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The definition of Harpiola (Vespertilionidae : Murininae) and the description of a new species from Taiwan.

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INTRODUCTION

Within the subfamily Murininae two genera are accepted generally, namely Murina and Harpiocephalus (Corbet and Hill, 1992; Koopman, 1993, 1994; Pav-linov, 2003; Simmons, 2005). The name Harpiola was introduced by Thomas (1915) as a separate genus with Murina grisea Peters, 1872 as its type species. Although Tate (1941) accepted this opinion and pro-vided a brief diagnosis of the genus (char-acterized by the attachment point of the wing and by dental features), Ellerman and Morrison-Scott (1951) listed this form as a subgenus of Murina, a taxonomic arrangement which was followed subse-quently.

The description of the genus Harpiola was based on a single specimen collected from Northwest India and both the wet specimen and its extracted skull are in a very bad condition. As Corbet and Hill (1992: 151) stated “the features upon which Thomas based Harpiola do not appear to be artificial, but to some extent the status of grisea must remain uncertain until further material is obtained.” Recently, alongside the description of the second known speci-men of H. grisea from Mizoram, India, Bhattacharyya (2002) re-evaluated Har-piola and raised once again the taxon to generic rank.

During a series of bat surveys carried out between 1998 and 2004 in mountain areas in Taiwan, 11 Harpiola bats were

PL ISSN 1508-1109 © Museum and Institute of Zoology PAS

The definition of Harpiola (Vespertilionidae: Murininae) and

the description of a new species from Taiwan

HAO-CHIKUO1, YIN-PINGFANG2, GÁBORCSORBA3, and LING-LINGLEE1, 4

1Graduate Institute of Ecology and Evolutionary Biology, National Taiwan University,

1, Sec 4. Roosevelt Road, Taipei, Taiwan R.O.C.

2Department of Biological Resources, National Chiayi University, 300 University Road, Chiayi,

Taiwan R.O.C.

3Department of Zoology, Hungarian Natural History Museum, H-1083 Budapest, Ludovika tér 2, Hungary 4Corresponding author: E-mail: leell@ntu.edu.tw

A new species of Harpiola from Taiwan is described based on 11 specimens collected between 1998 and 2004. Careful examination of these specimens and those of the genus Murina, revealed the valid characters distinguishing Harpiola from Murina, including the enlarged upper incisors, the well developed first premolars in both toothrows with their bulk subequal to canines and the second premolars in the corresponding toothrow, and the strongly bifid lower canine. The new species from Taiwan can be distinguished from Harpiola grisea in India by having different shape of second upper premolar and different structure of first upper molar.

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obtained that contributed to the richest col-lection of specimens of this genus in the world. In this paper, detailed comparisons of these specimens with those of H. grisea and Murina spp. were made to reveal their taxonomic status and to re-define the diag-nostic features of the genus Harpiola. Meanwhile, the special zoogeography of Harpiola as well as other mammalian gen-era was also considered.

MATERIALS ANDMETHODS

The following comparative material was used:

Harpiola grisea: India, BM(NH) 79.11.21.117

(holo-type); Murina aenea: Malaysia, BM(NH) 64.770 (holotype); M. aurata: China, MNHN CG1870-590a (paratype); M. cyclotis cyclotis: India, BM(NH) 9.4.4.4 (cotype); M. cyclotis peninsularis: Malaysia, BM(NH) 64.771 (holotype); M. florium: Indonesia, BM(NH) 63.12.26.14 (holotype); M. hilgendorfi: Ja-pan, HZM 1.2974; M. huttoni: India, BM(NH) 79.11.21.685 (holotype); M. leucogaster rubex: India, BM(NH) 16.3.25.111 (holotype); M. puta: Taiwan, ZMNTU 1998.7.3; M. rozendaali: Malaysia, BM(NH) 83.360 (holotype); M. silvatica: Japan, HNHM 2001.38.1; M. suilla: Java, HNHM 2000.13.2; M. tubinaris: Pakistan, HNHM 99.14.6;

M. ussuriensis: Russian Federation, ZMMU 96368

(paralectotype).

