by lateral pressure. This appears to be the proper place for considering the manner in which the results of the deposition of sediment over a limited area would enter our equation. Geologists believe thick deposits to have been accumulated upon sinking areas, and attribute the subsidence of the area to the weight of the deposit itself. If rocks plastic from heat, or any other cause, come within a moderate distance of the surface this is possible. But the displacement of such plastic matter from beneath the new deposit must cause the rise of an equal volume somewhere else, so that if the terms of our equation were in any way affected (which would depend upon whether the effect extended below the datum level) it would be by an addition to the term %(8), and a consequent equal addition to E(a), and their sum would be zero. So that, as already intimated, our equation will include any such effect of denudation and the resulting deposition. Such an action as has been just referred to would produce a slight amount of crumpling in the strata, but only a very slight one, unless the new deposit sunk through some subjacent strata, and pressed them sideways out of its way; as I have shewn has happened in the contorted drift of Norfolk*. But it is only possible for this to have happened where the deposit produces a very much greater pressure upon the area which it occupies, than the rocks displaced by it did; that is when it is very much thicker than * Geol. Mag. Vol. v. p. 550. 418 Mr oO. FISHER, ON THE INEQUALITIES OF THE EARTH'S SURFACE they were. And it is clearly impossible that it should elevate any displaced strata to a greater altitude than its own, which is necessarily limited to that of the sea-level where the deposit is going on. The test of such a mode of action would be to enquire whether any portion of the strata which ought to lie beneath the thick deposit can be discovered in a contorted condition abutting upon it. Thus far, for the sake of perfect generality, it has been assumed that the disturbed upper strata rest upon a liquid, or at least a plastic, substratum. In what follows, the enquiry will be made whether this condition of the subjacent rocks is probable. And the method taken will be to consider whether the contrary supposition, that the sub- jacent rocks are solid, will account for the amount of corrugation which the earth actually exhibits. If the earth had cooled as a solid body, the outer layers at any epoch having attained their complete amount of contraction sooner than the interior, would have been too large to fit the interior after the cooling had proceeded further. They would there- fore become corrugated. But in this case that corrugation would have necessarily taken place wholly in an upward direction; and there .could be no places where any portion of the surface could have become depressed below the datum level. Hence upon this hypothesis we may introduce into our datum-level equation the supposition that =(B)=0. And it becomes Srrke == (A). A little consideration will give the following geometrical relation: The volume of the Sea above the datum level =the area of the whole surface of the globe x the depth of the datum level below the sea-level—the volume of rock displacing water between those levels. Let S=area of the Sea. D=its mean depth. L=area of the Land. W=volume of the Land. d=the depth of the datum level below the surface of the Sea, considering these as parallel. Now the volume of rock displacing water between the datum level and the sea- level will be =(A)—W, and since in the case supposed = (A) =87r*ke, we shall obtain the following relation : SD = (S+ L)d ~ (8mr*ke— W). But it is to be observed that this relation assumes the surface of the ocean to be everywhere parallel to the datum level, which is itself supposed to be parallel to the original surface of the globe when it was first covered with 4 solid crust. But the distribution of the great continents and oceans, as well as observations on the plumbline, render it probable that there is some cause which makes the density greater beneath the great oceans, and tends to accumulate the water upon them, so that their depth is greater VIEWED IN CONNECTION WITH THE SECULAR COOLING. 419 than it would be were that solely due to a relative depression or locally diminished curva- ture of the sea-bed. Let us then call X the accumulated volume of water in excess of that, which the oceans would contain, were their surface parallel everywhere to the datum level; retaining D as the observed mean depth of the sea, but using d as measured from the supposed surface as just defined. In order to allow for the accumulated water, we must correct our equation by sub tracting Y from the actual volume of the ocean, so that SD—X=(S+L) d—(87r*ke— W). Observing that S+Z is the whole area of the globe, and therefore equals 477°, and that SEL would be the depth of a layer of water of the volume of X if it were equally spread over the whole globe, which depth we will call 6. The above equation gives vol. of sea—vol. of land ee as area of the globe The area of the ocean is estimated to be 146 millions of square miles, and that of the land 51 millions*. Mr Carrick Moore+ has shewn that Sir John Herschel was misled by an inaccurate expression in Zhe Cosmos in his estimate of the mean height of the land being 1800 feett, and that it ought to be put at half that, or 900 feet. The average depth of the ocean is reckoned at three miles, and its volume at 438 millions of cubic miles. By substituting these numbers we obtain Qhe =d + 6 — 2°2, AB the datum level. SS the sea level supposed parallel to it, O the actual surface of the ocean. S'S’ the surface of the accumulated water levelled down, 0,D=d, O'D=d+8. Our next step must be to fix upon a value for d+6. This is the depth of the datum level below the surface of an imaginary ocean, which is rather less deep than the actual one. Now it is not probable that there is any place where the bottom of the ocean coincides with the datum level. Nevertheless there seems reason to think that this will approximately be the case in the deeper parts. If this be so, d+6 will not differ * Herschel’s Physical Geography, second edition, p. 19. + Nature, Vol. v. p. 479. t Physical Geography, p. 118. Wor, Xl) Parr it 54 420 Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE greatly from the measure of the deepest parts of the ocean, and is not likely to exceed it by much, because, as already mentioned, the surface from which it is reckoned is some- what lower than the actual surface of the ocean. If the deepest parts of the ocean do not coincide with the datum level they must be one of two things. They must be either depressions below it, belonging to the series S(B), or else they must be the places where the ocean bottom is least raised above the datum level. That they should be depressions beneath that level is scarcely possible under our present assumption that the earth has cooled as a solid; whence we have concluded that >(B)=0. But if we take the other alternative, and suppose the ocean-bottom to be raised above the datum level, even where the depths are very great, then we are taking d+6 too small, and our estimate for 2ke will be too small. And this appears by far the more probable case. On the whole, then, we may feel pretty well assured that we are within the mark if we put for d+6 the measure of the more profound parts of the ocean. The mean depth being taken at three miles we shall not be much in excess if we put d+6 as equal to four miles. Our value for 2ke then becomes Qhe = 4 —2°2 miles, =1°8 miles, = 9504 feet (about), a value which is more likely to be too small than too large. We will now proceed to seek for a probable value for 2ke on physical grounds, in order to compare it with that just found. For this purpose we must estimate the compression, Ave, corresponding to the present thickness & of the crust and to a length J upon the present surface. ; Let ABCD be a section of a portion of the crust of length 7 and depth &. Then e is the mean coefficient of compression for the whole, which will depend upon the depth to which the cooling down to the melting temperature under pressure has advanced. The area which by its having become compressed will have gone to form the corruga- tions AFB will be BDE, in which BE is the quantity by which AZ has been com- pressed, while CD has not been compressed at all. Then if BM=a, and c= the compression at the depth a, MQ=lIe. VIEWED IN CONNECTION WITH THE SECULAR COOLING. 421 k And the area in question will be | Iedx, 0 which must be equal to the area AFB, or to kle, k Hence ke = | oda. 0 We have now to express ¢ in terms of a. For this purpose I avail myself of Sir W. Thomson’s paper “On the Secular Cooling of the Earth,” using the symbols and a modification of the diagram there given, but not neces- sarily adhering to his values of the constants. In the event I think it will appear that we must adopt some modification .of his views as to the condition of the globe at the time at which a rigid crust commenced to form, because we shall find that they lead us to a value of 2ke which is not in accordance with natural appearances. Sir William Thomson’s views are thus expressed. “The earth although once all melted, or melted all round its surface*, did in all probability really become solid at its melting temperature all through, or all through the outer layer which had been melted, and not until the solidification was thus complete or nearly so did the surface begin to cool}.” The point which is material in the application of his investigation is, that cooling is to take place by conduction and not by convection. Diagram modified from Sir W. Thomson’s paper on the secular cooling of the Earth. ON the depth below the surface=x. NP the excess of temperature at that depth above the temperature of the surface=v. Og the excess of the melting temperature above that of the surface=V, 7000° Alar. V is the melting temperature of rock; which Sir William Thomson places pro- visionally at 7000° Far., a very high estimate. Mr Mallet has shewn that the tempera- ture of slags run from an iron furnace is less than 4000° Far.t V _ 7000° = —— = —___ = 7900° Far, lvq “88622 * Dr Sterry Hunt supports the latter view. American | Royal Society of Edinburgh, 1862, Also Thomson and Journal of Science, Vol. v. p. 264. Tait’s Nat. Phil., p. 711. + Seeular Cooling of the Earth, § (s). Trans. of the t Trans. Royal Soc. 1873, p. 198, 54—2 422 Mer O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE The quantity a is thus defined: a=2Vx«t, where « is the conductivity of rock, found by Sir W. Thomson to be “400 for unit of length a British foot and unit of time a year.” And ¢ is the number of years since the Earth became solid, which must have elapsed in order that the rate of increase of temperature in descending into the Earth should be what it is observed to be. We learn (§ p.) that yp=! | ae dz. 0 When then the rigid crust was commencing to form, the whole globe, or at any rate its outer portion to a considerable depth, was at the melting temperature V° Far. represented on the diagram by Og or Nn. At a subsequent time the temperature at the depth ON or « was 2’, represented by the ordinate NP. Hence at that depth cooling has taken place through (V—v)’, represented by Pn. Now the compression which each layer of the crust undergoes is caused by the contraction of all the matter beneath it. Let E be the cubic contraction for 1° Far. Hence the layer of the sphere at the depth x whose area is 47° and thickness 82 will have contracted by EH 4a7* Pn dz, and consequently the whole contraction at that depth will be H4ar’f Pnda, where the integral is to be taken from 2=a to that depth to which contraction has taken place. We may neglect the change in r for a moderate depth, since it is multiplied by the small quantity Z. The diagram shews that the difference between the actual and the melting temperature at the depth 2a becomes small. It is Vx 00468." Li? == * Generally Pn=V- als & da: b =i a 1 a 2 1 ‘a? 3 Sal 1-3 tra) -3(e) 3 OS Win =) + we.) The general term of the series is Z 1 1 tr) 2n+1 (1) ot mrils) : Suppose the superior limit of the integral to be 2a. Then this becomes Me b 1 panty (=}) |n In + 1 And the next term to this will be or gzintiti Fe aoa ean al pants Se aiya z jnt1 2n+3° And the sum of the two terms eA eater 2° sa aes bee ( 1 2n+ 3) gett In* — 3n—-1 in+1 (Qn +1) (2n+3)" =(-1)'b VIEWED IN CONNECTION WITH THE SECULAR COOLING. 423 In order to obtain a value of the contraction at the depth 2, which we are assured is larger than the truth, we will take the integral from a= to «=2a for the contraction to that depth, and add the contraction of all below on the assumption that it continues to the centre as great as it is at the depth 2a. We will therefore consider the whole cubic contraction at the depth a to be Ean? | Bee a Vx 00468, where r’ = r — 2a. But what we want is the linear compression of the shell in a horizontal direction at this depth, and that will be the same as the linear contraction of a spherical surface at the same depth, We will therefore find the mean contraction of the whole below that, as if the contraction was equally distributed throughout its volume. Suppose £’ to be the mean coefficient of contraction of the sphere below the given depth 2. Air Therefore E ar = HArr’ a Pudz+E —_ V x 00468. Hence B= ef " Pn dx + EV x ‘00468 nearly, and the coefficient of linear contraction, which is that of compression for the next overlying shell, is one third of this, JOM i 1 or ¢=— | Pn dz + 3 EV x ‘00468 nearly. k The entire compression of the shell, or ke, which equals | edz, will therefore be less than 0 k 2a k al (| Pade) de+% EV x:00468 [ Wade. @ lle. Z 3 P And because the melting temperature is nearly reached at the depth 2a, we may take that for the thickness of the corrugated crust. So that he< ZE(. “Pa de) de+ EV x ‘00468. In this expression x=2 will give the third and fourth terms of the series; n=4, the fifth and sixth, &e., &e. The sum of the two first terms of the series is —0-666666. Making m successively 2, 4,...18 (for which last value the first significant digit will not occur before the 7 place of decimals), we get for the sum of the first 20 terms 1548749 — 0666666 = 0-882083, whence at the depth 2a, Pn=V— 5 x 0°882083, (where b= Ai a) ’ 2 us 882 cae (/m = 177245), = V x 0:00468. 424 Mer O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE In which expression 2 =V-2['« ne a ° After a somewhat tedious calculation the value of the factor under the sign of integration comes out | V2a? — a*b x 1:549800.* * Tt has been already proved that eS +13 +CW inal ailel a, dager [Pade = Ve oe 5 ee 55 Gan +( Veee = 1 1 aint ~"! In In+ 1 2n+2 a —=t}: The general term of the series is ,0 1 1 (2 2n+2 (—1)"c in Gav) Gned)? (‘) “CY carp GarDe) Taking 2a as the superior limit of the integral, the general term becomes “ab 1 Oro es jn (22 + 1) ae 2) a gent =e face 1 genth pats nas 2n+1° And putting n+1 for m the next term to this will be 1 pats Fa a n+ 3” And the sum of the two terms 1)" ab geatt 1 93 (-)) wai est- Gy eo} n+l 9 aS gf li na sO ERS ons (AN n+ 2(2n+1) (2n+3) Giving to m the series of values 2, 4, 6,...,14, we obtain for the sum of the terms from the 3™ to the 16", both inclusive, ab x 0:0606654. Hence we have for the value JPndx, when «= 2a (or at the superior limit), V2a- ab (2-5 -,+0 606654) = V2a—ab ge au VIEWED IN CONNECTION WITH THE SECULAR COOLING. 425 Taking the observed rate of cooling at ;4th of a degree Far. per foot in descending, the time since the supposed solidification at 7000° took place will have been 100,000,000 - V years, and @=2V«t=400,000 feet, or about 80 miles. And }=—= =7900°. ior 2a 2: I Pndx = V (2a—x) —ab x 1:273320 b (a 1 lat 1 1 Lat ail omava: [2° 5° Cat 1 1 gente Ves ae) “ssp ou ( if “Pnilz) de = V (2ue -$) - 1-273320abx gof[blanl lie 1g qs oe bial eee + 5 1 ] =) 7 We ir 2n+22n+3 a” : The integral has to be taken from «=0 at the surface to that value of « which expresses the thickness of the rigid crust beyond which corrugations will not have been formed. Now the melting temperature V is reached within an extremely small error at the depth 2a. We may therefore take 2a as the superior limit, and the above quantity becomes 2a 2a / (| Pnda) dee = V (4a? — 2a%) — 1-273320 x 2a" inp eels see ual, AI Mth VL ile al? 3 Gees (Dig G a ae eel 1 (Qa)\ns re pa armel Fe ase of which series the general term after simplification becomes gat 1 [w+] (2m +1) (Qn +8) ° gant4 4 Dpers Qn? +n+6 n+2 (2n+1)(2n+ 3) (2n+5)° (—1)"a%b and the next to it —(-1)"a@b and their sum, after reduction, (—1)"a*b This is the sum of the two corresponding terms in the former series (A) multiplied by the factor 2 1 fe 2 (n+ 2) ” In+5 2n? = (n = 2) Giving to m the values 2, 4, 6,...14, we obtain for the sum of the terms of the series from the 3™ to the 16%, both inclusive, 1 ab x 0:193840. Hence I ([ ‘Pade dx = V2a?—a’b x 2°5436404+ (sy = V2a?— ab x 1:549800. — — : 3) + a’b x 0°193840 426 Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH'S SURFACE With these values of the constants we obtain for the value of our Integral, 281100000000000 sq. feet = 2811 x 10" sq. feet. In order to fix upon a value of Z£, the coefficient of contraction for 1° Far. I refer to some experiments on a large scale conducted by Mr Mallet*. And upon reducing his results I find for slag from an iron furnace H = 0000215," whence = the linear contraction ="000007 ; which does not differ more than might be expected from the estimates made at much lower temperatures for rock by Mr Adie+. It appears that the integral = 4xtV (2 a —) = 4xtV x 02533, 4 T which varies as the time multiplied by the melting temperature, as the true value of ke would also do, if it were fully integrated. If we take the Earth’s mean radius at 20,890,000 feet,? we obtain as the final result for the two terms that ke is less Se ee 1 To obtain the coefficient of contraction for 1° F. from Mr Mallet’s experiments. The cones of slag began to solidify when the containing iron moulds were at 450° F. At that time the mean contents of the moulds was 8231-9229 cubic inches, and the volume of the slag when measured at 53° F. was 7700°2303 cubic inches. The slag entered the cones at a temperature of 3680°. By Mr Mallet’s estimate, founded on the time in which solidification commenced, the solidifica- tion commenced at about “3062°, or say 3000° F.” Hence the number of degrees through which the temperature fell was 3062°—53°=3009°. We have therefore to determine Z, the coefficient of cubical contraction between incipient con- solidation and 53° F. 7700 = 8232 (1— #3009), jg pu Oe naa = B232 x 3000 3000 = = 0000215 ; and = 0000071. ? The Earth’s mean radius in feet = 20890000. Hence the first term of the quantity to which ke is approximately equal will be _ 2811 x 10" x 215 ~ "2089 x 10* x 107 = 289-30 feet. 9 The second term in the value of ke is > EV 00468 (=mV, suppose) * Loc. cit., p. 201, + Trans. Roy. Soc, Edin. Vol. xu. p. 370. VIEWED IN CONNECTION WITH THE SECULAR COOLING. 427 than 289+4183 feet. Probably half the value of the latter term will be a full estimate. For among other reasons for its being over estimated, it will be perceived that the contraction for the decrease of temperature, V x 0:00468, has been attributed to the crust whose depth is 2a in addition to what is proper to it. We may say then that ke = 380 feet, and 2ke = 760 feet. In other words, if all the elevations which would be produced upon the above supposi- tions were levelled down, they would form a coating over the whole globe of less than 800 feet in thickness. This is an exceedingly small result, especially when we consider that our calculation has been conducted so as to make the result larger, rather than smaller, than the exact value would be, if the nature of the analysis admitted of that being found. But now if we introduce for V the temperature which Mr Mallet has determined for melting slag, viz. 4000° F., instead of 7000°, and the corresponding value for ¢ of 33,000,000 years’, the result becomes surprisingly small indeed. For then, by the rule of proportion proved on the last page, 2he = 144 feet only. In this case 2a would be about 100 miles. In what direction are we to look for an explanation to account for the discrepancy between a value for 2ke anywhere between these very wide limits, and the value 9504 feet, which we have previously obtained by estimating the existing irregularities of the earth’s surface; for the latter is from 11°8 to 66 times as large as the former? It does not seem probable that any error in the values of the melting temperature between those wide limits, or in the value of the conductivity on the one hand; nor in estimating the mag- nitude of the irregularities of the earth’s surface on the other; can explain the difference. = 800000 x :000007 + 7000 x -00468 7 468 =8 x 10°x7x 10° x THe * 10° _ 8x7 x7 x 468 ks 10° =183 feet. Probably half this value will be a full estimate; for among other things in the calculation the contraction for mV is attributed to the depth 2a in addition to what is proper to it. 1 To find the time since the Earth was all melted, on the supposition that it has cooled as a solid, and that the temperature was then 4000° Far. as determined for slag in Mr Mallet’s experiments. Referring to Thomson’s paper loc. cit., he a Taking 7 of 1° as the rate of decrease at the surface where x=0, we get from the above t= 33 millions of years. Vou. XII. Parr II. 55 428 Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE Very likely it will be replied that the great basins, shallow as compared to their areas, which are occupied by the oceans, are not the troughs of great corrugations, but are de- pressions caused by vertical contraction in the direction of the radius; in which case they would not be contemplated by our datum-level equation. It is possible that this may be so to some extent, and I have accordingly made a correction which will help to meet the case. But if the elevated tracts which form continents are due to lateral compression of the crust, of which I think there can be no doubt, then the depressions which are their necessary correlatives must be sought beneath the oceans. Were the strata of the earth in the main horizontally disposed, vertical movements, however difficult to be accounted for, might nevertheless be appealed to. But the generally corrugated condition of the more ancient rocks forbids us to ignore the effect of lateral compression in elevating and relatively de- pressing the surface. It is thus that we believe the areas to have been raised, out of which denuding agencies have modelled the present land, while the depressions have served as the necessary receptacles for the waste materials so abundantly removed. In passing I may observe, that I attribute the greater amount of corrugation observ- able in the older rocks simply to their exhibiting the accumulated effects of repeated compression, to which the newer deposits have not been so often subjected. But as regards the original crust of the earth, formed from the cooling of the melted surface, I have shewn that the compression, and therefore corrugation, if such has been produced, must diminish towards the lower parts. The observed corrugations alone seem sufficient to shew that the amount of compres- sion, to which the earth’s surface has been subjected, must have been very much greater than the result of our calculation can account for. For instance, in Professor Ramsay’s restored sections, figured in the first volume of the memoirs of the Geological Survey, the compression appears to lie between 4; and z4, while the country through which they are carried is by no means violently disturbed: the rocks consisting of the Old Red and Car- boniferous systems. Were the subjacent strata exposed, they might be expected to exhibit greater flexure. It has been shewn that the compression of a shell at the depth a is less than =| Pade++ EV x 0.00468. rj. 3 The compression at the surface will be obtained by putting e=0. And then, with the high value of 7000° F. for the melting temperature, we get the compression less than (0058, which is nearly ten times as little as that exhibited in the sections.’ nn 1 The compression of the shell at the depth a is less than @ [Pade TB x 00468, the first term of which =” { V (20a) ~ ab1-273320 + (series in a; VIEWED IN CONNECTION WITH THE SECULAR COOLING. 429 —_* It is easy to find a relation between 2ke and the compression at the surface, on the supposition of the law of cooling adopted above. It is, compression at the surface 3.6371 , Qhe ae For the compression at the surface put «=0, and this becomes = (V 2a —ab x 1:273320) 27339 = sav (2-7 a) ; We 2 _£ a (2 2:546640 re = 2 a¥ (18426073) = 400000 x 7000 x a _ 4x7 x 18964 se 10° 53 =a 0053 ; and the second term = -00023, the two together being less than 1000° 1 . E y Compression at the surface = zi Deh Vl SA 2607 Suswersse Meeted cee nicks Sos datas arose anaes (1) And ee ei (2 ee! oe) ee es Ma (2). ; i Substituting its value for ,/z, viz. 1°7467, and dividing (1) by (2), we obtain the ratio in the text. Putting 2khe = 9504 feet, t = 100,000,000 years, a=2,/xt = 400000 ; Qkhe x 3°6371 2 / Kt _ 9504 x 36371 400000 we get the compression at the surface, which = If, however, we take the lower value for V, viz. 4000° Far., then t = 33,000,000 years, a/xt = 114890 ; 55—2 430 Mr 0. FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE But if we give 2ke the value of 9504 feet, as estimated, then, with the value 7000° Far. for V, the compression at the surface comes out 7, and with the value 4000° Far. for V, it comes out nearly double that or +. These values are so large that although they agree well enough with Geological appearances, they would involve so great an accompanying diminution of the earth’s radius, amounting to from 600 to 300 miles, and such a different former condition of the interior, represented by the proportionally increased value of Z, as to render the steps of the above investigation manifestly inapplicable. Nevertheless from appearances, I think we must be prepared to give up our contraction theory altogether, or to admit a change of dimensions approaching to such an amount. Provided the formule used were applicable, then if we could discover the mean com- pression at the surface by observation, knowing the value of 2ke, we should have data sufficient for determining the melting temperature, the time, and the coefficient of con- - traction in the three equations: compression at surface = aa speatdladisna saab esimselemaes (1) NIG du_ Ve di Jaret in which at the surface e=(0 and aa ae da pile Z oe 2 it reais ame 2 E E ke = - AKERVEXO;25Od) senaceessecoreres (3) Can we then attribute the intense corrugations which meet our observation to any other source than that of lateral pressure caused by a shrinking globe? It appears to me difficult to conceive any other. No local change in the condition of the superficial strata seems competent to produce a sufficient amount of extension in the superficial strata. I cannot perceive, as I have already endeavoured to explain, how the deposition of thick beds upon the sea-bottom could effect it by their weight, and I have elsewhere, I think, proved that they could not do so by causing a new distribution of heat within the crust*. There is likewise much reason to believe that consolidation and metamorphism are accom- panied by a contraction in volume. : 9504 x 3°6371 and the compression = ——j59739 9000 x 4 ~ 230000 36 ~ 230 ='15 nearly nearly 1 == nearly. 7 * Geological Magazine, Vol. x. p. 248. VIEWED IN CONNECTION WITH THE SECULAR COOLING. 431 The supposition that the earth became solid throughout, or nearly so, before a crust began to be formed, necessitates the consequence that the contraction, out of which the compression would arise, must be almost wholly confined to the cooled upper portion: and it is upon this supposition that we have calculated its amount, and found it so much smaller than is warranted by natural appearances. If, however, we could suppose that a solid crust was formed upon the surface, long before its interior parts had fallen to the melting temperature, it seems that a much greater amount of compression might result, through the contraction extending far below the cooled upper portion. The objection made to this view is, that the crust would break up as fast as it formed, and sink into the underlying fluid until the whole was brought to the melting temperature. But Mr Mallet has shewn, in the paper already more than once referred to*, that the difference between the specific gravities of solidifying and molten rock is so small, being scarcely 6 per cent., that when we consider the intermediate condition of viscosity, we need not assume this breaking up and sinking of the crust. And if in its early stages shrinkage cracks did form, it seems likely that the fluid which welled up into them would immediately solidify and seal them up. Scrope tells us that “the interior of a lava-stream often retains a very high temperature for a great length of time after its emission, continuing to send forth vapour from its crevices and fumeroles, and probably remaining liquid, and even more or less in motion, throughout its central and lower portion for years+.” Sir W. Thomson clearly contemplates the mode of solidification from without inwards as not impossible, for he says “If experimenters will find the latent heat of fusion and the variation of conductivity and specific heat of the earth’s crust up to its melting point, it will be easy to modify the solution given above so as to make it applicable to the case of a liquid globe gradually solidifying from without inwards in consequence of heat conducted through the solid crust to a cold external medium.” If this supposition is admissible, there may have been a considerably larger nucleus enclosed within the crust in early times than we have at present, and a great portion of that nucleus may have consisted in superheated water, the rocks being in a state of igneo- aqueous solution’: and much of this water may have been blown eff in steam during ‘ The following remarks upon the above passage were received from a quarter which disposes me to place great reliance on them : “Tt is probable, or rather certain, that water substance, if it exists at great depths under great pressure and at high temperature, is neither a gas nor a liquid, being above its critical point. “Tn this state substances are easily dissolved in it, not however so much on account of a greater tendency to combine with water, as on account of a greater tendency of their own to dissipation. At still higher temperature the water substance becomes itself dissociated into oxygen and hydrogen. But it does not follow that the dissolved substances will be precipitated. The magma may be all the more complete the higher the temperature, because, though the bonds of affinity have fallen away, the prison-walls prevent the elements from escaping. But of all the known regions of the Universe the most unsafe to reason about is that which is under our fect.” * Phil. Trans. 1873, p. 160, § 50. + Volcanos, 2nd Edition, p. 84, § 8. 432 Mr O. FISHER, ON THE INEQUALITIES OF THE EARTH'S SURFACE yoleanic eruptions, by that means materially contributing to the diminution of the volume. I have suggested in my former paper read before this Society, that Mr Sorby’s observa- tions on the water enclosed in granitic crystals, along with crystals of chlorides, renders it probable that the steam emitted in eruptions may be a constituent part of the deep- seated rocks, for it is probable that but a small part of the water contained in any magma would become confined in the interior of the crystals. Here, however, the question arises whether it would be possible for a crust to form over a layer of molten rock in a condition of igneo-aqueous fusion. Would not the escape of the water cause a state of constant ebullition which would prevent the forma- tion of any crust until it had ceased through the escape of all the water? The idea of igneo-aqueous solution involves a condition of chemical combination between the water and the elements of the rock such that while that condition lasted there would be no tendency to separation between the two, and evaporation would be in abeyance. We can therefore conceive that at depths where the heat and pressure were sufficient there might be no tendency to evaporation and consequent ebullition, so that after the water had escaped to a certain depth ebullition would cease, and a crust be formed; but that more water would be ready to separate to a greater depth when its affinity for the rock became lessened through the abstraction of heat, or diminution of pressure owing to the crust being partially supported by corrugation. If such was the condition of the interior in the early stages of the cosmogony, a large portion of the oceans now above the crust may once have been beneath it, and thus we gain a novel conception of a sense in which the fountains of the abyss may once have been broken up. A somewhat analogous escape of elastic vapour from beneath a denser envelope is I believe considered to be now taking place in the Sun. The comparatively small specific gravity of the earth as a whole, considering the great pressure to which its interior parts must be subject, has been held to prove that it is even now in a state of expansion through intense heat*. But the question of its true condition is surrounded with difficulties. The suppo- sition made by Sir Wm. Thomson of a cool nucleus covered by a sufficiently deep layer of molten rock is adopted by Dr Sterry Hunt+, and would afford the conditions required by the argument of this paper. It would also afford the mean rigidity required to meet the objection to a fluid interior drawn from the absence of internal tides. But it would not account for the small specific gravity of the whole, nor yet would it meet the argument from precession in the form in which it was originally advanced by Mr Hopkins: for a layer of fluid beneath the crust would destroy a rigid connection between it and the interior. This form of the argument however has been attacked, and apparently with some success}. The late Archdeacon Pratt, in defence of the general * Herschel’s Phys. Geography, 2nd Bait. p. 7. + American Journal of Science, Vol. vy. p. 264. tt General Barnard on ‘‘ Problems of Rotary Motion,” Smithsonian Contributions, No. 240. VIEWED IN CONNECTION WITH THE SECULAR COOLING. 435 argument from precession, placed the matter in a simple light in a letter to Nature in 1871*, and in that form I think it would be met by the supposition of a cool nucleus covered by a molten ocean with a solidified crust. But I would invite attention to the fact that the conclusions I have arrived at, concerning a highly fluid condition of the interior, have reference to an early period of its history; while the tests of the tides, and of precession, are confined in their application to the present. Nevertheless I am disposed to think that, at any rate, what may be termed a superheated condition of the mass still exists at no very great depth below the surface. By which I mean that if it be solid the solidity is due to pressure, * Nature, Vol. tv. p. 344. III. On the Inequalities of the Earth's Surface as produced by Lateral Pressure, upon the hypothesis of a liquid substratum. By Osmonp Fisuer, Clk., M.A., F.G.S. [Read Feb. 22, 1875.] So long ago as in 1831, the late Professor Sedgwick wrote, “As the earth has apparently diminished in temperature, we have a right to look for some indication of a contraction of its dimensions. May not some of the great parallel corrugations of the older systems of strata have been produced by such a partial contraction *?” This theory is now commonly accepted, and there seems no necessity for limiting its application to the older systems of rocks. The principal difficulties connected with the inequalities of the earth’s surface appear to be with regard to the basins of the great oceans. Were the earth a perfectly smooth spheroid, without any inequalities in its surface, even in that case an excess of density in particular regions would determine a flow of water towards them, and it is conceivable that dry land and oceans might exist, even although the radial distances of the land-sur- faces and of the sea-bottoms from the centre of figure might be perfectly equal. That the distribution of the oceans is to some extent actually due to such a cause appears certain; for otherwise a whole hemisphere could not be almost entirely covered with water. On this point Archdeacon Pratt remarks, “There is no doubt that the solid parts of the earth’s crust beneath the Pacific Ocean, must be denser than in the corresponding parts on the opposite side, otherwise the ocean would flow away to the other parts of the earth.” And after explaining the reason for this statement he adds, “There must there- fore be some excess of matter in the solid parts of the earth between the Pacific Ocean and the earth’s centre, which retains the water in its place. This effect may be produced in an infinite variety of ways; and therefore, without data, it is useless to speculate regarding the arrangement of matter which actually exists in the solid parts belowt.” And Herschel considers that the prevalence of land and water in two opposite hemispheres “proves the foree by which the continents are sustained to be one of twmefaction, inasmuch as it indicates a situation of the centre of gravity of the total mass of the * ‘On the general structure of the Cumbrian Mountains.’ + Figure of the Earth, 4th ed., p. 236. 1871. A great Trans. Geol. Soc., Jan. 5, 1831, I am indebted to Mr | ice-cap would also have its effect. See Croll’s Climate and Bonney for this reference, Time, Chaps. xx1it, Xxrv. Mr FISHER, ON THE INEQUALITIES OF THE EARTH'S SURFACEH, &c. 435 earth somewhat eccentric relatively to that of the general figure of the external surface— the eccentricity lying in the direction of our antipodes: and is therefore a proof of the comparative lightness of the materials of the terrestrial hemisphere *.” A like conclusion as to the greater comparative density of the bed of the Ocean, was arrived at by Archdeacon Pratt from the fact that at seven Coast Stations out of thirteen, six being in the Anglo-gallic, and one in the Russian arc, it has been found that a deflection of the plumb-line exists towards the seat. “In fact,” he remarks, “the density of the crust beneath the mountains must be less than that below the plains, and still less than that below the ocean-bed. If solidification from a fluid state com- menced at the surface, the amount of radial contraction in the solid parts beneath the surface of the mountain-region has been less than in the parts beneath the sea-bed. In fact it is this unequal contraction which appears to have caused the hollows in the external surface, which have become the basins into which the waters have flowed to form the Ocean.” Latterly, however, he attributed the formation of mountainous regions to horizontal compression f. Mr Mallet, in his paper on Volcanic Energy, takes a similar view. He thinks that the land- and sea-boundaries were shaped out by radial contraction during the first great stage of the operation of refrigeration, while the crust was thin and flexible, owing to the rapid contraction of its viscous portion, which must then have been much thicker than the solid sheet above it §. Mr Hopkins appears to have been opposed to such views as maintain a difference of radial contraction; and to have held that lateral compression was the cause of the formation of the greater inequalities as well as of the lesser ones, for we find him in discussing M. Elie de Beaumont’s theories to have used these words: “The physical cause to which our Author refers the phenomena of elevation—the shrinking of the earth’s crust —is that which appears to me most unlikely to produce that paroxysmal action which his theory so essentially requires; and most likely to produce those slow and gradual movements which it scarcely recognizes. The actual depressions of the great oceanic basins, and generally the more widely extended geological depressions of the present or former periods, may, I think, be referred with great probability to this cause.” The theories respecting the formation of the larger features of the earth’s surface are discussed by Professor Le Conte in so lucid and unprejudiced a style, that his papers are well worthy of study. The followmg passage conveys his conclusions. “ Moun- tain chains and mountain ranges are therefore, I think, beyond question produced by horizontal thrust, crushing the horizontal thrust being * Physical Geography, § 13. 1862. + Figure of the Earth, pp. 200 and 206. 4th Ed. 1871. t In the third edition of the Figure of the Earth no mention was made of horizontal compression, but moun- tains were attributed to vertical expansion. In the fourth the author’s estimate of the horizontal force of compression is referred to. Vou. XII. Parr II. together the whole rock mass, and swelling it up vertically; the necessary result of secular contraction of the interior of § Trans. Royal Society, 1872, § 52 and § 60. Mr Mallet’s theory has been discussed by Mr Scrope in the Geological Mag., Dec. 11. Vol. 1. pp. 28, 127, and by the author, Journal of Geol. Soc., Vol. xxxt. p. 469, and Phil. Mag., Oct. 1875. || Presidential Address to the Geol. Soe., 1853. Geol. Journ., Vol, rx. p, Ixxxix, 56 436 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, the earth. The smaller inequalities, such as ridges, peaks, gorges, and in fact nearly all that constitutes scenery, are produced by subsequent erosion. I feel considerable confidence in the substantial truth of the foregoing statement of the formation of mountain-chains. As to the mode of formation of continents and sea-bottoms, I feel less confidence. It is possible that even these may be formed by a similar unequal yielding to horizontal thrust, and a similar crushing together and upswelling. If so, it would be necessary to suppose the amount of horizontal yielding in this case much less, but the depth effected much greater than in the case of mountain-chains. But as we find no unmistakable structural evidences of such crushing, except in the case of mountain-chains, I have preferred to attribute the formation of continents and sea-bottoms to unequal radial contraction*.” The last sentence of this passage appears to invite the remark that we cannot expect in general to have evidence of crushing except in those areas which are open to investigation, viz. on dry land. But there it is not confined to mountain-chains. Contorted strata are to be also found in what would be termed level countries, often covered with horizontal deposits of later date}: and this fact in itself proves that these contorted strata have been once covered by an ocean, offering a strong presumption that there are contorted strata now at the bottoms of the oceans. I have lately shown, in a paper read before this Society in December, 1873, that if the earth be supposed to have become solid throughout simultaneously, and if the sur- face of the ocean be now parallel to what would have been the surface of the solid globe if no inequalities of contour had resulted from its contraction (and I call such a surface the datum level), then the inequalities produced by the contraction through cooling would have been very much less than those which actually exist: for I have proved them to be from sixty-six to eleven and a half times as great as they would have been under the supposed circumstances, according to what assumption we make regarding the surface- temperature at which a permanent crust began to form, between the limits of 4000° Fahr. and 7000° Fahr. respectively. And I have suggested, as a mode of accounting for this discrepancy, that the earth need not have become solid throughout simultaneously, and consequently may have been considerably larger than it is now at the time when a solid crust was first formed. But it is obvious that in strictness this reasoning fails if the surface of the water, to which the inequalities are referred, be not parallel to what would be the surface of the solid globe, if no imequalities of contour had resulted from its contraction, that is to the “datum level”; its increased depth in the great oceans being in that case supposed due to greater density beneath them. And therefore I have in my investigation allowed a considerable margin to meet this supposition. But I suppose that the soundings at the edges of the oceanic basins, and indeed over their entire areas, will be admitted to prove that they are in the main essentially depressions. It also fails—and this case is more important—if, in accordance with some of the * American Journal of Science, 3rd series, Vol. rv. p. 462. 1872. + For example the highly contorted carboniferous strata of parts of Belgium. UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 457 views already referred to, the oceans occupy depressions in the earth’s surface produced by unequal contraction in the radial direction. Upon such a supposition the view of Archdeacon Pratt. and Professor Le Conte appears to receive support from my result; for if lateral compression from the cooling of a solid globe will not account for the mequalities of the surface, it seems natural to call in to our assistance difference of radial contraction. But it appears to me that the explanation of the formation of oceanic areas by this means (if mere cooling be its sole cause) is encompassed with more difficulties than upon the other view of compression. And these appear to have struck Professor Le Conte himself: for later im his paper* he says, “I am fully aware that there are some phenomena of movement of the earth’s crust, which are not explained by the foregoing theory. I refer especially to those great and wide-spread oscillations which have marked the great divisions of time, and have left their impress in the general unconformability of strata. The last of these great oscillations took place during the Post-tertiary period. I cannot explain these oscillations.” That there have been such oscillations of level seems incontestable. The same areas have risen and sunk, again and again, during the lapse of ages. The deposits of the Appalachian Chain we are told atta a thickness of eight miles+. Consequently the rocks now exposed at the localities where the lower beds of the series are at the surface, must at one time have been some miles at least below the sea-level. If that which is now dry land was once so far below the sea, it is not only probable from analogy that an area which is now sea was then correspondingly raised, but it is necessary that the area now occupied by sea must have presented land-surfaces, in order to have furnished the detritus out of which the rocks of the present dry land were formed. And in the case of the Appalachians, that land has been proved to have been situated to the Northward, and also the detritus to have come “especially from a continental mass to the Eastward },” where the Atlantic now rolls. The Islands of New Zealand, occupying a central position in the aqueous hemisphere, contain a series of deposits analogous to those of the northern hemisphere§, whence it follows that they must have been often submerged during periods when land-surfaces existed in what is now an extended ocean, from which lands the materials of their rocks must have been derived. In considering this great subject, we are at once brought face to face with the vexed question of the condition of the earth’s interior, whether it be rigid or not. On the one hand we have Sir W. Thomson’s arguments || for the rigidity, as deduced from the amount of precession. And although General Barnard has demonstrated that a fluid’ nucleus enclosed in a rigid crust would behave as a solid under the action of the precession producing forces‘, yet for practical purposes this will not assist the view which * Loe. cit., p. 472. p- 27, note, See also Hochstetter’s Geology of New Zealand. + Hall’s Paleontology of New York, p. 67. | Auckland, 1864. t Le Conte, American Journal of Science, Vol. tv. | On the Rigidity of the Earth,’ Phil. Trans., 1864. p. 464. { Smithsonian Contributions to Knowledge, No. 240. § Capt. F. W. Hutton, Geol. Mag., Decade u., Vol 1., | Washington City, 1871. 56—2 458 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, requires a fluid nucleus, for no crust can be supposed so rigid as not to yield to the pressure of tides internal to it. On the other hand, the close accordance of the figure of the earth with the present values of the radius, and of the angular velocity, neither of which can be considered to have been always exactly the same as they are at present, shows that there is probably some capability in the matter of which it is composed for readjusting itself when the values of these elements are changed. Combined with these considerations, we have those which point to a very high temperature in the interior, resting not only on the observed increase of heat in descending, but also, as Herschel has remarked, on the mean density, so much less than might be expected when the great interior pressure is taken into account. These facts taken together appear to give probability to Sir W. Thomson’s suggestion, that the high rigidity of the interior may be due to pressure, which may be sufficiently great to enable it to resist the deforming influence of the disturbing forces of the Sun and Moon. But although we cannot conceive of unequal radial contraction from cooling merely apart from rigidity, yet, on the other hand, rigidity will not of itself render unequal radial contraction probable. If the interior of the earth consists of matter so highly heated that it owes its rigidity to pressure, this seems to require that if there be mequality of con- traction, it must reside within the limits of the crust, cooled below the temperature of fusion for the pressure; that is, in that portion in which liquefaction would not result upon a reduction of the pressure. If unequal radial contraction from cooling has taken place, it seems that it must have arisen from one of two causes, or from those two eo-operating, viz. firstly, from a difference in the conditions of cooling under which any two areas in question have been placed, or secondly, from a difference in the constitution of the materials of the globe at those two places. Now if we take 0:0@002 as the coefficient of contraction of rock for 1° Fahr., which is what Mr Mallet informs me to be the mean, and if we take 4000° Fahr. on the same authority for the temperature of melting rock, then 008 is the coefficient for contraction from the melting point to 0° Fahr., and one-third of this, or 0°027, that of linear contraction. Hence it would require a thickness ef 400 miles cooled from 4000° to zero to give rise to a contraction of 10 miles upon any given radius, and if upon some other radius we conceive the contraction to have been from some cause or another one- tenth more, then on these suppositions we might have a difference of contraction of one mile upon the two radii in question. The excessive extravagance of these assumptions, and the inadequacy of the result even if we ignore the phenomenon of oscillation of level, sufficiently prove that the existing imequalities of the surface cannot be explained by difference of radial contraction through cooling. Nor is this all: for in regard to the conditions of cooling, it seems likely that those areas which are sea bottoms would cool more rapidly, on account of the heat being conveyed away by the ice-cold water which is found at the bottom of deep seas. So that the tendency from mere cooling would be continuously to deepen the seas, without any corresponding cause to re-elevate their floors. The supposition that the thick deposits which go on at the bottom of some seas may act as a “jacket” and prevent the escape of heat, meets this objection only to a small UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 439 extent ; first, because these deposits appear to have taken place over sinking areas, and to be so taking place still (and we can hardly attribute the stoppage as well as the reversal of the movement to a cause of such slight potency as difference of cooling has been shown to be). And secondly, because the heat conducted into the new deposits, and by some relied upon to re-elevate them, must be abstracted from the couches beneath, so that there can be no absolute increase in the amount of heat beneath the area in question*. If we now consider the second cause capable of producing a difference of radial contraction, viz. a diversity of the materials of the globe at the two places in question, it is palpable that this cannot explain oscillations of level. For that would require the materials to become changed in their properties from time to time, in a manner highly improbable. Humboldt’s suggestion of secular currents in the interior to explain the oscillations of level+ is directly opposed to the condition of rigidity of the nucleus. In short, it seems that no modification of the theory of difference of radial contraction, arising from cooling merely, can be relied upon; and it becomes important to enquire how far the theory of lateral compression, combined with the hypothesis of a liquid substratum, can account for the phenomena of elevation and depression of areas of the earth’s surface. And here it is proper to notice a paper of much interest by Professor Shalert, who does not appeal to radial contraction, but only to compression. He insists upon a distinction, which he thinks ought to be drawn between the corrugations exhibited in mountain- ridges, and the broader features of “continental fold and oceanic depression §.” “We find in continental folds broad curves of the surface, which narrow without exception towards the south, and which exhibit in no part of their structure the evidence of powerful lateral thrust, which are the most conspicuous phenomena of mountain-chains.” He seeks an explanation of this circumstance by attributing the larger folds of continental and oceanic areas to the adaptation of a solid crust to a diminished nucleus, and suggests that the cause of the corrugations, producing mountain-chains, must be sought in changes going on within the crust itself, and in no way connected with the regions below. If I under- stand his meaning, he considers that there has been formed a solid crust, of which the lower layers were formerly much hotter than they are at present, and that the tendency to equalization of temperature between the lower and upper portions of the crust, has produced corrugations mest intense in its upper layers. But besides this he thinks that the nucleus within the crust has shrunk away from the crust as a whole, so that corrugations of a wider and less abrupt character have been produced in the crust as a whole; these wider folds forming the broader features ef continental fold and oceanic depression. In support of his view, as regards the contraction within the crust itself, he adduces Sir Wm. Thomson’s estimate of the secular diminution of the rate of increase of heat m descending. But he does not fully explain why the nucleus should contract away from the crust as a whole. If the contraction be simply due to loss of heat from a solid * See a paper by the author, Geol. Mag., Vol. x. p. 251. | June 6, 1866, in Geol. Mag., Vol. v. p. 511. + Cosmos, Vol. tv. p. 19. Sabine’s Ed. § Loc. cit., p. 514. t Reprinted from Proc. Boston Soc. of Nat. Hist., 440 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, earth, the temperature increasing continuously from that of the surface to that of solidity due to pressure, there ought not to be any abrupt passage from the rate of contraction of the crust itself to that of the matter inside it, so that the series ought to be main- tained which this author denies to exist; “at one extremity of which could be placed the greatest relief of continental fold and oceanic depression, and passing gradually. to the most inconsiderable flexures.” Now it is remarkable that while, upon calculating the amount of inequalities of the surface, large and small, which can be got out of the compression arising from the con- traction due to cooling, taking the temperature in descending to increase according to Sir W. Thomson’s law for a solid globe, I have found that it falls very far short imdeed of that which exists; yet I have shown above, that we must, for all that, look to causes connected with lateral compression and not to difference of radial contraction to account for these inequalities. I conclude, therefore, that Professor Shaler’s idea is in the main correct, although he does not distinctly explain the reason of the crust being left behind by the contracting globe. He holds that the earth first solidified from the centre, and after a time from the surface also, so that there is probably at present a thin layer of molten material between the solidified erust and a solidified interior. According to this view all the contraction required to make up the deficiency between the inequalities as calculated for a solid globe, and the inequali- ties as they exist, must be obtained from the amount by which this intervening layer has contracted since the time when the crust first solidified. I do not see how this can suffice, unless we invoke some further cause of diminution of volume, besides the contraction which would be due to mere cooling; such as the escape of gases, and of water in the state of steam, from volcanic vents*. But in any case the conclusions of Professor Shaler and myself, formed on totally distinct grounds, lead to the same general result, viz. that there is a solid crust, resting in corrugations upon a liquid or quasi-liquid heated layer. The existence of such a layer of liquid or viscous matter seems to be rendered probable by the following consideration. The increase of temperature, though rapid near the surface, becomes less and less as we descend, so that, if the earth were once wholly melted, the temperature near the centre is not very greatly above what it is at a depth which, compared to the earth’s radius, is small. Consequently, if it requires great pressure to solidify the materials at such a temperature, it is probable that the melting temperature may be reached before the pressure is sufficient to solidify. Of course this reasoning is worthless, unless we admit, as I hold we may, that the crust need not break up and sink. A further argument of great weight in favour of the existence of such a layer is that the contraction of the sphere has been evidently relieved by corrugations, not uniformly distributed over the surface, but locally along lines of elevation, often separated by considerable intervals. ‘There must therefore have been a lateral movement of the crust interior in early times as well as down to the present, has wrote to the author in October, 1875; ‘There is one of the | been, and is, the cause of those subsidences of the crust, to points you put forward which never struck me before, but | which the basins of seas and oceans, and the crumplings of which now appears to me most valuable, namely that the | the terrestrial rocks are owing, far more than to any general enormous amount of steam that has escaped from the | contraction of the nucleus by cooling.” * The eminent vulcanologist, the late Mr Poulett Serope, UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 44] over the nucleus towards these lines of elevation, which can hardly have been the case unless there exists a more or less fluid layer upon which it rests*. layer exist, it would be subject to tides. Should such a liquid But they would be analogous to the oceanic tides rather than to tides in the solid earth, and their period would probably be greater than half the diurnal period, as is the case with the ocean-tides. Consequently the crest of the wave would not be opposite the disturbing body, and would not affect precession as much as if it were. But it might have an effect in causing disturbances in the crust, and inducing earthquake action, as the observations of M. Perry show to be probably the case. Assuming then that a solid crust rests in corrugations upon a liquid or viscous layer, which is capable of yielding to such forces as the gravitation of the crust exerts, we have to consider the conditions of equilibrium of such a crust. In the first place, the disturbances which we see that it has experienced, to the greatest depths which denudation exposes to our observation, prove that when once dis- turbed in lengths of any extent it may be considered flexible. Moreover, when it rests in corrugations upon the subjacent liquid, it must be in unstable equilibrium; that is, the corrugations can have no particular relation to the places where they occur, but might exist equally well at some other. Here we at once encounter a condition which renders oscillations of the surface possible, provided we can account for the disturbance of the position of equilibrium. We are not able strictly to investigate by means of mathematical formule the form which such a crust as I have supposed would assume. be developable into a spherical one. The surface of equilibrium would not But, nevertheless, it appears to me that we may obtain some insight into the general character of the corrugations, by enquiring what form would be assumed by a heavy flexible crust, resting upon a liquid within a rectangular trough shorter than the crust, for this would give an approximate idea of the contour of the surface upon the course of a section of the sphere perpendicular to its surface and cutting a set of corrugations at right angles. It may be assumed that the trough and crust were originally of the same length, and that the corrugations have been produced by the ends of the trough having been made to approach each other. * Captain Dutton, U.S.A., in an interesting paper ‘‘ On the Contractional Hypothesis” published in 1874, in which he uses a line of argument similar to that followed by the author in his paper ‘‘On the Inequalities of the Earth’s Surface,” of Dec. 1, 1873, and equally concludes that no contraction from cooling can adequately explain the plica- tions of the Strata, has the following important remark: “‘The determination of plications to particular localities presents difficulties in the way of the contractional hypo- thesis which have been underrated. It has been assumed that if a contraction of the interior were to occur, the yielding of the outer crust would take place at localities of least resistance. But this could be true only on the as- sumption that the crust could have a horizontal movement in which the nucleus does not necessarily share. A vertical section through the Appalachian region and westward to | the 100th meridian shows a surface highly disturbed for about two hundred and fifty miles, and comparatively undisturbed for more than a thousand. No one would seriously argue that the contraction of the nucleus had been confined to portions underlying the disturbed regions: yet if the contraction was general, there must have been a large amount of slip of some portion of the undisturbed segment over the nucleus.” He does not consider such a ‘*slip” possible, even with a substratum of ‘liquid lava.” American Journal of Science and Arts, Jan. 1874, p. 113. The author has however shown that the coefficient of friction required to allow of such a moyement of the crust would not be smaller than might be thought admissible. See ‘“‘Mr Mallet’s theory of Voleanic Energy tested,” Phil. Mag., Oct. 1875, Vol. 50, p. 317. 442 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, yy eae = Weel Suppose COX to be a vertical section of the trough, CPA the section of the crust whose thickness is ec, OX, horizontal, the axis of a, OY, vertical, that of y, Oiilan, APS, TEIN =) ry the radius of curvature at P, p the density of the crust, o that of the liquid, p the pressure of the liquid at P, t the force of compression in the crust in the direction of the tangent at P. We may suppose for the present that the flexibility of the crust is not impaired by its thickness. Then applying the conditions of equilibrium we obtain the following equations: (1) t=gpe(C—y), where C is an arbitrary constant. (2) p=gpc cos —~, Let h be the value of the ordinate at the highest point A, and f the depth of a layer of fluid of the density of the crust, which upon a unit of area would produce a pressure equal to the compression at A. Let 6 be the depth of a layer of fluid of the density of the crust, which would produce a pressure equal to the fluid-pressure at A. Then from (1) we have compression at A = gpef=gpe (C—h); . f= Ch. Also p=the pressure due to the depth below A+the pressure at A; “ p=ge (h—y) +9p8. UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 443 Therefore, substituting in (2), -C go (h— y) +9p5 = gpe (cos @ + a ). Change the origin to O' by putting y/=y—-C; yay t Cay tht+f. Hence when y=h, y'=—/f, so that f is the depth of A below the new origin, and h—-y=—(y' +f); whence, by substituting for y, ; ‘dx / 72 Gh te 68 — oc a =) , ; dx yf or —oy = (af = p8) =pe (7 +4). Integrated this gives, y , A! == — (af — po) of! = pey te: id: ere or, pey 7, + (of - pd) y +5y°=-D he patra (A). If this equation be solved with regard to y’ it will give two values of y’ for one dz value of — or cos@. ds And if it be solved with regard to = or cos @, it will give the same value for cos @ whenever y' is the same. The curve is therefore of an undulating form as in the figure. If the two values of y’ be made equal, they will give the points of contrary flexure. If we put = =1, this will give the maximum and minimum ordinates. Now we know that one of these is —f. Hence we may find D; s D2 LCA OF Substituting this value we get for the said ordinates, Pipe Pes sien (C 6) +7 (¢ 8), and their difference, 2 £ (c—8). Vor, Xi Pary IL or “J 444 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, We have from equation (2), by making r infinite, since ¢ is not infinite, at a point of contrary flexure, p=gpe cos, or —geo (y' +f) +9p5 =gpe cos 0. Substituting the corresponding value of cos @ in the general equation, we get yaty/ P+22 0-OF for the depth of the point of contrary flexure. The negative sign must be taken. The radii of curvature are, at an anticlinal, of a and at a synclinal, of a= 5 Boe o To determine the constant f we must discuss the conditions at an anticlinal. Consider any element ds of the curve, and let the horizontal and vertical components which act upon it from beyond at one extremity of it be R and Q, P? p the fluid pressure, Q ¢ the compression, @ the inclination of the tangent to the horizon, as before. Resolving vertically, we have for equilibrium, p cos ds + t sin 8—gpcds + Q=0, and horizontally, p sin 6ds —t cos 0+ R=0. Eliminating ¢, pds —gpc cos Ods + Q cos 0+ F sin 9=0.......cecececcseceeeeseeenees (1). Now this being true when ds is indefinitely diminished, since Q and F# are not thereby altered, we have generally, Q cos8+ Rf sin@=0. But at an anticlinal Q and & must be the vertical and horizontal components of the compression at one side of the anticlinal. From the nature of the case the vertical com- ponents of the compression on either side of the anticlinal must act in the same direction ; and therefore they cannot be in equilibrium unless they are separately zero. And Q is one of them; (Ves *. also & sin 8 = 0, whence either sin @=0, or R=0. UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 445 Let us consider these two cases in order. When the element is under the action of the forces Q@ and & at one extremity of it, and ds is indefinitely diminished, observing that Q cos @+ & sin @=0, we have from (1) in the limit pds — gpe cos Ods = 0 ; pds ** gpc cos Ods— Now at an anticlinal we know that p=gpé, and (in the case we are now considering) cos @=1, and these are the values which these quantities assume there when ds is indefi- nitely diminished. Hence at an anticlinal the above becomes in the limit gpe _ 1, that is, when the curve is horizontal at an anticlinal, o=c. This condition evidently belongs to the case of a crust lying horizontally upon the liquid, which is a particular case of the general problem. In this instance the difference of the >) —f (c—68) as found p. 443 vanishes, as it ought to do. maximum and minimum ordi Next consider the second case, viz. when R=0. In this case f=0 since f measures R. The relation f=0 shows that there is no compression at the anticlinal. In this case tand=— © takes the form of - The origin of co-ordinates is by the same relation for f brought to the level of the anticlinals. Our equation (A) now becomes, suppressing the dash over the y, dx Ua — py + =_D; dx _ eS oy ee "ds pcy j : NOE afd This expression shows that D=0, otherwise As (or cos@) would be infinite at an DS) anticlinal where, reckoning from the newly determined origin, y=—f=0. Dividing the numerator and denominator by y, or “I w 446 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, Integrating this equation, and taking the origin at an anticlinal, we obtain 4p ; 4p 2 Tae is ara cee = x+y 5 et Ne 6 > OY 0. This represents a circular are. If X be the distance from one anticlinal to the next, we obtain for 6 the value sea ee =. ae oe Je Tép"* ? and for the final equation 2 2 2 2 6-3) +(p-by Wea =a ~ which gives the co-ordinates of the centre Xr 1 i'Gpree ie > nd 5 = C—N; SD and the radius —e. o : ‘ : 4p F : These conditions are possible so long as is less than —~¢; m which case the festoon between the anticlinals becomes a semicircle. It can easily be shown, by introducing the supposition that the curve is a circle into the equations of equilibrium (1), (2), that the difference of pressure varies as the difference of the ordinate. This variation would of course be true for any curve, but it would not be consistent with giving equilibrium of form to the curve unless it satisfied the equations of equilibrium, It appears therefore that, when no extraneous force acts upon the crust, it will assume the form of a series of equal circular arcs, the radii of which depend solely upon the mass (pc) of a unit of length of the crust, and upon the density @ of the liquid on which it is supported. The degradation of the undulating curve into a series of circular ares takes place through f becoming zero. The curvature (p. 444) at an anticlinal in this case becomes infinitely great. The next step is to conceive a long trough, bent lengthwise into a circular form of large radius, and that the crust and liquid are acted upon by a force gravitating towards the centre, and so we approximate to the case of a section of the earth’s surface taken along a great circle. In this case we must suppose the curve reentering. No extraneous force acting, the circular festooned form will approximately represent the curve of equi- librium. But for simplicity of conception it will be better to return to the idea of the rect- angular trough, and to suppose that the curve is in the phase of an anticlinal where it meets the trough at either end; which condition is necessary for equilibrium if there is no friction there: and that there are mm festoons. UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 447 Having found the form of equilibrium we have now to enquire what relation it holds to the length of the trough. We will consider that the trough was originally of the length Z, and that it is 5) compressed until its length becomes Z(1—e). Then since “Pe is the radius of every o circular are which can admit of equilibrium, and that the chords of the m festoons, or ares, must equal the reduced length of the trough, putting @ for the angle subtended by each festoon, we must have and because the whole length of the crust is that of the trough before compression, Oy : SD Fe .m™ oP sin? whence eel en A ARs et Nene ni ae ORR 1), $ (1) 2 aL and re Radhacsdaneeeconen einen scenes Ns ae (2) Any value of e¢ less than unity substituted in (1) will give a corresponding value of ¢, and thence from (2) a value of m. But none but integral values of m will be compatible with equilibrium, because the curve must meet either end of the trough at an anticlinal. ¢ diminishes as m increases, and when m is infinitely great and ¢ infinitely small, sin 5) a =l-e bo He. becomes unity, and therefore e=0, or there is no compression, m also increases as ¢ diminishes. Hence, ceteris paribus, the festoons are more numerous with a thin crust than with a thick one. The geometrical relations show that the lengths of the festoons must increase, and their number diminish, as the compression is increased. Consider now the crust to be in equilibrium with a certain pair of valves of m and e, m being integral, and let the trough then be shortened, but not sufticiently so for the next integral value of m to be reached. What will happen? The festoons cannot alter their curvature so as to adapt themselves to their lessened chords. It seems then that the compression at the anticlinals must become finite, instead of bemg zero, and that the ends of the festoons will be pushed up against each other, by which means material will be accumulated there, and Q and # will become finite at the anticlinals, 448 Mr FISHER, ON THE INEQUALITIES OF THE EARTH'S SURFACE, This will accord with the undulating form of the curve, but it seems that the equilibrium must become unstable, on account of the weight resting on the anticlinal in the highest possible position. It will therefore be liable to slide down upon the surface of the liquid on one side or other of the anticlinal. Let us suppose then that a portion of the crust PQ, whose weight is Q, has become thickened by the accumulation of material, and let P be the mean fluid-pressure upon this portion of crust. And let it be supposed that the curve is cut off above Q. Then we have for the equilibrium of PQ, QsinOd=t and Q cosd= Px PQ. The first of these equations combined with (1) p. 442, and with the equation to the circle, will give the position of the point P to which PQ will descend, balancing the curve below it in the same manner as it would be balanced if the curve were continuous above instead. The fluid-pressure need not be altered by the above supposition, because the general pressure will keep the anticlinal filled, although some of it should tend to escape above P@. But if the erust which was upon the other side of the anticlinal, surmounts it, and follows PQ down the incline, it will produce compression above @Q, and it seems that @ must, in that case, descend to the synclinal, where if p be less than o@ it will float and not dis- turb the general form of the curve beyond it. PQ, while thus descending, will force up the crust on the other side of the festoon, which in its turn surmounting the anticlinal, will descend the next incline. These changes, however, will take place rather by a relative movement of the liquid, than by an absolute movement of the crust, and will cause alter- nate elevation and depression of any given point in the crust. It seems probable that the thickened portion of the crust may tend to descend more rapidly than it is followed by the crust surmounting the anticlinal, in which case the compression will be more or less relieved, and fissures formed part of the way down the side of the festoon, through which the liquid would escape. This seems to offer a possible explanation of the situation of volcanic vents along lines parallel to mountain-chains. The tendency to heap up matter about the anticlinals, would, in the case of the earth’s crust, cause the strata to be accumulated in positions in which the tendency would be for them to slide among themselves and descend superficially to suit the angle of repose, thus causing the successive foldings and secondary corrugations, even on the smallest scale, such as occur on the flanks of the primary or central chains. And the folds might even tend to slope away from the crest, producing inverted bedding. Some amount of secondary corrugation would also be produced by the bending of the crust into the festoon-like form. But for several reasons it is not possible to say how UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 449 much. In the first place, this would increase with the thickness of the crust, and diminish with increase of radius of curvature. But in nature the curvature would have been greatest when the crust was thinnest, and would go on diminishing as the thickness increased. In short, if any point in the curve could be conceived to be at no time at an anticlinal, at such a point the curvature would go on perpetually decreasing as the crust grew thicker, and the compression of the general surface increased. It will also be noticed that the under side of any festoon would be subject to tension from the bending, and disposed to open cracks; while the compression (f) to which it is exposed would tend to keep them closed. The upper side would hence be more corrugated than if there were no such compression as (f). From the above remarks it does not appear that such an estimate as follows is of much service. Yet it may be as well to notice it. Let ECDF be a section of a portion of the crust originally a rectangle and bent into such a form that EF and CD are arcs of a circle whose centre is O. And suppose AB, an are situated between HF and CD, to be the original length of this rectangle. Let EF=nAB; CD=mAB; OC=r. Then we have aioe C uP y—-AC= nN m- The greatest crumpling along HF would occur when m=1. In this case n=1—— If c=p this would be }, which is a very large amount. But owing to the probable sliding of the parts over one another, the points H and F would not be upon the radii OA and OB, but fall beyond them, so that the crumpling would not be so great as that. If the conditions of perfect fluidity in the liquid and perfect flexibility in the crust were fulfilled, and the crust were of insensible thickness, then our result would be true, and the festoons be all equal circular ares. But under such circumstances as may be supposed to exist in nature, these conditions will be so imperfectly fulfilled, that all we can assert is that the above considerations will serve as a rough guide to what we may expect to occur. The conditions of equilibrium would be satisfied from point to point not very widely distant from each other, and we might expect the compression to be distributed unequally. Hence the festoons would be larger, and the anticlinals higher, in some regions than in others. The local thickenings of the crust would also render the forms of the curve locally to differ from the normal form. The effect of the hydrostatic pressure of the oceans must also be taken account of. 450 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, For suppose that any portion of the