The museum acronyms mentioned in the text are as follows: BM(NH) — Natural History Museum, London, formerly British Museum (Natural History); ESRI — Endemic Species Research Institute, Nan-tou; HNHM — Hungarian Natural History Museum, Budapest; HZM — Harrison Institute, Sevenoaks, formerly Harrison Zoological Museum; MNHN — Museum National d’Histoire Naturelle, Paris; NMNS — National Museum of Natural Science, Taichung; ZMNTU — Zoological Museum of National Taiwan University, Taipei; THU — Tunghai University, Tai-chung; ZMMU — Zoological Museum of Moscow State University, Moscow.

The forearm (FA) measurements were taken from dry or alcohol preserved museum specimens with 0.1 mm accuracy. The following craniodental measure-ments were taken to the nearest 0.01 mm by the au-thors with digital calipers under a stereomicroscope: total length of skull — from the anterior rim of alve-olus of the first upper incisor to the most projecting point of the occipital region; condylobasal length — from the exoccipital condyle to the posterior rim of alveolus of the first upper incisor; upper canine width

— taken across the outer borders of upper canines; upper molar width — taken across the outer crowns of the last upper molars; zygomatic width — the great-est width of the skull across the zygomatic arches; mastoid width — the greatest distance accross the mastoid region; postorbital width — the least width of the postorbital constriction; maxillary toothrow length — from the front of upper canine to the back of the crown of the third molar; upper canine–premo-lar length — the canine–premo-largest distance from the front of the upper canine to the back of the crown of the posterior premolar; length of mandible — from the anterior rim of the alveolus of the first lower incisor to the most posterior part of the condyle; mandibular toothrow length — from the front of the lower canine to the back of the crown of the third lower molar; lower ca-nine–premolar length — the greatest distance from the front of the lower canine to the back of the crown of the posterior premolar; height of the coronoid process — taken perpendicularly from the extremity of the coronoid process to the ramus mandibulae. EVALUATION OFTHE HARPIOLACHARACTERS

One of the main diagnostic features at-tributed to Harpiola is the attachment point of the plagiopatagium. According to Tate (1941), Corbet and Hill (1992), Koopman (1994), and Bhattacharyya (2002), it is at-tached to the base of the first toe in Harpio-la whereas it is attached to the base of the claw of the first digit (the distal phalanx) in Murina. However, when describing M. hil-gendorfi, Peters (1880) noted and figured the attachment of the wing as being close to the base of the first toe. This view was sup-ported subsequently by Wallin (1969) and Yoshiyuki (1989). The latter described the Japanese population of M. hilgendorfi as having a “plagiopatagium attached to basal portion of first phalanxes of the first toe” (Yoshiyuki, 1989: 216). Ognev (1928) not-ed a similar point of attachment for M. og-nevi and M. sibirica (these latter taxa are considered to be synonyms of M. hilgen-dorfi — see Simmons, 2005). However, this character was overlooked by subsequent au-thors and curiously Tate (1941: 579, 580), although citing Ognev’s (1928) statements,

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erroneously listed the attachment point as a differentiating character of Harpiola.

For dental characters, the main distin-guishing feature of Harpiola cited by Tate (1941), Corbet and Hill (1992), and Koop-man (1994) was the reduction of the meta-cones of the anterior and middle upper molars. However, these authors were only able to investigate the holotype of H. grisea (the single known specimen of the genus at that time) the teeth of which are extremely worn. Bhattacharyya (2002) in his descrip-tion of a second specimen of this species did not mention the cusp pattern of the molars. Nevertheless, in Harpiola from Taiwan (de-scribed herein as new species), the meta-cones of the first (M1) and second (M2)

mo-lars are well developed and clearly higher than the paracones. Furthermore, Thomas (1915), Corbet and Hill (1992), and Bhat-tacharyya (2002) all mentioned the high-ly reduced canines of Harpiola, which are in contrast with the usual specialized ca-nines seen in Murina. However, even high-er degree of reduction in the bulk of upphigh-er canine can be observed in M. aurata (Mae-da, 1980; Corbet and Hill, 1992). Although quantitative comparison and statistical analysis were not applicable because very few specimens of relevant species are avail-able in the museum collections, detailed examination of comparative material in this study revealed that the upper canine of the new species from Taiwan is clearly the highest one in the upper toothrow with the basal area subequal to that of the second premolar, and is not especially reduced as compared to those of M. aurata. As for lower canine, the new species of Harpio-la from Taiwan shows higher degree of re-duction in both bulk and height than most species of genus Murina except M. aurata. The new species of Harpiola from Taiwan has both basal area and height of lower canine subequal to those of lower second premolar, similar proportions are also found

in M. aurata. In conclusion, the degree of reduction of canines in both upper and low-er toothrows can not slow-erve as diagnostic characters distinguishing Harpiola from Murina.