curve be covered by a heavy fluid (the ocean) of density ». Let k& be the height of this ocean above the level, upon which the origin was taken in the first instance (on p. 442). Then, in forming the equations of equilibrium, we shall have as before t=gpe(C—y). But for p we shall have, p=pressure at A + pressure due to the depth below A — pressure due to the ocean, =go(h—y) +9p5—gu(k—y), =-9(- #) y+ (ch— pk) + 9p6. This is an equation of the same form as before. We cannot find the values of the constants exactly as we did before, but we can see that the curve must consist of two portions, one above the ocean and one below it. At the point where these two portions mect one another, the direction of the curve must be continuous. The portion above this point will evidently follow the law already investigated in all respects. For the part below we have expressions for p and ¢ of the same form as before. This part will, therefore, follow the law of a curve, which may be supposed to rest upon a liquid of the density o—p instead of o. The law of form will, therefore, be the same as before, the constants being determined according to this suppo- sitious case. The compression (t) of the part of the curve beneath the ocean will, however, be ereater than in the suppositious case, the crest being higher. But that will not alter its form, because (f) in any case varies according to the height of the anticlinal. The radius of the portion beneath the ocean will be 2 c. o— pf To estimate how much the curvature would be affected by the ocean, we may con- sider the density of the subjacent liquid somewhat greater than that of granite (say) 3, and that of sea-water (say) 1. Then o—p=2, and the radius will be increased in the ratio of 3 : 2. This decrease of curvature will be independent of the depth of the ocean, and the change will occur at its margin. It may possibly have the effect of opening channels of communication between the surface and the subjacent liquid, and contribute somewhat to the formation of voleanie vents near the seaboard. % x x by * >< Ey b d f DCA 2 z ice F The diagram illustrates the effect of increased general compression in elevating and depressing points in the crust. The point A is supposed for simplicity to remain fixed. Then the points B, C, D, EB, F, @, upon further compression assume the positions b, ¢, d, e, fi g. UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 451 One reason (p. 448) has already been suggested for alternate elevation and depression at any point. Another cause of the same thing may be seen in the fact that the length of the festoons, and height and distance of the anticlinals, will be increased, both by additional compression, and by a thickening of the crust. The first of these will result from secular cooling. The second from the same cause, and also from the distribution by denudations of the material accumulated, as shewn at p- 447, about the anticlinals. The latter would be more local than the former in its effects. If we now pass to the case of any rectangular vessel, and suppose it to be compressed in two orthogonal directions, then the form of the corrugations in these two directions would be approximately governed by the laws we have investigated, provided the plane crust be supposed capable of adapting itself to a form not developable into a plane. And even in the case of a contracting sphere, the character of the corrugations may be assumed to be on the whole similar to that investigated, and they would be arranged about polygonal areas. The consequence of the crust not being absolutely flexible would be, that it would rest within certain limits in forms either more or less curved than the proper surface of equilibrium, and it must be confessed that the great ratio, which the thickness of the crust bears to the radius of the festoon, renders the result of the investigation much more difficult of application to the case of nature than it would otherwise be. But we may accept it, and use it for what it is worth. Still further, the general compression along a great circle will very inadequately represent what would occur when the compression over the whole surface has to be taken into account. In short, the case of nature is so extremely complicated, that it is very difficult to reason satisfactorily upon it. Nevertheless I think that the conclusions we haye arrived at concerning the case of a thin heavy flexible crust resting on a perfect liquid, have some value as far as the results may be summed up in the following propositions :— (1) A vertical section of the lower surface of the crust, where it meets the liquid, when carried across a series of the corrugations, would present a series of festoon-like arcs approximating to a circular form, concave upwards. (2) The anticlinals would be of cusp-like form, resulting from the intersection of contiguous festoons. There would not be found anything of the character of rolling undulations with flat-crested anticlinals. (3) The law of curvature of the festoons would not depend upon the amount of the general compression, though their amplitude, or the distance from crest to crest, would be increased, under circumstances of equilibrium (7.e. where m, p. 447, is integral), upon the compression being increased. The curvature would diminish upon the thickness being increased (~ = a °) (4) But the curve of equilibrium, even if it were once exactly established, could not be strictly maintained during successive contractions of the sphere, because we have Vou. XII. Part II. 58 452 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, seen that, for a trough of given length, a flexible crust and a perfect liquid, equilibrium can only subsist for certain special amounts of contraction having relation to the length of the trough. No doubt the limits within which this would be possible would be enlarged by defects in flexibility and liquidity, and they would also be enlarged by a great increase in the length of the trough as compared to the thickness of the crust: for then m would always be nearly an integer. This last seems to be the condition of things to which we may best compare the earth’s crust, so long as it was thin, when there may also be supposed to have been a state of more complete liquidity in the subjacent matter. But as the crust became thicker, and the layer of subjacent matter probably more viscous, the diminished freedom of movement between the crust and the liquid would tend partially to assimilate the condition of things to the case of the trough of limited length. The behaviour of the crust might therefore be expected to partake of the two explained as belonging to these two cases. There would be a pushing up of the anticlinals against each other owing to the defect of freedom of movement which would assimilate the conditions to those of a trough of limited length, and there would be an alternate elevation and depression of different parts of the curve, the liquid flowing from one anticlinal into the adjoining one, in virtue of such limited freedom of motion as existed between it and the crust. And these conclusions seem to agree pretty well with the phenomena. I have already remarked on the difficulty introduced by the large ratio which the thickness of the crust bears to the radius of the festoons. But, on the other hand, their curvature will have gone on diminishing as the crust has increased in thickness, to which thickness the material accumulated about tbe anticlinals will have contributed through the operation of denuding agency, still further diminishing the surface-curvature by its outspreading. The basset edges of the crust situated above the anticlinals of the liquid might occupy a considerable area on the upper surface, on account of its thickness. And throughout this area great disturbance of the strata might be expected to occur, owing to the approximate yerticality which the beds would tend to assume, and from which they would settle into positions of repose. The result noticed at (p. 447), and frequently since referred to, that there would be a tendency for material to accumulate about the anticlinals, which by its weight would tend to descend along the inclined side of the festoon, seems to agree very well with the phenomena. It would produce a fan-like arrangement of the strata about the crest itself, UPON THE HYPOTHESIS OF A LIQUID SUBSTRATUM. 453 and a crumpling of those on its flanks more intense in its immediate neighbourhood, and diminishing in intensity as we recede from it. The effect would be local, and the crumplings might be on either on a large or a small scale, which would depend chiefly upon the nature of the rocks themselves. This is exactly what occurs, and it must be admitted that, however much it seems necessary to attribute the contortions of rocks to a general contraction of the earth’s volume as a primary cause, yet the foldings which offer themselves to view in nature are of such a character, and often of such small dimensions, sometimes only a few feet from bend to bend, that it seems impossible to attribute them directly, to what must have operated on so grand a scale as the contraction of the globe itself. Causes have been pointed out why the crests might descend, and lose their pristine elevation. They would then become subject to be planed off by denu- dation. But if our theory is true, intensely crumpled rocks, wherever found, must have been at some time or another in an elevated position with reference to the neighbouring surface. And where an anticlinal with its accumulated load has sunk, additional com- pression and crumpling may have been produced by the alteration of the curvature there, but a ridge would nevertheless be preserved, if strong enough to resist denudation, more sharp in its outline perhaps than previously to sinking. Owing to the anticlinals shifting their positions by a gradual motion parallel to them- selves, elevations and depressions where they define a seaboard would be followed rather by the land encroaching upon the sea and the sea upon the land, than by new conti- nents rising abruptly out of mid-ocean or old ones sinking bodily into it. Not the least important result of the investigation is that, the more ready we conceive the crust of the earth to reply by flexure to the forces brought to bear upon it (and many facts seem to show that with time given it does so with considerable facility), the more does the compression (é) within it become limited to that due to the depth of any point beneath the crest of the anticlinal, whilst at the anticlinal itself the compression tends actually to vanish. This agrees perfectly with the investigations of the late Archdeacon Pratt upon the plumb-line in India, where he found the density of the mountainous tracts less, and of the ocean bed greater, than the average. But it does not support Mr Mallet’s views regarding the source of volcanic energy being derived from the crushing of the rocks*. For although it is certain that the contraction of the earth might originate stresses within a rigid crust if unrelieved of practically infinite amount, yet if the crust is capable of readily yielding to them, the stresses become comparatively small. Geologists have often observed that volcanic vents, and outpourings of basaltic rocks, have frequently occurred without much disturbance of the neighbouring strata ; almost as if the crust had opened of its own accord and let them escape. A notable instance of such an outpouring of igneous rocks has lately been described by Professor Joseph Le Conte in the American Journal of Science, 3rd Series, Vol. vu. p. 167. 1874. He calls it “The great lava flood of the West;” and supposes it to have been poured forth from fissures. It covers nearly 300,000 square miles of the North-Western portion of the States, an area larger than the * The author of this paper has, as he believes, shown | for October, 1875, entitled ‘Mr Mallet’s theory of Volcanic the inadequacy of this cause in his article in the Phil. Mag. | Energy tested.’ 58—2 454 Mr FISHER, ON THE INEQUALITIES OF THE EARTH’S SURFACE, é&c. whole of France, and has an average depth of about 2000 feet. Such fissure eruptions require the compression of the crust locally to vanish in the manner which has been shown in this paper to be not improbable. And what is the essential character of volcanic vents? They appear from description to be orifices from which superheated gases escape. Molten lava often fills the bottom of a crater for a long period. What keeps it hot? If it were a supply of lava from below, there need be a continuous escape above. But that does not appear to be requisite. It must then be the high temperature of the gases which pass through it, and in so dog support its temperature. If this be a correct explanation, then any place in the crust, which is not steam-tight, may originate a volcano. The gases would fuse their own way through, converting the rocks melted in their passage into lava. But some of the subjacent liquid layer would finally find an exit, on account of the pressure beneath being in general greater than that due to the thickness of the crust. It must be admitted that the difficulty of accounting for the grand features of oceanic depressions and continental elevations has not been directly removed by the investigations above made. Our theory adapts itself more immediately to the features of smaller dimensions, such as are exhibited in a mountainous country. The remark however which has been already made, that the conditions of equilibrium would be satisfied from point to point, not very widely distaat from one another, leads to the conclusion that the more elevated tracts, which form our present continents, may probably be those regions by whose elevation the contraction of the earth has been more recently relieved. The fluid pressure beneath the crust, which is what would tend to burst it upwards elsewhere, being due to the depth of the base of the crust below the highest point of the fluid, would be only slowly conveyed to great distances by a viscous couch; so that compression having been freshly relieved throughout a continental tract, the upward pressure beneath the far-off ocean-bed may be for a long while in abeyance. But after a long interval, during which the continental area has by the cooling of the crust become newly strengthened, the general lateral thrust may reassert its influence, and the ocean-floor, bemg now the weaker area, and exposed to the greater pressure from beneath, may in its turn become the scene of disturbance, and of elevation, the land in its turn sinking. That irregularities of contour, such as affect the land-surfaces, are continued under a modified form beneath the ocean, has been proved by recent soundings*; leading to the conclusion that its bed consists of old land-surfaces submerged. There is reason however to think that the escape of water from beneath the crust in volcanic eruptions through protracted ages has borne a share in the subsidence of the crust}. And it is a matter of common observation that voleanic areas where the energy is now well-nigh spent are almost always areas of depression. * See ‘A section of the bed of the Pacific from Yoko- | Earth’s Surface,’ in these Transactions, Dec. 1873; also hama to Cape Flattery,’ Nature, Vol. x. p. 484. note, p. 440. + See the author's paper on the ‘ Inequalities of the TV. Exercises in Curvilinear and Normal Co-ordinates. By the Rev. J. W. Warren, Caius College. [Read May 22, 1876.] EXERCISE THE FIRST. CURVILINEAR CO-ORDINATES. 2, y, 2 representing the rectangular Cartesian co-ordinates of a point in space, and U bemg any function of these variables, suppose we write “=, (, ¥, 2), U=>, (x; ¥, 2), U—?, (% ¥%, 2). U, by aid of these equations, may be expressed in terms of w,, 2,, w,. Bee. dx _ dye dz Write du," du, du," dx: dy _ dz ames du, b,, du, =6,, (da du, “1 du, oe du, Cs: Hence dx = a,du, + b,du, + ¢,du,, dy =a,du, + b,du, + c,du,, dz =a,du, + b,du, + c,du,. a, My, a: | Write Bialep™ Wie Dee. | Cio F Ca7), Cay) | dB dB dB Hence Bdu, = Tp dx ee dy + Ee dz, dB dB dB Bdu, = db, da + db, dy + db, dz, d dB dB Bdu, = d daz + a dy + He dz. 456 Rev. J. W. WARREN’S EXERCISES IN Hene 1 (dU be dU dB dU dB a ~B \du, da," du, db," du, de’ dU_1 (dUdB dU dB dU al dy B \du, da,” du, db,” du, de,S’ dU_1 = dB dU dB dU dB dz B \du, da,* du, db, * du, de,S Write ap Payee equal to C,,du, + C,du,? + C,,du, + 20,,du,du, + 2C,,du,du, + 2C,,du,du,. 2 adUy* dU\* dh 0S? Write also = + aa ae equal to dU dU aU? , au av aU aU 4, au au As (ee ee +4n(G,) tes da dat ddan * 2 It is sometimes convenient when we wish to compare our results with those arrived at by previous Authors to write ChE CLG, SCn— ae C, =F, Cy, ee J. CS aC. 1 22? 83? A; A,,; A,,; A,,, A A,,- C.,, C,,, C.. are functions of It is clear from the preceding equations that C GE aGe 23 The converse being also true, in fact we clearly have C,, = E= a? + a} + as, C,,=G= b? + b? + b;, CO.= T= cf + co + Css C, = F=a,b, + a,b, + agbs, C,, = H= b,c, + b,c, + dyes, Cy = J = Cd, + Cx, + C5. We also have Roe ene, su (Ch) +R) +8) 4u= (ie) +(e) +a) 1 dB dB dB dB dB ab db, * da, db, * da, db, ee e dB dB dB dB a Se eres do, * db, do, db, de, 1 (dB dB @B db a8 dB ~ B \de, da, de, da, de, da,S ” CURVILINEAR AND NORMAL CO-ORDINATES. Hence clearly we can write | &, A, as, E, F, sa Ci, CS Cs | lh. Bh os a ee aa Ce Cle Cate 1G, Co, C5 J, H, I \ C3, Cc Cry | | | r 2 ! 1 | ie i TRA ee va va tera | Bl dace |e gua? |e dB dB aB pe) am Ax | C0. — Csr AL BAe A. 43) 6, CL08 CS Been BARA Ae) On C,C,—C2=B" - A,: BY(AgA,— 4,2) = Ca; C022 6.6, = BAA SE AAG AeA.) = Os: C0, — C,0j=B'A,; B*(ApA,—4g4x) = Cy; 0,0,=0,0,=BA,; B (4,4,= ALA.) = Cy; _ 1 (/dB\? , /dBY , (dB\*) An =F ‘ae ae wer [? _1 (dB dB, dB dB , dB aBy =~ B (da, db," da, db, da, db,J ’ _ 1 (dB dB , dB dB , dB dB) and since ae =~ B da, de,‘ da, de, * da, de,\’ it clearly follows that 1dB B da, = A,,a, + A,,b, + Aly 1 dB Baa =A a, + Ab, + A\.c,, 1 dB RB da, — Anas + Aba + ARE Cr and symmetrically we also have 1 dB B db, a A,.p, ig Ac, aL A,,d,, 1dB B db, a A,b, + A,.c, ite A,.4,, 1dB B db, = A,p, + A, Cs a A.M, ’ 1 dB = SSS. p / A -/ B de, AC, Tt A,,a, aF Ah 1 dB R aes = A,o, + Aya,+ A,,b,, 1dB teat A ae tees 3 i | 453 It Ce cS Cs, C 12? C. 23? is clear that we can obtain from '., nine formule similar to the above nine expressing the values the of a,, 4, a, 6, b,, 5, ¢,, ¢) ¢, linearly in terms of These formule clearly are w— Ch a,= C, a,=C,, (yet Gk 168 n= (OL = (0% P= (OA 0 = (C5 1 dB LedB sas 15) Gh B da,’ B da, previously Rey. J. W. WARREN’S EXERCISES IN written down values of , &e., &e. We also clearly have the nine formule A, Cy + A,C,+A,C,,=1, A,C 15 ALCS te A,C, =1, A OLA, + AQC,=1; A, C,,+A,,C,,+ A,,C,,= 0, A uC\s ar A.C, “ A,,C,,= 0, eee eee eee eee ee eee A, 0; + AO) 4. A, 0.0, 21 ~ 31 A,.C,, + A.C, + A,, Cy = 0. A,,C,, +A,,C, + 4n0,,=0, A,,C,, + A,,C,, + AnC, =0. 81~ 21 p P40, 5 apt eB Ge pint Ca RE Cw Bae 3 Cn Get Co Bae 3Bt On Bae Ou-5 ae Bt Cog Et Ce Be pant Oe Bae Cs Ge Bde t Ou B-ae t CoB GE Bet CB get Ca Heap Figg Ong t On Bae CURVILINEAR AND NORMAL CO-ORDINATES. 459 We have also the formule 5) - B HB= 6h > CRAG. OGM OR eC Ss UO dA. +20¢,,.daA,,, +2C,,.dA,,, 5) + 5dB= A, dC, + Ay. dy + Ag, Ay, +2A,,.daC,, +2A,,. dC. 13) +2A,,.dC,,. All these formule as yet given express relations between the quantities U, u,, u,, u,, x, y, 2, and the differential coefficients of the first order of one or more of these quantities with respect to one of the quantities u,, u,, u,, 2, y, 2; now, however, we proceed to determine the values of and relations between the differential coefficients of THe MSCCONCMOXOEE:-secence-teeeeens ifiwe: refers toi ‘the valueswofeC), Cs, Cal. C2. 0. . Co in terms of a,, a,, a,; 0,, b,, b,; &, ©, C3 it is clear that we may write du, 8 di) Oe edb. 2 da." da da da, dF 1 dE = = Oe ‘ 1 du, 7 du” du, du, 2 dus da, eye da, ne da,_dJ_1 dE. Sue Gitar? ain ds ge Cin, Wadia aia. it is clear that we may also write da, 4 da, aE da,_1 dE a: Tu, 2° du, oe ieee du,’ da, da, da, _1 dG or Fie * du, + by. du, 2 du,’ da, da, da,_1 (dH | dJ a “4 du, uae du, By du, 2 = du, du,)" Hence by elimination we arrive at the equations de _da_1 dB 1dE dus dies Bde,” 2 du 1 dB (dF _1 dk Papi db, ~ |du, 2 du,)’ 1 dB (dJ_1 dé. * Be de, ° \du, 2 du,)’ VoL. XII. Part IT. 59 460 Rey. J. W. WARREN’S EXERCISES IN dx _da, db, ae du,du, du, du, q 1 dB idk 1 dB 1 aG@ Bday Wadi,” Be abe 2 «dee, 1 dB 1 (dH dJ dF B* de, 2 (du. du, du ? 3. and clearly there are in all eighteen of such equations; viz. nine of the first species, and nine of the second. By the previous written values of =, Te &e., these two equations may clearly be written + (WEA § 92) [aah + dae. + duah adj, AGE: Ur du, ~9 = {Ane + A,,a, + Ay} eee eee eee eee eee eee eee ee ey {ana + A,,d, + Aye} = — {Aub Wen Aya,} 1 2 dH dJ dF —— + ——-—+ 4A,¢,. + Aa,+ A,,b,t - ee du, dus; aren za We might now proceed to calculate aU @U €U &U dz** dy’ dz” dady’ and the various geometrical quantities that are expressible in terms of these differential coefficients might thus be expressed in terms of Curvilinear co-ordinates. We might thus calculate a radius of normal curvature of any one of the three surfaces u,, w,, u,; and thus in particular we might calculate the six radii of normal curvature due to the inter- section of the surfaces u,, u,, u,;; such investigations, however, we shall find it unnecessary to undertake, inasmuch as they can be more simply arrived at by a change of the independent variables in the corresponding formule in Normal Co-ordinates. &e. ; CURVILINEAR AND NORMAL CO-ORDINATES. 461 EXERCISE THE SECOND. NorRMAL CO-ORDINATES. THE general system of curvilinear co-ordinates involves aa a which are functions of A,,, A,,, A, 4, Asn, A,,, and therefore of C,,, C,,, Cu, Cy, Cy, C5, a8 well as of their various differential coefficients with respect to w,, ig Uz; it is obvious that such a system is a little complicated, and therefore we may enquire whether any particular systems exist less cumbrous in form and capable of being applied to obtain useful results. One such system of co-ordinates is obtained by writing A,,, A,,, A,,, each equal to zero. It is clear from the values found for du,, du,, du, in he previous chapter that such a system of co-ordinates are orthogonal, that is to say w,, v,, and w,, mutually eut at right angles. I shall not pause at present to consider this particular system of co-ordinates, but I shall return to their consideration in a future chapter. A second simplified case of the general system of co-ordinates is got by taking as our independent variables not the surfaces u,, u,, %, but the length of the intercepted normals between w,, w,, %, and three fixed surfaces parallel to u,, w,, %, respectively. Let us call the lengths of these three normals n,, n,, »,; then x, y, 2, representing the co-ordinates of a point common to w,, u,, u,, and therefore to n,, n,, n,. Let a, ¥,, %, denote the co-ordinates of the remaining end of.n,; 2, y,, 2, the co-ordinates of the remaining end of n,; and z,, y,, 2, the co-ordinates of the remaining end of n,. We clearly have therefore ae (a —2,)° cu (y =a a0 (2 = 2,)"> (@ = 5)" te (y = Yo)” at (2 im Ze) n, = (a — a,)* + (y—y)° + (2 —2,)*. We clearly have therefore dn Euan: 3 /anNs (ze) +(7) +(3: ? (Si a =) di ‘a1 diz Dee us ( aes mR) + a1 (ae) +(e) + dn, dn, a, diy dn, d ae = co 6, dz’ de ' dy dy dz'de dn, dn, , dn, dn, , dn, dn, _ ae < dx * dy* dy * dz’ dz = cos ¢, dn, dn, , dn, dn, dn, dn, dx" dx t dy “dy t dz dz od ‘i co) 462 Rev. J. W. WARREN’S EXERCISES IN s where @ equals the angles between n, and n,, ¢ equals the angle between n, and n,, and yw between n, and n,. The system of co-ordinates I have just mentioned I shall call normal co-ordinates. It is obvious that they introduce much simplicity into our general formule inasmuch as for such a system of co-ordinates we clearly have by the above equations A= An— Antal A,, = cos 0, A,, = cos ¢, A,, = cos Wp. I shall for shortness in the case of normal co-ordinates write for A,, A,, Aw; A,, A,, A, respectively, retaining 4,,, A,, A, for the general system alone. I shall also denote by 28,, 28,, 25,, Py», Py, Py what Cy, Cy, Cy, Cy, Cy, Cy respectively become when we pass from general co-ordinates to the particular case of normal co- ordinates. It is obvious that S,, S,, S, are proportional to the squares of the sines of the angles between n, and n,, , and ,, m, and n,, whilst P,,, P,, Ps are as the cosines of the angles between the three curves formed by the intersection of w,, u,, Us. It is hardly needful to write down all the formule of the general system again thus simplified by the use of normal co-ordinates; but the ten following formule I shall so frequently make use of that perhaps their exhibition will be useful : P,+24,.S,+A,.P,,=0, P+ A. B+ 3A,. 8.=0, eee ee rs P, + A,. Py + 2A, . 8,= 9, = 28, : dA,+P, . dA,+P,, s dA, ; we have also obviously 1 yp 244A; + 1-4? - A} - A} 3 I should here perhaps mention the theorem that if normals be drawn at every point of a given surface, and if equal lengths be taken along these normals, the surface locus of these points has also these normals of the first surface for its normals, this CURVILINEAR AND NORMAL CO-ORDINATES, 463 theorem which is easily demonstrated gives a clear conception of the manner in which the three normals become independent variables. The three normal co-ordinates of a point in space having been supposed to be given, and the three fixed surfaces from which the normals start and to which w,, u,, u, move constantly parallel having been assigned, it is obvious that the angles between the normals, and therefore 4A,, A,, A,, are fully determined; accordingly there must exist certain equations connecting the values of A,, A,, A, with those of n,, n,, n,, and the object of the remainder of this Exercise will be to obtain six differential equations of the second order between A,, A, and A,, and n,, n, and x,. I proceed to determine these six differential equations, first pointing out a few formula and geometric facts in relation to normal co-ordinates useful for our after calculations. Taking unit lengths along the three normals that meet at any point, we clearly form a spherical triangle whose sides are 6, ¢, , and whose angles are 180—90, 180—®, 180—‘Y, where ©, ® and WV are the angles between the three curves formed by the mutual intersection of the surfaces u,, u,, wu, to which m,, n,, », are normals. We write sn@=k.sn@; sn®=k.sind; snV=k.sinw; by the ordinary formule of spherical trigonometry we therefore have 1, cosy, cosh 1, cosV, cos® k= | cos, ie costGaalie k= cos, 15 cos. ©) |); cosg, cos@, 1 cos ®, cos®, 1 sin’ @.sin’¢. sin* Wy, sin’ @ . sin’. sin’; 1 je ~ P- sin’ @. sin’ p. sin yp. If we now consider the parallelopiped formed by the three surfaces u,, w,, u, and three others very close to these three and parallel to them, and if we denote by ds, a side of this parallelopiped formed by the intersection of the surfaces w, and w,, by ds, a side formed by the intersection of w, and wu, and by ds, a side formed by the intersection of wu, and u,; we have then, by the known formule of solid geometry, for the area of our parallelopiped, the formula | 1, cos VW, cos ® [2 cos VW, 1, cos © | x dsds.ds.: cos ®, cos ©, 1 but this volume also equals dn, multiplied by area of face of parallelopiped to which dn, is perpendicular; i.e. the volume equals dn,.ds,.ds,.sin ©, whence 464 Rey. J. W. WARREN’S EXERCISES IN il, cosV, cos ® at dn,.sn@=j|cos¥, 1, cos® ads cos ®, cos 0, rin . dn,.k.sn@=k'.sin @.sin d.siny.ds,; whence we obtain ds,= Bsin Odn,. Similarly we obtain ds, = B.sing.dn,; ds,= B. siny.dn,. Now by our notation we have dx dz dx Sa a Egan with similar formule for CL, By On; C., and G,3 0.5.5 Now ee. dn, is clearly the change of 2 caused by us going a distance ds, along the intersection of the two surfaces wu, and w,; calling this change Az, and the cosine of the angle that ds, makes with the axis of 2, cos a, we clearly have ’ . cos a, = Ar + ds,, @L whence C08 cides — tee 7 =a,dn. 1 Whence we obtain a,=Bsin@.cosa,/; and thus, if f,, y,, &c. represent symmetric angles to a,, we easily obtain = B.sin@.cosa’; a,=B.sin@.cosB’; a,=B.sin8.cosy ; b,=B.sing.cosa,’; b,=B.sing.cos8/; b,=B.sind.cosy, ; ¢,=B.siny.cosa,; c,= B.sinw.cosB/; c,=B. siny.cosy,. Let now a,8,y, be the director angles of n,, 2,8,y, of m,, and 2,,y, of n,. We know that sin © . cos a, = cos 8,’ . cosy,’ — cos f,'. cosy,’ ; . k.sin @. cosa, = cos B,'. cosy,’ — cos 8,’ . cos ¥, ; COS a, ; , , ' er ae ae cos 8,'. cosy, — cos 8,’ . cosy,’ ; . Bos a, = b,c, — b,c,. Hence by symmetry we clearly have B.cos a, = b,c, —b,c,; B.cos a, =¢,a,—¢,4,; B.cos a, = a,b, —a,),; B.cos 8,=b,c,—b,; B.cos 8, =c,a,-—¢,a,; B.cos 8, = a,b,—a,b,; B. cosy, =b,c,—b.,; B.cosy,=ca,—¢,a,3 B.cosy, = a,b,—a,),. CURVILINEAR AND NORMAL CO-ORDINATES. 465 It served as an exercise to obtain these last nine formule thus; but we might also easily obtain them by simply solving the equations de = a,dn, + b,dn, + ¢,dn,, dy = a,dn, + b,dn, + ¢,dn,, dz = a,dn, + b,dn, + c,dn,, for dn,, dn,, dn,, and then comparing the result with the formule n,. dn, = (w—x,)dx+ (y—y,) dy + (z—-2z,)dz, . dn, = (w—ax,) dx + (y — y,) dy + (z — z,) dz, 4 an, = (x —a,) dx + (y— y,) dy + (2 —2,) dz. If we now consider such an expression as da, da, da, COS @,. FB + cos By. 7 + COS 5-7 3 2 2 2. we clearly see by the preceding formule that it may also be written in the form da, da, da, dn,’. dn,’ dn and hence obviously we have da da, da,)* foos Cho a + cos B,. Ti COS Y, + mt ead Colmar Ye ested dn,’ dn,’ dn, 1 ~ B a, a,, a, b,; b,, b, da, da, da, db, db, db, dn,” dn. dn), dn,’ dn,’ dn, + = a; a,; a, x a,, a, a0 (Dg Ae Us by 4 bis b, Now by means of the values of a , &e., implicitly given at the close of Exer- 2 cise the first on Curvilinear Co-ordinates, the first three lines of the preceding equation can obviously be expressed in terms of d,, A,, A, and the first differential coefficients of 466 Rey. J. W. WARREN’S EXERCISES IN S,, 8,, S,, Pes ES Es multiplied clearly only contain the terms or constituents zy (s) () - (Ge (ae T \dn, db, =) = = db, \, ae ae le - 2 with respect to n,, ”,, 7,3 also the determinants when squared or (a) Ou Gee ee FO ae di BTA 4b, Tad, a= oe 0G +a, Gta, a nid Beat, a 0 G40, T+), Te ae a Ge ta Ge a, Fe aie ae Of the constituents here written down the first two have the same coefficient and may therefore be united, and it is easy to see that when thus united these combined terms equal PS, @P,, aS 1 2. 2 ~ dnZ s dndn, dn?’ 1 thus clearly the equation that we have indicated is a differential equation of the second order involving A,, A,, A,, and the differential coefficients of S,, S,, S,; P,,., P,,, P.» with respect to n,, m,, m,. It only remains now actually to work this equation out; 3 before I do so, however, I observe that although in the case of normal co-ordinates the adoption of the angles ¢,, 8,, y,3 2, By, %13 % By % Is very useful, and tends much to shorten our future calculations, yet most of the results so arrived at might be obtained by a purely analytical process without the introduction of any trigonometric conceptions whatever; for example, we might so obtain the previous equation, for clearly we may write it CURVILINEAR AND NORMAL CO-ORDINATES, 467 dB da, ,oB da, dB da,)* / dn de. dn, S de. ban { da, ,adB da, dB st f \ dc, dn, dc, dn, de. dn, i dB db, dB db, dB db; dc dn, des dn. ade. dn: da\* (da\ a (dan. da, da, da, | da, da, da Ga) e @) a Gal # ‘dn, ee dn, ae dn,’ Ge dn, + 6,. dn, +b, cdi da d 1 — Oy Tet Me TE Os a a +a; +02; ab, +a), +a,), da, da, da, ' Reyes 2 2 By. ans +b,. i +b,. ae a,b, +a,b,+a4,; b? +b? +b, da, db, da, db, da, db, da, da, da, . da, da, da, dn, dn, dn, dn, * dn, dn,’ ee dn, pees dn, ages dn,’ ue dn, +6,. dn, eee: dn, db, Ib, db, + a,. = +a,. a +a,.9 3 ap +a)t+as; ab, +a,b,+ a,b, db, db, db, | ie ee dn, *dn, ye dn,’ Gb, + dab + a,b,5 6, +b" +b, da ely eae : If we look back now to the values of Bae ue , &e, implicitly given at the close of the previous Exercise, it is clear that the above in the case of general co-ordinates may be written 1 1dzy 1d@. 4 1/dH dJ aa, ot 2am Sit wl am eet (Fa du, du, GH dF 1d adi 1 df = fda reece =) lool ae a al (4,.(@F 222) 4, 240, 4 (at_1 aay “ yan du, 2 Aes 2 tu, * Ae (au. 2 du, 1@H. dF re a’G i1dE 1 dG 2 du, du,du, 2 du?’ 2 du,’ 2 du, 1 1 d# -B 2 du, ae ie 1 dG 0 _ 1 dz. Le | dE eee du,’ du, 2 du, 1jidF 1 dG@ Splat) Fl Weal 1 dG 2 du ij’ G Vou. XII. Part IL. 60 468 Rey. J. W. WARREN’S EXERCISES IN so that in the particular case of normal co-ordinates we clearly have dS GS, Wide CHR GHe- NP Vos lk San Ga ss Ti 7 dn, ) ds, Ne. GSS dB as. -y {4s- dn, Hebe. ae ~ dn, : 5 ta eo) aR ds: aS (aR cas, x Ae es Sie eos Hee ==) 2 d*s, a dP ais. 28, as, dn,’ dn,dn, dn? d dn,’ dn, 1 dS, -B ae pg es dS, . dn, 5 F328; ji 2 aie Bas ae CLI OID CLIO 1 GLEE nt Se, ; is Sl dh, ae ee ee dS. ae . >) dn, 5 Pas 28, Let us write the above equation in the form (11) + (22) + (33) + (23) + (13) + (12) = where (11) equals the sum of all the terms that have two 1’s in their suffix; (13) equals sum of all the terms that have a 1 and a 3 in suffix, and so on. (11) equals pean e aan \2 dn, scr as, (aP., ape dS.) raat ican aT 1 dS.(5, a8, 90 ds, P Heel ) + dn, |" ay dn, ety 8) edn)” 25, = B*.(1—A,); 28,=B*.(1—4,"), P= (AA,—A4,); but therefore (11) equals F dP. aP., dP, ds, a dn, A, dn, cig dn, i AG IF dn 1 ds, dS, dn dn, dn, but S,+8,+5,+4,P,,+4,P,,+A,. P,, = P.,.dA, +P dA, +P. dA,=— eet CURVILINEAR AND NORMAL CO-ORDINATES, 469 so that clearly (11) equals it 2) _ dS, dS, 2 iL GS, GH 4 ( dn, dn, dn, B° dn,* dn,” (22) is obviously symmetrical with (11), hence (22) equals 1 dP," dS, dS,,1 dS, dB 4 (en) dn, dn, ° B* dn,” dn,” (33) clearly equals 1 i dS, d8, +t ( dn, dn,” dn, (12) clearly equals asa, Aide St lide) of4 %% 1dP, Or ees 7 3s dn, a dn, ) {4, “dn, 2 TF i —(tPsg 4, Pag 4, Sh (dn, bee dn, ai (Ne © SOIR ds, * | dn, Sia dn, Te rat ds, ds, moles ae ai (ee ) 1 | = hp, ~ Cho. 2) dns dn; | °F pee, di, ds, a GP, AP, ia Ci ar dite Guay “ane aan es dn, “dng J which equals ve dS, dS, _dS, ds, S* (dn, dn, dn, “dn, dP, dS. dP. ds, 1a | dn, ‘dn, dn, * it (dP., dS, dP, ds, aay (dn, dn, dn, * dn, GUE GIES -A,. Tre ais Wale. (a) dP. ak. a 2° dn, . dn, wet eee (a) aP dee Botan Te acooke (2) GHEY CASS — dn, . dn, ac eccccccces (a) ale ds, a ‘dn, . dn, a eee a eeeneee (a) bape lee McRae Nar i: 2 dn,° dn Ge Gi (9). al 1 60— i) 470 Rev. J. W. WARREN’S EXERCISES IN The five terms marked (a) equal and hence they equal and hence these terms marked (a) equal 2 dn, dn, s dn, ~ dn, ae. dP, dP aS. ~ 2 da, {4, dn, can E oa : owl GN GHP. A A GP SP Dae” dines: "ditt een” sala 2 idee dA, dA dA, +3 dn, {Ps dn, + Pas i a aa IdE(p @As dA, dA *3 dn, Es dn, cae dn, ae i dae (as aie dA, waa. 2 dees oat (ae see dA, ee dn, ea ; dn, + P,, = 2 L@Es (GA, dA, dA, 2 dn, his dn, nt + Paget 1b ge {p dA, dA, dA, +3 dn, es 1° dn, "dn, a Ides (suas, dA, hapa a ed) 2 dan, ie ee | PeadP {5 aa, dA, (Hes +4 de Pant Pont tA eh mE EN re CM ge dA,) —_— —* se = Sa 2 4 dn, | + dn, 8” ny) Gs 27 fee Ft dA,) —5 =" }4,.— 84 P,. 4 dn, ( *" dn, a ian) The last line in (12), that is to say the terms marked 8, equal 3 fadP,, dP., 3 dP. dP, 4 (dn, dn, dn, dn, ee 4 (dn, * dn, © dn, ° “dn, CURVILINEAR AND NORMAL COORDINATES. (12) therefore may be written in the form _1 (dP, dP. ,aP., dP, 4 (dn, ~ dn, diy dn, u ds, d its ds, dP 12 2 (dn, dn, dn, dn, 1 @P, aB PIB i dn, ° dn, _i @P, dB 2B dn dn, 2 (aS, aS, dS, dS) + A,. |e. aie. ea amg conan co aBtaacboceapcbs ial gleam eras Sale he ee dns adnt Std) « dnk) (dP, dS, _ dP, a8, +A,. Vals, © GEL ae, © Gi see me monenenees + 3 (dP dP. dP,, dP.) TERM Ort, Sich Hip, | oo mepeencen ceo 1 CU mane Sen. een -G Vil ee ea SEs, © Ge os 1 di dBW aPE + 4 A, 5 - dn, dn, — dn, ‘dn, cocconacsapecosconos 1 (dP. dAn Rap. dar + 7A IPE. ld 6 am = dn, = TA anos pdacsaccedacagune 1 ar, GA. GHP. dA,) + 4 Je . i dn, . dn, =: dn, dn, J eee eee ee es My object is now to express each of the first differential coefficients of A,, equals 471 of these last eight determinants in (12) in terms A,, A, with respect to n,, m,, n,- A (8, 8, _ a, Fe S\dn, ‘dn, dn, * dn, BAMA dng da” an) +2B.8,4,A ‘- : int Te oat +2B .8,4,A, ae a = ir ae 472 . Rev. J. W. WARREN’S EXERCISES IN which equals dA, dA, dA, dA, as a = ‘ ae dn,” = (oe dA, dA, dA f + 2B°S PAA 3 2° 3 dn, ~ (dn, ~~ ‘dns ~tdn ; dA, dA, dA, dA + 2BS,P,,4,4, (dn, * dn, ea dn, ~ mal dA, dA, dA, a4, 9 — + 2B'8,P A.A, = ‘dn, dn, ~ dn, dA, ddA, dA, dA, dn,“ dn, dn, ~ dn +2BSP AA { =; 13,-1--<3: The next term in (12) that we have to reduce is {iP dS, dP ds ; 13 dn, ° dn, z dn, ~ dn It equals B‘A,A,A, ea ae = = ; ek Be ae aa ae 4+ 2BAS,, ir 5 s = . fe +2BAAS, {fa ' re 4 i ai +2BACAPa- 9) {iy Sao ~ dh,” dah or in this writing = = —{f,d4,+P,dA,+P,dA,}, we clearly transform it into 1 = 2 BAA, {7 “dn, dn,” dn - dA, dA, dA a te le j dn, 7 an, * dn, dA, dA, dA, dA 2 2 = 3 1 = 3 u FEBS Seles es * dn, di, ; a} + 2B°A3S,P. i= a 48. dA, dA, dA aa dn, * dn, - dn, © dn, _ dA, dA, dA, dA 2 Re {4 1 = : : + 2B°A,A,S,P., (dn, * an, dn, : a (dA, dA, dA, dA (dn, 9 LoS bY Ss + 2B'4,4,8,P,, m, dn, dn,” ik CURVILINEAR AND NORMAL COORDINATES. +2BA P.,.(A,P,— 8) on ad z = ; ah +2BA P,,. (A,P,—S,) ee 7 ei ae Thirdly, we reduce (e. dS, dP. ct 2 dn, dn, dn, ° dn,)- This clearly equals w4,AA, A: ahs Ad + Baas (oo. Gh oh ohh Pe ae eee +2BA,AS, ie a = ie = sepas, {2 o_o aA} + 2BA,A,P,, (rr = = ir . ae which clearly equals BAAA, SF! et — de dey es Gs a a + 2BALSP,, ie a4 a oa = + 2BA28P., cs : = I +2BA ASP, eas a - = at + 2BA,A,S,P., Vie oe - = ai + 2B°A,P,,(A,P.— S,) = - : a at + 2B°A,P,,(A,P.— 8,) en : ae = at 474 Rey. J. W. WARREN’S EXERCISES IN The four terms marked respectively (v), (vi), (vii), (viii), im the expression for (12) we must now reduce. It is clear that they may be written 1 (dB dP, dB dP, oe (Ae oa) (dn, dn, re 7 1 (ie Gi Gis) GHe. goons Caer Bo ae Candas a (dA, dP, dA, dk, ae [BA A, sldn,° dn, dn,~ dn, 4 1 , (G40 eRe dA de) +4 B4,A, (dn, : dn, dn, ~ dn, J 1 4 if (a4, dP, dd, @P gPaAs (dn, ~ dn, dn, ~ dn, 4 (24, dP, dA, dP.) 2A» 2 12 2 LG +5 PAA a ydn Wary, ary) 2 and the above clearly equals 1 Bla _4» (2 dP,, dB =| (dn, ° dn, dn,* dn ib (dA, GP.) dA, dP. 152A As) (dn, “dn, ~ dn,“ Gn } ite. (dA, pdb, aA, dP) top A a aie “dn, wan, ~ “dng\- The first of these three lines equals dB dA dB dA Cpe 8.) if 2 0 2 se BA (8, Ge, * dn, dn,~ dn (dB dA, dB dA, + BA, (8, —'S,) j (dn, 1 ans ~ dn, * dni “aB dA, dB -dA; pe a ie "dn, dn,” aa ; whilst the last two develope into (dB dA, dB aA, ees te (dn, * dn, dn, * dn } dB «aA YdB edAs + BA,A,P, AR n, dn, ~ dn, “dn k (dA, dA, dA, dA, 2B oa dn, ° dn, dn, ° dn Z (dA, dA, dA, dA ga (dn, "dn, dn, ° dn r {ee dA, dA, dA naa er (dn, ‘dn, dn, * dn (dA, dA, dA, aA, ede awe ae ec a ar cl Sti? ie ms CURVILINEAR From these nine lines it is now AND NORMAL CO-ORDINATES. easy to pick out the coefficients of dA, dA, dA, dA,) dn, °.dn, dn, ~ dn)? ga, a4, G4, a4, dn, °.dn, dn, *- 7h z a4, dd, a4, aA). dn, “-dn, “dn, * =| y the result is easily seen to be equal to 1 5 REATAS, + BA A,P.,, ae dA, dA, a : ig es di dn, din, . din, e 2 BAP, : (A;’ a A,’), > BYP,,(A?— A; + 2 a 28 ), 1 3 BAA, + (e . aA, _ aA, al + BA,A, C Ee dn, dn, dn, ~° dn, + BA,P,, . (S,-8,), + B’P,, . (S,-S,), BA AA; Ae Wipe at CAA Sa e tetra 1 Pee i im * dn, ee : 4 — BA,A, . Py. Ps, ar BA,P., (S, Fe S,); + B’A,P., (S,— 8). It now only remains to reduce the term marked (iv) in 12, that is to say 3 (@P Ee Gi Stee 4 |dn,° dn, dn, * dn,)’ on expansion this clearly becomes equal to 43 oe dA, dA, I 4 dn, “dn, dn, ° dn, F aah a (CAnwteA. oaA, oar gat: ae Ls Gi Oe Vor, Xi Parr Ii, 61 Ut 476 Rev, J. W. WARREN’S EXERCISES IN Sine dA, dA, dA, dA, tig oe ae ‘dn, dn, ° a Seer oe dA, dA, dA, > ges dn, ° dn, adi, : ay Sic dA, dA, dA, .dA, ge aos ae * dn, nae , a Siva, (dA, dA, dA, dA gay Sigat dite dae wat 3 dB dP. dB: oF. +3B(4,4,-4) in a a a ay 3 dB dP. dB ‘dP, +3.B(4,4,-A) { ee a viet (a: These two last lines marked (a) may be replaced by 3 dB dA, dB dA +3 B. {P+ AsPas} S > OO ~ dn," dn, i 3 dB dA, dB dA, 138 {Pat ASPs} im "dn, dn," at eR dB dA, dB dA, +58 pe eee ae eget, = ‘dn, dn, Ft And hence by picking out coefficients we find that ie ha ae ot dn, : dn, dn, * dn, may be written na, 2 Tee dA, dA, a4) a dn,” dn," dn, 3 ; | +58. (4,P, 3 ges) © 13? 3 | = ok ie (Ps aoe ea) Fe 12? L 3 ay BA,A,, _3 py dA, dA, dA, sr + a8 a dn, dy ‘dn, Foe +58 Pa Agee, a — A,P,,) ba 33 +58 UP agree CURVILINEAR AND NORMAL CO-ORDINATES. 477 ee | +53. (43-9, | g (aa dA, dA, dA, 3 3? dn, * dn, dn, * dn, 5B. (2+ A,P,,) P, 3 3B’. (P,, +A a IBA We are now clearly in a position to calculate what may be called the “Determinant” portion of (12), accordingly .we first calculate the coefficient of dA, dA, dA, dA, dn, ° dn, dn, ° dn, ? it equals (1) + 2B A, ASP. 2 3-112 @) 3) ) — B'A?A,—2B'S,P,A?+2B°A, [A.P,—8} P, 2 = 137k 2°13 G) (6) +2BA28,P.,-2BA,A,S.P., ae S 6) Dec! — 5 BAA, + 5 BY (4; — A,) (P,,+ 4,P,)— BAAP,? Ear See 122 (20) (11) 2) (13) = BYA,A,+ 2 4,43 Bt Ee BigP en Pe Sree We now must make use of the ten equations given at commencement of this Exercise: by their aid the four terms marked (3), (5), (8), (12) become a : B Cs 3: A,A,A,) Tee ate : B (4 on A,’) A,P.,, this equals = =¢ 2 BALL st By (Sh S,) Loe to this join on the three terms marked (4), (9) and (13), and we get | AS, ee 34,8, 3 A,P,, + 2A,A,P,, | | 3 | 2 | .-4,4,P,.—5 Pa j 2 BP, 1s 32 5 C which clearly equals ; 2 S,—34,S,+34,8, B 1 1-2 + * (424,4,P Sa i— =- 38 vs 12 which clearly equals 34,8, -34,8,+34,8,4+34,4,P BP,,| ie 12 a if 61—2 478 Rey. J. W. WARREN’S EXERCISES IN which clearly equals 3 pt AP,, {1-A?+42-A24+24,4,4,— 245} - BYA,P,,, 1 12 ~ ; il ; which finally equals aS: BAP. Now the terms marked (1) and (6) mutually cancel, there therefore remains only to consider the four terms marked (2), (7), (10), (11); these may be written in the form B {4 i= A; =F (4,4, a 4) ? which clearly equals B (248,43 2.) BN Vt iis | 4 13)? but 2A S,=— P,,-A,P : 1 : Whence joining on the previous found term +5 BAP,» we finally obtain for the coefficient of dA, dA, dA, (Fr. “dn, dn; dnp? in (12), the term = eee Bo ae and therefore by symmetry the coefficient of dA, dA, dA, a dn, * dn, dn, * dn, B equals =z {2A,P..+ 3P,,}. The remainder of the “Determinant” portion of (12) is the coefficient of dA, dA, dd, dA, dn, ° dn, dn, ° dn,}’ which is, by picking out, found to equal (1) @) (3) B‘A,A,A,+2B°A,A,8.P,, + 2B4,AS,P,, (4) (5) — B‘A,A,A, ~2B°4,A,8,P, (6) +2BA.(A.P,,—8).P., (7) (8) (9) — B‘A,A,A, —2B*A,A,6,P,, + 2B°A, (A,Py,— S,) P,, 20) _ aD (12) + B‘A,A,A,— B°A,A,P,,P,,— B*A,A,PgP a 8),1%,- 25 (13) (14) 1 + 2 BA, (AS se +5 BA, .(A— et DN se (15) (16) (17) 3 2 3 2 3 +5 BY. (42-1) -5 BY, (Pt AsPu) Py 5 BY (Pat AyPa) Py: 3” 23: The terms marked (6), (9), (11), (12), (16), and (17), since P,, + 24S, +A, Pt 24,8, +4,P..= 0; CURVILINEAR AND NORMAL CO-ORDINATES, may clearly be written But and B’A,P,, {24,P,, — 28, —A,P,, + 38,} + BA,P,, {2A,P.,, — 2S,-A,P,, + 38,} 28,—28,-—A,P,,=B {A*—A,A,A 28,-28,—A,P,,= B{A?—A,A,A, Hence clearly the two lines (a), (2) equal BAP, {A,P,,+8,} + BA,P,, {A,P,, to this join the terms marked (13) and (14), and we sigan BAP. {A,P,,+5,+5,—-S8} which equals to this join the term marked (15), and we finally get for the coefficient of (ia dA, dA, mal the term a BAP, {ae + S, tr S.— S}, +5 BAP, +4,P}=5B.(1—28) 5 2" 13 P 33) dn, dn, dn, dn, +5 5-5 BS,. 0, sf =— BA,P,,, [aA ? 479 We can now write down the complete value of (12); referring to page 469, we find it equals 1 (@e. dn, ° “4 1 2 1 dP, e dP, , a8, dP, 4 Pa a dn, dn, ° dn, dB 1 dP, Pn 4S, AP dn, * dn, dn, * dn, dB ~ 2B dn, * dn, 2B dn, * dn, +2 pap 4+ a Be = (24,P dA,dA, dA, dA r wt 3P,,} ize dn, iz dn, dn, dA, dA, dA, dA,) Put BPol ind * dng ~ in day} B aA, dd, dd, aA, Saas {2 — 58,} ie . dn, dn, . dn, 450 We must now calculate (13); if we refer back to page 468, we easily find that it equals Jee Ane dS, ~ dn, “dn, dP, +7 TS 2 se 1aP,} 2 dn, dP. dn, eeeene dS,_d8, cat “dn, dn, : . dn. teen ewe The last four lines marked (a) ak os be written dS, i x dP., “dn, Sa dA, "dn, = dA, a = SP dA 1 a2 dn, dP, dP) 1° dn, 2 dn,} dA 1 aati } a, a8 an, dP, ee dA, Pes on 2 = 2 : 2 2 1dP., intl dA, 2 dn, ‘“ 1dP.,) “+5 2 dn, ) dA, Zip 1 2 | 2 FE ay 1 dA, "dn, +5 Fa: dn, Ss 2— 8). Rev. J. W. WARREN’S EXERCISES IN in the form CURVILINEAR AND NORMAL CO-ORDINATES. But this is easily transformed into + 8, i dS, dA, _ a8, a dn,’ dn, dn, * dn, 1g (Fs 24,_aP, 24, 9 ld 3 dn, dn, dn, 1 aS. dA, dS, dA; +2 Pus = aii e aa al 1 dP dA, yd Pay dA i 472 _ Ea dn, "4 And this clearly equals 2 og dB dA, dB om B:)9(da, > dni” dn dn, 1 dB dA, dB dA, $5 Pa, aan a 1 dB dA, dB dA, +3 SPs - ema aA Lp dB dA, dB iy + 2R* Ydn,* dn, dn,* dn Pe (dA. “dA, dA.» dA; ag rR a : a, di, St S, (dA, dA, dA, dA iD ae ‘dn, dn,’ al dA, dA, dA, aA Ae APs + BG : Hee dn, rates a Hence remembering that 48,8, — P,,’= B’, and that — a &, ~ APs + = dA, dB _ dA, Uae ~ dn," dn, Bs, a : dA, dA, dA, 4 |dn,° dn, dn,* dn, Bip. (da yaa. Pon dA + —=—s ° ° 4 (dn, ae ~ dn, * dn 4 BS, (dA, dA, dA, dA, x ee dn, dn, ih, Hence referring back to page 480, we clearly see that (13) equals Lids. sabe re Fat 4(dn, ° aay + an, * dn 1 dS, dP,, _— ds, 2 \dn,.° dn, dn,” dn, 481 *8 above equals 482 Rev. J. W. WARREN’S EXERCISES IN BP, es dA, dA, iA, 4 dn, “dn; dn, — dn, 3B’P., (dA, dA, dA, dA, i + dn,“ dn, dn, ~° dn, 2S, (04, dd, ad, dA 4 dn, dn, dn, ~° dn, (23) is clearly symmetrical with the above value of (13), and since we now know the value of (25), (13), (12), (11), (22), and (33), we can clearly write down the form of their sum, that is to say, the differential equation of second order indicated at commencement of this exercise, before we do so, however, it may conduce to clearness and conciseness to adopt the following cyclic notation: write dA, dA, dA, dA, /(23\ dn, ie = Fe : Inia | day dd, dy a, (12) dn, ° dn, dn,* dn, & Similarly dA, dA, dA, dA,_ = ,dn, “-dn, .dn, ° dn, =(51 ? dA, dA, dda, dA, _ =) dn, ° dn, .dn,° dn, = (51 ? dA, dA, dA, dA, _ a dn, dn,” dn, dni (s1 Similarly dA, dA, dA, dA, _ z) dn dn. Oma ame ay” d4:, dA,, dAx, dA, Ga) tn, ; dn, , ‘dn, ° dn, \12/’ dd, dyad, a4,_ (12) : dn, * dn, * dn, * dn, =( : We also write or denote by A the expression : (day _dS, aS, ‘dn dn,’ dn, 1 (eB) aS, as 4 Ae % dn, * dn, 1/dP,,\’ dS, dS, . 4 eo ~ dig “dn, Liides a 2 CPE) dP: vt 4 dn, ° Ti, ‘dn, CURVILINEAR AND NORMAL CO-ORDINATES. 4 (00) i) 1 (dS, dP, dS, aP,, 2 \dn, dn, — dn," dn, ~1{dBy AP. WP Ae ae ss dn, dn) an. 41 dS, dP, , ds, oul 2 \dn,° dn, dn, dn, L(dPe de alan ma 4 (dn, ° dn, laa * dn, 41 dS, dP.) dS, Pal 2 |dn, ° dn, * dn, ° dn, Adopting this expressive notation the differential equation of the second order in- dicated at page 468 may be written - aS, Le adPs t aS, dn~ dn dn, dn, + A dS, dB dS, dB \ dn, ° dn,“ dn, ~ dn | hap dbo WaP a” L [Dian dn, 59) dn, dn, (24,P,,+3P,). (15) —@A,Pat 3Px). 12) ~ 2-108). Gs) | | | i ( Na Cee | ( r + nz B oo all equal zero. Of course we have two more differential equations symmetrical with the above, we do not at present proceed to write these down, but go on to calculate the three re- maining differential equations of the second order. For convenience we use 1,5 1, M, ID place of the correct symbols u,, u,, u,. We have already seen that da, da, dag| | @, db, db, dn,’ dn,’ dn, dn,’ dn,’ dn, i; O05 o, a Gj Gs, 1G, be, 8: 0.8 108 D. <0n 8b: Minus dn,’ dn,’ dn, 1 Sa Sela bs,) OS bu: VOI ART TL 62 484 Rev. J. W. WARREN’S EXERCISES IN admits of being written in two forms, and of course the difference of these two forms is zero, hence we obtain three differential equations of the second order. Now the like remark applies to da, da, da, db, db, db, dn,’ dn,’ dn, dn,’ dn,’ dn, Br bh, Sb calnealitee o.rome Os Cis tiG yy tate Cin Oe © CF Minus da, da, da, db, db, db, dn,’ dn,’ dn, dn,’ dn,’ dn, 6, 16, Bll bs ate Ca Gee ey C, Men &: Clearly we may either multiply together our determinants first and then interpret the symbols, 7.¢. substitute other values for them, or we may substitute the values for our symbols first and then multiply afterwards; subtracting these two equal results we get a differential equation of the second order, and two more can be formed in a similar manner; thus we have, multiplying our determinants, da, db, , da, db, , da, db,_, da, da, da,. . da, da, da dn, dn, dn, dn, dn,” dn,’ e dn, +6, dn, +h dn,’ “ dn, po dn, sats ae db, db, db, 2 Bi. ae +5,. TF +6,. fee 67+6,'+b, ; be, + b,c, + b,c, db db db A Scag ied, 8 dn,” b,c, + b,c, + b,c, sep teste, Minus da, db, , da, db, , da, db, , da, da, da, . da, da, da dn, dn, dn,’dn, © dn, dn,’ *dn, +2, dn, #?, dn,’ “ dn, Fits dn, Hi di, db db db b+ Tn, + Os an, ts dn, bY + bP +b, ; b,c, +4,¢, + b,c, db, db, db, . . 2 2 2 dit, Cdn, + 8" dn,” b,c, + b,c, + b,c, 3 +O, +6, G5 Any Uy dB but B=| 6, 6, 6, | 5 . 5,¢,—4,¢,= ia, &e., &e. CC, 6 ‘?) a? 3 and so this same result may be exhibited in the form (a dB da, dB | da, dn, da, 4 dn,’ da, dn, “da, ct {at dB , db, dB, db, al dn, da, dn,’ da, dn, “da, CURVILINEAR AND NORMAL CO-ORDINATES. Minus da, dB , da, dB | da, a dn, da,° dn, da, dn,’ da i dB db, dB , db, ct x i=—. ater = c é dn, da, ° dn, da, dn, da, Now to simplify all this observe that da, db, _ da, db, , da, db, dn, dn, dn, dn, dn, “dn, _da, db, da, db, da, db dn, dn, dn,’ dn, x dn, dn, 3. 2 equals a La Ae ian {a,c, + 4,C, + a,C,} 1 2 as = ) dn.dn {a,b, ats a,b, at aps} 1 3 ice +5 an? {b,c, + b,c, + b,c,} 1 Deas bs eV suiat “ 2 dn.dn la, +a,’ + 4,4, 2 ] which equals aL (ead & ar Pee Cig 2 2 |dn,dn, ° dn,dn, dn dn,dn,) © We have also Pe ee er ee oe da, da, da, _dJ 1dk “dn, + %* dn, 5° dn, dn, 2dn,’ cB tO GE eB dn Ti +b, ot 40.92 os T+), ee ere c, ee 3 c= 5a or 486 Rev. J. W. WARREN’S EXERCISES IN We have also of course b7+67 +b" =G, cite +e, =I, b,c, + 4,¢, + b,c, = 7. If we refer back to “Exercise the first” we find that da, dB i1dE dB { sae} ee dn, da,* 2dn,° db, ° \dn, 2dn, ay _1dB) + de, * dn, 2dn,§ ° and we may change a, into a, or a, in this, if we at the same time change 6, into 6, or b,, and ¢, into ¢, or c,. We have also db, _dBidE dBidG dn, da,2dn,° db,2dn, dB1 (dH | dJ cat 3) dc, 2 \dn, ‘dn, dn, And here also we may change a, into a, or a,, b, into b, or b,, c, ito c, or c,, and leave all else the same. We can easily obtain in a similar manner pir oB 1ldzE dB1i(dF dH o} dn, da, °2dn,° db,2\dn,‘ dn, dn, ,@B 1dr de, ° 2 dn,“ db, dB i(dF dJ d dn, da,'2\dn, ‘dn, dn, ap 1dG dB dl db," 2dn,° de, ° 2dn," 3 1 And B + Hence we can form the equation i dJ 1@F .1@H 1 dk. dF idk dJ 1dk 2 dn,dn, t9 dn,dn, t3 dn2~ 2dn,dn,’ dn, 2dn,’ dn, 2dn, 1 1dG@ B 2 dn, > G > H 1ldlI 2 dn, oO ee 0, Afah a _ aN ar tal why dn, * dn, dn)* 2dn, 1 1dG@ — 3 an, : G : H Li(din vas aoe 3 lin * das tant it aoe of CURVILINEAR AND NORMAL CO-ORDINATES. , idk 1/dF dH dJ I CHE {4 2 dn, art 2 ea + dn, dn, dn, | ee iat 1dE 1 dG HE CAE GHA: «14, dn, * saan t4ua lan tama) A 4, 32 a, (F1) 44, (1B) t Zia me (om, 2 dn, tae 2 din, -)f s Se i cy Co a ae aD = dn, ~ dn, a" 2 dn, eee 2 7) : all equal zero, so that in the particular case of normal co-ordinates we obtain ' GPS hd Pedy aS =a 2 31 12 3 Z 2B CRS lara % dn, dn, ~ dn; z amet ; 0: adP., 9 dS, Ky dP, a ds, 2 lie Ch? eho, dn, UW GIS + BE dn, 28, > Pp dS. 3. A 28 dn,’ Pry 2 28; 0 1 (Re a dP, = _ a8, 7 UNdn) sd, “dn, dn, 1 ds es Zieh : a8 : BE dn ? 25, ? Ee 1 dP, dP. Fa) ; 2 ( dn, dn, ay ie il Ge A dP, _ =) a as,\ dn, °2\dn, dn, dn, > dn,) ed. ae) - {+4 Ee -) a Gi , as “ alae a ean) ae a all equal zero. It is obvious that, as at page 468, we may write this last equation (11) + (22) + (33) + (12) + (13) + (23) = and we must now as there develope separately each of these terms. “N Rey. J. W. WARREN’S EXERCISES IN 488 Firstly, for (11) it equals 1dP,, (dS, Jee v2 ies = ae quae dn, at +34 dP, dseret ~ {4,5 SL oetagy if GIS ables dS. war. 41) 1 +8, 3 * dn, 5 dn, an. 5 a P dS, d8,_ 1p yi | og dn, “dn, 4 Ps. (Ge J But this equals 1 dP,, 1 dP, dP., ~ 2 dn, a eas dn, oe dn, 1d8, dP,, 1d8, 4P,, ~ 2dn,° dn, 2dn,~ dn, dS, dS, 1 =) eign, ‘ ee es i But this equals 1aPe (ds. ds. oH, a dP. — 2 dn, ane ae. ee ade which finally equals Next (22) equals ae which clearly equals (33) and (22) must be symmetrical, and hence i IP, ds, dS, +4, 0 (52) - dn, ae pa aP,) _ ds, dS, “14 (on dn, © dn, lap a, Bide Orns ds, dS, ,1dP,, Sdn, | * dn, 2 dn, 1 AM dP, (dS, 1 , 4F,,) 2 dn, iS 2 “2 dn, J 1 fp (@Pait_ pa a8 B 13 as ae ai ne ats 1 es) _ AWS, = 4 ( dn, dn, © dn,)~ 4 : Fs) > a a aaa dn, ‘ dn) * aP,, 2 dn, (83) equals dP. ° dn, a) CURVILINEAR AND NORMAL CO-ORDINATES. 489 Having thus calculated the values of (11), (22), and (33), we shall now proceed to calculate (23), it equals , chsh, CASE LGW e Sa dn, {4, "dn, | 2 ai ds, aS: eae as dn, {4, “dn, 2 dn, ‘ dP, — fA, ae dn, Bishan: Tel oes at \dn, 2 *° dn,) \dn 2 dn dS, ds, dS, dS, } 1 + 28, dn, dn, 2 dn, dn, B tedP ee. L 2 Pane can. J all which may be written ely tes aE + dn, ‘adn, _1d8, (dS, , dS, 2a dP, dP, dP 2, 2 dn, a ven. dn, ee dn, tA dn, A; dn, } 1 dS, (dS, _ dS, _ o, dP,, & oer 2dn, et dn, + in, ee dn, Peat t dn a} 1dS, dA, A, dA, me2vans ea ‘dn, oe = pie aa 1 dS, dA dA, dA, 9 dn, ‘IP. ine dn, aoe Tat 1d8 dA, dA, dA, +3 an a Ps: “dn, pears dn, st Pe = 1d, dA, dA dA, ue) dn, {P.. “dn, oe Te ee sa aS, dS. 4 Led Sead al es, ee bee 2 dn, * dn 82 ane * dn, +3 ‘dn, ° dn, 1dS8, dS, 14 GIRS die “iste “HE. 2Qdn, ° dn, 2 *dn,° dn, “dn, ~ ‘dn, 1dS, dS, 2m ds, ~2dn,° dn, 2dn,° dn, i 7G, Ge. il dS, dP,, ne ae dn,° dn, g4s: dn, ° dn,” 490 Rey. J. W. WARREN’S EXERCISES IN 1 (dS, dS, dP) Bsc A, lan, * as + on, * dn.) (dP, dP, 28 iy Ene " dn,” dn,” dn, 1 dB dS, 1° atB ds, nOBe dn,° dn, 2B dn,’ dn, if Ps d'P., = ae dP. vi + dn, © dn dns 5 Vdn 2 2 3 1 (dS, dS. 8S, I - —— : 2 | dn, ~ dn, \ dns dn; : 1 (dS, dS, d'S, ds, + 2 Yan, dn, dn, ° dn, Ax GES: bE eel Sy ee Pee + : - : 2! dn, ~*dn, “dns i Vdni ee! GS, GP a8, dF. 2 Wan, * dns day wan ie We are now going to reduce the ten terms marked ', and the ten marked ”; firstly, we shall reduce the ten marked ’, and then the ten marked ” can be deduced from these by changing dn, into dn,, and dn, into dn,...._ To proceed, the ten terms marked ’ equal (a8, 4, Pi, 4, dP, 14s, | de Bn dt 2 > dn. 2 dn, Pas dA, , P., dA, 3 2 de edi Se dA dA,) | ~ 4 dn, aie oe _4, a, dP 4° dn,° dn,” ah, A. ay, At are 1d8,|dn,* 2° dn, * 2 * dn, 2 dn, P, dA) Pda 2° dn, -2 5 dn 1d8, dA dA, re ee, 4, a8, dP, 4 dn, dn CURVILINEAR AND NORMAL CO-ORDINATES. 491 +2A ds, Sh /3| aP,, dP,, 1dP., 2° dn, 1" dn, * dn, 8 aR. dA / ogee po rs CON 2 Gh J Wee dA Ana tee a 13 }9 2 1 13 8 dn, 128, * da, ay 4 aly dn, zi a dS. aE. Py } 2 1 : Z ,1dP, [Pe dn, * 8 ing * dn 8 dn, dA dA ( +28,, 5 i | L nN, nr J 1 dP, s dA, dA, aE. ~ 8 dn, 126, . dn, +P "dn, ara Te | , It is easily seen therefore that if we omit all the terms that are destroyed either by equal terms in " or else by virtue of the ten equations given at commencement of this Exercise that ‘ may be written Ste eas er 4 dn, “dn, ate dn, 1 dS, dA, dA, Se ee +P,. Gh laP s (- dA, dA —8 dn, 126, . Fie ae =a dee dA, dA.) . ~8 7 128, * dn, fees 7 : hence we easily see that if in ’—” we neglect all the terms that mutually destroy that, *—" equals P,, (dS, dA, ds, “a rare dn, ° aie ~ dn,” dn, 4 Pas dS, dA, dS, dA,) 4 |(dn,* dn, dn,” dn,) me Bs dP, dA, = dP,, dA.) 8 (dn, ° dn, dn, ~ dn,J Pa dP,, dA, i dP., aA,) 278 dn, “dn, dn, ° dn} Node G4, eke 24, ed dn, ‘dn, dn, “ dn, & S, (22, dA, dP, dA, 4 (dn, ~ dny dn» dn, Worn Xin) PART ile 63 492 But this may be written Rev. J. W. WARREN’S EXERCISES IN a eae dA, see dB dA, dn, dn,° dn, S,P,,-S,P,, (dB dA, dB at “ : a 2B dn,° dn, dn, dn, P.P.— P,P (@B dA dB a + —_ B Ch Gps Cie, GD. B er dA, Ad dAb gay | ae = poe = dn. ans ~ “dn, B 7s 8 P, a ZA AS, | 2 a fad ae ar ar. ¥ {ea dA, dA, a 8 dn, dn, dn, dn, Bp 8 12 B +748, ‘Be —— AS, R Le dA, dA, a 4 es dn, ‘dn, dn, ° dn, Eg 4 Lit. And thus we obtain for the full developed form of (23) the expression A, (aS, dP, a8, ab, 2 (dn, ~ a * an, * dn, ony A, dP... dP, , dP, dP.) 4 (dn, ° dn, ~ dn, ~ dn.) 1a et ds Tee dB ds, 2B°* dn,‘ dn, 2B‘ dn, ‘dn, AB + | [+ Pa+ (Ss) + Pa (25) - 28 - (5)p We now proceed to calculate (12) which is clearly symmetrical with (18). (12) equals -+ \4 3 dS, 1laP ‘Th, * Sn, ‘A, dS, dP., DAS dn, ° dn dS, at, ak. dn, "es dP,,) dn, } CURVILINEAR AND NORMAL CO-ORDINATES. =i dS, _A, dP. ds, A, abe a hes, = pha a dn, 2 dn, Am aes ds. Ae aE. oe tae oe “atk Which equals dS, 4, By dP, , as) dn ; dn, OT mare. 1 ds, dex ~ Gh. +5 aS tA. MG eae ai 1 ee dS, _ dS, 2 dn, |dn, dn, i dS, ds, So OUP dn, Which equals fi 1 Hh, dS, , dS, dP dP. dP + an, + an, dn, tah aie? ae ee dn dS, dP 5 bas Tare AD ae 2 dn, dA dA dA +P,, dn, +P, an, +P. dA, dA dA [ Bi eer peers dS. ds, ab ares oe Ce eee a dS, ds. Ser a Which equals 1dP, dA dA dA a ae ae Fat Pa. Git aP, (dS, A, dP, i dn, 2° dn, oe dS, 8, 8) dn We. ~ dn, dP. dS, 493 63—2 494 Rey. J. W. WARREN’S EXERCISES IN Which equals Baye igo" * 2B dn, * dn, Pads, pn ahs A, a2, dn, dr, * an ae an Ps dS, A, aP is Ay Ch ee + Tin, ar 2 1 One ie Te AY dP,, dP, dP, dS, Loe: dn, © dn Say ee dn, dS, dS, dS, ds, 37 Ae = . dn, a dn, . dn, . Which equals 1 dB dP, 1 @B dP, + OB ° dn,” dn, 2B dn,° dn, i GS GHEE Be Sy dn, * dn, Po (ae, “as, P, wae ee: as "dn, dn,“ dn, A, dP, dP, 4 dP, a8, 2. dni dnt *" dae dey dS, dS, dS, d&, Alan * do zm oa Which equals A (dP (OS. Ee . = 2 \ dn, “dn, -". dn,— dn, LA, (db n Gin GPs dP 4 (dn, ° dn, dn,~ dn, ¢ ie db~ ae. L .dBY ab. 4B° dn,° dn, 4B° dn, dn, A, (dP,, dS, dP, oI (2) ? ahs tin. ea” Tan dS, dS, dS, -d8, + A, fie te ta a sree (2) Lae (dP, dA, dP; dA, ae \dn, “dn, dn, * dn, (8) 1 (2P., dB GP. aB + 3B ee dn, ans , at) _ Ay (dP, Gh aE aP.. (7) rm dn, ° dn, dn, * dn,j" mu? the last five lines marked (a), (a), (8), (8), (y) I now proceed to reduce. CURVILINEAR AND NORMAL CO-ORDINATES. The two lines marked (a), (2) equal dS, A Agee eat ~ aBx dn, ) ° dS, A, aP,) . Niinan 2S sane} which equals dS, dP, ds, dn, © dn, me (dS, dA, dS8, yp dn, dn, ; (dS, dA, _ dS, dA, | dri Vadnk ” To the first of these last three lines add die (dS eAa d + F 2 dn, \dn, 2 dn, alge: GS ae aR 2 dn, | dn, 2' + dn, eg {> aA. ~dS, P., (dS, dn, dn, Be dn, * 2 (dn, aida dn, the line (y), then (a) + (2) + (y) equals aaa dA, dn, dAt) - ain) or slightly modifying first two lines by aid of the ten formule given at commencement of this Exercise and adding on (8), (8), we obtain that (a) + (a) +(8)+(8)+(y) equals GAG Eas. oF. dn, + P,, (dP,, -dA, dP, aA, 4 |dn, * dn, ‘dn, * dn; + ieee {adP,, dA, = Cher ‘ at 4 (dn, dn, dn, * dn, Ihe (dP. dA, e CAE. at 2, | dn, : dn, dn, 5 dn, 1 (@P. @B i dP,, at is 4B dn, ° dn, dn, ° dn, s dS, dA, ds, “A Bar = * dn, a dn, dn, P,, (dS, dA, — d&, a 2 (dn, i dn, dn, dn, f But the sum of these first four lines clearly equals aE Ps 4 (dn, ab B remembering that — {es dA, dP,, ral ; dn, ane Wah) * =P,,.dA,+P,,.dA,+P,,.dA,; and by picking out coefficients 496 Rey. J. W. WARREN’S EXERCISES IN it is thus easy to see that in (a) + (a) + (8) +(8) +(y) the coefficient of dA, dA, dA, dA, dn, * dn, ~ dn, "dn, 2 storie +o P,— AS +, P.Po—2P8s, F P which equals — B = B4A 5}. The coefficient of dA, dA, dA, dA, dn, * dn dn, ~ dn, 2 equals oe i APs SP Piper Ss, 3 12 12, 13 23 12 32. B? which equals +74 Bae. The coefficient of (re dA, _ dA, ai dn, ° dn, dn, * dn equals eres 13 28,8,P,, 2 which equals — BA,S,. Hence finally (12) equals : ic dS,...aP. of 3 = + 2 (dn, “dn, ~ dn, dn, A; dP. GP44 dbs —# = ; 1 dn, ~ dn, dB dP 1 dB dP, + ZB ° dn,’ dn," 4B° dn,* dn, fs (acest). @)-ata +448 (3h (13) must be symmetrical with this, therefore it equals A,(dP,, dS, , dP,, d8,) ost + | di, dn, dn, © dn } dP., ie. at 7 dn, a * dn lL dab. dP. lk. dB a 4B dn," dn, 4B ° dn, “+ dn, Efe 489 (52)— AP (ge) 448 QD CURVILINEAR AND NORMAL CO-ORDINATES. 49 Hence and by symmetry our second three differential equations are i (, EE a. EP ie {+ “dndn,* dn dndn, dndn, +A pee 9A, qBy = dB ,4P., dB as 2 dP, GB dP a dB dP,, Ot 4B “dn, dn, , an, — dn, ‘dn, dn, * dn, dn,* dn, dn,” dn, * dn 23 31 12 ae 24,8, c (ea) ata AAs S “@) aS He . Gs) B? 12 | F] + (P+ 44,8). (5 = nee aes Cay seers.) 0 23 31 12 | + (2, +44,8). (5 3) rahe (G5) . 444.8, Ga 1 + 2 aS, a*P —! CL a*P., \ 2 dnn,* dn2 dndn, dnidn,! +A,A & 1 {_,dS, dB_,dS, dB dP, dB Ps = 2 ee dB +o dB 4 oP rg dB 4B | dn, dn, \< dn, dn, 2 dn, “dn, dn, ding ‘dn, “dn, Es “dn, f 31 12 23\ 1 | ~24.8, C) + AP... Gy eA a BY 31 12 23 yt Patt, Ss) +44,8,.(} a) ene eS +0, 31 12 23 es +(B,+44,8).(53) -4,Pa-(Gg) + 44s5,- (58) : Gate Bene ase, d*P., | 42 sae 120 ssi 31 dn,dn,' dn dndn, dndn,) + AA 1 > dS, dB_ dS, dB_,dP,, dB dP,, dB i dP,, dB e dP,, dB | aP,, a 4B | eae ‘dn, *dn,'dn, ~ dn, dn, dn, dn,” dn,~dn, dn, °dn, dn, ° dn, f 12 /23 31 | —2 A,S, * lea Te ps Jee c Ga) ate Ales 7 Ca) | 8) ) =) a 1h +7 ]t es + 44,8). (5, +44,8,.(55 -A,P,,. (53) += 0. 12 23 31 | E (Py +44,8).(51) -A.Py- ei, + 44,8. (oi) | 498 Rev. J. W. WARREN’S EXERCISES IN For the sake of convenience of reference I also write down the three remaining differential equations of the second order; one has already been given, and the other two are deducible by symmetry. GS. GP. 20'S; — 347-8 -_— dn,* © dndn, dn,’ s eS 2 Be Bee f-2-109 (2) are ars (8) ety 29.) EL au (4) (os cae cae) a Paar) Sea ae MS ES eee, dn, dn dn, dn, ary) te pre Bee ee eae a | -@-108). (53) _ (24,P,, + 8P,). (3 4) ~ (24,P,+3P,) - (55) AB Bede ames eye eat ifs)» | +8Pue(i9) - P,-(is) -28,-(15) | _@5,, @P, as, dn~* dndn, dn? +A ic 1 {455 dB a dB dP. dB Vidk, =| B dn, ° dn, ‘dn, 2 dn, ‘dn, 2 dn, ° dn, 31 12 23 -~(2- 108,). (51) — (24,Py + 3Px)-(35) — 24,Pa + 3P.).(31) +G 1 +82-(ts) -28,.(;3) ~2,.( ( 12 31 12 +3Py.(35) =P a. (55) ae: CURVILINEAR AND NORMAL CO-ORDINATES. 499 These six Equations for Normal Co-ordinates as well as the Determinant Equations for Curvilinear Co-ordinates on pages 467 and 486 are now given for (I believe) the first time. an; “ais ~ dn ae dn, aie dndn, 4 Pu BEV eES TPa 9 d*S, \ 2 dndn, dndn, dn, dn,dn,} = a qP al v ( dn, Gx ) (aa 42S: | dS, _ 4, = _ 448, dS, dn, dn, Bis n, dn, © dn, | ce, edboeee es dS, dP,, 9 WS, aP,, aie dn, dn, dn, dn, Fe, i dn; dn,° dn, dP OPE, We, ae. 9 US: dP, 99S, ab, ~ dn, © dn, dn, ~ dn, i > dng > dn, dn, dP Gh dey dk. 2 dS, dP, ,48, dP. “dn, ° dn, dn,” dn, dn," dn, dn, dn, fie, Pet Big Pa BPp Ey oP | 4 "dn," & “dn, 2 dn, | | 4 Fs Ca a ds, een dP. | ile Gus: 4 "dn, 2 dn, 4 adn | 7B inale eEo dk ee OES een Te: dn, eas dn, 2° da, | dS, SS, dP, +8 Ga,, Bsn, | 1 dB Ae Jen +R: aa into a similar quantity ; +5 Sr into a similar quantity. T But now obviously this may be written in the form ae == 3 a: 501 502 Rey. J. W. WARREN’S EXERCISES IN da ies a? oi dn, Ce a ,8,) + dn,dn, dn, a Pa 5 PaP,) + dn,dn, (sp oe 2 5 PaPa) + — dn ra, (% Ps 9) 5 PaPs) A ==. a Baad, 3 d == = (48,5,— Pt) +3, (PaPa—2P 8) + Fe (P,P, —2P,,8)} + a= into a similar quantity + == into a similar quantity ; but it is clear this may be written a: Fdae eB (ant ‘3 dnZ i in) 4+ reas ae d* aa ad? 7 eal 2 td 2 / dndn, ( 2 tf das A =>) Ble, ame) tae ae Ge) tame a dB d /A,B dB d {A 7} " dn, =)s m (CE) NES = d {A oe dB de 6 i *) : B + an, * dn AG + + dn, “dn 4 2 dB d a dB c A By) | + an, “dn, ( 4 + an, = = if we examine above we find all such terms as (=) : iz 2 mutually destroy, and 1 n, thus we see our symmetrical differential equation of second order equals TB @B eR d*B a°B d°B = : 2 dn,* a dn,* * dn, Hae dn dn, ee cere dn,dn, Toad dn,dn, + ad°A, Pe ad’*A, i ose dn,dn, dndn, dndn, 3 (dB Gs od dB (a ) aes =") dn 2 (dn, dn + Ga, dn, dn, + dn, dn, dn, +A+B 23 31 2 =28, .(55)+ Pu- (95) + Pa +(ss) B a ig 12\| | +34+ P,. (55) - 28, ie P,, =0 3 31 1 + Pu (1) + Pa (73) - 2% + (9) CURVILINEAR AND NORMAL CO-ORDINATES. 503 EXERCISE THE THIRD. NorMAL Co-ORDINATES. Ir we refer back to “ Exercise the first,’ we find that LO ig OS te a duct edie A dey dy _ dy dy _ qin ae. Gia 2 dagen dz 2 dz B dz _ di on de ue aa Hence it is easy to see a (b,¢, — b.c,) + a C,a, = Cy) ot a (a,b, a8 a,b,) =0, du, ke 4 du, d d d aT 2 = == —¢ —_ = @.b.) =0 a (b,¢, Cae a (Ca, — CM) Dai (a,b, — 4,),) = 9, d d d ; Set ipa === (ola = ed; == (0 \=0, a (b,c, — 6,¢,) + ali (c,a, — ¢,a,) + ait, (a,b, — a,b, These equations are true both in Normal and Curvilinear Co-ordinates, and may be written as follows: OB 5 cB eee du,da, du, db, ‘dude,’ a eo BS du,da, ' du,db, * du,de, iB « @B ieee a4 du,da, | du,db, du,de, 504 Rev. J. W. WARREN’S EXERCISES IN Now muitiply the first of these equations by = ee , the second by ~ = and the B da, 1 dB third by Baa then add together, and using Normal Co-ordinates we obtain ¢ 3 dB dB dB dA, , dA.) . dn, ets dn, ee dn, = = ie dn,)? equal to aaa Bas) Se é dB) B Fae da,/ db, dn, G da,) ‘db, dn,\B da,!\’ A ee a aut (de, dn, Ge da,/ de, dn, (3 da,/ * dc, dn, & da,/)~ This equation takes the simple form dB eS UBh EdBye aes dA} dn, - "dn, eS aoa ao 8 (dn, e dn, d cos @ a 200s B, d cos y, | [888 os cake US Oe eae ee ea =e}, d cos dcos y, | [ cos 4, 8, ——+ f dais? where 2, 8, ¥,, % By Yo % B, Ys, have the same significance as in Second Exercise. Hence, remembering that Cosa + 00 0s By cos 1 =O, cosa, =, cos 8, a cos 9, =, COs cos B= 7 008 9 =, cosa, =“, c08 B= 7 £08 94 = Or, we easily see that the above equation may be written Be dB a dA, , dA, B B \dn, t “2 dn, + “*dn,S * dn, * dn, dcosa, , deosB, , dcosy, mee Re ie But it is a well-known proposition in Solid Geometry that dcosa, , dcosP, 4 F008 YY, 1 i dic dy dz R CURVILINEAR AND NORMAL CO-ORDINATES. DOS where #, and &, have their usual geometric significance ; and hence we obtain the value of ee. mR, in Normal Co-ordinates, Ue (Ces dB dB) dA, dA, BR,’ R, 3B a ae AA a ~Ginet Che We may obtain this same result by a good many other methods. I shall give a few for exercise. If we pass from a point on the normal n, to a very near point on the same normal, it comes to the same in the end whether we do this directly by travelling a distance PQ along the normal », to the surface u,, or else first go a distance Pe along the intersection of the surfaces u, and u,, next a distance ef along the intersection of the surfaces u, and u,, and then a distance /Q along the intersection of the surfaces uv, and u,. Hence clearly we have d PQ.A, =(Aiz, + SO + (CPs an,” where: (ef),, (fQ),, (eP), denote the perpendiculars let fall from Q on the faces ePgh; efPM; PUgk; ((ef)=PQ; (fQ.=4,x PQ; (P),=4,x PQ} And A, denotes the operation of taking the difference of the values of a quantity at two very near positions P and Q on the normal n, to the surface u,, and dividing this difference by the distance PQ; hence Re C Sree d ta Sdn,” Gh d dtp. SGae 2 dn, ‘dn, a dn,’ tee 4 Ons god 2 ‘dn, 2dn, ‘dn, 506 Rey. J. W. WARREN’S EXERCISES IN Whence we obtain ee ASP Atl. as dn, ad —-=2S. .A.+ B.A, +P. Ay, dn, d os ame 8 (Bi oo Ag g/l cali 3 dn, Now let dS, be an element of the surface u,, R, and R, the chief radi of curvature of w, at the point where dS, is taken; also let X% and p equal the elementary angles that R, and R, make with the consecutive normals along the lines of curvature; hence clearly dS = OR xp OR Now for the parallel surface neither X nor # receives any change; and hence we clearly see that dS, — value of dS, for parallel surface equals 2X... £,. B, letRt 6 JAM) whichweduals 4a WS FO lat ae But we know that dS,=ds, . ds,xsin®@; therefore we have dS,=B.sin ©. sind. siny . dn, dn, ; and hence d8,=B . dn, dn,. We clearly therefore have dS, —value of dS, for parallel surface = (B-—value of B for parallel surface) . dn, dn, + (dn, . dn,—value of dn, . dn, for parallel surface) . B and by above also equal to B. dn,dn,. PQ. tat +f Now dn,dn,—value of dn, dn, for parallel surface is what we call in the calculus of variations § . (dn,dn,), which it is well known equals din, , dén dn, . dn, . ie =e oat, but n,= PQ x A, and én,=PQx A,, whence since PQ is of course supposed not to vary with n, or n, we have dA, aA ie 8 (dn,dn,) = dn, . 5 Ea {+ Ean : hence we obtain as before 1‘ oo dB dB dB) dA, , dA Bae a °dn, ate ‘ dae y res : RY Ee CURVILINEAR AND NORMAL CO-ORDINATES. 507 I shall yet mention a third way to obtain above formula. Suppose that we write Px ay az COS, . 72 + cos f, . dnd +0084,» 73 =f, 2 2 Wax dy Pz = COS a, . 73 +0088, . 5% + cosy, . 7 = G, 3 3 3 as dy &z ; COB a, . Tian . fegaem ot ; aie an It is a known result that eee is proportional to fi, h, EG +E'G —-2FF. (See Traité du Calcul Différentiel et du Calcul Intégral, Par §S. F. Lacroix. Tome second, Seconde Edition, Paris, 1814, 629, § 774)...... But by previous Exercises we see that dS, CHE. _ ds, die OS. Nea a an dn, a (Bee =| (an = Tn) ae adP., ds, dS, , (dP, ds, Rea Se ae ea = dBi. GBe TEP Md Pe aN. : aa * dn, ae cre = ae 3 Multiply first line by 28,, and add it to second line multiplied by 2S, and subtract from this third line multiplied by 2P,,, result clearly is p,.o29, 2529, dn, d8, IP, ds, +25, a4 = i 25,A, 5" aes ee Pe a +symmetrical terms differentiated with regard to n,. = But this clearly equals il Gi) = 2 dn, (P re 4S,S,) dS, , 4 aPy , oP, +28, (24,53 + At Ge 2) _ 94,8, ie _94,8, ie dP, aa 24,P,, i *4 Je ae +terms differentiated with regard to n,. Vor. XII. Part IL. 508 Rev. J. W. WARREN’S EXERCISES IN But the above equals dB dA ~ B. 5 48,8, Gt — BaP Ge =24,-— 7 S) as Lage ge ap — Py a oe A, dn, Fe) + 4APat +a similar term differentiated with a to n,. But this equals qB dA, —B. tina dn, A, d \ + dn, (28,P, —28,P,)+3 A dn, (2 a 45, 8;) a a 45,5,) +a similar term differentiated with regard to n.. Hence once more, we find that 1 AA Gal, Bal (ee dB dB R, ia dn, ding = a A A, dn, es a Having determined the value of —— in normal co-ordinates, I now proceed to find 1 2 1S: RR im terms of the same system. rt the value of ae 4 1 1 By page 505 we have — A, (e+z) equal to Rit Rp» hence clearly we have 2 1 Ns 1 1 a7 (eH) + Cet R): which equals 1 /dB. a d 2 # (ae + gy, AB) + a (4,B)) d d d dB es (an s Aa +A \s (an, s i Bia, Fr oe .B)). which equals iB PB &B } 42% 2 1 dn, eA, dn + 4, dn B 2 } ee giete FB &B Pie dndn,* 24,9 dn dn, + 24,4, dn,dn., dn, CURVILINEAR AND NORMAL CO-ORDINATES. 509 @A, aA, aA * Gdn, Sans) adn, aA, aA, dA, inida tas De c (AAs si (Fa a dn, 2 (dA, _ dA, (dB dB dBy | (ae + a) ia Page a) dB (dA, , dA, , dA) dn, ia Mi dn, ae dn, ) dB(dA,, , 44, , dA,) dn, \dn, ee dn, . rere ; B L B iy B 9 Such is a formula for =— in normal co-ordinates. I shall for conciseness refer to it RR, 2 > 1 1 1 =? o r 2 =—+—=—; => 1 ing A RR, 2M. The following formule for R, R, 3 RR, &e. are interesting, Write 2V=fP+fO+ fe +24, f,f,4+ 24, ff, +24,f,f,, then clearly, if after differentia- tion with regard to f,, f,, f, we write f=1, f,=0, f,=0, we shall have ~dV a@ dV ad ,adVad 1 df, dn," df, dn,” df, dn,’ B(F eS) d dV, a dV d dV A ieee! “Et Che MC Tee Let now r, s, ¢; a, b, c, be six such symbols, so that @ is always the companion of r, b of s, and ¢ of ¢ Moreover, let it be understood that each of 7, s, and ¢ are in- dependent of one another, as well as a, b, and c. Finally, let it be understood that wherever we see an 7, this 7 is in succession to be changed, first into f,, next into f,, thirdly into /, And wherever we see an a, let it be understood that first this a is to be changed into n,, next into n,, and lastly into nm, Let the same hold for s, ¢, and for 6 and c. This being understood clearly, we may write _ A,=2=- Ye (consisting of three terms), ..........00..0..00..-0. (a) i C 1 1 is d dV J eA Vi é [ a Boda re (@ilabxaey WETS) Sigone Looe sodsesaacneSeeeecsc- (8) | + | but did Lad, (av S i ds db B da\dr B) ‘ 510 Rev. J. W. WARREN’S EXERCISES IN Hence we see that we may write 2 Tec aa ve a iaVi DS IBS iene RR, B “db ds da dr because the right-hand side of this equation equals 1 Gh PA MN Gh GA B 7a te oe Bda deo! which equals d dV Bis (ese Aig sles a = (5 7 B)x (3; i) os ge ogee ds db "Bi da dn? which agrees with previous page, In the same way as we obtained the equation rae7(EtE)t4 (e+z) Lik, h, R, h, h, it is easy to obtain the equation 1 i eel il RR, Gem cia Gex)=° Hence by formulz (a), (8) and (y), ad a\aiee ee d dV d dV d aV or 2 a = aa B= Orncns teres: (8). de dt db ds da dr Equation (8) clearly consists of twenty-seven terms. Another formula for S may be obtained thus: Lh, Write = as, + DP = aS, F dP, ES dS, : dS, dn, dndn, dn;’ dn, dn,’ dn, 1 dS, AS ee eres ; 28, | Ps dP, _ W, ; B ; 28, dn, dn, CURVILINEAR AND NORMAL CO-ORDINATES. Am a Sane an, 1 dS. + B an, 5 28, 3 1p 23 ds, ical Bea ANE) dn, ’ Fi, 3 28, This clearly may be written BY xm= CHS GHZ &S Pu it - aaa A Fa dS, dS, _ 4P,, 2 +(e) dn, dn, dn, dn dn a s, as, dS, dP,, dS, + (28) | dn. dn , dn, dn, dn, dP,, aP,, i nf dS, d8, dS, dS, dP, dS, dP, dn, ~ dn, dn, ~ dn, dn, dn, dn, dn, Hence B*m equals +B {ila +h dn, dn, A + an? oe PSG t 8 ae Pod “48 fos Bhan 2, Mia {39,24 29,85 — p, Pal +1ABe (a9, 254 29,25 — 7p, Hence Bim equals B aS, a, ae id dn2 dn,dn, dn," B [es dB dS dB 4 dB 1dP,, dB i ian dn, dn, dn, 2 “dn, dn, 2 a dn, dn Dy | _ bo Rev. J. W. WARREN’S EXERCISES IN Dp A). Pa Ad Ate 3 l- a3 5 - i i ee ee A, Sas 4 eae 2 dns Sdn * 5 dn, ah, dA, | dA, dA, dA mas dn,’ as dn,’ 4p mt Ay dn, waa 2| @ (1 /dS, 1 aP,, d (1 /dS, 1 dP, iy E 17 a ~ 2 dn, )f + an, E (a 2 dn, dt . But 1 (dS,_14P,, , dB B(dA, 4 d4,_ 4 44, 94 44, Bla 2 in) = CS De a= sD) a aa = ae a ae a(S C A dP and therefore as | B : FRE as } equals wad Ay +3 re (4,— 4,A,) a ( d* @ A, £3 ig oA @A,) oe laa a a dn, ai. 2 dn, Fo 2 dn) 13 dA,dA, dA, dA, dA ‘i +54 1- dn, dn, ~ dn, a 2 2(F dA, dA,) di ae Aa Sle dA, dA, _ dA, _¢4 seat In. Sd a auo-2 dn, ° dn, CA Hence by symmetry we see that ad. {1 4dS, Gi-dP., d°4d 7a8,_ 1.-dP,, 5 E E ea = ao) 2 dn, i & ~2 dn )t] equals =. ‘B &B iB = Qe eA PE Ate TAG ye “dndn, CSAs. dn,’ es 2) aA 7A aA, aA, CA Wee if) 2 2. ek eee aS St Te 5s (ese! (ea. *dn,dn, *dn, dn, sda, Sans } dA, dA, ddA,dA, (dA ) =) +% \- dn, ang di, dn, (on * at } CURVILINEAR AND NORMAL CO-ORDINATES. 513 BaBidd, 4 dd, 4 ad, 2 dn, |dn, 2 dn, > dn, 3 @B (dA, dA, 4 dA,) 2dn,|dn, ~*dn, ~ * dn,J dB dA, dB dA, 2 dn, dn 5 dn, dn, ~ 3 8 2 2 — 34 We reduce the determinant on page 512 thus; multiply third column by A,, and add to it first column multiplied by A,, result is il 1 aia | 1 2 ea: 3 (1-4,/); 3 4rd, — 4,4, +5 A, | | dA dA dA dA = fps AL ate Be 2 pa vars |: 2 dn,” a dn,” A; Fe A, 7, : dA, As ga (ty. gy eR | se dna Budn, ” = Adres * dn, (this is finally to be divided by dn, PHB (EL 4 2, _ a 2dn,|dn, — ? dn, 5 dn, 3A dB dA, dB dA, *dn, ° dn, dn, * dn -B\+ C 3) 48, +(5 5) 408+ (55 3) 3 C8 Dh- Referring back to page 508 and adding above to B multiplied by expression at foot of that page, we find that B. (21+ m) equals @B @B as a°B a°B &B dn? Be dn, ss dn, er dn,dn, 124, Taran. dn, dn, Sea iane dn, dn, aA, i d*A, is oe f dn,dn, dn dn, dndn, 3 dB Go =) dB (f iu ca a dB (Fe “ dA =). 2 (dn, \dn dn, dn, \dn, — dn, dn, dn, dA, dA, dA, dA, + B ie . dn, = dn, . iat Pee ceecccccccceeceessecs (a) " A,(dB dA, dB dA, + >) ‘ae . ie = dn, . mt eee wereeeeee pteeeseeeces (8) As dB dA, dB dA, (y) - dn, . du, dn, dn, SSeS ee U's 1 (dB. dA, dB dA, + 5 ‘ae : dn, _ dn, a des daduswesteacuncewe (8) 1 (dB. dA, dB dA, + 3 e . dn, dn, I cece cece rereucceererceee (e) Pe B ‘i (Gs) 4,8, + ee A,S, + (ss) 3 @8,-1)} otlvete (0). CURVILINEAR AND NORMAL CO-ORDINATES. 515 Now let us consider for a moment the five lines marked a, 8, y, § and ¢, we clearly have a ee es) a PAP, /23\ BA,P,, a C= Gs) - 2 Ss ke an ( a8 BA,P,, 13) (= peace ae (53 BP, /12\ BP, /23 Sat 5 (es) ao i) ape BP. ta ane i 2 aD 12)" 23 31 ~2P., () +2P., Go) : | Hence adding the last line & we find that B. (21+ m) is equal to iB TB @B iB dB &B Ga> adi dae Oo umn ee aad go ane (del a dA. # wa, i (dn,dn Te, + Gn dn, dn dn, 7 dB | dB (Ge +) to ao) 2 ler dn,) * dn, \dn dn oa dn, fos, 2)+27.(2) +27. (3) +H) -2r.G)) + 42P. | [-s7eGal+era() = | Now from above subtract the symmetrical zero equation given at foot of page 502, w: clearly obtain B(2M@+m) is equal to Vou. XI. Part II. 66 516 Rey. J. W. WARREN’S EXERCISES IN But it was proved at foot of page 511 (see value of determinant given at foot of page 513) that B’m is equal to as, _ GED ass dn*, dn,dn, dn, _1, dS, dB dS, dB_14P,, dB_1 dP, ze Bl" dn, du, + dn, dn, 2 dn, dn, 2 dn, dn, ae lks {. (48, — ~2)(5 3) +448, (3) +448, es} Add to this the second zero equation on page 498, and then divide the result by B we clearly obtain Bm is equal to Zs [+*8 G)-% Ge)-?. G) +2) «0p, (2) —29,(31) 7, (3) [+37, (2 Now comparing this Equation with the immediately previous found value of B (2 +m) we clearly obtain m=— M, whence we have a second formula for 1 BR,” and then by the second equation on page 498 we obtain a third formula for The formula m =— is due to Gauss, who obtained it by a different process (see Disquisitiones generales circa superficies curvas, Gottingen, 1827). I conclude this Exercise with a remarkable formula, the proof of which I leave to the reader as an Exercise in the preceding methods. I call the formula remarkable on account of the curious destruction of terms, the coefficients, for instance, of the nine determinants 12 (a) yp &e. completely vanishing. The reader should compare this following formula with pages 467 and 487. ppl 2S du, du |{, dG { al (= dF a) : {a dG Ce e A, 0% dG 5 ave istrea Ne —— +=— -=— y lt a din du. U\d i, dt, ® du, : ares I wl 15 ny, dH di ad dJ dl dG dl dJ or Se) bie ‘(2 du, ae Aig du, * 4,(2 du, ara {a 2 du, + 4 adu, + 4u( ae ta a.) il 1 LR, ‘ kK, du, du, CURVILINEAR AND NORMAL CO-ORDINATES. o17 dF. ad ds dl d dG di dF dJ dH | elas we) du, " du, > {4age + Ange + An (ie + Fe at dF dE dJ dE dE GNP iahel GBS CHE dE {41 (2a a a tere (2 du, ii) ae ia : 14.5 + Ars a aE du, vi = ie a is equal to dA,, dH dA,, dI dA,, dG 2 1 So Seat, Seo Le cea 6 eee * du, du a du, d setae du, du, dA,, dJ GAY dl dAn wal co Wei ee 11 as 11 one du, du, ei. { du, du, du, Gat par ta OF, 4 a, {das dG dA, ge du, du, du, du, du, du, dA, dA, du, du, dA, [4s dE 2 4G dF © wa da, Ody hea =) dA,, 5 CHa 5 CHE dJ du, (4 1 dy +A 8 du, + 24,,4,, 7) February, 1875. 66—2 518 Rev. J. W. WARREN’S EXERCISES IN NOTE TO EXERCISE THE FIRST. (Pace 460.) I give here the values expressed in terms of general Curvilinear Co-ordinates of the radius of normal Curvature and the radius of geodesic Curvature of a Curve traced on a surface. I represent by the notation r, the radius of normal Curvature with regard to the surface xz=O of the Curve formed by the intersection of the surfaces x=0, y=0, whilst y,, denotes the radius of geodesic Curvature with regard to the surface «=0 of the same Curve; {(u, x), (u, y)} represents the value of the angle between two curves traced on the surface w=(, these curves being formed one by the intersection of the surfaces «=0, w=0, and the other by the inter- section of the surfaces w=0, y=0. All the formule that follow are derived by simple transformation of co-ordinates from the corresponding Homogeneous formule expressed by means of Cartesian co-ordinates. [ also assume the known or else easily demonstrated formulse 1 1 { 1 1) — = ———_ 1— — cos (2, y) —} Yo (HY) Ve OY gy? 2 es | : cos (a: 1 Yu a sin (x, y) Vy Ss ze ») ar : where (x,y) represents the angle between the surfaces x=0, y=0; finally r and y will represent the radius of normal and geodesic curvatures of some curve traced on the surface U(w,, U,) u,) = 0, a direction of an element of this curve being defined by the differentials du, :du,: du, These preliminaries being thus explained my formulz are: LG of ZS dU? aU? dUdU dU dU dU dU)\a Es A | — SW Oy te ee ae a) EO OYA Peal len +4,, ce we Ga) as sar du, 78 iy: du, * Ay, =o! -{C,, . du,’ + C,,du,’ + C,,du2 + 20, du,du, + 2C,du,du, + 2C,,du,du,} equals [SUR ALE Re wey an Mie Ca oF nay, (du? sal Gs du, Git du,’ a a du,du, idly du,du, mend du,du, due a +{K,, . du?+kK,,. du? + K,du, + 2K, du,du, + 2K,,du,du, + 2K,,du,du,}, (52eu) + 4 (6 1d0,\ . 4 (My _1d, ag} "\2 du, du, 2 du ) 7 es 2 =) * du, don) 4 ; (a1 ew 1dC,\) aU | j= 2 du, a: G a) * du, dC, _1dC,, 1 dC, dC,, 1d0,\) aU Aes 7a) +4, (3 I ie Ge 3 mal ct where | > | AE RRP ees? i oN CURVILINEAR AND NORMAL CO-ORDINATES. 519 and (sf, (20m 4 nM) 4 4, Bang 4, 0m) aU oe teas du, = 2 dw, 18 me * du, dC, dC, adC,, aC. dC, dU = 3 +14, ae du, Ary du, east du, a du, 7 we) % du, {’ dC, dC, aC, dC,, dC.) dU 4 eee aa, is age) otoee * du, K,, K,, K,, and K,, being obtained by a cyclic change of suffixes from above. There are eighteen = coefficients of aes &ec, in all, compare these with the eighteen functions of first Exercise. 1 : 4 24 Wy Cos (Ux) = 4 du, 2 du, = du, os (Uu,) = 3 {. v7) t4 Ta) A dU\* 24 dU RU dU dU dU dU)s° ee GE du, = (aa) Sdu, du, 8 du, du, * du, oat Cos (Uu,) and cos(Uu,) are derived by a cyclic change of suffixes, by the aid of these cosines we dU dU adU_ aU 1 du,? de,” du,’ du? &e. from the above value of = can obviously eliminate ee eer ere eee errr er eee eee eee ee eee eee eee ee eee eee eee ere ee Tee ee eee ee eee eee eee ee eee eee ee ee ere “6 : dU\? dU av avy } . {C,,. du,*+2C,, . du,du, + C,,du,} . {¢, Nae ore di, +C,, a } NOCCE= Gan Yu equals lic.c -0,4 {ae Fe le | | (Css i ee a cea = diy + Fa du} +f C 1dC,, , ‘aC, _ dC,,.\\ dU du2 "2 du, * (5 ah, du, JS du, (an eee Je dC,, dC, )} dU | | | is ( Cre 2 du 2 Ti rae j du, | dC, dC.) dU : du.du + 2 Bes =e du, J ; za | ; tee dC, dC.) dU | | +{ Gi du, Cs = C da, | f dC. 1 dC. 1 dC,,) dU +{ Cun (Fe ED eS du. ae é 1d0q_40,)) a0 ug du, * eee See J du, for since du,=0, and dU=O, therefore dU dU EAE _ ,@ —— du, + du,=0, or du,= = Ns and du,=—d 7; U du a du, 2 2 1 and if then we write ste equal to a constant or zero, we get a differential equation to deter- U Yus 520 Rey. J. W. WARREN’S EXERCISES IN mine the integral U=0 of Didonian or Geodesic curves traced on the surface u,=0. Write for shortness {(Tu,) (uu,)}=o, and {(Uu,) (ujm)}=o, dU _ dU i= dU? dU ri dU ahs BiG (Ness ; dU dU 1 du, 2 du, ifn (40 dU dU dU\\3° Ce (Caer) ag 2C;, du, du, aa \ C. cos o, = aU (C20. = C3 = sin w, = 2 ae ee 3 1 dU~\? dU dU dU\*)4 One {Cu = ’ + AX —II,=0, (8) 3°+AX-T1,=0, (Gy SAN SIME SSF SSO a and to these of course must be added six similar series of differential equations of the second order, derived symmetrically from the surfaces wu, and u,. Observe also that we may write the first (for example) of the Equations on page 497 1 dA, dA dA = 34,4, +A,A,; B SSS SS iy = z| as + Os .} ip ? dn, vile dn, = dn, Hee 0 from which probably interesting results might Ate but I must now bear in mind the warning of our great Novelist, that “We can do nothing safely without some judgment as to where we are to stop.” V. The Place of Musie in Education as conceived by AnristotLe (Politics V. [vu] ec. 3—7). By Professor JEss. [Read May 17, 1875.] THE object of education is to make the man a good citizen, and so to put him in the way of attaining happiness; that is, the conscious activity of the highest part of his nature in accordance with the law of his own excellence. Education should be the same for all the citizens; and in order of time physical training must come first, moral training second, intellectual training last. The State Education should aim more at the development of the contemplative than of the practical reason, since the legislator’s object is to fit the citizen, above all things, for the wise and happy enjoy- ment of peace. The particular branches of Education, as ordinarily recognised, are, Aristotle says, four in number:—Grammar, Gymnastic, Music, and (as some reckon) Drawing. Grammar*, Gymnastict, and Drawingt have evident practical utilities; but with what object is Music to be taught? This cannot be said to be either such a direct utility as is the end of Gymnastic, or such as is the end of Grammar and Drawing. Three objects, Aristotle says, might popularly be assigned: sraideéa, discipline ; Traidta, pastime; and d:aywyy, the rational employment of leisure. Classified more scien- tifically, the objects which are to be attained by the study of Music a.e, he concludes, these three:—zraideia, discipline; Svaywyy, rational amusement; and xa@apors, the purifi- cation of the emotions. It is of the third and last especially that I wish to say a few words, with a view to elucidating, if possible, the exact meaning which Aristotle attached to it, and which, as it seems to me, is as suggestive for our own day as it is significant of the Greek feeling towards art universally. But, before coming to xa@apous, it will be worth while to touch briefly on the two other objects—adéeia and Siaywyn. I. aideta. The disciplinary value of Music for youthful learners is twofold ; artistic and moral. Artistic, as educing and training those perceptions which will make * «For business—for economy—for learning—for political | artistic finish, which guide one (for instance) in purchases, actions.’ év tots lilos wrios: as making men connoisseurs of art ; + mpds vylecav kal ddxnv. above all, as training the sense of beauty in the human t As fitting to give accurate ideas of shape and of | form. Vou. XII. Part IL. 67 524 Pror. JEBB, ON THE PLACE OF MUSIC IN EDUCATION Music a delightful resource in mature life—as, in short, preparing Svayey7. Moral, since, as we listen to Music, we become zoioé twes: there is a definite affection of the soul; and, if the Music is rightly chosen, it disciplines the moral nature by establishing in us the faculty of rejoicing aright—é@ifovea SvvacGar yaipew opOds. For Music can give us images, duowmata, of certain feelings,—love, hatred, joy, sorrow; and pleasure in the imitations will create sympathy with the feelings represented. It is peculiar to the sense of hearing that it can thus be the channel of a moral imitation. The sense of touch and the sense of taste are not accessible to such suggestion. The sense of sight is so in only a slight degree. For though forms and colours are, in a way, ethical, or significant of character, they are so in a different manner from musical sounds or words. Musical sounds and words are imitative expressions of character and feeling. Forms and colours are not expressions, but only symbols; they are not opowwpata but onyeia. Granting, however, that Music as a discipline has potentially this double value, the artistic and the moral, what kind of Music is to be chosen as especially useful for the discipline of the young? According to a division which, Aristotle says, had been used by some scientific writers (tay év didocodia twas) of his day, médy, styles or genera of Music, were classified as 1. 7OcKa, 9 , 2. mpaktiKa, 3. evOovovactiKa. 1. The meaning of 70a is explained by the mention of the Dorian pédn as being 7@ueérata. It is a grave and manly character in music, remote alike from excitement and from a voluptuous languor. 2. ‘Practical’ Music is that which accompanies and interprets action; — stirring, vigorous, animated, like martial music, but, on the other hand, steady and _ restrained. In the Problems Aristotle says that the Hypo-phrygian mode—in which the enthusiasm of the pure Phrygian was tempered—has an 700s mpaxtuxoy*: and so the iambic trimeter, as compared with the saltatory tetrameter, is said to be mpaxtixov+—as Horace expresses it, natum rebus agendis. 3. ‘Enthusiastic’ Musie is such as the Phrygian—a wild, excited strain, fitted to stimulate the worshippers in the orgiastie rites of Dionysos or Cybele. Now, for raideia, the Ethical Music is of course to be used,—the Dorian chiefly; though the Lydian Music may also, Aristotle thinks, satisfy the three conditions — absence of excess, the limit of what is practicable, and propriety. But does Music, considered as a part of early training, imply the power of performing upon any instrument? Aristotle gives two reasons for answering Yes——(i) A measure of practical knowledge is necessary to make a good judge of Music. (ii) A musical instrument may be for youths what the wdAatayy of Archytas is for children— * Arist. Problem, xtx. 41. And hence the reason,"he | passive sympathiser—xndevr7s darpaxtos. adds, why 7 broppuy.orl was never used in Tragedy by the + Arist. Poet. c. 24, 76 32 lauBixby Kal rerpdmerpoy Kivn- chorus, which has no part in the action, but is merely a | rind, 76 uev dpxnorexdy, 7d 58 mpakrixdy. AS CONCEIVED BY ARISTOTLE. 525 a@ means of keeping them out of mischief. But here he states and answers an objection. May not the pursuit of executive skill in Music degrade the citizen into a Bavavoos or mechanic? Aristotle answers:—It may do so, if it is carried too far. We have to fix a limit up to which it may be studied by those who are being trained to the virtue of a citizen. This limit is determined by two things. First; the learning of Music must not interfere with other studies. Secondly, the body of the citizen must in no way be unfitted for war or those exercises which befit free men. No mechanic, any more than a slave, can do these actions which are according to virtue. Therefore youths must not enter upon such laborious musical training as is preparatory for the contests of artists (teyvixol aydves). Nor must they attempt those brilliant pieces of an extra- ordinary difficulty (ta @avyyacia xal ta mepitta) which have been brought into contests, and thence into education. In a word—the study of Music must stop short of what is teyvixy, professional. The feeling of the Greeks in and before Aristotle’s time towards artistic specialists seems to have varied with the eminence of the artist a good deal more than it does among us. ‘The artists of genius were recognised as great men. The ordinary artists were mechanics—men who had gone aside from the true political life, and whose moral natures were maimed. j Il. Scayoyn. The distinction between sadva and éssaywyy must be clearly seen. maidia is mere recreation: it is for the sake of rest (xapw avamavcews), and fulfils its end if it is pleasant. dsaywy) is something more: it has two elements, corresponding to the two chief constituents of happiness itself—rd xadov and 70 76v. It is the employ- ment of leisure in a manner befitting a citizen. Let it be remembered what is Aristotle’s view of this oyod)*. The soul is of two parts, Rational and Irrational; the Rational is divided into the Theoretic and the Practical Reason. As the Practical is subordinate to the Theoretic Reason, so useful or necessary actions are subordinate to noble actions. War leads up to peace. Work leads up to rest. Bravery and Patience are necessary for work, ie. ‘Philosophy,’ intellectual culture, is necessary for the right use of rest. Temperance and Justice are necessary both for work and for rest, The aim of education is to teach men first how they shall procure, secondly how they shall use, leisure. Greek civili- sation became more and more developed, the science of leisure—if one may use such a phrase—was more and more cultivated. Aristotle's word oyoXaotixos means neither exactly ‘leisurely, nor, of course, ‘scholastic,’ but rather ‘fitted for leisure,’ ie. qualified to use it intelligently: see Polit. VII. (v1) 8 § 22, and VIII (v.) 11 § 5, pate cyoras ... pate auANGyous oxOAacTiKo’s. ‘As they became more fitted for leisure, he says, ‘through their material resources, and of a loftier spirit towards virtue,—having already, too, after the Persian wars, been lifted up in spirit by their achievements——they began to lay hold on all learning, drawing no line—ovédév dvaxpivovres—but pushing their search onward t.’ Here, then, is the reason of the place held by Music in the mature life of the normal citizen—it is one of the noblest and most elevating forms of dsaywy7 or rational * Arist. Pol. v. 14. + v. (vu) 6, § 11. 67—2 526 Pror. JEBB, ON THE PLACE OF MUSIC IN EDUCATION recreation. And while it is thus a great general instrument of Saywy7, it ministers, in that quality, to two special purposes—the culture of the intelligence, ¢povnots, and the purification of the emotions, x«a@apors. 1. To the intelligence it renders, first of all, the service of relaxation, dveous: secondly, it affords a gentle exercise for the critical faculty in alliance with the imagi- nation, thus aiding to render the perceptions subtle and exact. Athene’s reason for throwing away the flute when she had found it, was not, Aristotle suggests, that it distorted the player’s face, but rather that it contributed nothing to this essential object of the best Music—culture of the intelligence, 2. What, however, is to be understood by «a@apeus, that purification of the emotions which is the highest and final moral function of Music? The word xa@apo.s, as applied to Tragedy in Aristotle’s Poetics, and here, in his Politics, to Music, has been variously explained. In the Poetics, Tragedy is described as effecting, by means of pity and terror, the purification of such passions: 6 édéov cal goBov Tepaivovca thy Taév TowwvTwr Trabn- patov Kkablapouw, The explanations which have been suggested are, so far as I know, four in number; for I set aside the notion, resting on verbal misconceptions, that ta@nwatwv and xa@apois could mean ‘removal of calamities —the prevention, that is, of such disasters as Tragedy represents. 1. «xa@apo.s =that moderation of the emotions which results from familiarity with the objects that excite them: as the passions of pity or terror might be moderated, through habit, in the physician or the soldier. This explanation is manifestly not only inadequate, but not specially applicable to tragic fiction. 2. xa@apois =chastisement of the bad passions, effected by pity and terror at what Tragedy represents. When we see in Tragedy what the bad passions entail, we restrain them, This view appears untenable when we observe that it excludes pity and terror from the passions thus chastened; whereas Aristotle says, tév toiovTwv raOnwatwv, such passions— such, namely, as pity and terror; e.g. love and hatred. Compare Polit. V. (vim) 7 § 5, tauTo 61) TovTO avayKaiov TacyeW Kal Tos EXeHuovas Kal Tos PoSnTLKOrs Kal Tods ONoUS ma@ntixovs: where tavTd Tacyew = kaBapcews Tuxeiv. 3. xa@apois =the separation from pity and terror of what is disagreeable or painful in such emotions when they are excited by real objects, and not, as in Tragedy, by fictions. Pleasure is doubtless attendant on xa@apots ta@nuatwv: but clearly «a@apors consists in something more than making an emotion pleasurable; and, moreover, the operation of xaQapors on the moral nature is manifestly something gradual, and when effected, lasting; it is not confined to a momentary impression; it is a process, and ultimately a healing of the soul. 4. xa@apois =‘the correction or refinement of the passions.’ This is Twining’s expla- nation; it is the nearest, I think, to the truth; but I do not think that he has found exactly the right point of view. TI will quote his own words * :— * Twining, Poetics, Vol. u. p. 17. AS CONCEIVED BY ARISTOTLE. 527 ‘The passions of savages, or of men in the first rude stages of civilisation, are fero- cious and painful. They pity, or they fear, either violently or not at all. With them, there is hardly any medium between ungovernable agitation and absolute insensibility. Suppose such a people to have access, like the Athenians, to theatrical representations, and to have their emotions kept in frequent and pleasurable exercise by fictitious distress ; the consequence, I think, would be that by degrees they would come to have more feeling and less perturbation. Instead of sympathetic emotions rarely excited, painfully felt, and soon extinguished, they would gradually acquire a calm, lasting, and useful habit of general tenderness and sensibility. The doctrine, therefore, of Aristotle, that tragedy purges the passions would perhaps only amount to this—that the habitual exercise of the passions by works of imagination in general, of the serious and pathetic kind (such as Tragedies, Novels, &c.) has a tendency to soften and refine those passions when excited by real ob- jects in common life.’ This view appears essentially modern. The idea of softening, refining, correcting, was certainly not, I think, attached by Aristotle to xa@aipev, cafapos in this relation. In order to explain what, as I think, he did mean, a few prefatory remarks will be needful. Winckelmann observes that the two great characteristics of the Greek ideal, whether in art or in action, are what he calls Heiterkeit and Allgemeinheit; cheerful self- possession, and generality. The first explains itself; it is repose without sternness or sad- ness or monotony. The second, generality, is to be understood as the very opposite of laxity or vagueness; it means the concentration of impressions into types. This generality is best exemplified in sculpture. There, the character necessarily predominates over the situation. The sculptor has to choose a type which is intrinsically interesting, indepen- dently of a special situation or a critical moment: and then he has to present this type in its broad, central, incisive lines*. This he effects, not by accumulating details, but by abstracting from them. All that is accidental, that distracts the simple effect of the supreme types of humanity, all traces in them of the commonness of the world, he gradually purges away. Now sculpture is not only the most Greek of Greek Arts+, it is the Greek soul and nature itself. ‘In its poets and orators, as Hegel says, ‘in its historians and philosophers, Greece cannot be conceived from a central point, unless one brings, as a key to the un- derstanding of it, an insight into the ideal forms of sculpture, and regards the images of statesmen and philosophers as well as epic and dramatic heroes from the artistic point of view; for those who act, as well as those who create and think, have in those beautiful days of Greece this plastic character. As Sculpture purges away the accidental, the common, the disturbing, so it is that Tragedy and Music xa@atpover ta rajwata. Tragedy moves pity and terror; and Greek Tragedy moves them by the simple, massive presentation of great issues, from which the vulgar, the spurious, the petty, the maudlin are excluded; it is the conflict of the human will with necessity, it is the antithesis between written and unwritten law, it is the stedfast endurance of suffering incurred in a good cause, it is the duty to Apollo prevailing over the dread of the Furies. The objects or the issues, however grave, which in real * See Pater, Studies in the History of the Renaissance, p. 188. + ib. 192. 528 Pror. JEBB, ON THE PLACE OF MUSIC IN EDUCATION life move such emotions, have seldom this netteté, this clearness of outline and freedom from alloy. What is the ultimate reason of that offence which persons of ordinary culti- vation experience when some tragic or horrible occurrence is made the subject of florid comment? It is that such comment is doing deliberately and violently the very reverse of what, according to Aristotle, Tragedy has to do; it is using pity and terror not to clarify but to adulterate the feelings by confounding an essential pathos with its most trivial or repulsive accidents. A living poet and novelist, whose fictions have some moral as well as some artistic affinity with Greek Tragedy, was once asked how these works had been influenced by the study of Sophocles; and the answer was, ‘in the delineation of the great primary emotions.’ Music also can move pity or terror or the feelings akin to them: And lo, with thee doth rise The lord of melodies, Sovereign of glorious sound, as thou of form: Love, hate, hope, fear, scorn, wrath, defiance, prayer, Each at Beethoven’s mandate thrills the air, The low, sad night-wind or the rushing storm. And Music, when rightly used, can effect the hkatharsis of these emotions. What Tragedy does in the sphere of action, 7pd&s, so far idealised as to be represented by great and simple themes, this Music does in the sphere of moral imitation, 7@c«7) opotwos, whieh 18 peculiar to it. It gives a scope to the emotions from which everything foreign, turbid’ vitiating or perplexing, is banished; it clarifies them; it presents them im a genuine sim- plicity, and at the same time with a majesty and a power from which even detail which was not impertinent would still necessarily make some detraction. In order to see distinctly that the modern and perhaps peculiarly English idea of toning down, refining, is not necessarily, and if at all, only in a secondary sense, connected with katharsis, let us look at a striking and suggestive passage in the Politics V. (vul.) 7, § 4. Aristotle has just been saying that Music is valuable for the three things, wavdeia, diaywyn, Kafapors, and has drawn the inference that all the modes, dppoviat, may be ad- vantageously applied for one or another of these purposes; the most ethical harmonies, for matdeia ; the practical and the enthusiastic for akroasis—for listening to while other people play *. But why is enthusiastic music thus universally available? Because, he says, the emotion which in some souls occurs vehemently, icyvpds, exists originally, daapye, in all human souls, in some degree: this is true of pity and terror; this is true also, he adds, of enthusiasm ;—just as Wordsworth says, that the poet is a man who has the common sensibilities in a higher degree. ‘Some persons,’ Aristotle continues, ‘are liable to the seizure by this tumult in the soul’—ird tatrns Tis Kunjoews xataxdymuor eiciv. Then he gives an illustration. So much depends on the exact rendering of the words here, that I must give the Greek. &« 6d tdv iepdv perav Cpduev tovtovs, Tay ypnowvtat Tois eEopyiatovet Tiv ux péreot, Kabictauwévous Womep latpeias tvysvtas Kal KaOdpoews, tavTd 8) TodTO * We must not alter mpds dxpéacw into mpds xd@apow as | of the use mpds dtaywyjv—the cultivation of the intelligence, Twining does. Aristotle has used the conveniently general | and xa@apots. word axpéacis precisely because it includes the two elements AS CONCEIVED BY ARISTOTLE. 529 avayKatov Tacyew Kal Tos €denovas Kal Tos PoBnTiKors Kal Tods GANovs TaOyTLKOUs, tos 8 dddous Kal’ ’cov émiBadrev Exactovy TovTwY, Kal Tao ylyvecOal Twa KaBapow Kai kouditecOar we 7 S5ovy5. The point is—Are we to identify ta lepa wédn with ta Tv ~uxnv efopyrafovta edn? I think it is clear that Aristotle meant to do so; and I render thus :-— ‘Now, as a result of the sacred chants, we see such persons (viz. those who are susceptible of enthusiasm in the higher degree), when they have experienced that music which brings the soul to a frenzy, becoming their true selves (kaOvorapévous*), haying met, as it were, with that which can heal and purify them, The same thing must needs be felt by the compassionate, by the timid, in a word, by the emotional, and by the rest of the world, in such measure as each shares this or that susceptibility. All must experience a purification, a relief attended by pleasure.’ If this is a right version, then the xa@apous, the tatpefa of these naturally enthusiastic natures, is nothing else than enthusiasm itself: it consists in their liberation, by the in- spiring and elevating power of the music, from all that restrains, obscures, or abuses that divine emotion. It is anything but a toning down or a refinement; it is an exalting, an intensifying influence. But an objection might occur. Why should not the sentence be translated thus :— “We see such persons, after that they have experienced the orgiastic melodies, brought into their normal state by the sacred melodies’? The ‘epa péAn would thus be tranquillizing melodies following the é£opyiafovta. Twining, in his introduction to the Poetics, is, however, clearly right in identifying the iepa with the é£opyafovta. xpnowvta is to be translated by the English perfect++,—‘when they have used’ or experienced : the catactaots, or pleasurable subsidence of the soul, is a consequence of the orgiastic music. To sum up:—Where does Aristotle place Music in Education? He says that, for the youthful, it is a discipline both artistic and moral; but differs here from the modern view in requiring such a measure of executive skill as will make the connoisseur a practical judge; a view inseparable from the Greek conception of art and art-criticism as part of the complete civic life, but one which has been partly superseded in modern days by the creation of a scientific wsthetic, bringing principles, ascertained through practice, within the appre- hension of those who have had no practical experience. For the mature, Music is a noble recreation, fitted to develope the intelligence, fitted also to purify the emotions; that is, to detach from the emotions whatever is accident or alloy, and to afford them a field for their clear and essential exercise. And this without loss of intensity—rather, to the heightening of true power. Compare two passages in which a Greek poet and an English poet, eminently not Greek, have made a husband and wife converse on a great distress. Compare the parting of Hektor and Andromache in the vith Book of the Ziad with the dialogue between Adam and Eve in the xth Book of Paradise Lost. No one would say, I think, that the scene in the Jiiad was less strenuous, less intense, less passionate, than * For xadicrapevovs, ef. Arist. Rhet. 1. 11, where jdovy is | the leading verb is future, by our future perfect: émeidav defined ‘as xlvyols tis THs Wuxis Kal Kardoracis abpoa Kal | rodro tdw, édebcouar: when I shall have seen this, T shall alcOyrh els tiv vrdpxovsay picw—‘a settling down, sudden | come: (2) when the leading verb, being in the present and sensible, into our proper nature.’ indicative, denotes a general truth, by our perfect: drav + The aorist subjunctive, after brav, éreddy, érav, isto | rovro tw, dwépxouar: when I have seen this, I (always) be translated differently in two different cases:—(1) when | depart, Cf. Goodwin, Greek Moods and Tenses, p. 26. 530 Pror. JEBB, ON THE PLACE OF MUSIC IN EDUCATION, &e. the other. But the xa@apois ta@nwator is effected by the Greek as it surely is not by the English, and why? Because, in the Jliad, the type is not merged in the person; because cheerful repose, and generality or typical concentration, are so perfectly preserved in the Greek*. How is Aristotle’s view of Music related to Plato's? Much has been said of the ‘antagonism’ between Aristotle and Plato as to music; but I cannot help thinking that the commentators have confused the thing by dwelling on trivial differences of detail. The most definite utterance in Plato as to the general power of music is in the Republic i. pp. 401, 402: where he says, that musical training is so powerful ‘because rhythm and harmony find their way into the secret places of the soul, on which they mightily fasten, bearing grace in their movements, and making his soul graceful who is rightly educated, and his ungraceful who is ill-trained; and also, because he who has received this true education of the inner being will most shrewdly perceive omissions or faults in art or nature, and with a true taste, while he praises and rejoices over, and receives into his soul the good, and becomes noble and good, he will justly blame and hate the bad, now in the days of his youth, even before he is able to know the reason of the thing; and when reason comes he will recognise and salute her as a friend with whom his education has made him long familiar.’ That love for the beautiful which is engendered by Music is, for Plato, an intro- duction to the more divine Erdés for the Ideas. Here, then, we have a clue to the essential difference, so far as there is one, between the Platonic and the Aristotelian view of Music. Plato connects music rather with religion, Aristotle with art; Plato regards it as a means of evoking, Aristotle of defining enthusiasm. Aristotle’s remark that Plato is wrong in allowing the Phrygian along with the Dorian mode, while he excludes that in- strument, namely, the flute, which goes best with the enthusiastic music, seems to show a certain insensibility to Plato’s spirit. The flute is the instrument which, in Greek music, marked the divergence of wa@os from pa@now: the use of Music, according to Plato, was to enlist a divine ma@os in the service of a divine padnots. There are phenomena of the present day which forcibly suggest the importance of Aristotle’s view regarding the wniversal moral importance of Music as an element in education,—I mean the ecstatic movements, whatever special form they may take, and the passion for excitement. What are these but instances of repressed, untrained, and therefore ungovernable sensibilities taking the first opening that is offered to them? Experiment has shown what Music, when it is good enough, can do in educating such sensibilities. There is one point, however, on which the modern world would differ from Aristotle—the only important one. The principles of music now rest on a really scientific basis, not on grounds fully possessed only by special experts, and merely touched by philosophers as part of a wider domain. Again, the generalisations of esthetic give critics who are not even specially musicians a stand-point of their own. Aristotle’s plea, therefore, that it is necessary yewpoupyeiv in order xpitas tév Epywv elvat orrovdaiovs has no longer its Greek validity, In all else, the modern world may still, perhaps, not scorn to hear the old. * (i) P. L, x. 720—844, 914—936: (ii) Iliad, yx. 475—571. + Arist. Polit. y. (vi11.) vi. § 9. CAMBRIDGE ; PRINTED BY C. J, CLAY, M.A., AT THE UNIVERSITY PRESS. I. Exereises in Curvilinear and Normal Co-ordinates. By the Rev, James Wiirram Warren, Caius College. [Read May 7, 1877.] EXERCISE THE FOURTH, NoRMAL Co-ORDINATES, LET us suppose that the equations of three surfaces each contain such suitable parameters that when these parameters change in value each of the three surfaces may be made continuously to pass through all space. Clearly the Cartesian co-ordinates of any point in space may be supposed to be expressed in terms of the equations of three such surfaces. U 2? Now imagine two distinct systems of three such surfaces, call one the system w,, w,, Us, u, and and the second the system U,, U,, U,, we shall denote functions of w,, u,, u, similar functions of U,, U,, U, by the same letters, but we shall dash these letters in the case of U,, U,, U,, thus we shall have such corresponding symbols as C,,, C,,, &e. It is clear that im general a surface of the system, w, say, will not coincide with any surface of the system U,, but we shall now suppose the surfaces u,, u,, u, and U,, U,, U, to be such that one surface of the system wu, coincides with one surface of the system U,, and this common surface we shall for the sake of conciseness call the surface (U, and w,). We make the same hypothesis for the surfaces u, and U,, u, and U,, so that we have the surfaces (U, and u,) and (U, and w,). 3 It is clear that in general U, is a function of u,, w,, w,, whilst uv, is a function of U,, U,, and U,: the same remark is of course true for U, and U, and w,, w,; this being the case we clearly have du du du es Hk LEUNG p Get os te aU a eos d _du, d , du, a, du, a uc Hob du, * aU, du, dU, du, Vou. XII. Parr III. 68 5 o\ 32 Rey. J. W. WARREN’S EXERCISES IN Or Suppose now the surfaces u, and U, to take up the position (U, and w,), then clearly we have all over the surface (U, and w,) both du,=0 and dU,=0; therefore since the relative value of the differentials dU, and dU, are quite arbitrary, we clearly have all over (U, and wu,) both = =0 and ct =0, and hence clearly we have at the point in space where the three surfaces (U, and w,), (U, and w,), (U, and u,) intersect (which point for shortness we may call the “common point”), du, a du du, du du aU. 1 _— pooeel 2 — —_2 = -_ = dO aU” ates ata ape do @u, De eae qos aa oe du, dU? =0, &c., d _ du, a> ad du. d ad du, d iu dae di aah an eG eal a oy It is clear however that we have no reason for asserting that at the common point any one of the nine quantities ive aa. , &e. are zero, in fact-it is not true that this is the case, and one of the first things we shall do is to express these nine quaint ' : 5 : : 3 dA as linear functions of the nine differential coefficients a Wie ‘ 1 2 The notation of this Exercise is the same as that used in my first three Exercises ; with the following additions, we write Ass =i: Beni ,=A,; we ,=A,. (A, SB, (Ay A w! (Ay, Ay)? Ay,C, = 23,5 Aj Og = 2353 Aggy = 255: (Anant ¥ Crs iz TT 3 5 (A,, A w! -C,, 7 T; > (4,,4,,)8 7 Cre ad H,.. 25, aE TI, | 1 A, A, 1 T, 23, U,,|=4" [Ay 1 A) =o. tS ross Me The same symbols dashed will denote the same functions with regard to the dashed or U system of co-ordinates, and it is clear that at the “common point” we shall have A,=A,’, &., =,=,', I,=TII/, &, A =A’. When the dashed or U system of co-ordinates are normal co-ordinates (as we shall generally suppose them to be), we write in place of A/, =, II,,, &e, the letters A,, S,, P,, of my former Exercises, we then have of course at the “common oe A,=A,, &e.; where however it must always be carefully borne in mind that A,, S,, P,,, &c., and the normal B are supposed to be expressed in 29? terms of n,, 7,, 7,, a “the ten quantities that are strictly equal to them at the CURVILINEAR AND NORMAL CO-ORDINATES. 533 common point, ze. A,, = u,, U,, and u,. p I,,, &e, A are always supposed to be expressed in terms of du.d dud du, d dn, du,’ dn, du,’ ‘dn, du,’ (supposed to operate always on functions expressed in terms of U,, Us, Ug, never of m,, Ny, Ny) neo Oe ae le by the symbols of abbreviation ra wali ; these symbols, mind, are mere symbols of 1 2 abbreviation, and wherever we see one of ile: it is always supposed to be a mere We shall also represent the three operating symbols : d representative symbol for te ae &e. The object of this Exercise may be now clearly dn, du, ce ee < . : a @5 a2, dAe ah stated, it to deter f poss . 2 1 1 spe it is to determine if possible the thirty quantities 7 Te ae dB dA Be 5 an as linear functions of the nine quantities ue &e.; and if we can find thirty ny ~ dy, du, such equations true at the “common point” it is tolerably clear that the general equa- , &e., 1 tions of normal co-ordinates can be immediately transformed into the general equations of Curvilinear Co-ordinates, and that in both cases they are similar in form. A few more matters with regard to notation only need to be explained; we write, for the sake of conciseness, Ay Ft An (2 ae — Fil) + As es aie — i) =2 Ky) Ae oe ‘ (2 Fa ae) +40 (2S dCs, -53)= rs pee (2), ee ee 6 Fn) = 2%) 4 (UA) allem A, (3 1, al) 4 Ma 4 nas Lan ena 3) Age ie ge) age? oe 2 Twelve more equations exist similar to (2) and (8), and of course we have eighteen similar equations dashed. Lastly let us write du,=P,,.dn, +P, .dn,+ P,,.dn, Oe PO Lae Og Fags eb pices de eattaln aa ae ie (4) ; du, =P,,.dn,+P,,. dn, +P,,. dn, we have, by what was already stated, at the “common point” P,=0, P,,=0, P= P,,=0, P,,=0, P,,=0, and it only requires a little consideration to see that we also have at the common point 68—2 BSE: Rey. J. W. WARREN’S EXERCISES IN 1, a A,} | P,,=A,; ' PRES ROE EOE «| (5) 1 JES oa ad) Observe also that the ten quantities A,, =,, II,,, &c, A are all functions of angles. Our system of notation being thus all clearly explained, I now go on to the direct object of this exercise. I divide what follows into sections for the sake of clearness; and our first section will have for its object the problem to express the nine quantities Cu, du, F : % Sin (AL. DALE &c. linearly in terms of the nine quantities , &e., where, as already dn,’ dn dn,’ dv,’ “dv, explained, Te, &e. are mere abbreviations. dp, 15 du, du, dA,, dA,, = dn,dn,’ &e. and dv, > dv, "3 se . V being any function of w,, w,, u,, and therefore of n,, m,, m,, we in “Exercise the first” have defined the feta Ags) Aavs) OkGs, sOly/ eas Aen Cees ine 2 (ae) + (Gy) + Ge) ~40 (Ge) +24 Ge ae, FO hence we clearly must have «Ge dV av dV dV a - &e. = ( ) 24 re (>) 2 el poe een pee i te is du, du, dn, ‘dn, dn, meet In Equation (6) write V equal to u,, or u,, or u,, and we immediately obtain three equations, one of which is ue (sa) (3) 3) 2A du, du, od du, du, 9 DA du, du, 0; dn, dn,) * \dn, 7 dn, fie ae in, dn, °dn, dn,” differentiate now both sides of the equation (7) in succession with regard to »,, n,, and n,, we easily obtain dA, _ of Uy ig du, my du, du 2 ane ~~? dn, dn, ° dn, ¢ dig? 1 bo dA,, (4") du, du, AS dn, dn, dA, 2(As y, au, dis, A. dn, dn, CURVILINEAR AND NORMAL CO-ORDINATES. D390 We can clearly therefore write down the following nine equations: du, ss 1 dA,, aa AN dA,, ee dA,, dn? Ee Oni AE da A id, ) 7 Mu, _ (da, a A,, dA, = AR dA) dnt 2, aus (AL dun Ax dias : du, 1 {is eas ase dA.) dno” 2 \duk. 7A, du,” As du.) ’ ‘ Sie Te Be Wl sda oie (8). du, 1 (42) dA,, . du, _1 Au! dA,, Gnvdn,-2\A_) aie? , dngdn,~ 2 ae ~ du,’ du, _1 ral dA,, du, _1 ao) dA,, dn,dn, 2 ‘Fe du, dn,dn, 2 ae * G” du, _1 ay dA,, dite al (42) dA,, dn,dn, 2 . a du dn,dn, 2 Ae du, Thus we have accomplished the object of this article. I now proceed to article the second. 1D. dA, dA, - dA, LS) i One : dA, = > in) : @ Tea See Car a as 0r dy, » Sie.r . Let us in equation (6) write V equal to w,—X.w, where X is any indeterminate, then equate the coefficients of ’ on both sides, we instantly obtain Ap ee Nie ata i ena, 3 dn, dn, dn, dn, dn, dn, +A (du, du, fe du, oe 1 (dn, dn, dn, dn, (du, du, | du, au 2 (dn, dn, dn, dn, du, da, du, du) 8 \dn, dn, ~ dn, dn, +A Now differentiate both sides of this equation (9) in succession with regard to n,, n, and n,, and then suppose that we are at the “common point,” we clearly obtain 3 du, dA,, _ du, du, dA, du, du, n du, du, dn, du, dn, dn, dn, ‘dn, dn,dn, '~ ‘dn, dn,dn,’ 536 Rey. J. W. WARREN’S EXERCISES IN or substituting from system (8), a ( A. ie As f e2) Tiny} (AyAe) daa) dn, \A_ A.) dug Oa A= A du, 2 i 22 33 22 therefore clearly we have dn, ~ "du, \(4,,4,)3 dA, dA, or finally, a amd wcie nie cen p00 6060 0.0.00 60D 000.0 00 Be c10\01000 VOC C eee 0.000.000 cvcnenedcciescnecicevvedesecveviciace ravce We also clearly have du, dA,, du, du, GA, du, du, " du, du, dn, du, ~ dn, dn, dn, dn, ° dn,dn, * dn, dn, dn, e du, d*u, 4 Ms ios * (dn, dn,dn, ° dn, dn,*) ’ or substituting from system (8), dA, = pe dA,, —_ A, dA,, gE As dA,, dn, ~ A, du, 2A, du, 2A,,.A,2 du, eo ee esas Roe — dA, Ay “e)| 2A,,.A,, |4,,% du, = \ du, AZ du, “Agrdu, }}* and this clearly may be written in the form dA, = 4 d A 1 dA,, hos te - dn, = A,.? du, ee. =r 2A,2 A, {(4u4n = AA) “du, (4. AA) du, ’ but by the formule given in Exercise the first, (A, A,, = A,, A.) = 2 = B A= A? ? toes 93. (A,, a A,,. A,,) = a = Ay fs. As: Oy = z : AeA ’ so that clearly we may write d\ 1 es 1 SEO an cy, 2a ae {t, ana CURVILINEAR AND NORMAL CO-ORDINATES. or “I Hence by symmetry we may obviously write down the nine formule dA, _ dA, dn, ~ dv,” dA, dA, dn, dy,’ dA,_ dA, dn, dv,’ dA, > dA, a a Sr dA,, _ os das din dy) 2A AS 18 dy, Oh \\ dA, dA, 1 ( dae os dA, qn eta ae os es =| tt eee (10). dA, dA, 1 WA ia ten) dn, dv, 2A? A, {0, dv, aaa dv. J 4 Apes alae n eatadh lee coal due dv QAP ASS 3 dv, 720 z dA, dA, iL dA,, dA, an, du, ZASYA- {t, dv, a = ; dA, dA, 1 f dA,, dA,, die diag a2 Asean ps dy 2%, a | We have thus accomplished the object proposed in the second section, and accord- ingly go on to third section. dil,, dy, ? (8-5 ) The operating symbols aP, es . and & dn, dn, d a +P + P. iil. 11 ) be. linearly in terms of = 1 5 we} = d d ie Lp? 31 (P are clearly equivalent. following operational formulz are true, a ™ div, 21 du, d ae (Pa aoe da) da) 8 Hence it is easy to deduce that at the common point the & du, d dn? = i du, dn, du, oe eee eee eee eee ee el 1), (a , @ du, d@, du, ad Z dn, dn, — (Ax A,,)? du, du, + dn, dn, . du, dn, dn, du, eee eee ewan eee eneee (12 538 Rev. J. W. WARREN’S EXERCISES IN Operate with these symbols on the Cartesian co-ordinates of a point and substitute in such expressions as dz da , dy Ty dz @z dn, dn? dn, dn? * dn, dn,’ de dx dy dy dz dz dn, dn, dn, , dn, dn, dn, Zs dn, dn, dn,’ we easily thus deduce the four following typical equations : 7 ae a Ba, aeOe | nénasdooneaocangsoc* (13), dP... as. pa eel an) } Cu, Fe gata An Aad Ge — 3 Tan) tA Cu Gaede RSH. (14), dS, e ell dC, 3 { Tu, Pu, : dn, ut A . 2d Uy + A,3 ou dn, ia + GR dn, dn, eee enw ewe eens (15), dP. dP. ae dC,, , aC, v2) } du, du, aay 7a aie cee Aya)? Ca ‘ du, du, : a a Now first let us consider formule (13). Suppose that we substitute in it for = from me | the group of formule (8), we clearly obtain d&,_ Ae, dy, My _ Cy4y dy _ CyAn Ay dn, 2 "1 du, uy Ay as i ail which clearly may be written in the form dS, _d>,_ Cy, dA, = tae aa {a5 d A, a shcTUnnaSGbSabeHUAASGeSbOADoR Doce (17). Again, let us consider the formula (15) and substitute from the group of formule (8), we clearly obtain dAje Cea Oa aay. 2A, dy,’ so that clearly we have dS, _d>,, Il, dA, dn, dv, 2A,,° dp Again, from the formula (16) and its analogues we can easily deduce the result dP i dC, 7 5 Ue zal) du, s(n eu, Pu i Pu, du, ) . as {¢ dn,dn, + 1s pas + 4's [On dn, dn, +O dn “an Sips and making use of formula (8) as in ee two cases we easily obtain the result Hig dit 1 ( dA. ahi tng aah HS shea deo Vepes sa uasebaancee este 19). dn, dv, +34, ths = +I], ae } dei (19) CURVILINEAR AND NORMAL CO-ORDINATES. 539 Lastly, suppose that we take the sum of formule (14) and (15), we then clearly obtain 2 , < } Mu, Tu, Gu, dn, ~ 0" * ” du, Arn gi Oe es pao Cx aa dazt Cy dn,dn,’ hence substituting from the group of formule (8) we clearly obtain that ,dA, Add, A,*A,, set Fe enone oe du, rae Oy, Ap Fe 2 dy 1(/A,\3dA 1 (dA, +63 (a2) “Goth # Goa (ah hence clearly we obtain that CIRCA ee dA dA, eo e+ 3a, (OB All) Ge A It is clear that the group of formule of which (17)...(20) form a part are eighteen in number; it is perhaps needless to write them all down, but I shall give six for the sake of reference, and the others can be formed by a cyclic change of suffixes. aS, _ a, =, f dA,, dA,,) ) dn, dy ~ Ay" Mey T Ae an | dS, _ dz, 1 dA,, dn, ‘a dv, a 2A., 4 Ts ; dy, aS, d& 1 dA, ) fg ae T 94 { ies a J (21) dP,, dil, 1 a HA A ET ee Se SS dn, dy, 2A,, ” dy, dv, dP., = dll, 1 ( os a “ot dn, dv, BAe NOs A,M,,). 7" —A,U,.- i. CEs dii,, it os as a dn, dv, — OOM pe; {\@s,-A,0,) Pe AI, ”, There are six more formule similar to the first three, and six similar to the last three, which the reader can easily write down. If we examine the groups of formule (10) and (21), we find that they are governed by a series of remarkable symmetrical laws; firstly, none of the three differential co- dA, dA, dA efficients —"; du, due? du, that we differentiate with, the same double suffix enters the A, ze. m, and A,, are associated, but m, and A,, are never associated; thirdly, A,, A, and A, enter m a logarithmic form. Other laws may also strike the reader. I now proceed to the fourth %3 enter our formule; secondly, whatever suffix appears in the x section. Vou. XII. Parr Iii. 69 540 Rey. J. W. WARREN’S EXERCISES IN I CBS MGAN by a : CAT: i( a = &e. linearly in terms of ae &e, j B we have defined in “Exercise the first,’ A we have defined in this Exercise, the relation between A and B is shortly stated by the formula PB’ is the B of normal co-ordinates, at the “common point” we have of course A4=8’ Now it is clear from the definitions of A and B that we have \ ao ee oA, +P, . dd, +Py. a4,| as | a oe eee (23). C4 a1, . dA +T1,. dA, +1, - dA, | Now the differentiation may be performed with regard to n,, or n,, or n,; take for example n,, then by substitution from the group of formuli (10) we easily obtain that _i dB _ 1 dA My M1, {tas gs GA, Bin A ty, 2A : a idvs aca dv, ae It dA,,_ os dA,,) tom A. eau HS) But II, 11,, — 22,- fi, = toe DET 2 See We Anes hence clearly we may write down the three formule, qB dA A dA,, dA,,) } dn, dv, 2A, {A, dv, +A, dy, +} dB a A dA,, dA, 9 dn, - = v, — aye { 3 av + A, dv. er (24), ies es A dA,, dA,, dn, dy, 2A,, 1, ‘dv, TAs dy, I The reader will observe that the three symmetrical laws mentioned at close of section II. hold also for these last three formule. I now proceed to the fifth section. Wi {Connection between (K,,)', &c., (K,,), Key (,), &e} We have already defined (K,), &c. and (K,,),, the only difference between them being that the former has reference to normal co-ordinates, the latter to general CURVILINEAR AND NORMAL CO-ORDINATES. 541 curvilinear co-ordinates. It renee to define (i,,); &e. We do so thus: let («,,), &e. be the same function of A,A,A,, >,>,>,, I,,[;,I1,,, and A, that (K’,,), &e. is of A,A,A,, 88,8, 2™3? IP._IP Je os RIL Ee Gonsbec cone cob conbe 0b 660e 0oag0aco CECCOSOSE CHE OOET SI COBaCOSEOGUEO DEG qpECoD COSI COnerccroosOcccpogar 237° 13 Now let us make use first of the four typical formule (13) (14) (15) and (16), and next of the group of formule (21); substituting the values in formule (2) and (3) we easily obtain the followmg group of formule: : 1 @¢ K,,), =4,2. (K,),+ zy. ( ap ll ( Cia)s A, s dn,’ ——— fake (ia) + A! .(K, ay (K,,). = A, any (K,) 3 c&,),= (4948), il Ih ker A,} ‘dn,dn, lL du, | 2'8 " A 3 dn,dn, (a); —4, - (K+ (ee There are twelve more similar formule. We also have eighteen formule comnecting (K,,), &c., and (x,,), &¢, but as we shall make in this exercise no use of these last formule it is hardly needful to write them down. Consider now for a moment the group of formule (25); we can clearly deduce from them the following two important equations: ()e— Bead) = (A (Bade — Bia « Bada veceee (6), {(K.), Ci (K, vie (kK, sat = (A, . A,,)3 {(K, ia (K, i ia (K, he (K.,),} A..\3 Tu au, } Cu, a ae eal 5 Kas ind Fs (4 y (K, 31 dn dn dn, — (An. Ass) . (K, Qh. dn= = eal (20)> Formula 26 shows us that the formula m= — pa page 516, “Exercise the third”, being 12 proved to be true by normal co-ordinates must be true in general, and formula (27) enables us to derive last example given in “Exercise the third” from the corresponding equation proved for normal co-ordinates, I now proceed with the sixth section. 69—2 542 Rey. J. W. WARREN’S EXERCISES IN Vile (37 SS BaP ees Li, Ge eal os [ane anda, ~at) Aaa 3 eb Maga, 2 dt) & | a | | expressed i in terms of ia dn, ze J : 4 Tu, « : In order to make our formule as concise as possible we use here Ga &e., in place of 1 dA, &e.; the group of formule (8) however give us the former nine quantities as linear U functions of the last nine. Write for conciseness the quantities in brackets in first line of this section equal to m, and m, respectively. It has been mentioned in the previous Exercises that it oe Fat ge. dy dz dz SeNdres “ane dn,” 2 tan? dn,’ Px it d*y y- dz iS Saree = (aoe (aa 3 Hence substituting the values of aa &e., found by means of formule (11) and (12), 1 we easily obtain that ° : ° 1 TS, CP an, 4n{ 1 = On dt Cate lh. oot | ~ dn2 ee en 2 du, dhé, 2 du, dC, 1dC,\ Gu, ao a dC, du 7 ao ae 7) ( 5 ) 120, du, 34C,, du, Ee te dain, Ana) ag daar -C (au a (5 Pu, eu, Pu, ) du, \* 4 \dn,dn, dn,**dn ~ dndn,*dn,dn,/ — = (Gada) We now use this same method to transform the quantity > &8, E dP, cP. Cid aoe “dn, dn,° v dn,dn, re dn,dn, dite ae - 2 du, 2 “dn, (28). +C 12 | J This quantity for conciseness we shall represent by the letter U’,. We then have, as already stated in “Exercise the second,” fo dz, dy ty | te a ~ dndn,* dnidn,* dndn, dn dn, + dn,dn, * dn,dn,’ ta Wx - <4 Gy Ge ae dn, dndn, dn? dnjdn, dn? dn dn,’ CURVILINEAR AND NORMAL CO-ORDINATES. 545 Now make use of formule (11) and (12). We easily find by mere substitution that il {2 as, ie Ci ee aP. ) 2 ("dn dn, * dn? dn,dn, dn,dn,J is equal to a Aad) ae PCy aie a aC., du, aie + du? du,du, du,du,)- ,4 way { du, dC, 4 PU, aC, aC, aC, dn, ae an cede ( du, du, y du, ) ete (A,,A,,)° du, dC, + du, (" Gc; dC, as a) 2 dn,dn, du, dn,dn, du, du, & (ARAL ot jel dC, a a) 2 (dn? (aa du, du, ay, du, (dC,, me! 2) ze du, (ac... 1 =) 4 (dn dn, ( du, 2du,/ dn,dn, lea 72 du, du, au, du, @u, 1 dn, di, dn, oy *dn,dn, dn,dn, +0. jy @u du, du, Pu, 2 dn} di, dn,¢ dn, dn,’ * dn, dn, (du, du, du, du, a a2 Obes aie ie ay as diet dan lc ae ee ae es (29). We shall now conclude this Exercise with one more section; and as a partial verifi- eation of the formule we have arrived at, it shall be devoted to a deduction of the Equations of Orthogonal Curvilinear Co-ordinates from the corresponding formule of Normal Co-ordinates. Vi {Passage from Normal Co-ordinates to Orthogonal Curvilinear Co-ordinates.} A system of Normal Co-ordinates it is clear cannot in general be also a system of Orthogonal Co-ordinates; that is to say, for every position they assume in space, yet one position of our normal co-ordinate surfaces can clearly be always made orthogonal. We assume such to be the case, and shall now substitute in the Equations given on pages 497 and 498 of Exercise the second from formule 10, 21, 24, 28 and 29 of this Exercise. 544 Rey. J. W. WARREN'S EXERCISES IN First, consider the formule on top of page 497, Exercise the second. It consists of four distinct portions, so to speak, and bearing in mind that for orthogonal co-ordinates we must have 25,=1, 25,=1, 25,=1, I,,=0, II,,=0, U,,=0, A=1, a little considera- tion shows that the last three of these portions in the case of orthogonal co-ordinates are zero, there remains therefore only the first line, ze. 1, transformed, or in fact, our Equation 29. Let us write then in Equation 29, C,, = 0, Oh = (0) C., = 9, AiG =1, A,.C,. = 1, Ale (OL =1; du, ail 4m)! dA,, > oP (Gal A,,)? dC, dn,dn, 2 Ge 7 dae 2Cw eda. du, iJ (A,, A,,)} dC, dias Se" aa B00 2° oa,” oes _(4,4)* dC, dn, dn, = cecccvcccccsecessccese 2 j C., = dit, : dru, _ eae aC,, PATE ay vee Cee eae Now substitute these values in equation 29, multiply by 2, divide by 4,,. (A,,.A,,)!; we then obtain ‘ OC a Cd, ele BOOS aC Gand Come du,du, 2C,, du, du, 2C,, du, du, -2C,, du, du, 0.0 GOr and this is the same as the second equation given on page 499 “Exercise the second.” Lastly, we give a short sketch of how we are to transform the equation on top of page 498, Exercise the second. It also consists of four portions or lines; the Jirst line gives us, by aid of formula (28) after dividing by 4, A,, and multiplying by 2, POO Wile dG iod Cams Oh ane — C= 22 du,* du, 2C, du, du, 20, du, du, 3 (2) A (86a | 2 an aa. (ae 5 Sas teeee nese Cesceameete tee (a); the second line, that is to say A, contributes the terms : 1 aC, dG. i dO. d0, «dO. aC. 16C,,0,C, du, ' du, d } POR du, du, LeCim du, du, du, the third line it is easily seen by aid of formula (24) can contribute no terms. And lastly, by aid of formula (10) it is easy to see that the fourth or determinant portion contributes the terms 1 aC, dC, d0,, d0,, .d0,, dd, , 60,0 mae du, * du, du, mon du, y <: DAG (7). 11 ~ 22 CURVILINEAR AND NORMAL CO-ORDINATES. 545 Now divide (8) and (y) by 4A,,A,,, multiply them by 2, add them to (a), and we obtain Oye Ounemnd Cs aCn at 2d0) dG. eS east pees eit es 2 Py 6Yal p 5Yal : du, du 2C, du, du, 2C,, du, ° du, 22 Sel a) = LT /aGeNe 2C,, ( du, 2C 10 ae ) ld, aCe , > oy Tin ae cere ccnnnnsecccerecceseversecees (31) ; 33 and this agrees with the first equation of page 499 “ Exercise the second.” March, 1877. aed ~ = wes oF ra nee, a a6 | ao If. On Boltzmann's Theorem on the average distribution of energy in a system of material points. By Professor J. Crerk Maxwett, [Read May 6, 1878.] Dr Lupwic Boirzmany, in his “Studien iiber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten” [Sitzb. d. k. Akad. Wien, Bd. Lvur, 8 Oct. 1868], has devoted his third section to the general solution of the problem of the equi- librium of kinetic energy among a finite number of material points. His method of treatment is ingenious, and, as far as I can see, satisfactory, but I think that a problem of such primary importance in molecular science ought to be scrutinized and examined on eyery side, so that as many persons as possible may be enabled to follow the demon- stration, and to know on what assumptions it rests. This is more especially necessary when the assumptions relate to the degree of irregularity to be expected in the motion of a system whose motion is not completely known. Mr H, W. Watson, in his Treatise on the Kinetic Theory of Gases*, has developed with great clearness the steps of the investigation of the distribution of energy among a set of particles which are supposed to act on each other only at very small distances. The particles may be acted on by external forces such as gravity, but it is expressly stipulated that the time during which a particle is encountering other particles is very small compared with the time during which there is no sensible action between it and other particles; and also that the time during which a particle is simultaneously within the distance of molecular action of more than one other particle may be neglected. Now this method of treating the question, however necessary 1t may be in the subsequent investigation of the processes of diffusion, &c. in gases, is inapplicable to the theory of the equilibrium of temperature in liquids and solids, for in these bodies the particles are never free from the action of neighbouring particles. It is true that in following the steps of the investigation, as given either by Boltzmann or by Watson, it is difficult, if not impossible, to see where the stipulation about the shortness and the isolation of the encounters is made use of. We may almost say that it is intro- duced rather for the sake of enabling the reader to form a more definite mental image * Clarendon Press Series, 1876. WO, SOU, we UL 70 048 Pror. CLERK MAXWELL ON BOLTZMANN'S THEOREM ON THE AVERAGE of the material system than as a condition of the demonstration. Be this as it may, the presence of such a stipulation in the enunciation of the problem cannot fail to leave in the mind of the reader the impression of a corresponding limitation in the generality of the solution. In the theorem of Boltzmann which we have now to consider there is no such limitation. The material pots may act on each other at all distances, and according to any law which is consistent with the conservation of energy, and they may also be acted on by any forces external to the system provided these also are consistent with that law. The only assumption which is necessary for the direct proof is that the system, if left to itself in its actual state of motion, will, sooner or later, pass through every phase which is consistent with the equation of energy. Now it is manifest that there are cases in which this does not take place. The motion of a system not acted on by external forces satisfies six equations besides the equation of energy, so that the system cannot pass through those phases, which, though they satisfy the equation of energy, do not also satisfy these six equations, Again, there may be particular laws of force, as for instance that according to which the stress between two particles is proportional to the distance between them, for which the whole motion repeats itself after a finite time. In such cases a particular value of one variable corresponds to a particular value of each of the other variables, so that phases formed by sets of values of the variables which do not correspond cannot occur, though they may satisfy the seven general equations. But if we suppose that the material particles, or some of them, occasionally encounter a fixed obstacle such as the sides of a vessel containing the particles, then, except for special forms of the surface of this obstacle, each encounter will introduce a disturbance into the motion of the system, so that it will pass from one undisturbed path into another. The two paths must both satisfy the equation of energy, and they must inter- sect each other in the phase for which the conditions of encounter with the fixed obstacle are satisfied, but they are not subject to the equations of momentum, It is difficult in a case of such extreme complexity to arrive at a thoroughly satisfactory con- clusion, but we may with considerable confidence assert that except for particular forms of the surface of the fixed obstacle, the system will sooner or later, after a sufficient number of encounters, pass through every phase consistent with the equation of energy. I shall begin with the case in which the system is supposed to be contained within a fixed vessel, and shall afterwards consider the case of a free system, or of a system contained in a vessel rotating uniformly about an axis which itself moves uniformly in a straight line. | I have found it convenient, instead of considering one system of material particles, to consider a large number of systems similar to each other in all respects except in the initial circumstances of the motion, which are supposed to vary from system to system, the total energy being the same in all. In the statistical investigation of the motion, DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 549 we confine our attention to the number of these systems which at a given time are in a phase such that the variables which define it lie within given limits, If the number of systems which are in a given phase (defined with respect to con- figuration and velocity) does not vary with the time, the distribution of the systems is said to be steady. It is shown that if the distribution is steady, a certain function of the variables must be constant for all phases belonging to the same path. If the path passes through all phases consistent with the equation of energy, this function must be constant for all such phases. If however there are phases consistent with the equation of energy, but which do not belong to the same path, the value of the function may be different for such phases. But whether we are able or not to prove that the constancy of this function is a necessary condition of a steady distribution, it is manifest that if the function is initially constant for all phases consistent with the equation of energy, it will remain so during the motion, This therefore is one solution, if not the only solution, of the problem of a steady distribution, Now we know from the empirical laws of the diffusion of heat that the problem of the equilibrium of temperature in an isolated material system has one and only one solution. But we have found one solution of the problem of equilibrium of energy in a system of material points in motion, If, therefore, the real material system in which the equilibrium of temperature takes place is capable of being accurately represented by a system of material points (as defined in pure dynamics) acting on each other according to determinate, though unknown, laws, then the mathematical condition of the equi- librium of energy must be the dynamical representative of the physical condition of the equality of temperature. It appears from the theorem that in the ultimate state of the system the average kinetic energy of two given portions of the system must be in the ratio of the number of degrees of freedom of those portions. This, therefore, must be the condition of the equality of temperature of the two portions of the system. Hence at a given temperature the total kinetic energy of a material system must be the product of the number of degrees of freedom of that system into a constant which is the same for all substances at that temperature, being in fact the temperature on the thermodynamic scale multiplied by an absolute constant. If the temperature, therefore, is raised by unity, the kinetic energy is increased by the product of the number of degrees of freedom into the absolute constant. The observed specific heat of the body, expressed in dynamical measure, is the in- crement of the total energy when the temperature is increased by unity. The observed specific heat cannot therefore be less than the product of the number of degrees of 70—2 550 Pror, CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE freedom into the absolute constant, unless the potential energy diminishes as the tem- perature rises, Dynamical Specification of the motzon. We shall begin by supposing the material system to be of the most general type, having its configuration determined by the n variables q,,q,-..g,, and its motion deter- mined by the corresponding momenta p,, p,...p,. The state of the system at any instant is completely defined if we know the values of these 2n variables for that instant. We shall suppose the forces acting between the parts of the system to be of the most general kind consistent with the conservation of energy. This may be expressed by defining JV, the potential energy of the system, as a function of g,...q,, the variables which define the configuration, The kinetic energy of the system is denoted by Z. We shall suppose it to be expressed in terms of the g’s and p’s as in Hamilton’s method. The total energy is denoted by BV $F De wtavekerseevtsss uxgeroeorcdevsaneererae bostaeeas (1), and is a constant during the motion of the system. Hamilton’s equations of motion for this system are oq, dE 2 oa dj Ree Se aha 9 ae =), Op, _dE 3 Ot dq," eee tee cove cceccenerecasevtececeserncesavesbevsace ( ); where g, and p, are the co-ordinate and the momentum corresponding to each other. Let us now consider a finite motion of the system. Let the initial co-ordinates and momenta be distinguished by accented letters, and the final co-ordinates and momenta by the same letters unaccented. To define completely such a motion requires 2n+1 variables to be given. These may be the n initial co-ordinates, the n initial momenta, and the time occupied by the motion. There is another method however in which the 2n+1 variables are the mn initial co-ordinates, the n final co-ordinates, and the total energy. When these quantities are given there are in general only a finite number of possible motions. i o Ot DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. Definition of the “ Action” of the system during the motion. Twice the time integral of the kinetic energy, taken from the beginning to the end of the motion, and expressed in terms of the initial and final co-ordinates and of the total energy, is called the “Action” of the system during the motion. If we denote it by A, and is expressed as a function of q,’...g,’, 7,-..g,, and F#. It is shewn in treatises on dynamics* that dA ; z qe Mie caciecsecacesseberaccabecsvacveuseeetsanthestotoebeieeace (5), dA = and aaa I/D soeeosadubaqoadODnSODNObeRonoabondoDaboD90e saoNcdosOLE sccocd (9): dp, _ a Gane dps = Hence aa ap =— CPi reat oe Meer as (7). The indices r and s in this equation may be the same or different. Also if ¢ and ¢ are the values of the time at the beginning and at the end of the motion, dA , WE =] ic a0donocsonsdccaenccpan coddeocouugsedtonoeoacnd (8) dp,__ dt’ dp, __ dt Hence dE = dq, (9) and dE = dq. einiele/ere/s'o wis\eib/ela\ctn'efaiethis|utslelv\e aisicinatals (10). In the course of our investigation we shall have to compare the product of the differentials of the co-ordinates and momenta at the beginning of the motion with the corresponding product at the end of the motion. We shall write for brevity ds=dg,...dq, for the product of the differentials of the co-ordinates, and ds=dp, ...dp, for the product of the differentials of the momenta, and we shall use the product ds'dsd# as a middle term in comparing ds'do'dt’ with dsdedt. ng ong , dp, dp,, dt’ j = > oe eeee aoe I acu de saicaanwaeatecans os Now dsdo‘dt = ds dsdE & + ( ae 7 (1), dp, dp,, x) = 1 a where S+ ( agers dq, di denotes the functional determinant dp, dp; dp, Gos dy? di dp, py Ug. | weve vesetreceereeesnereeenees (12) ihe i de lige haiti a dq, > teeeee dq, , dE * Thomson and Tait’s Natural Philosophy, § 330. 552 Pror. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE Substituting for the elements of this determinant their values as given by equations (7), (9), and (10) it becomes ie De i See ahs pets ot dq,’ dq, _ dp, i dp, = dt | ‘Sc Sin aves de HERA ESO Eee (13) qqe ae de dp, DL A | — pe dB? ~ dE Now the rows in this determinant are the same as the columns in the former one; the accented and unaccented letters being exchanged and the signs of all the elements changed. We may therefore express the relation between the two determinants in the abbreviated form dp, dp, sn) a dp dp, dt Z+(—2 —,)=(- ee a a) evict e deaercteier ). Se ( de dee dE) TON" St ( Teton get a) (14) a es 3 dp; dp at Hence ds'do'dt = ds'dsdE + Ga ome a 7m) j dp dp, at =(-)" DH (4 o.. a =(-)"'ds'dsdB'S + ( ode a 7) = (—)"" dodsdt = USAT Ak. demas scan sedecadnaies nae Rene onan tase a aCe (15) If we suppose the time, ¢—t’, to be given, dt=dt' and CS AG = GAG as. eciecadeysyucusbeoasens cinsteaeaeeseeee (16), or GQ. snatee dn’ Up,’ ssse-s dp,) = dg, scans dq, Op, ..... OD sastescxcdatoestt (17). The initial state of the system is a function of 2n variables. We have hitherto supposed these to be the m co-ordinates and the n momenta, but since the total energy is a function of these variables we may substitute for one of the momenta, say p,’, its value in terms of the n co-ordinates, the »—1 remaining momenta, and JZ, and thus express every quantity we have to deal with in terms of the latter set of variables. Then since by equation (2) dE _0q,_., dp ar qi, ei dlujevoimce ‘aipracs al claveimteimia/aelbya\ecasareletainy ai aiaralcrnaratetaeeen (18), UC ian laser QQ Opi. seckns Op, =. dg," 0-04. dg.) dps ...00. dp, dE 7 scaenee Sane (19) DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 553 The left-hand members of these equations have been proved equal, and in the right- hand members d# is the same at the beginning and end of the motion. Dividing out dE we find This equation is applicable to the case in which the total energy is supposed not to vary from one particular instance of the motion to another, and in which, therefore, the 2n variables are no longer independent, but, being subject to the equation of energy, are reduced to 2n—1. Statistical Specification. We have hitherto, in speaking of a phase of the motion of the system, supposed it to be defined by the values of the n co-ordinates and the n momenta. We shall call the phase so defined the phase (pg). We shall now adopt a wider definition by saying that the system is in the phase (a,b) whenever the values of the co-ordinates are such that q, is between 6, and b,+db,, g, between b, and b,+db,, and so on; also p, between a, and a,+da,, and so on. The limits of the first component of momentum, p,, are not specified, because the value of p, is not independent of the other variables, being given in terms of E and the other 2n—1 variables in virtue of the equation of energy. The quantities a, b are of the same kind as p and q respectively, only they are noi supposed to vary on account of the motion of the system. In the statistical method of investigation, we do not follow the system during its motion, but we fix our attention on a particular phase, and ascertain whether the system is in that phase or not, and also when it enters the phase and when it leaves it. Boltzmann defines the probability of the system being in the phase (a,b) as the ratio of the aggregate time during which it is in that phase to the whole time of the motion, the whole time being supposed to be very great. I prefer to suppose that there are a great many systems the properties of which are the same, and that each of these is set in motion with a different set of values for the n co-ordinates and the n—1 momenta, the value of the total energy H being the same in all, and to consider the number of these systems which, at a given instant, are in the phase (a,)). The motion of each system is of course independent of the other systems. Let N be the whole number of systems, and let the number of these which, at the time f, are in the phase (a,b) be denoted by NV (a,, }, t). The aim of the statistical method is to express N(a,, b, t) as a function of N, of the co-ordinates and momenta with their limits, and of ¢ It is manifest that N can only enter the function as a factor, for the different systems do not act on each other. Also any differential as da or db can only 504 Pror. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE enter as a factor, for the number of systems within any phase must vary in the ratio of the interval between the limits of that phase. We may therefore write N (i) = Nf (@g-4.-.. 65 Gosden Didag Nc. daldb ase (22), where we have to determine the form of the function f. We shall now follow the motion of these systems from the time ¢’, when we begin to watch the motion, to the time ¢ when we cease to watch it. Since the systems which at the time ¢ form the group N(a,, 4, t) are individually the same systems which at the time ¢ formed the group NV (a,,, 0’, ¢’), we have IN (G5 bj SIN GUS) ete cg roe (23), or NF (G3 s.<-55 t) da, ....+ bs = NF (aserace. B) dae xcs. GD. scene cu seeeernee (24). But by equation (21) res db (Balt ose Cb OI Selene Ae (25) Hence Pf (Ga cccrib) OF (Gs aoe eb nO oka eee (26), and KE (a,, 0; 2) = NOG) Ades ces2 baa meet aeca en ae (28). n If the distribution of the N systems in the different phases is such that the number im a given phase does not vary with the time, the distribution is said to be steady. The condition of this is that C must be constant for all phases belonging to the same path. It will require further investigation to determine whether or not this path necessarily includes all phases consistent with the equation of energy. If, however, we assume that the original distribution of the systems according to the different phases is such that C is constant for all phases consistent with the equation of energy, and zero for all phases which that equation shows to be impossible, then the law of distribution will not change with the time, and C will be an absolute constant. We have therefore found one solution of the problem of finding a steady distribution. Whether there may be other solutions remains to be investigated. Let N(4) denote the number of systems in which gq, is between 0, and 6,+db,, q. between b, and b,+db,, and so on, and g, between b, and b+db,, the momenta not being specified otherwise than by their being consistent with the equation of energy» then N() =| * i Byker oR (29), DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. DDO the integration being extended to all values of the momenta consistent with the equation of energy. To simplify the integration let us suppose the variables transformed so that the kinetic energy is expressed in terms of the squares of the component momenta, of (Tt PO TAG J Se OY) A el) RR ee ERM SEALE (30), where a,...a, are the transformed momenta, and y,...u, are functions of the co-ordinates, which we may call moments of mobility, and which, in the case of material points, are the reciprocals of the masses. Now let us assume Bp AA Be OV, oso aus sisewes owns oan ocienieStna(snuswesenae (31), fens CA Ife (Ana Os) ncsincsian sac nise ssaneaneer at iatemecenet (32), [ieee ie OCS Toe cibaameeaaneecb reso non onosEciid Aor (33), Pegs pg (Ay 0 ic coca cencertentn ct tavice-uunsng aes mae (34) Then by the equation of energy Of these quantities, A, is a function of the co-ordinates only, because HE is given and V is a function of the co-ordinates, A,, is a function of the co-ordinates and a,, A, , of the co-ordinates and of a, and a,,, and so on. Also by equation (2) Ba (wpe AS GSP ots sects os a etene store (36). [ff--f Gr aa...da, we begin by integrating with respect to a,, thus To integrate the expression i C(b,)*da, = | Cl qe andes ©. hans cee toa (37), the limits of integration being + A,. The result is EGG) a ae 5 Sad (I(T Ts 2. eel a ele nescence SBE Sabre (38). Vou. XII. Parr III. (Al 5096 Pror. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE For the next integration we have A; 2 [.4'da, = i Medea Ntda ee ae (39). - A; 2 Hence after 7 integrations, r being any number less than n, the result is rt1 Eead NOT ty hea) War Anat] % ggg 40, Oy DDgecrcvessisee (40). RSS 2 Putting r=n—1 and remembering that p,4,°=2H—2V, we find De Gauls ue Ane 0d eh, ee (41). P (3) This is the number of systems whose configuration is specified by the variables b,...b,, while the momenta may have any values consistent with the equation of energy. N(b)=NC The quantity H—V, which occurs in this equation, is, by equation (1), equal in magnitude to 7’, the kinetic energy of the system. The quantity Z, however, is defined explicitly in terms of the velocities or the momenta of the system, whereas H — V does not involve these quantities explicitly, but is expressed as a function of the configura- tion. We shall find it convenient, however, especially in the study of more complicated problems, to remember that the number of systems in a given configuration is a function of the kinetic energy corresponding to that configuration. If the kinetie energy is not expressed as a sum of squares, but in the more general form, T =3[11) a? + [12] a, a, + Ke. $F [B2] ot A [23] fea, Fides. 1. cc ccc seascenseccunecosceasanse (42), where the quantities denoted by [11] &c. are functions of the co-ordinates, which we may call the moments and products of mobility of the system; then since the discrimi- nant eee eee eee ee eee ig an invariant, its value is the same when 7 is reduced to a sum of squares, in which case all the elements except those in the principal diagonal of the determinant vanish, and we have DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 557 and we may write the value of WV (bd), N@)= LG aos = If the system consists of n’ material particles, whose masses are m,...m,, then the number of degrees of freedom is n=3n’ and 1 Ee ea, hg = By, — Hg Mee 2 AM SO Ol «a -'seccse cu sens pneseneee (46). Hence in this case we may write Te sn a (ee mo yh (OB 2) dh ab, te aces (47). r ee) These expressions give the number of systems in a given configuration only when £—Y is positive for that configuration, for since the kinetic energy is necessarily posi- tive, the potential energy cannot exceed the total energy. For configurations specified in such a way that if they existed V would be greater than £, the value of W (6) is zero. The value of V(b) is also zero for configurations which, though they make V less than #, cannot be reached by a continuous path from the original configuration without passing through configurations which make V greater than £. We shall return to this expression for the number of systems in a completely specified configuration, but in the mean time it will be useful to consider how many of these systems have one of their momenta, p,, between given limits. In this way we shall be able to determine completely the average distribution of momentum among the variables without making any assumptions about the nature of the system which might limit the generality of our results. In order to find the number of systems in the configuration (b) for which one of. the momenta, say p,, lies between a, and a,+da,, we must stop before the last inte- gration. Putting r=n—2 in equation (40) al n-1 ees (H, ++ Pn) (rea hes ) y (— F( 5 The whole number of systems in configuration (4) is given by (45). Hence the proportion of these systems for which a, les between a, and a,+da, is n- 2 N(,a,)=NC en Db dite mao en eee (48). 2a0(3) (evan, rar ( >) (-vi? 298 Prov. CLERK MAXWELL ON BOLTZMANN'S THEOREM ON THE AVERAGE If we write then &, denotes the part of the kinetic energy arismg from the momentum a,. The proportion of the systems in configuration (6) for which k, is between k, and k,+dk, is rG) @- Posey ae cles so baeaae (51) rayr(*) (e-vy? Since any one of the variables may be taken for g,, the law of distribution of values of the kinetic energy is the same for all the variables. The mean value of the kinetic energy corresponding to any variable is R= GSH 17. nee eee (52) The maximum value is De WK oon. ola dina ce seat osaecote = oe sciee sco ae eaaeee ER (53). The mean value of k’ is 1.3...2r—1 oor s RW HD com HOP DO eet ete tenet eee etna (54). When ~ is very large, the expression (51) approximates to = re SER HEs, ee eee are ee (55) Recapitulation. The result of our investigation may therefore be stated as follows: (2) We begin by considering a set of material systems which satisfy the general equations of dynamics (2) and (3), and the equation of energy (1). If in these systems the distribution of configurations satisfies equation (45), and the distribution of motion satisfies equation (51), these equations will continue to be satisfied during the subsequent motion of the system. One result of equation (51), to which we shall have to refer, is that the average kinetic energy corresponding to any one of the variables is the same for every one of the variables of the system. (8) We now turn our attention to a system of real bodies enclosed in a rigid vessel impervious to matter and to heat. We know by experiment that in such a system the temperature cannot remain steady in every part unless the temperature of every part of the system is the same, and that this condition is necessary in whatever manner the configuration of the system may be varied by altering the position and mean density of the portions of sensible size into which we are able to divide it. DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 559 Now if the system of real bodies is a material system which satisfies the equations of dynamics, and if equations (45) and (51) are also satisfied, the condition of the system will, as we have shewn, (a), be steady in every respect, and therefore in respect of temperature. Hence by (8) the temperature of every part of the system must be the same. Therefore if equations (45) and (51) are satisfied, the condition of equality of temperature is also satisfied. But the condition of equality of temperature does not depend on the configuration of the system, for though we can alter the configuration by external constraint we cannot prevent the temperature from becoming equalized. It does not depend, therefore, on equation (45). We must therefore conclude, that if equation (51) is satisfied, the condition of equality of temperature is also satisfied, or, in other words, that equation (51) is the condition of equality of temperature. Hence when two parts of a system have the same temperature, the average kinetic energy corresponding to any one of the variables belonging to these parts must be the same. If the system is a gas or a mixture of gases not acted on by external forces, the theorem that the average kinetic energy of a single molecule is the same for molecules of different gases is not sufficient to establish the condition of equilibrium of temperature between gases of different kinds such as oxygen and nitrogen, because when the gases are mixed we have no means of ascertaining the temperature of the oxygen or of the nitrogen separately. We can only ascertain the temperature of the mixture by putting a thermometer into it. We cannot legitimately assert that the temperatures of the oxygen and of the nitrogen must be equal because they are in contact with each other, for the only way in which we can conceive the oxygen or the nitrogen as existing in the mixture is by picturing the medium as a system of molecules, and as soon as we begin to see the molecules dis- tinctly, heat becomes resolved into motion. But since our investigation is equally applicable to a system of any kind, provided only it satisfies the equations of dynamics, we may suppose it to consist of pure oxygen and pure nitrogen separated by a solid diaphragm, the solid diaphragm consisting of molecules capable of motion, but acting on each other with forces which are sufficient to prevent any molecule from getting far apart from its neighbours except under the action of disturbing forces greater than any which would occur in a system at the given temperature. In this system, though the oxygen and the nitrogen cannot mix, each can make an exchange of molecular energy with the surface molecules of the diaphragm,. and exchanges of energy can go on within the solid diaphragm itself without any exchange of molecules between distant parts of the diaphragm. Hence, in this system, the average kinetic energy of a molecule of oxygen will become equal to that of a molecule of nitrogen in the final state of the system, that is to say, 260 Pror, CLERK MAXWELL ON BOLTZMANN'S THEOREM ON THE AVERAGE when the temperatures of all parts of the system have become equal, and since in that final state we have pure oxygen on one side and pure nitrogen on the other, we can verify the equality of temperature by means of a thermometer, and we can now assert that the temperatures, not only of oxygen and nitrogen, but of all bodies, are equal when the average kinetic energy of a single molecule of each of these substances is the same. Approximate value of the probability when V is small compared with EL. To find the number of systems the configuration of which is specified as regards the limits of certain of the variables while the other variables are left undetermined, we should have to integrate the expressions in equations (41), (45), or (47) with respect to each of the undetermined variables in succession, the integrations being extended to all values of these variables which are consistent with the equation of energy. These integrations cannot be performed unless the potential energy of the system is a known function of the variables which determine its configuration. We cannot therefore in general continue the integration so as to determine the number of systems in which the limits are specified for some, but not all, of the variables. But when the number of variables is very great, and when the potential energy of the specified configuration is very small compared with the total energy of the system, n-2 we may obtain a useful approximation to the value of [Z2—V]¥* in an exponential form, for if we write, as in equation (53), H=nK, nearly, provided m is very great and V is small compared with #. The expression is no longer approximate when V is nearly as great as H, and it does not vanish, as it ought to do, when V= #. Hence when the potential energy of the system in the given configuration is very small compared with its kinetic energy, we may use the approximately correct state- ment, that the number of systems in a given configuration is inversely proportional to the exponential function, the index of which is half the potential energy of the system in the given configuration divided by the average kinetic energy corresponding to each variable of the system. If we divide the system into any two parts, A and B, we may consider V, the potential energy of the whole system, as made up of three parts, V4 and Vg, the potential energy of A and B, each on itself, and W, that of B with respect to A. DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 561 When, as in the case of a gas, the parts of the system are in a great degree in- dependent of each other, the average values of Vy and Vy may be treated as constants, and the variations of V will be the same as those of W, so that the variable part of the exponential function will be reduced to If we suppose that A denotes a single molecule of a particular kind of gas, and that B denotes all the other molecules, of whatever kind, in the system, then, since there are many molecules similar to A, we may pass, from the number of systems in which A is within a given element of volume, to the average number of molecules similar to A which are within that element, or, in other words, the average density of the gas A within that element. We may therefore interpret the expression (57) as asserting that the density of a par- ticular kind of gas at a given point is inversely proportional to the exponential function whose index is half the potential energy of a single molecule of the gas at that point, divided by the average kinetic energy corresponding to a variable of the system. We must remember that since the centre of mass of a molecule is determined by three variables, the mean kinetic energy of agitation of the centre of mass of a mole- cule is three times the quantity AK which denotes the mean kinetic energy of a single variable. Part If. A Free system. In a material system not acted on by external forces the motion satisfies six equations besides the equation of energy, so that we must not include in our integration all the phases which satisfy the equation of energy, but only those of them which also satisfy these six equations. In what follows, we shall suppose the system to consist of n particles, whose masses are m,...m,, and whose co-ordinates a, y, z, and velocity-components wu, v, w, are distinguished by the same suffix as the particle to which they belong. Let us now consider a system consisting of s of these particles, and write Mt 1 Fs OSC. M1 — oan see Seer en eevee ae (58), mx, +m,x,+ &e.+m,x,= V,X,, | my, + my, + &e. + my, = M, 3 WA ahead sae, Roos ee m,2z,+m,2,+ ke. + m,2,= M,Z, J then M, will be the mass of the minor system and X,, Y,, Z, the co-ordinates of its centre of mass. If we also write mu, + &.+m,u,= M,U,, m2, a &e. + m, U, = M, UBS eis (60), mw, + &e.+m,w,= M, W,, °62 Prov. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE m, (yw, — 2,2) ate &e. a5 m, 3, Ay 2,,) a F, ae M, CY; W, — Z, UA m, (2, — 2,10,) + &e. + m, (,u, —2,,) = G+ M, (ZU, X,W), b vessescceee (61), m, (2% >, Y.%,) + &. + ™, (x,v, a, Y.U,) == H, a M, (X, Vi Fe, Y,0); then U,, V,, W, will be the velocity-components of the centre of mass, and /, G,, H, the components of angular momentum round this point. We shall also write $m, (uo +92 +,7) + &e. thm, (ur + v7 + w,7) = T, ....ccecceceecereeeenees (62). The seven conditions satisfied by the whole system are that the seven quantities U,, V,, W,, F,, G,, H, and £ are constant during the motion. Under these conditions the 3n momentum-components are not independent. We shall therefore transform equation (17) into one in which the differentials of the first seven velocity-components are replaced by the differentials of the seven constants. The functional determinant is found by differentiating the seven quantities U,, V,, W., F,, @, H, and EF with respect to the momenta m,u,, m,v,, M,W,3 Ml, M,V,, MW, ; and m,u,. We thus obtain is 0, 0, 0, 2, —Yp % 0, ip Wb =65 ime Oh 0, 0, iy, th Sie 0, Ww, 1, 0, 0, 0, Pay Maes, || NU anton ce eeeeeee (63), 0, i 0, -—2z, 0, ae ap 0, 0, ie Yo ae, 0, w, 1. 0, 0, 0, Pe Yan which we may write ASQ e Ti, cossalsocquecsicceissseiissesis esssceinuseeeeeee (64), where A= (Y,— Yo) (Zp — Zp) — (Ya— Yo) (%y— Bq) ceveereceeccereceacseescees (65), or twice the projection on the plane of yz of the triangle whose vertices are m,, m,, and m,, and Tia Pig = (U, — U,) (@,— 2) + (Y, —%) (Y, — Ya) + (W,— W,) (2, — 2) creeree eens (66), or the rate of increase of the distance between m, and m, multiplied into that distance. In a system composed of material particles, each component of momentum is equal to the corresponding velocity-component multiplied into the mass of the particle. We may therefore write p,=m,u, and so on, and since the masses are invariable we may omit them from both members of equation (17), and write it Ge) 50 OS Ot, oye GO. = Ei, os. AS OW, AU, m (e— FZ) — 2) cscs cceeue (76). CO =m [fx — X)’+(y—- Y)] N=Xm (x-X) (y— Y) Writing for the sake of brevity A, —N, -— MM | a@,—n,-m DENS Ne CBr G | d=! =a, B, =U] osseeseeepneonsees (77), — WM, -L, c| '—m, —l, c Vor. XII. Part III. 72 564 Pror. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE the relations between the moments and products of mobility and those of inertia will be given by equations of the forms aD=BC-L’ Ad=be—P ID=—-MN- AL, Lid = Sit — Gh, oesese cnet (78). Dd =1. If we write f=u—-Uigqe-ry N= 9 = V LTE — PZ provrererxveriocaseosesredansasaneraces (79); S=w—W+ py— qe then &, », € will be the velocity-components of a particle with respect to axes passing through the centre of mass of the system and rotating with the angular velocity whose components are p, q, ™ We may therefore call £, », § the velocity-components of the internal motion. If the system were to become rigid, the internal motion would become zero. The energy of internal motion may be expressed in terms of & 7, § thus :— Mee 7 (Sac ee Ot eee rte Bee ese cor eer oameacceE rere: (80). We have now to express the energy of internal motion of a system of s—1 particles in terms of the quantities U, V, W, F, G, H and T belonging to the system of s particles, together with the position and velocity of the s™ particle. To avoid the repetition of suffixes we shall distinguish quantities belonging to the minor system of s—1 particles by accented letters, and quantities belonging to the com- plete system of s particles and the particle m, by unaccented letters. We shall also write Mm ane We thus find M=M-—m MX’ =MX—-—mz see (81). M'U’ =MU— mu, | . 5 i ‘ Fi =F-p(y—YV)(w-— W)4+p(2-Z) (v- V) | A’ = A=p(y—Y)-p(2-Z) = | L=L—-ply— Y) (2-2) T'=T=-tm(wi+v4+w’) | K'=K+ihy[(U—u)? + (V—0)? + (W-w))]— dm (4+ +’). 5 Since the choice of the axes of reference is arbitrary, we may simplify the expres- sions by taking for origin the centre of mass of the system J/, and for the axis of < the line passing through the particle m. We may also turn the axes of a and y about that of z till A becomes a maximum, the condition of which is ILM + CN =0. DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS 565 We shall also reckon velocities with reference to the centre of mass of the system JZ With these simplifications we find F’ =P + prz Gi Gee H=H 4 A'= A —p2? B=B-p2 C'=C | ey. M=M AGENT S| pate en | 1 = apz*’ 1 — bya?” | Sat steKe (82) ieee les oS T= daz’ 1 — apa?’ o=ct+p (igs at a) een, | 1— bus 1—apz = D(1 = apa’) (1 — byz’). We are now able to calculate the energy of rotation, J’, of the minor system 27’ =a FF? +04" + cH" — 2 GH —2m'H'F’ — WF'G : = 2) + al" aps” — 2vpz (Fa— Hm) + wz? (Fa— Hm)’) |} + sare (84). txt (Gb— Hl) + pz’ (Gb — Hl)" j Combining these results and reducing we find for the energy of internal motion of the system J’ T= Ly (1 — bye)" (uw — Gb + Hl)? —dy (1 ~ ape’)? (v — Fa + Hl)? —dpw’.....(85). +2 a (P@))" Meo ie a ate 1 q+8 42 oh q ~2\2 7s ~a\s 2 (ee Hence [[lz dudvdw = re) | (LE = apzs)* l= bye)” [aa 2 eesneesee( 80); the integration being extended to all values of u, v, and w which make Jf’ positive 2 2 D . - b. . . Now (1—aypz’) (1 — byz*) = D’ and this is an imvariant, Hence in general, whatever axes we choose, “ ray (4) as [f/m i DLJ dee du,dv,dw, = ane mJ? [Af,” Dy i. : plo hak (87). gt5 Hen) For the system consisting of the two particles m, and m, the energy of rotation is M, =} He : han conecen aera omreosad (ste) SO ae 2 (f+ G+ A,’ (88) 72—2 566 Pror. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE and the energy of internal motion is a | ad a eres cath ohh (89) Hence we may write equation (70) z r3n-7 x. M. N@®=]" C(mimgmyay,) ‘2, =) [Adin cadet ee ee (90). We have first to express J, in terms of quantities having the suffix ,. If we make the plane of yz pass through the three particles m,, m,, m,, so that the origin coincides with their centre of mass and has the same velocity, and the axis of z passes through m,, then a is twice the area of the triangle whose vertices are m, m, and m,, Mn, Mam F, = F,+ 5 U, 520s, GF Gar sagt ds aad o FAB ay etrs tees LOr .(91), OD NG A Fe) = ED, ar) eve enccn ner: cat Exe cede (92), Am, M, Ms *) M,G.z 3 M, Ore : Mm yf f ve es el 3 a aa uw? LTS M, (1+ m m, r [ (x mmr, + M,m,z, ) * (o+ m mgr oe =| z u, re Rieke ate a en eeree Ga veelinek Ramenniinttc ..«(93) We have now to integrate extending the integration to all values of v, and w, which make J, positive, and remembering that equation (92) shows that u, is independent of v, and w, The result is =a Iz I>? dv dw, = @y emg p CE 12 ek Manis "It Sacck ne vs (94) : PO | oes MMT 5 ; Now for the three particles m,, m,, m,, ™m i. gt, 2 2 2 foe D,=— ‘WL. Bo [Mop MQNg Mg, Me, Eps Mm) Ga, de cas ss vee Where r,,, r,, and r,, are the distances between the particles, and z is the area of the triangle m,m,m,. - 3 + 2, ~ 2 | M, ’ ‘ 96 Also 4, MM, +1,,° mm, + 7,," mm, = — (m,m,r",,? + Mym,2,).-0.eeees AML eIpon (96). 2°12 et We may now write equation (90) in the form °3n-9 3 N (b) =| 0} 8 [3 m,mmeMD,) 41,3 Qe UD sade Nan ein meee (97). DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 567 Continuing the integration by equation (87) we find -8 sn—-8 1)Sn—6 3n-8 ea ecient PD st Rae ane ae N(Q)=27 C Temes) = (98), where J, is what we have defined as the energy of internal motion of the system, or the work which the system would do, in virtue of its motion, against the system of internal forces which would be called into play if the distances between the parts of the material system were in an insensibly small time to become invariable. | In order to determine the number of systems in a given configuration for which the velocity-components of the particle m, lie between the limits u+4du, v+4dv, w+ idw, we must form the expression for NV (b, u,, v,, w,) by stopping short before the last triple integration. We thus find 3n—11 \sn-3 : 5 4 8n-11 : N (6, Un» v,> w,) a 2 : Cc . — 9 (m, oor Mn OM Dales ; du,dv, dw,. be (99). : (Cr If, as in equations ($2) to (86), we suppose the origin of co-ordinates to be the centre of mass of the whole system, the axis of z to pass through the particle m,, and the axes of 2 and y to be in the directions of the principal axes of the section of the momental ellipsoid normal to z, then writing ES — Ge, QUE PZ, CHW 2-2. cncssea+crmmenarcqonenen- (100), so that £& 7, € are the velocity-components of m, relative to axes moving as the system would do if it were then to become rigid, with the angular velocity whose components are p, q, 7, we may write T= 1, —4p (1 — bye’) — du (1 — ape’) "7? — hp’... (ishceaueenes (101). The sum of the last three terms of this expression, with its sign taken positive, represents the part of the internal motion of the system which is due to the fact that the particle m, is moving with the relative velocity whose components are & 7, C. We may also define it as the work which would be done by the particle m, against the internal forces of the system, if these forces were suddenly to become such as to render the whole system rigid in an infinitely short time. Comparing this result with that obtained in equation (48), we see that the law of distribution of the velocities of the particle m, is the same as what it would be m a fixed vessel containing n—2 particles, provided that we substitute for u*, v’, w® the quantities (1 —byz*)'&, (1—apz2*)*y’, &° respectively. Hence the mean square of the velocity in the direction cf the line joming the particle with the centre of mass is the same at all points of the system, but the mean 568 Prov. CLERK MAXWELL ON BOLTZMANN’S THEOREM ON THE AVERAGE square of the velocity in other directions is less than this in the ratio of 1—ayz* to 1, where z is the perpendicular from the centre of mass on the line of relative motion of the particle, and a is the moment of mobility of the system about an axis through the centre of mass and normal to the plane through that centre and the line of motion. When the product of the mass of the particle into the square of its distance from the centre is so small that it may be neglected in comparison with the moments of inertia of the system, then quantities like ay2* and bu2z* may be neglected in respect of unity, and we may assert that the mean square of the relative velocity, for a particle of given mass, is the same in all directions and at all points of the system; but that for different particles it varies inversely as their masses; so that the average energy of motion relative to the moving axes is the same for particles of all kinds throughout the system. We have already learned from equation (98) that in a free system of x particles the number of cases in which the system is in a given configuration, or, in other words, 3n— 5) 2 ae : 5 3 8 ; the probability of that configuration, is proportional to the power of the energy of internal motion corresponding to that configuration. We have next to consider the manner in which this probability depends on the position of a particular particle, say of the last particle, m,. Let J,° denote the energy of internal motion of the complete system when m, is at the centre of mass of the system and is without any velocity relative to that centre. It is manifest that in this case m, contributes nothing towards the energy of internal motion. Now let m, be carried from the centre of mass to the point (0, 0, 2) and left there without any velocity (that is, let w=v=w=0). Let W be the work which must be done against the forces of the system to effect this transference, then since the total energy of the system and the three angular momenta must be maintained constant, we shall have after this displacement, for the energy of internal motion of the remaining n—1 particles, FO Wo wcaein sanan anne cdantenses ater cveodege (102). But by equation (85) ID, = I. + 4m (1 — bye?) (u — gz)? + du (1 — apez*)* (v + pz)? + du’... (103). Substituting the value of J,, from equation (102), and remembering that u=v=w=0, we find for the energy of internal motion in the new configuration DL, = T° — W+ Sue (1 — bya”) * ga* + dye (1 — ayer?) pra? ec ceccec esses (104). The probability, therefore, of a configuration in which, the positions of all the other . . . . . . . 3n-8 particles being given, that of m, is varied, is proportional to J, 2 , J, being given by equation (104). DISTRIBUTION OF ENERGY IN A SYSTEM OF MATERIAL POINTS. a69 When, as in the case of a gas, there are a great many particles similar to m,, we may speak of the density of the medium consisting of such particles in the element dxdydz. In this case, however, for reasons already given, neglect the quantities op* and buz*, and we may write m for ~. We may also choose our axes in the manner which is most convenient. We shall therefore make the axis of z that round which the system, if it were rendered rigid, would rotate with velocity w, and we shall suppose this axis to be vertical, as otherwise a steady motion under the action of gravity could not exist, and we shall denote the horizontal distance from this axis by r. We may now write for the density of the gas at-the point (z, r) 3n—8 =, (Le (2) neat rage) eects adeacteoi.wsesse 105), P Po n ME where p, is the density at the origin. When n is a very large number and when the second term of the binomial is very small compared with unity, we may write for this the exponential expression e at (wr? — 292) p= pre" dasusboonodoaaunccecoaceencosboconece oe (106). If m, is the mass of a molecule of hydrogen, wm, will be the mass of a molecule of the kind of gas considered, where « is the chemical equivalent of the gas. Also if 7 is the temperature on the centigrade scale, and «@ the coefficient of dilata- tion of a perfect gas, then since the “velocity of mean square” of agitation of the molecules of hydrogen at 0°C. is 1:844x 10° centimetres per second, the kinetic energy of agitation of a system containing n molecules of any kind will be 3m,n (1'844)? 10” (1 + a7), and the difference between this and the energy of internal motion may be neglected. We thus find for the density at any point a wy? — 292 p= poet EB LONGEED sete ae cee cere ceee (107). Let us now consider a tube of uniform section placed on a whirling table so that one end, A, of the tube coincides with the axis while the other end, B, revolves about the axis with the angular velocity . The linear velocity of B is ar, and we shall suppose, for the sake of easy calculation, that this velocity is one-tenth of the velocity of agitation of the molecules of hydrogen. The velocity of the end B would be 1844 metres per second. If the tube contains hydrogen at 0°C., the ratio of the density ot the gas at B to the density at A will be e790, or approximately 14 535. be If it contains a gas whose chemical equivalent is w, the ratio will be 1+ 300° 570 Pror. CLERK MAXWELL ON BOLTZMANN'S THEOREM. If the tube contains hydrogen and carbonic acid, and if a certain volume of the tube at A contains 200 parts of hydrogen and 200 of carbonic acid, then an equal volume of the tube at B will contam 201 parts of hydrogen and 222 parts of carbonic acid. The time during which the experiment would require to be continued in order to obtain a given degree of approximation to the ultimate distribution of the mixed gases varies as the square of the length of the tube. Thus in Loschmidt’s experiments on the diffusion of gases he used a tube about a metre long, and continued his experiments from half an hour to an hour in order to obtain the results from which he could best deduce the coefficient of diffusion. In these experiments the inequalities of distribution of hydrogen and carbonic acid were reduced to less than a third part of their original value in half an hour, and if the experiment had gone on for two hours the differences from the ultimate distribution would have been reduced to a hundredth part of their original value. We may therefore consider two hours as ample time for an experiment on the ultimate distribution of these two gases in a tube one metre in length. - But if we make the whirling tube 20 centimetres long, the differences of distribution from the ultimate distribution would be reduced to a hundredth part of their original value in a twenty-fifth part of the time, that is to say in 4 minutes 48 seconds. If it were found more convenient to have bulbs on the ends of the tubes, so as to be able to secure the gas at each end before it got mixed up by the violent com- motion arising from the stopping of the whirling tube, we should have to allow a longer time for the whirling. In order to obtain a similar distribution of the two gases in a vertical tube by the action of gravity the tube would require to be 1720 metres high, and in order to obtain the same degree of approximation to the ultimate distribution we should have to let the experiment go on for 675 years, carefully preserving the tube during that time from all inequalities of temperature, which, by causing convection-currents, would continually mix up the gases and prevent their partial separation. CAMBRIDGE; PRINTED BY C, J. CLAY, M.A. AT THE UNIVERSITY PRESS. INDEX TO THE TRANSACTIONS OF THE Cambridge Philosophical Society. VOLS. I—XII. I. NAMES OF AUTHORS. Apams, Prof. J. C., Note on Sir G. B. Airy’s memoir on the resolution of a certain Trinomial: Noy. 23, 1868; x1. 444—445, Airy, Sir G, B., On the use of silvered glass for the Mirrors of Reflecting Telescopes; Nov. 25, 1822; 11. 105—118. — On the Figure assumed by a Fluid Homogeneous Mass, &c., &c.: March 15, 1824; 11. 203—216. —— On the Achromatism of the Eye-pieces of Tele- scopes, and of Microscopes: May 17, 1824 ; I. 227—252. — On the defect in the Eye, and a mode of cor- recting it: Feb. 21, 1825; 11. 267—271. —— On the form of the teeth of Wheels: May 2, 1825; II. 277—286. —— On Laplace’s investigation of the Attraction of Spheroids differing little from a Sphere: May 8, 1826; 11. 379—390. —— On the Spherical Aberration in the Eye-pieces of Telescopes: May 14, 21, 1827; 11. 1—63. —— On Pendulums and Balances, and the Theory of Escapements: Noy. 26, 1826; m1, 105—128. —— On the Longitude of Cambridge Observatory : ; Nov. 24, 1828; m1. 155—170. —— On a means of correcting the length of a Pen- dulum by a Ball suspended by Wire; Noy, 16, 1829 ; 111. 855—360. — On the conditions under which Perpetual Motion is possible: Dec. 14, 1829; m1. 8369—872. Voi. XII. Atry, Sir G. B., On the Nature of the Light in the two Rays produced by the Double Refraction of Quartz: Feb. 21, 1831; rv. 79—123. —— Addition to this memoir: Apr. 18, 1831; Iv. 199 —208. — Ona remarkable modification of Newton’s Rings: Nov. 14, 1831; rv. 279—288. —— On a new Analyzer, and its use in Polarization : March 5, 1832; rv. 318—322. —— On the phenomena of Newton’s Rings with Sub- stances of different refractive Powers: March 19, 1832; rv. 409424. —— On a calculation of Newton’s Experiments on Diffraction: May 7, 1833; v. 101—111. —— On the Latitude of Cambridge Observatory : Apr. 14, 1834 ; v. 271—281, — On the Diffraction of an Object-Glass with cir- cular aperture: Noy. 24, 1834; v. 283—291. See EARNSHAW. —— On the Intensity of Light in the neighbourhood of a Caustic: May 2, 1836: March 26, 1838; vi. 379—402. —— On Triple Algebra: Oct. 28,1844; vim. 241—254, Supplement to this memoir: May 8, 1848; vit. 595—599. —— On a new construction of the Going-Fusee: March 2, 1840; vir, 217—225. —— On an Eye affected by a mal-formation: May 25, 1846 ; vii. 861—362, 73 ii INDEX TO TRANSACTIONS I—XII. Ary, Sir G. B., Further observations on the same: Feb. 12, 1872; x11. 392—393. —— On the substitution of Ordinary Geometry for the general Doctrine of Proportions: Dec. 7, 1857; x. 166—172. —— Suggestion of a Proof that every Equation has a Root: Dec. 6, 1858; x. 283—289. —— Supplement to this memoir: Dec. 12, 1859; x. 327—330. — On the factorial Resolution of the Trinomial a" —2 cosna = : Nov. 9, 1868; x1. 426—443. AxIn, C. K., On the origin of Electricity: Dec. 7, 1863; xI. 6—20. ALDERSON, Jas., On a Spermaceti Whale, stranded in Yorkshire: May 16, 1825; 11. 253—266. — On an Artificial formation of Plumbago: Feb. 21, 1825; u. 441—443. Awnstep, D. T., On some Fossil Multilocular Shells found in Cornwall: Feb. 26, 1838; v1. 415— 422. — On a portion of the Tertiary Formations of Switzerland: May 20, 1839; vu. 141—152. — Onsome Phenomena of the Weathering of Rocks: March 2, 1868; x1. 387—395. BaBpaGe, C., On the notation employed in the Calculus of Functions: May 1, 1820; 1. 63—76. —— On the General Term of a New Class of Infinite Series: May 3, 1824; m. 218—225. —— On the influence of Signs in Mathematical Reason- ing: Dec. 16, 1821; m1. 325—377. Baxter, H. F., On Organic Polarity: March 8, 1858; x. 248—260. Bevan, B., Experiments on Percussion: Noy. 10, 1825 ; Ir. 444, Bonn, Prof., Statistical Report on Addenbrooke’s Hospital for 1836: March 13, 1837; vi. 361 —377. The same for 1837: Apr. 30, 1838; v1. 565—575. 300LE, Prof. George, Of Propositions numerically defi- nite: March 16, 1868; x1. 396—411. Brewster, Dr., On the Brazilian Topaz; its colour, structure, and optical properties: May 6, 1822; Ir, 1—9. sropIE, P. B., On Land and Freshwater Shells, and Bones of Animals found near Cambridge: Apr. 30, 1838 ; virr. 138—140. Cay ry, Prof., On the Theory of Determinants : Feb. 20, 1843; vur. 75—88. —— On the Theory of Involution: Feb. 22, 1864; xr. 21—38, —— On a case of the Involution of Cubic Curves: Feb. 22, 1864; x1. 39—80. —— On the classification of Cubic Curves: Apr. 18, 1864 ; x1. 81—128. Cayxey, Prof., On Cubic Cones and Curves: Apr. 18, 1864; x1. 129—144. — On certain Skew Surfaces, otherwise Scrolls: Nov. 11, 1867; x1. 277—289. —— On the Six Coordinates of a Line: Noy. 11, 1867; XI. 290—323. —— On a certain Sextic Torse: Noy. 8, 1869; xt. 507—523. — On the Centro-surface of an Ellipsoid: March 7, 1870; x1. 319—365. — On Dr Wiener’s model of a cubic surface with 27 real lines; and on the construction of a double-sixer: May 15, 1871; x11. 366—383. —— On the geometrical Representation of Cauchy’s theorems of Root-limitation: Feb. 16, 1874; XII. 395—413. Cecit, W., On Hydrogen Gas, as a moving power in Machinery, &e.: Nov. 27, 1820; 1. 217—239. —— On an apparatus for grinding Mirrors and Object Lenses: Dec. 11,1822; m1. 85—99. CHautis, Prof., On the extension of Bode’s Law to the distance of Satellites from their Primaries: Dee. 8, 1828; 11. 171—183. — On the small Vibratory Motions of Elastic Fluids : March 30, 1829 ; 111. 269—320. —— On the general Equations for the Motion of Fluids, &c., &c.: Feb. 22, 1830; 111. 383—416. —— Researches in the Theory of the Motion of Fluids: March 3, 1834 ; v. 173—203. — On the Decrement of Temperature depending on the Height above the Earth’s surface : Feb. 13, 1837; v1. 443—455. — On the motion of a small Sphere, acted on by Vibrations of an Elastic Medium: Apr. 26, 1841; Vil. 333—353. —— On the Differential Equations applicable to the Motion of Fluids: Apr. 11, 1842; vm. 371—396. —— Ona new Equation in Hydrodynamics : March 6, 1843 ; vin. 31—43. — On the Theory of Luminous Rays on the hy- pothesis of Undulations; May 11, 1846; vt. 363—370. —— On the Theory of the Polarization of Light, on the same hypothesis; May 25, 1846; vir. 3871— 378. — On the Transmission of Light, and on Double Refraction; on the same hypothesis: May 17, 1847 ; vil. 524—532. — On the Mathematical Theory of Luminous Vibra- tions: March 6, 1848; vin, 584—594. —— On the Aurora Borealis of Nov. 17, 1848: Nov. 27, 1848; vill. 621—632. —— On the Determination of the Longitude of Cam- bridge Observatory by Galvanic Signals; May 15, 1854; 1x. 487—514. I. NAMES OF AUTHORS. ii Curistrs, 8S. H., On the Laws by which Masses of Iron affect Magnetic Needles: May 15, 1820; 1. 147—173. Crark, Prof. W., On a case of Human Monstrosity : May 16, 1831; Iv. 219—255. CiarkKe£, Prof. E. D., Inaugural address at the first general meeting of the Society : Dec. 13, 1816; I. (l—7). —— On the Purple Precipitate of Cassius: May 15, 1820; 1. 53—61. —— On a deposit of Natron in the tower of a Church: Noy. 27, 1820; 1. 193—201. —— On the Crystallization of Water, &c.; March 5, 1821; 1. 209—215. Cuirron, R. B., Note on Prof. De Morgan’s Memoir on the history of Signs + and —; Feb. 13, 1865; XI. 213—218. Copprneton, Rev. H., On the improvement of the Microscope: March 22, 1830; m1. 421—428. Cox, HomersHam, On Impact on Elastic Beams: Dec. 10, 1849; Ix. 73—78. —— On the Deflection of Imperfectly Elastic Beams, and on the Hyperbolic Law of Elasticity: March 11, 1850; rx. pt. 1. 177—190. Cumine, Prof, On the connexion of Galvanism and Magnetism : Apr. 2, 1821; 1. 269—279. — On Magnetism as a Measure of Electricity: May 21, 1821; 1. 281—286. — On a large Human Calculus in the Library of Trinity College: Nov. 26, 1821; 1 347— 349. — On the development of Electro-Magnetism by Heat: Apr. 28, 1823; 11. 47—76. — See Alderson, J. Der Moreay, Prof., On the general Equation of Curves of the Second Degree: Noy. 15, 1830; Iv. 71—78. — On the General Equation of Surfaces of the Second Degree: Nov. 12, 1832; v. 77—94. — On Discontinuous Constants, as applied to Infinite Series: May 16, 1836; vi. 185—193. —— On a Question in the Theory of Probabilities: Feb. 26, 1837; vi. 423—430. —— On the Foundation of Algebra: Dec. 9, 1839; vu. 173—187. — Do. Do. Do. Noy. 29, 1841; vil. 287—300. —— Do. Do. Do. Noy. 27, 1843; vit. 139—142. — Do. Do. On Triple Algebra: Oct. 28, 1844; virt. 241—254, —— On Divergent Series, &c., &c.: March 4, 1844; vu. 182—2038. —— On the Structure of the Syllogism, and its ap- plication, &c.: Noy. 9, 1846; viz. 879—408. De Moran, Prof., On Integrating Partial Differential Equations: June 5, 1848; vim. 606—613. —— On the Symbols of Logic, the Theory of the Syllo- gism, &c., &c.: Feb. 25, 1850; 1x. 79—197. —— On some points of the Integral Calculus ; Feb. 24, 1851; 1x. pt. ii, 107—138. —— On some points in the Theory of Differential Equations: March 27, 1854; rx. 515—554. —— On the singular points of Curves, and on Newton’s method of Co-ordinated Exponents: May 21, 1855; Ix. 608—627. —— On the Solution of a Differential Equation: Apr. 28, 1856; x. 21—26. —— On the Beats of Imperfect Consonances: Novy. 9, 1857; x. 129—145. —— On the Syllogism, No. m1, and on Logie in general: Feb. 8, 1858; x. 173—230. —— On the Syllogism, No. 1v., and on the Logie of Relations ; Apr. 23, 1860; x. 331—358*. —— On the Proof of the existence of a Root in every Algebraic Equation: Dec. 7, 1857; x. 261—270. —— On the General Principles of which the Composi- tion of Forces is a Consequence: March 14, 1859; x. 290—304. — On the Theory of Errors of Observation : Noy. 11, 1861; x. 409—427. —— On the Syllogism, No. y., and on some parts of the Onymatic System: May 4, 1863; x. 428— 487. — On Infinity: and on the sign of Equality ; May 16, 1864; x1. 145—189, —— Ona Theorem relating to Neutral Series : May 16, 1864; x1. 190—202. — On the Early History of the Signs + and—: Noy. 28, 1864; x1. 203—218. — On the Root of any Function: and on Neutral Series, No. u.: May 7, 1866; x1. 289—266. — Note on the same: Oct. 26, 1868; x1. 447—460. Denison, E. B., On Clock Escapements: Noy. 27, 1848; VII. 633—638. — On Turret-clock Remontoirs: Feb. 26, 1849; vu. 639—641. — On some recent Improvements in Clock-Escape- ments: Feb. 7, 1853; 1x. 417—430. Donawpson, Dr. J. W., On the Structure of the Athe- nian Trireme: Noy. 6, 1856; x. 84—93. —— On the Statue of Solon mentioned by /®schines and Demosthenes: Feb. 22, 1858; x. 231—239. — On Plato’s Cosmical System: as exhibited in The Republic: Book x. : Feb. 28, 1859; x. 305—316. — On the Origin and Use of the word ArGuMENT; Noy. 28, 1859; x. 317—326. EarnsHaw, Rey. 8., On Fluid Motion, as expressed by the Equation of Continuity: March 21, 1836: VI. 203—233, (3—2 lv INDEX TO TRANSACTIONS I—XIE. EARNSHAW, Rev. S., On the Diffraction of an Object- Glass with a triangular Aperture: Dec. 12, 1836; vi. 431—442. See Airy. —— On the Nature of the Molecular Forces of Lumi- niferous Ether: March 18, 1839; vi. 97—112. — On the Values of the Sine and Cosine of an Infinite Angle: Dec. 9, 1844; vim. 255—268. — On two great Solitary Waves of the First Order: Dec. 8, 1845; vu. 326—341. Exuis, R. L., On the Foundation of the Theory of Probabilities: Feb. 14, 1842; vu. 1—6. — On the method of Least Squares: March 4, 1844; vu. 204—219, —— Remarks on the Theory of Matter: May 22, 1848; vir. 600—605. — On the Fundamental Principle of the Theory of Probabilities : Noy. 13, 1854; rx. 605—607. FarisH, Prof., On Isometrical Perspective: Feb. 21, March 6, 1820; 1. 1—19. FENNELL, C. A. M., On the First Ages of a Written Greek Literature: Nov. 23, 1868; x1. 461— 480. FIsHER, Rey. Osmonp, On the Purbeck Strata of Dorset- shire: Noy. 13, 1854; rx. 555—581. —— On the eleyation of Mountains by lateral pres- sure, &c., &c.: Apr. 27, 1868; x1. 489—506. —— On the Inequalities of the Earth’s Surface viewed in connection with the secular cooling: Dec. 1, 1873; xu. 414433. —— On the same, as produced by lateral pressure, on the hypothesis of a liquid substratum: Feb. 22, 1875; x. 434—454. GLAISHER, J. W. L., Tables of the first 250 Bernouilli’s Numbers, and of their logarithms: May 29, 1871; x11. 384—387. —— Supplement to the same memoir: March 11, 1872; XII. 388—391, GopFray, Hueu, On a Chart and Diagram for facilitat- ing Great Circle Sailing: May 10, 1858; x. 271—282. Goopr, Henry, On a peculiar Defect of Vision: Noy. 9, 1846; May 17, 1847; vim. 493—496. Goopwiy, Rey. H., On the Connexion between Me- chanics and Geometry: Feb. 10, 1845; vu. 269—277. —— On the Pure Science of Magnitude and Direction: May 12, 1845; vu. 278—286. —— On the Geometrical Representation of the Roots of Algebraic Equations: Apr. 27, 1846; vir. 342 —360. GREEN, GEORGE, On the Laws of Equilibrium of Fluids analogous to the Electric Fluid: Noy. 12, 1832; v. 1—63. —— On the Exterior and Interior Attractions of Ellip- soids, &c., &c.: May 6, 1833; v. 395—429. GREEN, GEORGE, On the Reflexion and Refraction of Sound: Dec. 11, 1837; v1. 403—413. — On the Motion of Waves in a variable Canal of small depth and width: May 15, 1837; vi. 457—462. —— Note on the motion of Waves in Canals: Feb. 18, 1839; vu. 87—95. —— Memoir on the Laws of Reflection and Refraction at the common Surface of two non-crystallized Media: Dec. 11, 1837; vir. 1—24. —— Supplement to the memoir: May 6, 1839; vit. 1138—120. — On the Propagation of Light in Crystallized Media: May 20, 1839; vir. 121—140. Greeory, Dr. OLryrHus, On some Experiments to determine the Velocity of Sound: Dee. 8, 1823; m1. 119—137. Haitstone, Rey. J., On an extraordinary depression of the Barometer in Dec. 1821, &., &c.: Feb. 25, 1822; 1. 453—458. Havitanpd, Dr., On the solution of the Stomach by Gastric Juices: Dec. 11, 1820; 1 287—290. Haywarp, R. B., On a direct method of estimating Velocities, &e., &e., with reference to Axes moveable in Space: Feb, 25, 1856; x. 1—20. HeEnstow, Prof., On the Geology of Anglesea: Nov. 26, 1821; 1. 359—452. —— On a hybrid Digitalis: Novy. 14, 1831; 1v. 257— 278, . —— On the Monstrosity of the Common Mignionette : May 21, 1832; v. 95—100, Herscuet, Sir J. F. W., Deviation in Crystals from Newton’s scale of Tints, with Polarized Light : May 1, 1820; 1. 21—41. —— Planes of Polarization, as affected by Plates of Rock Crystal: Apr. 17, 1820; 1. 43—52. — Functional Equations, Reduction of, to Equations of Finite Differences : March 6, 1820; 1. 77—87. —— Apophyllite, On the Refraction of coloured Rays in: May 7, 1821; 1. 241—247. —— Ona Machine for resolving by Inspection Trans- cendental Equations: May 7, 1832; rv. 425— 440, Hiern, W. P., A Monograph of Ebenacew: March 11, 1872; x11. 27—300. Hotpircw, Rey. H., On Rolling Curves: Dee. 10, 1838; vit. 61—86. —— On Small Finite Oscillations: May 15, 1843; vrir. 89—104. Horkrys, W., On Aerial Vibrations in Cylindrical Tubes: May 20, 1833; v. 231—270. Researches in Physical Geology; May 4, 1835; vi. 1—84. —— On the Motion of Glaciers: May 1, 1843; vir. 50—74. I. NAMES OF AUTHORS. Horxrys, W., On the Motion of Glaciers (second me- moir): Dec. 11, 1843; vim. 159—169. — On the Transport of Erratic Blocks: Apr. 29, 1844; yr. 220—240. — On the Internal Pressure of Rock Masses, &c., ke. : May 3, 1847 ; vil. 456—470. — On the External Temperature of the Earth, and the other Planets of the Solar System: May 21, 1855; Ix. 628—672. Humpury, Prof. G. M., On the growth of the Jaws: Noy. 9, 1863; x1. 1—5. Jarrett, T., On Algebraic Notation: Nov. 12, 1827; m1. 65—103. Jess, Prof. R. C., On the place of Music in Education as conceived by Aristotle in his “ Politics :” May 17, 1875; xm. 523—530. Jenyns, Rey. L., On the Ornithology of Cambridge- shire: Noy. 28, 1825 ; 1. 287—324. — On Pennant’s Natterjack; with a list of the Rep- tiles of Cambridgeshire: Feb. 22, 1830; m1. 373—381. — Monograph on the British species of Cyclas and Pisidium : Noy. 28, 1831; Iv. 289—312. KELLAND, Prof., On the Dispersion of Light, on the theory of Finite Intervals: Feb. 22, 1836; v1. 153—184. — On the Motion of a System of Particles, with reference to Sound and Heat: May 16, 1836; VI. 235—288. — On the transmission of Light in Crystallized Media: Feb, 13, 1887; vi. 323—352, —— Supplement to the same: May 1, 1837; v1. 353— 360. —— On Molecular Equilibrium: March 26, 1838; vu. 25—59. — On the Quantity of Light intercepted by a grating placed before a Lens; and on the effect of the Interference: March 30, 1840; vu. 153—171. Kewp, GEoree, On the Nature of the Biliary Secretion: March 6, 1843; vit. 44—49. Kuve, J., A new demonstration of the Parallelogram of Forces: Apr. 14, 1823; 0. 45—46. Lz, Prof., On the Astronomical Tables of Mohammed Abibeker Al Farsi: the mss. of which are in the Public Library: Nov. 13, 1820; 1. 249— 265. Lesttz, Prof., On the Sounds excited in Hydrogen Gas: Apr. 2, 1821; 1. 267—268. Lowe, R. T., Primitie Faune et Flore Madere et Portus Sancti: Nov. 15, 1830; 1v. 1—70. Piscium Maderensium Species (see M. Young): Noy. 10, 1834; vr. 195—201. —— Noyvitie Flore Maderensis: or Gleanings from Madeiran Botany: May 28, 1838; vi. 523— | dol, | Vv Lussock, Sir J., On the Calculation of Annuities, and on some points in the Theory of Chances: May 26, 1828; m1. 141—154. — Comparison of various Tables of Annuities: March 30, 1829 ; 11. 321—341. Lunn, F., Phosphate of Copper from the Rhine, Analy- sis of: March 5, 1821; 1. 203—207. ManDELL, W., On the improved methods of proctring Potassium: Nov. 26, 1821; 1. 343—345. MaxweE t, Prof. J. Cterk, On the Transformation of Surfaces by Bending: March 13, 1854; Ix. 445—470. — On Faraday’s lines of Force: Dec. 10, 1855, Feb. 11, 1856; x. 27—83. — On Boltzmann’s Theorem on the average distri- bution of energy in a system of material points : May 6, 1878; x11. 547—570. Minter, Prof., On the Crystals of Boracic Acid: Noy. 30, 1829; m1. 365—367. —— On Crystals found in Slags: March 22, 1830; mt. 417—420. —— On the position of the Axes of Optical Elasticity in certain Crystals: Dec. 8, 1834; v. 431— 438. March 21, 1836; vi. 209—215. —— On spurious Rainbows: March 22, 1841; vm. 277—286. Moorg, A. A., On a difficulty in Analysis noticed by Sir Wm. Hamilton: May 1, 1837; vi. 317—322. Morton, Pierce, On the Focus of a Conic Section: March 2, 1829; m1, 185—190. Mosetry, Rey. H., On the Equilibrium of the Arch: Dee, 9, 1833; v. 293—313. — On the Theory of the Equilibrium of Bodies in Contact: May 15, 1837; v1. 463—491. Munro, Rey. H. A. J., On a Metrical Latin Inscrip- tion at Cirta in Algeria: Feb. 13, 1860; x. 374408. Murpny, R., On the General Properties of Definite Integrals: May 24, 1830; m1. 429443. — On the Resolution of Algebraical Equations: March 7, 1831; Iv. 125—153. — On the inverse method of Definite Integrals, with Physical Applications: March 5, 1832; tv. 353—408. — Second memoir on the same: Noy. 11, 1833; v. 113—148. —— Third memoir on the same: March 2, 1835; v. 315—393. —— On Elimination between an Indefinite number of unknown Quantities : Nov. 26, 1832; v. 65—75. — On the resolution of Equations in Finite Dif ferences: Noy. 15, 1835; vr. 91—106. O'BRIEN, Rev. M., On the Propagation of Luminous Waves in the Interior of Transparent Bodies : Apr. 25, 1842; vil. 397—437. vi INDEX TO TRANSACTIONS I—XII. O’Brien, Rev. M., On the Reflection and Refraction of Light at the surface of an Uncrystallized Body : Nov. 28, 1842; vi. 7—26. —— On the possibility of accounting for the Absorption of Light, &c., &e.: Feb. 14, 1843; viz. 27—30. — On a New Notation to be used in Geometry, &c., &c.: Noy. 23, 1846; vir. 415—428. —— On a System of Symbolical Geometry and Me- chanics: March 15, 1847; vu. 497—507. — On the Equation for the Vibratory Motion of an Elastic Medium: March 15, 1847; vim. 508— 523. OxEs, J., On the remains of a Fossil Beaver found in Cambridgeshire: March 6, 1820; 1 175—177. —— Ona dilatation of the Ureters, &c.: Nov. 12, 1821; I. 851—358. Owen, Ricuarp, F.R.S., Description of an extinct Lacertian Reptile: Apr. 11, 1842; vir. 355— 369. Pacet, Prof. G. E., On some remarkable Abnormities in the Voluntary Muscles: March 8, 1858; x. 240—247. Paey, F. A., On Homeric Tumuli: March 12, 1866; XI. 267—276. —— On the Comparatively Late Date, and Composite Character of our Iliad and Odyssey: Nov. 26, 1866; x1. 360—386. Pear, J. B., On the Geology of some parts of Suffolk, particularly of the Valley of the Gipping: Feb. 27, 1854; 1x. 431444. Pierson, R., On the Theory of the Long Inequality of Uranus and Neptune: 1852; rx. Appendix, pp. lxvii. Porter, R., Mathematical considerations on the Prob- lem of the Rainbow: Dec. 14, 1835; v1. 141— 152. —— On a new correction in the Construction of the Double Achromatic Object-Glass: Apr. 30, 1838; vi. 553—564. —— On the Heights of two Aurore Boreales, &c., &c.: Dec. 8, 1845; vim. 320—325. Power, Rey. J., On the principle of Virtual Velocities: March 21, 1825 ; 1m. 273—276. —— On the Theory of Residuo-Capillary Attraction, &c., &. : March 17, 1834; v. 205—229. —— On a Railway Accident; and on a Principle of Motion involved in precautions against Col- lisions : May 29, 1841; vir. 301—317. —— On the Truth of a Theorem in Hydrodynamics; May 9, 1842; vir. 455—464. Ricavp, Prof., On the relative Quantities of Land and Water on the Globe: Feb. 13, 1837; v1. 289 —300. Rours, J. H., On the Oscillation of a Suspension Chain : Dec. 8, 1851; 1x. 379—398. Rours, J. H., On the Motion of Beams, and thin Elastic Rods: Apr. 23, 1860; x. 359—373. —— On the Strains to which Ordnance are subject, and on the Vibrations of Solid Bodies: Apr. 18, 1864; x1. 324—359. Roramay, R. W., On Variations of Magnetic Intensity, as computed and observed: Noy. 10, 1825; 1. 445. — Onan Ancient Observation of a Winter Solstice: Noy. 30, 1829; m1. 361—363. —— An account of Observations of Halley’s Comet: Dec. 11, 1837; v1. 493—506. Sautrer, J. W., On Crotalocrinus rugosus, Miller: a Crinoid in the Woodwardian Museum: Feb. 8, 1869; x1. 481484. —— Diagram of the relations of the Univalve to the Bivalve: and of this to the Brachiopod: Feb. 8, 1869; x1. 485—488. Srpewick, Prof., On the Primitive Ridge of Devonshire and Cornwall: March 20, 1820; 1. 89—146. —— On the Physical Structure of the Lizard district in Cornwall: Apr. 2, May 7, 1821; 1. 291—330. —— On some Trap Dykes in Yorkshire and Durham: May 20, 1822; 1. 2144. — On the Association of Trap Rocks with Mountain Limestone in Tees-Dale: May 12, 1823; March 1, 15, 1824; 1. 139—195. —— Note on a memoir by Dr Brodie on Land and Freshwater Shells, &c.: Apr. 30, 1838; vt. 139—140. SuitH, ARcHIBALD, On the Equation to Fresnel’s Waye- Surface: May 18, 1835; vi. 85—89. Spinspury, F. G., On the Magnetism evolved by a single Galvanic combination, &c., &c. : Noy. 25, 1822; I. 77—83. SrerHens, J. F., Description of Chiasognathus Grantii, a Lucanideous Insect: May 16, 1881; rv. 209 —216. Strokes, Prof. G. G., On the Steady Motion of Incom- pressible Fluids: Apr. 25, 1842; vir. 489—453. —— Memoir on some cases of Fluid Motion: May 29, 1843; vu. 105—137. —— Supplement to this memoir: Noy. 3, 1846; vm. 409—414. — On the Internal Friction of Fluids in Motion: and the Equilibrium and Motion of Elastic Solids: Apr. 14, 1845; vit. 287—319. —— On the Theory of Oscillatory Waves: March 1, 1847; vu. 441—455. — On the Critical Value of the Sums of Periodic Series: Dec. 6, 1847; vir. 583—583. —— On the central Spot of Newton's Rings: Dee. 11, 1848; vu. 642—658. —— On the Variation of Gravity at the Earth’s Sur- face; Apr. 23, 1849; vit. 672—695. I. NAMES OF AUTHORS. Stokes, Prof. G. G., On an Equation relating to the breaking of Railway Bridges: May 21, 1849; vut. 707—735. — On the Dynamical Theory of Diffraction: Nov. 26, 1849; rx. 1—62, — On the numerical Calculation of a class of Definite Integrals and Infinite Series: March 11, 1850; Ix. 166—187. — On the effect of the Internal Friction of Fluids on the motion of Pendulums: Dec. 9, 1850; Ix. pt. ii. 8—106. — On the Colours of Thick Plates: May 19, 1851; 1x, pt. ii. 147—176. — On the Composition and Resolution of Streams of Polarized Light from different sources : Feb. 16, March 15, 1852; 1x. 399—416. — On the Discontinuity of Arbitrary Constants in Divergent Developments: May 11, 1857; x. 105—128. —— Supplement to the same memoir: May 25, 1868; xI. 412425. THompson, Prof. W. H., On the genuineness of the Sophista of Plato, and on some of its philoso- phic bearings: Nov. 23, 1857; x. 146—165. TopHunterR, I, On the Method of Least Squares: May 29, 1865; x1, 219—238. — On the Are of the Meridian measured in Lapland: May 1, 1871; x11. 1—26. —— On the equation determining the form of the strata in Legendre’s and Laplace’s Theory of the Figure of the Earth: Oct. 16, 1871; x1. 301—318. Tozmr, J., Mathematical Investigation of the effect of Machinery on the Wealth of a Community: May 14, 1838; v1. 507—522. —— On the effect of the Non-residence of Landlords, on the same: March 16, 1840; vi. 189— 196. —— J., On the Force of Testimony in Legal Evidence : Nov. 27, 1843; vin. 143—158. WatLAck, WILLIAM, Geometrical Theorems and For- mule, as applicable to Geodesy: Nov. 30, 1835 ; vi. 107—140. Warsurton, H., On the Partition of Numbers; Com- binations and Permutations: March 1, 1847; vill. 471—492. — On self-repeating series: May 15, 1854; 1x. 471 —486. Warren, Rev. J. W., Exercises in Curvilinear and Normal Co-ordinates: May 22, 1876; May 7, 1877; xl. 455—522; 531—545. Wepewoop, H., On the Knowledge of Body and Space : March 11, 1850; 1x. 157—165. WueweEt., Dr, On the Apsides of Orbits of great excentricity : Apr. 17, 1820; 1. 179—191. vil WueEwELL, Dr., On the double Crystals of Fluor Spar: Noy. 26, 1821; 1. 331—342. —— On the Rotatory motion of Bodies: May 6, 1822; 1. 11—20. — On the Angle made by two Planes, or two straight lines, referred to three oblique Co-ordinates : Nov. 24, 1823; 1. 197—202. Note on Mr Cxctt’s memoir on Grinding Mirrors, &c.: Dec. 11, 1822: 11. 100—103. — On Crystalline Combinations: Noy. 13, 1826; 1. 391—425. —— On a Notation to designate the Planes of Crys- tals: Feb. 11, 1826; 11. 427—439. — A Mathematical Exposition of some doctrines of Political Economy: March 2, 14, 1829; m1. 191—230. —— Second memoir on the same subject: 1x. Apr. 15, 1850; 123—149. —— Third memoir on the same subject : Noy. 11, 1850; Ix. pt. i. 1—7. — Ditto, Ditto, as applied to Ricardo’s Political Economy: Apr. 18, May 2, 1831; Iv. 155— 198. — On the Nature of the Truth of the Laws of Motion: Feb. 17, 1834; v. 149—172. —— On the results of Observations with a new Anemometer: May 1, 1887; vi. 301—315. — Demonstration that all Matter is heavy: Feb. 22, 1841; vi. 197—207. — Discussion whether Cause or Effect are simul- taneous or successive: March 14, 1842; vu. 319—331. — On the Fundamental Antithesis of Philosophy; Feb. 5, 1844; vu. 170—181. — Second memoir on the same subject: Nov. 13, 1848; vu. 614—620. — On the Intrinsic Equation to a Curye, &c., Xe. : Feb. 12, 1849; vir. 659—671. —— Second memoir on the same subject: Apr. 15, 1850; 1x. 150—156. — On Hegel’s Criticism of Newton’s Principia: May 21, 1849; vin. 696—7U6. | ___ On Aristotle’s account of Induction: Feb. 11, 1850; x. 63—72. —— On the Transformation of Hypotheses in the History of Science: May 19, 1851; 1x. pt. 11. 139—146. —— On Plato’s Survey of the Sciences: Apr. 23, 1855 ; Ix. 582—589. —— On his notion of Dialectic: May 7, 1855; 1x. 590—597. —— On the Intellectual Powers, according to Plato: Noy. 12, 1855; 1x. 598—604. —— Of the Platonic Theory of Ideas: Nov. 10, 1856; x. 94—104. vill Wiis, Prof., On the pressure produced by a stream of Air on a flat plate, &c. &.: Apr. 21, 1828; mr. 129—140. — On Vowel Sounds; and on Reed Organ-Pipes: Noy. 24, 1828; March 16, 1829; m1. 231268. —— On the Mechanism of the Larynx: May 18, 1829; Iv, 323—352. INDEX TO TRANSACTIONS I—XII. Youn, J. R., On the Principle of Continuity, in refer- ence to Analysis: Dec. 7, 1846; vim. 429— 440. Younec, M., Piscium Maderensium Species, Iconibus illustrate: Nov. 10, 1834; vr. 195. Ii. INDEX OF SUBJECTS. Aberration in the Eye-pieces of Telescopes : May 14; 21, 1827; m1. 1—58. Achromatic Eye-pieces, and Achromatism: May 17, 1824; 1. 227—252 ; I. 59—63. —— Object-Glass, New Correction for: Apr. 30, 1838; VI. 553—s64. Addenbrooke’s Hospital, Report on, for 1836: March 13, 1837 ; vI. 361—377. Ditto, Ditto, 1838; VI. 565—575. Al Farsi, Astronomical Tables of: Nov. 13, 1820; 1. 249—265. Algebra, Foundations of, Part I.: Dec. 9, 1839; vu. 173—187. Ditto, 287—300. Ditto, 139—142. Ditto, 241— 254. Algebraic Equations, Geometrical representation of their Roots: Apr. 27, 1846; vm. 342—360. Notation: Noy. 12, 1827; m1. 65—104. Algebraical Equations, Resolution of : March 7, 1831; Iv. 125—153. Analysis, on a Difficulty in, noticed by Sir W. Hamil- ton: May 1, 1837; v1 317—822. Analyzer, on a new: March 5, 1832; rv. 313—322. Anemometer, Observations with a new: May 1, 1837; vi. 301—315. Angle, Memoir on, as referred to three oblique Co- ordinates: Nov. 24, 1823; um. 197—202. Anglesea, Geological description of: Novy. 26, 1821; I. 359—452. Animals, Occurrence of Extinct, near Cambridge: Apr. 30, 1838; vir. 138—140. Annuities, Calculation of, with Tables: May 26, 1828; mm. 141—154. Comparison of various Tables of ; March 30, 1829; lt. 321—341. Antithesis, Fundamental, of Philosophy: Feb. 5, 1844; vi. 170—181. for 1837: Apr. 30, Ditto, II.: Nov. 29, 1841; vit. Ditto, III.: Nov. 27, 1843; vim. Ditto, IV.: Oct. 28, 1844; vin. Antithesis, Second memoir: Nov. 13, 1848; vu. 614 —620. Apophyllite, Refraction in Rays from: May 7, 1821; I. 241—247. Apsides of Orbits of Great Excentricity : April 17, 1820; I. 179—191. Arbitrary Constants, Discontinuity of, &e.: May 11, 1857; x. 105—128. —— Ditto, Ditto, Supplement to this memoir: May 25, 1868; x1. 412—425. Areh, Equilibrium of the: Dee. 9, 1833; v. 293—313. Argument, Use and meaning of the word: Nov. 28, 1859; x. 317—326. Aristotle’s account of Induction: Feb. 11, 1850; rx. 63—72. Atmospheric temperature, Decrement of : Feb. 13, 1837 ; vi. 443—455. Attraction, Residuo-Capillary: March 7, 1834; v. 205 —229. Attractions of Ellipsoids, Determination of: May 6, 1833; Vv. 395—429. Aurora Borealis, Height of : Dec. 8, 1845; vu. 320— 325. — of Noy. 17, 1848: Noy. 27, 1848; vir. 621—632, Axes, Moveable, On Velocities, &c., &c., relative to: Feb, 25, 1856; x. 1—20. Barometer, Extraordinary Depression of: Feb. 25, 1822; I. 453—458. Beams and Elastic Rods, Theory of: Apr. 23, 1860; x, $5937: Beaver, Fossil Remains of: March 6, 1820; 1. 175—177. Bernouillis Numbers, Tables of, &c., &c.: May 29, 1871; x1. 384—391. Biliary Secretion, Nature of: March 6, 1843; vu. 44 —49. Bivalve, see Univalve. Bode’s Law, Extension of, to Satellites : Dec. 8, 1828; 1m. 171—183. Boltzmann, see Material points. Boracic Acid, on the Crystals of; Nov. 30, 1829; 11. 865—367, Il. INDEX OF SUBJECTS. ix Brachiopod, see Univalve. Brazilian Topaz, Colour, Structure and Optical Pro- perties of: May 6, 1822; 11. 1—9. Calculus of Functions, Notation employed in: May 1, 1820; 1. 63—76. — Human, specimen of : Noy. 26, 1821; 1. 347—349. Cambridgeshire, On the Ornithology of : Noy. 28, 1825; IL. 287—324. — List of Reptiles found in: Feb. 22, 1880; ut. 373—381. Cassius, Constituents of Purple Precipitate of : May 15, 1820; 1. 53—61. Cause and Effect, simultaneous or successive : March 14, 1842; vir. 319—331. Caustic, Intensity of Light near: May 2, 1836, March 26, 1838; v1. 379—402. Ditto, Supplement to this memoir: May 8, 1848; vit. 595—599. Chances, Some points in the Theory of : May 26, 1828; ti. 141—154. Chiasognathus Grantii, Description of : May 16, 1831; Iv. 209—217. Clock Escapements: Nov. 27, 1848; virt. 6833—638. —— Improvements in: Feb. 7, 1853; 1x. 417—430, Turret Remontoirs: Feb. 26, 1849; vii1. 639—641. Colours of Thick Plates: May 19, 1851; rx. [147— 176.] Combinations and Permutations: March 1, 1847; vit. 471—492, Comet, Observations of Halley’s: Dec. 11, 1887; v1. 493—506, Composition of Forces, General Principles of : March 14, 1859; x. 290—304. Conic Section, Focus of : March 2, 1829; 11, 185—190. Consonances, Beats of Imperfect: Noy. 9, 1857; x. 129—145. Continuity, Principle of, with reference to Analysis : Dec. 7, 1846; vu. 429—440. Co-ordinates, Six of a Line: Nov. 11, 1867; x1. 290— 323. —— Curvilinear and Normal: May 22, 1876; x11. 455 —522. — Ditto, Ditto, May 7, 1877; x11. 531 —d45, Copper, Analysis of Phosphate of; March 5, 1821; 1, 203—207. Cornwall, Fossil Shells: Feb. 16, 1838; v1. 415—422. Lizard District of: Apr. 2, May 7, 1821; 1. 291— 330. Primitiye Ridge of : March 20, 1820; 1. 89—146. Crotalocrinus rugosus: Feb. 8, 1869; x1. 481—484. Crystalline Combinations, on their Classification, Nov. 13, 1826; 1. 391—425, Crystallization of Water: March 5, 1821; 1. 209—215. Vou. XII. Crystallized Media, Propagation of Light in: May 20, 1839; vir. 121—140. Ditto, Transmission of Light in: Feb. 13, 1837; vi. 323—352, Ditto, Supplement: May 1, 1887; v1. 353 — 360. Crystals, Axes of Optical Elasticity in certain: Dee. 8, 1834; v. 431—438. Position of Axes of Optical Elasticity in: March 21, 1836; vil. 209—215. —— Planes of, on a Notation to designate: Feb. 11, 1826; 11. 427—439. —— as affecting Planes of Polarization; Apr. 17, 1820; I, 43—52. — found in Slags: May 22, 1830; m1. 417—420. Variation in Tints developed by: May 1, 1820; I. 21—41. Cubic Cones and Curves: Apr. 18, 1864; x1. 129—144. Cubic Curves, Involution of: Feb. 22, 1864; x1. 39 —80, — Classification of: Apr. 18, 1864; x1. 81—128. — Surface, with 27 lines, by Dr Wiener: May 15, 1871; x11. 366—383. Curve, Intrinsic Equation to: Feb. 12, 1849; vir. 659 —671. Second memoir: Apr. 15, 1850; rx. 150—156. — Rolling: Dec. 10, 1838; vir. 61—86. —— of the Second Degree, General Equation to: Nov. 15, 1880; tv. 71—78. Singular points of; May 21, 1855; rx. 608—627. Cyclas and Pisidium, on the British Species of : Noy. 28, 1831; Iv. 289—312. Cylindrical Tubes, Aerial Vibrations in; May 20, 1833; v. 231—270. Decrement of Atmospheric Temperature : Feb. 13, 1837 ; vi. 443—455. Definite Integrals, Inverse method of, with applications ; March 5, 1832; Iv. 353—408. — Ditto, Ditto, Ditto, Noy. 11, 1833; v. 113—148. — Ditto, Ditto, Ditto, March 2, 1835; v. 315—393. — Numerical calculation of: March 11, 1850; Ix, 166—187. Properties of: May 24, 1830; mt. 429—443. Determinants, Theory of: Feb. 20, 1843; vir. 75—88. Devonshire, Primitive Ridge of: Mar. 20, 1820; 1. 89—146. Differential Equations, Theory of: March 27, 1854; rx. 515—554, Supplement to this paper: Apr. 28, 1856; x. 21—26. Diffraction, Dynamical Theory of; Noy. 26, 1849; 1x. 1— 62. 74 D4 INDEX TO TRANSACTIONS I—XII. Diffraction, Newton’s Experiments on: May 7, 1833; v. 101—111. of an Object-glass with Circular aperture: Nov. 24, 1834; v. 283—291. Ditto, Ditto, 1836; vi. 481—442. Digitalis, Hybrid: Noy. 14, 1831; Iv. 257—278. Discontinuous Constants, Use of: May 16, 1836; VI. 185—193. Dispersion of Light, Hypothesis for: Feb. 22, 1836; vi. 153—184. Divergent Developments, see Arbitrary Constants. — Series: March 4, 1844; vu. 182—203. Double-Sixer, Construction of: May 15, 1871; xm. 366—383. Durham, see Yorkshire. Triangular Ditto: Dec. 12, Earth and Planets, External Temperature of: March 21, 1855; 1x. 628—672. Earth, Theory of the Figure of; Oct. 16, 1871; xm. 301 —318. —— Inequalities in the Surface of: Dec. 1, 1873; xu. 414—433. Ebenacew, Monograph of: March 11, 1872; x1. 27— 300. Elastic Beams, Deflection of, &c.; March 11, 1850; Ix. [177—190.] —— Impact of: Dec. 10, 1849; Ix. 73—78. —— Fluids, Vibratory Motions of: March 30, 1829; II. 269—320. —— Medium, Effect of Vibrations on a Sphere: April 26, 1841; vil. 333—353. —— Medium, Vibratory Motion of: March 15, 1847; vil. 508—523. — Rods, see Beams. —— Solids, Motion of: Apr. 14, 1845; vu. 287— 319. Electric Fluid, Equilibrium of Fluids analogous to: Nov. 12, 1832; v. 1—63. Electricity, Origin of: Dec. 7, 1863; x1. 6—20. Electro-Magnetism, Development of by Heat: Apr. 28, 1823; 1. 47—76. Elevation of Mountains by Lateral Pressure: Apr. 27, 1868; x1. 489—506. Elimination between Unknown Quantities: Noy. 26, 1832; v. 65—75. Ellipsoid, Centro-Surface of : March 7, 1870; x11. 319 —365, Ellipsoids, Exterior and Interior attractions of: May 6, 1833; v. 395—429. Endosmose and Exosmose, Explanation of: March 17, 1834; v. 205—229. Equality, Sign of: May 16, 1864; x1. 145—189, Equation, Algebraic, Proof of a root in every: Dec. 7, 1857; x. 261—270. Equation, Algebraic, Another proof: Dec. 6, 1858; x. 283—289. Ditto, Supplement to this memoir: Dee. 12,1859; x. 327—330. —— relating to the breaking of bridges: May 21, 1849 ; vil. 707—735. —— to a Curve, The Intrinsic: Feb. 12, 1849; vit. 659—671. Ditto, Second memoir: rx. 150—160. —— General, to Surfaces of the second degree : Noy. 12, 1832; v. 77—94. —— Integration of Partial differential: June 5, 1848; vu. 606—613. — Machine for resolving by Inspection : May 7, 1832 ; Iv. 425—440. Equilibrium of the Arch: Dec. 9, 1833; v. 293—313. —— of Bodies in Contact: May 15, 1837; v1. 463—491. —— Molecular: March 26, 1838; vir. 25—59. Erratic Blocks, Transport of: Apr. 29, 1844; VII. 220—240., Errors of Observation, Theory of: Noy. 11, 1861; x. 409—427. Escapements, Theory of: Noy. 26, 1826; m1. 105—128. Exponents, Newton’s method of: May 21, 1855; rx. 608—627. ‘ Extinct Lacertian Reptile, Traces of: April 11, 1842; VII. 355—369. Eye, Change in the State of: May 25, 1846; vim. 361 — 362. — Defect in, and how cured: Feb. 21, 1825; 11. 267—271. -- —— Further observations on: Feb. 12, 1872; xm. 392, 3. Fauna and Flora of Madeira: Nov. 15, 1830; rv. 1—70. Figure assumed by a Fluid Homogeneous Mass: March 15, 1824; rr. 2083—216. Finite Differences, Resolution of Equations in: Noy. 15, 1835; vi. 91—106. Flora of Madeira, Notes and Gleanings: May 28, 1838; Vi. 523—5dl. Fluid Motion, on: March 21, 1836; v1. 2083—233. — Ditto, On some cases of: May 29, 1843; vim. 105—137. — Ditto, Supplement to this memoir: Nov. 3, 1846; vir. 409—414., Fluids, Equilibrium of Certain: Noy. 12, 18382; v. 1—63. —— General Equations of the motion of, &e., &e.: Feb. 22, 1830; m1. 383—416. — Motion of Incompressible: April 25, 1842; vit. 439—453. —— Theory of the motion of: v. 173—203. —— in motion, Internal Friction of: Apr. 14, 1845; vill. 287—319. II. INDEX OF SUBJECTS. Fluids, Effect on Pendulums of Internal Friction of: Dee. 9, 1850; 1x. [8—106.] Fluor Spar, Double Crystals of: Nov. 26, 1821; 1. 331 —342. Focus of a Conic Section: March 2, 1829; mm. 185—190, Force, Faraday’s Lines of: Dec. 10, 1855, Feb. 11, 1856; x. 27—83, Forces, Principles of the Composition of: March 14, 1859; x. 290—304. Fossil remains of Beaver, found in Cambridgeshire: | March 6, 1820; 1 175—177. —— Shells, New Genus of: Feb. 26, 1888; vr. 415— 422, Fresnel’s Wave Surface, Equation to: May 18, 1835; VI. 85—89. Functional Equations, Reduction of, to Equations of Finite Differences: March 6, 1820; 1. 77—87. Galvanism, as connected with Magnetism: Apr. 2, 1821; I. 269—279. Gas, Hydrogen, as a moving Power in Machinery: Noy. 27, 1820; 1. 217—239. — on Sounds excited in: Apr. 2, 1821; 1. 267, 268, Gastric Fluids, Solvent effect of, on the Stomach after Death: Dec. 11, 1820; 1. 287—290. Geodesy, Geometrical Formule applicable to: Nov. 30, 1835; vi. 107—140. Geology, Researches in Physical: May 4, 1835; v1. 1—84. Geometry and Mechanics, Symbolical: March 15, 1847 ; vil. 497—507. — Substitution of, for the doctrine of Proportions: Dec. 7, 1857; x. 166—172. Gipping, Geology of the Valley of: Feb. 27, 1854; rx. | 431—444. Glaciers, Motion of: May 1, 1843; vir. 50—74. —— Ditto, Dec. 11, 1843; vir. 159—169. Feb, 18, 1837; vr. 289—300. Going-Fusee, New Construction of: March 2, 1840; vu. 217—225. Gravity, Variation of, at the Earth’s Surface: Apr. 23, 1849; vim. 672—695. Great Circle Sailing: May 10, 1858; x, 271—282. Greek Literature, First Ages of written : Noy. 23, 1868 ; x1, 461—480. Halley’s Comet, Observations of: Dee. 11, 1837; vi. | 493—506. Heat, see Motion of Particles. Hegel’s criticism of Newton: May 21, 1849; vim. 696 —706. Homeric Tumuli: March 12, 1866; x1. 267—276, Human Monstrosity, Case of, with Commentary: May 16, 1831; Iv. 219—255. xl Hybrid Digitalis, Examination of: Nov. 14, 1831; rv. 257—278. Hydrodynamical Theorem, Investigation of: May 9, 1842; vit. 455—464. Hydrodynamics, New Fundamental Equation March 6, 1843; vir. 31—43. Hyperbolic Law of Elasticity: March 11, 1850; rx. [177—190.] Hypotheses, Transformation of: May 19, 1851; Ix. [139—146.] aniee Ideas, Platonic Theory of: Noy. 10, 1856; x. 94—104. Iliad and Odyssey, Late date, &c., &., of: Nov. 26, 1866; x1. 360—386. Induction, Aristotle’s account of: Feb. 11, 1850; 1x. 638—72. Infinite Angle, Sine and Cosine of: Dec. 9, 1844; vit. 255—268. Series, Use of Discontinuous Constants in, &c., &e.: May 16, 1836; vr. 185—193. —— Ditto, General Term for a new Class of: May 3, 1824; 0. 217—225. Infinity, On: May 16, 1864; x1. 145—189. Inscription, Metrical Latin, from Algeria: Feb. 13, 1860; x. 374—408. Integral Calculus, On some points of: Feb. 24, 1851; 1x, [107—138]. Integrals, General Properties of Definite: May 24, 1830; In. 429—443, —— Inverse method of, with Applications: March 5, 1832; Iv. 353—408. Internal friction of fluids: Apr. 14, 1845; vim. 287— 319. Involution, Theory of: Feb. 22, 1864; x1. 21—38. Involution of Cubic Curves: Feb. 22, 1864; x1. 39—80. | Jaws, Growth of: Nov. 9, 1863; x1. 1—5. Globe, Relative Quantities of Land and Water on: | Knowledge of Body and Space: March 11, 1850; 1x. 157— 165. Laminated Pressure of Rock Masses; May 3, 1847; vu. 456—470, Land, see Globe, Laplace, on his Theory of the Attraction of Spheroids: May 8, 1826; 1. 379—390. Lapland, Are of the Meridian measured in: May 1, 1871; x11. 1—26. Larynx, On the Mechanism of: May 18, 1829; Iv. 323—352. | Latitude of Cambridge Observatory: Apr. 14, 1834; y. 271—281. Least Squares, Methed of: March 4, 1844; vir. 204— 219. —— Ditto, _ Ditto: May 29, 1865; x1. 219938. (4—2 xil Light, Nature of, from the Double Refraction of Quartz: Feb. 21, 1831; 1v. 79—123. — Nature of, from the Double Refraction of Quartz: Apr. 18, 1831; Iv. 199—208. —— on the Dispersion of: Feb. 22, 1836; v1. 153—184, —— Transmission of, in certain Media: Feb. 13, 1837 ; VI. 823—352. Ditto, 353—360. —— Intensity of, near a Caustic: May 2, 1836, March 26, 1838; v1. 379—402. Ditto, Supplement to this memoir: May 8, 1848; vill. 595—599. —— Reflection and Refraction of, &c.: Dec. 11, 1837; vil. 1—24, Ditto, | Supplement to this memoir: May 6, 1839; vir. 118—120. —— Propagation of, in Crystallized Media: May 20, 1839; vu. 121—140. —— Quantity of, &c., absorbed by a Grating placed beforea Lens: March 30, 1840; vir. 153—171. — Reflection and Refraction of: Noy. 28, 1842; vin. 7—26. —— Absorption of, &c.: Feb. 14, 1843; vir. 27—30. —— Transmission through Transparent media: May 17, 1847; vil. 524—532. Polarized: Dec. 8, 1851; 1x. 379—398. Lines of Force, Faraday’s: Dec. 10, 1855, Feb. 11, 1856; x. 27—83. y Liquid Substratum of the Earth, Theory of ; Feb. 22, 1875; xm. 434—454. Lizard district of Cornwall, Physical Structure of: Apr. 2, May 7, 1821; 1, 291—330. Logic, in general: Feb. 8, 1858; x. 173—230. of Relations: Apr. 23, 1860; x. 331—358. Symbols of, &., &c.: Feb. 25, 1850; 1x. 79—127. Longitude of Cambridge Observatory: Nov. 24, 1828; mn. 155—170. Supplement: May 1, 1837; v1. — Ditto, Ditto, May 15, 1854; Ix. 487—514. Luminiferous Ether, Constitution of: March 18, 1839 ; vit. 97—112. Luminous Rays, Theory of: March 11, 1846; vut. 363—378. — Vibrations, Theory of: March 6, 1848; vii. 584 —594. ——- Waves, Propagation of: April 25, 1842; vu. 397 —437. Machine for resolving Equations: May 7, 1832; Iv. 425—440, Machinery, Influence of, on the Wealth of a Com- munity; May 14, 1838; v1. 507—522. Madeira, Fauna and Flora of: Nov. 15, 1830; rv. 1—70. —— Fishes of: Nov, 10, 1884; vr. 195—201, INDEX TO TRANSACTIONS I—XiII. Madeira, Flora of, Notes and Gleanings: May 28, 1838 ; VI. 523—551. Magnetic Intensity, observed Variations of: 1825; 1. 445. —— Needles, as affected by Masses of Iron: May 15, 1820; 1. 147—173. Magnetism, Connection of, with Galvanism: Apr. 2, 1821; 1. 269—279. —— as a Measure of Electricity: May 21, 1821; 1. 281—286. —— evolved by a single Galvanic Combination, Ex- tract from Memoir on: Noy. 25, 1822; 1. 77—83. Magnitude and Direction, Pure Science of: May 12, 1845; vir. 278—286. Material points, Energy in a system of : May 6, 1878; xi. 547—570. Mathematical Reasoning, Influence of Signs on: Dec. 16, 1821; 11. 325—877. Matter, Demonstration that it is heavy: Feb. 22, 1841 ; vit. 197—207. —— Remarks on the Theory of: May 22, 1848; vit. 600—605. Mechanics and Geometry, Connection between: Feb. 10, 1845; vill. 269—277. Microscope, Improvement of: March 22, 1830; II. 421—428, Mirrors and Object-Lenses, Apparatus for Grinding : Dec. 11, 1822; 11. 85—103. — Use of Silvered Glass for: Noy. 25, 1822; 11. 105—118. Molecular Equilibrium: March 26, 1838; vi. 25—59. Monstrosity, Human, Case of: May 16, 1831; Iv. 219 —255. of the Common Mignionette: May 21, 1832; v. 95—100, Motion of Fluids, on the: Noy. 24, 1828; m1. 383—416. Ditto, Ditto, March 3, 1834; v. 173—203. —— of Fluids, Differential Equations to: April 11, 1842; vil. 371—396. —— Incompressible: April 25, 1842; vit, 439—453. — Truth of the Laws of: Feb. 17, 1834; v. 149— 172. —— of Particles, as affecting Sound and Heat: May 16, 1836; vi. 235—288. — of Waves in a small Canal: May 15, 1837; v1. 457—462. — of Waves in Canals: Feb. 18, 1839; vir. 87—95. Motive Power, Hydrogen Gas as a: Nov. 27, 1820; I, 217—239. Mountains, Elevation of, by Lateral Pressure; Apr. 27, 1868; x1. 489—506, Second memoir: Feb, 22, 1875; x11. 484—454. Music in Education, place of, according to Aristotle : May 17, 1875; x11. 523—530. II. INDEX OF SUBJECTS. Natron, remarkable deposit of: Nov. 27, 1820; 1. 193 —201. Natterjack, Habits and Character of: Feb. 22, 1830; 1. 373—381. Neptune, see Uranus. Neutral Series, Theory relating to: May 16, 1864; x1. 190—202. —— Ditto, Note on this paper: Oct. 26, 1868; x1. 447—460. Newton's method of Co-ordinated Exponents: May 21, 1855; 1x. 608—627. — Experiments on Diffraction: May 7, 1833; v. 101—111. —— Principia, Criticism of; May 21, 1349; vir. 696 —706. — Rings, Remarkable change in: Nov. 14, 1831; Iv. 279—288. On some Phenomena of: March 19, 1832; Iv. 409—424. Central spot of: Dec. 11, 1848; vu. 642 — 658. —— See Hegel. Non-Residence of Landlords, Influence of: March 16, 1840; vu. 189—196. Notation employed in the Calculus of Functions: May 1, 1820; 1. 63—76. —— Algebraic: Noy. 12, 1827; m1. 65—103. —— to designate the Planes of Crystals: Feb. 11, 1826; 0. 427—439. — a New, in Geometry, &c., &.: Nov. 23, 1846; vir. 415—428. Numbers, Partition of: March 1, 1847; vi. 471—492. Object-Glass with circular aperture, Diffraction in: Nov. 24, 1834; v. 283—291. —— with triangular aperture, Diffraction in: Dec. 12, 1836; vi. 431—442. — Achromatic, Correction for: Apr. 30, 1838; VI. 5538—564. Observatory at Cambridge, Longitude of: Noy. 24, 1828; m1. 155—170. ar Ditto, Latitude of: Apr. 14, 1834; y. 271—281. ~— Ditto, Longitude of, by Galvanic Signals: May 15, 1854; 1x. 487—514. Odyssey, see Iliad. Onymatic System, on various points of: May 4, 1863; x. 428—487. Optical Elasticity, Axes of in certain Crystals: Dee. 8, 1834; v. 431—438. — Ditto, Ditto, (second memoir): March 21, 1836; vir. 209—215. Orbits of great Excentricity, Position of their Apsides: | Apr. 17, 1820; 1. 179—191. Ordnance, Strains upon: Apr. 18, 1864; x1. 324—359. xill Ornithology of Cambridgeshire: Noy. 28, 1825; 11. 287—324. Oscillations, on small Finite: May 15, 1843; vir. 89 —104. ; —— of a suspension Chain: Dec. 8, 1851; 1x. 379— 398. Oscillatory Waves, Theory of: March 1, 1847; vu. 441—455. Parallelogram of Forces: New Demonstration of: Apr. 14, 1823; 1. 45—46. Partial differential Equations, Method of integrating : June 5, 1848; vi. 606—613. Pendulum, Correction of, by a Ball suspended by a wire: Noy. 16, 1829; m1. 355—360. Pendulums, Disturbances of: Noy. 26, 1826; m1. 105— 128. — Effect of Internal Friction on: Dec. 9, 1850; 1x. [S—106.] Percussion, Experiments on; 1825; m1, 444. Periodic Series, Critical Values of: Dec. 6, 1847; vu. 533—S83. Perpetual Motion, How possible: Dec. 14, 1829; m1. 369—372. Perspective, Isometrical: Feb. 21, Mar. 6, 1820; 1 1—19. Philosophy, Fundamental Antithesis of: Feb. 5, 1844; vin. 170—181. — Second memoir: Noy. 13, 1848; vu. 614—620. Phosphate of Copper from the Rhine: March 5, 1821; I. 203—207. Physical Geology, Researches in: May 4, 1835; v1. 1—84. Piscidium, see Cyclas. Piscium Maderensium Species, &., &c.: Nov. 10, 1834; vi. 195—201. Planets, see Earth. Plato’s Survey of the Sciences: Apr. 23, 1855; Ix. 582—589. Notion of Dialectic: May 7, 1855; 1x. 590—597. —— Ditto, of the Intellectual Powers: Noy. 12, 1855; Ix. 598—604. —— Genuineness of the Sophista of: Noy. 23, 1857; x. 146—165. Cosmical system: Feb. 28, 1859; x. 305—316. Platonic Theory of Ideas: Noy. 10, 1856; x. 94—104. Plumbago, on the artificial formation of: Feb. 21, 1825; mi. 441—443. Polarity, Organic: March 8, 1858; x. 248—260. Polarization, Use of a new Analyzer in: March 5, 1832; Ivy. 313—322. Polarized Light, Certain effects in Crystals exposed to: May 1, 1820; 1. 21—41. Ditto, as affected by Rotation: Apr. 17, 1820; 1. 43—52. Xiv Polarized Light, Composition and Resolution of: Feb. 16, March 15, 1852; 1x. 399416. Political Economy, Mathematical discussion of: March 2, 14, 1829; um. 191—230. Ditto, as expounded by Ricardo. First memoir: Apr. 18, May 2, 1831; Iv. 155— 198. Ditto, Mathematical Theory of, Second memoir: Apr. 15, 1850; rx. 128—149. Ditto, Ditto, Third memoir: Nov. 1, 1850; rx. [1—7.] Potassium, Apparatus for procuring: Noy. 26, 1821; I. 343—345. Pressure on a flat Plate opposed to a Stream of Air: Apr. 21, 1828; m1. 129—140. Primitive Ridge of Devonshire and Cornwall ; March 20, 1820; 1. 89—146. Probabilities, Question in the Theory of : Feb. 26, 1837 ; vi. 423—430. Foundation of Ditto: Feb. 14, 1842; vu. 1—6. —— Fundamental principle of the Theory of: Nov. 13, 1854; 1x. 605—607. Proportions, see Geometry. Propositions numerically definite: March 16, 1868; x1. 396—411. Purbeck Strata of Dorsetshire: Noy. 18, 1854; rx. 555—581. Quartz, Nature of the Light produced by: Feb. 21, 1831; 1v. 79—123. Ditto, Ditto, 1831; rv. 199—208. Ditto, Apr. 18, Railway Accidents, Causes of Fatal, &c.: Noy. 29, 1841; vir: 301—317. Railway Bridges, Equation relating to their breaking : May 21, 1849; vi. 707—735. Rainbow, Problem of, Mathematically considered: Dec. 14, 1835; vi. 141—152. Rainbows, Spurious: March 22, 1841; vi. 277—286. Reed Organ Pipes: Noy. 24, 1828, March 16, 1829; im. 231—262. Reflection and Refraction of Light: Dec. 11, 1837; vit. 1—24. —— Supplement to this memoir: May 6, 1839; vu. 118—140. —_—s— Ditto, ~_ Ditto, &e.: Noy. 28, 1842; vil. 7—26. Refraction, Theory of Double: May 17, 1847; vu. 524—532, ticardo, see Political Economy. tock Masses, Internal pressure of; May 3, 1847; vin. 456—470. Rocks, Weathering of: March 2, 1868; x1, 387—395. Root of any Function: May 7, 1866; x1. 239—266. INDEX OF TRANSACTIONS I—XIL Root-limitation, Cauchy’s Theorems of: Feb. 16, 1874; xi. 395—414. Rotatory Motion of Bodies: May 6, 1822; 1, 11—20. Secular Cooling of the Earth: Dec. 1, 1878; x11. 414 —433. Series, on Divergent: March 4, 1844; vim. 182—203. — Critical Value of Periodic: Dec. 6, 1847; vu. 533—583. — Numerical calculation of Infinite; March 11, 1850; 1x. 166—187. — Self-repeating: May 15, 1854; 1x. 471—486. — Theorem on Neutral: May 16, 1864; x1. 190—202. —— Ditto, Part II.:-May 7, 1866; x1. 239—266. —— Note on Ditto: Oct. 26, 1868; x1. 447—460. Sextic Torse, On a certain: Noy. 8, 1869; x1. 507— 523. : Shells, Occurrence of, in Gravel: Apr. 30, 1888; vil. 138—140, Signs, Influence of, in Mathematical Reasoning: Dec. 16, 1821; 1. 325—377. — + and -, Early History of: Noy. 28, 1864; x1. 203—212. — Note on this Memoir: Feb. 13, 1865; x1. 213—218, Skew Surfaces, or Scrolls: Nov. 11, 1867; x1. 277—289, Slags, Crystals found in: March 22, 1830; m1. 417—420. Solid Bodies, Vibrations of: Apr. 18, 1864; x1. 324359. Solitary Waves, Mathematical Theory of: Dec. 8, 1845 ; Vill. 326—341. Solon, Statue of: Feb. 22, 1858; x. 231—239., Sound, Experiments on the Velocity of: Dec. 8, 1823; mu. 119—137. see Motion of Particles. — Reflection and Refraction of: Dec. 11, 18387; vi. 403—413. Spar-Fluor, Double Crystals of: Noy. 26, 1821; 1. 331— 342. Spermaceti Whale, account of: May 16, 1825; 11. 253—266. Sphere, Motions of, acted on by Vibrations of an Elastic Medium: April 26, 1841; vit. 333—353. Spherical Aberration in Eye-pieces of Telescopes: May 14, 21, 1827; m1. 1—63. Spheroids differing little from a Sphere, on Laplace's Theory of: May 8, 1826; 11, 379—390. Squares, Method of Least: March 4, 1844; vir. 204 —219. Ditto, Ditto ; May 29, 1865; x1. 219—238. Suffolk, see Gipping. Surfaces of the second degree, Equation to: Noy. 12, 1832; v. 77—94. — Transformation of, by Bending; March 13, 1854 IX. 445—470, Suspension Chain, Oscillations of: Dec. 8, 1851; 1x. 379—398, II. INDEX OF SUBJECTS. Switzerland, Tertiary Formations of: May 20, 1839; vit. 141—152. Syllogism, Theory of the structure of: Nov. 9, 1846; vu. 879—408. — Ditto, Ditto, Pt. IL.: Feb. 25, 1850; Ix. 79—127. — Ditto, Ditto, Pt. III.: Feb. 8, 1858; x. 178—230. — Ditto, Ditto, Pt. IV.: Apr. 23, 1860; xX. 3381— 358. — Ditto, Ditto, Pt. V.: May 4, 1863; x. 428—487. Symbolical Geometry and Mechanics: March 15, 1847 ; vin. 497—507. Tertiary Formations of Switzerland: May 20, 1839; vir. 141—152. Testimony, Measure of Force of: Noy. 27, 1843; VIII. 143—158. Theory of Probabilities, Question in: Feb. 26, 1837; vi. 423—430. Topaz, see Brazilian Topaz. Transcendental Equations, Machine for resolving: May 7, 1832; Iv. 425—440. Trap Dykes in Yorkshire and Durham: May 20, 1822; mW. 21—44. —— Rocks, as associated with Mountain Limestone: May 12, 1823: March 1, 15, 1824; 11. 139— 195. Trinomial, Resolution of a certain: Nov. 9, 1868; xt. 426—443. —— Note on this memoir: Noy. 23, 1868; x1. 444, 445. Trireme, Structure of the Athenian: Nov. 6, 1856; x. 8493. Tumuli, Homeric: March 12, 1866; x1. 267—276. Undulations, Theory of, applied to Luminous Waves: May 25, 1846; vir. 371—378. Univalve, Relations of, to the Bivalve, and to the Brachiopod: Feb. 8, 1869; x1. 485—488. Uranus and Neptune, Long Inequality of: 1852; rx. Appendix. Ureters, Dilatation of: Nov. 12, 1821; 1. 351—358. XV Velocities, &c., referred to Moveable Axes: Feb. 25, 1856; x. 1—20. Velocity of Sound, Experiments on: Dec. 8, 1823; 11 119—137. Vibrations in Cylindrical Tubes: May 20, 1883; v. 231—270. —— Theory of Luminous: March 6, 1848; vu. 584— 594. — of Solid Bodies: Apr. 18, 1864; x1. 324—359. Vibratory Motion of Elastic Medium: March 15, 1847; vir. 508—523. Virtual Velocities, Demonstration of their principle: March 21, 1825; 11. 273—276. Vision, Peculiar defect in: Nov. 9, 1846, May 17, 1847; viir. 493—496. Voluntary Muscles, Abnormities in: March 8, 1858; x. 240—247. Vowel Sounds, On the: Nov. 24, 1828, March 16, 1829; In, 231—268. Water, Crystallization of: March 5, 1821; 1. 209—215. — see Globe. Wave Surface, Equation to Fresnel’s: May 18, 1835; VI. 85—89. Waves, Motion of, in a small Variable Canal: May 15, 1837; vi. 457—462. —— in Canals, Motion of: Feb. 18, 1839; vir. 87—95. —— Theory of the two great Solitary: Dec. 8, 1845; vul. 326—341. Wealth of a Community, Influence of Machinery on: May 14, 1838; vi. 507—522. Ditto, Effect of Non-Residence of Landlords on: March 16, 1840; vir. 189—196. Weathering of Rocks: March 2, 1868; x1. 387—395. Wheels, On the Forms of the Teeth of: May 2, 1825; 11. 277—286. Wiener, see Cubic Surface. Winter Solstice, Ancient Observation of: Nov. 30, 1829; I. 361—363. Written Greek Literature, First Age of: Nov. 23, 1868 ; x1. 461—480. Yorkshire and Durham, Trap Dykes in: May 20, 1822; 11. 21—44, CAMBRIDGE : PRINTED BY C. J. 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