The characters of Harpiola (Figs. 1 and 2) which do support its generic distinction are: 1) the upper incisors (I2and I3) are

ap-proximately two-third that of the C1 in

height; the basal area of second upper inci-sor (I3) is more than two-third that of C1(in

Murina, the height and crown area of I2and

I3 are at most half that of the C1); 2) the

basal area of C1, P2 and P4 are subequal,

their height are gradually decreasing; the corresponding teeth are similar in bulk and height in the lower toothrow (in Murina the canine always greatly exceeds the first pre-molar in height in both toothrows; P2almost

always, clearly smaller than P4in height); 3)

C1is strongly bifid, and the additional cusp is well developed (in Murina only a small secondary cingular cusp is present). These diagnostic characters can also be seen in the type of H. grisea; however, the cusp structure of C1can hardly be detected due to the very bad condition of the specimen.

SYSTEMATICDESCRIPTION

Harpiola isodon sp. nov. Holotype

ZMNTU 1998.5.3, adult Y, dry skin and skull. Collected by T. S. Ding and K. Y. Wang on 2 May 1998.

Type locality

Hualien County, Jhuosi Township, Yu-li WildYu-life Refuge, 23°32’N, 121°15’E, 2,000 m a.s.l.

Paratypes

ESRI B0358, adult X, in alcohol, skull extracted, Taichung County, Tahsuehshan Forest Recreation Area, Tahsuehshan #210

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logging road, 1,950 m a.s.l.; ESRI B0359, adult X, dry skin and skull, Taichung Coun-ty, Tahsuehshan Forest Recreation Area, Tahsuehshan #210 logging road, 1,950 m a.s.l.; THU B050016, adult Y, dry skin and skull, Chiayi County, Alishan Township, Lulinshan Major Wildlife Habitats, 2,400 m a.s.l.; ZMNTU 2003.8.4, adult X, dry skin and skull, Ilan County, Nanau Township, Nanaunan logging road, 1,000 m a.s.l.; NMNS 4858, adult X, dry skin and skull, Taichung County, Heping Township, Wul-ing Farm; NMNS 5741, adult Y, in alcohol, skull extracted, Nantou County, Renai Township, Tsueifong; THU 7385, adult Y, dry skin and skull, Chiayi County, Alishan Township, Lulinshan Major Wildlife Habi-tats, 2,400 m a.s.l.; HNHM 2003.36.31, adult X, in alcohol, skull extracted, Nantou County, Renai Township, Meifong Farm, 24°06’N, 121°11’E, 2,100 m a.s.l.; HNHM

2004.19.13. adult Y, skin and skull, Taitung County, Taimali Township, E-ma logging road, 22°37’N, 120°56’E, 1,000 m a.s.l.; HNHM 2004. 19.15, adult X, skin and skull, Chiayi County, Alishan Forest Re-creation Area, Sister Ponds, 23°31’N, 120°48’E, 2,200 m a.s.l.

Etymology

The name isodon (‘equal-toothed’ in English) refers to the subequal basal area of the canines, first and second premolars typ-ical for the genus.

Diagnosis

This is a medium-sized tube-nosed bat with the forearm length between 31–36 mm (Table 1). Guard hairs of dorsal fur are with shiny bright golden tips; the uropata-gium is densely furred on both sides; wing membrane is attached at the base of first toe.

FIG. 1. Lateral views of skulls of: A — Harpiola isodon sp. nov. (paratype HNHM 2003.36.31.) from Taiwan;

B — Murina suilla (type species of the genus, HNHM 2000.13.2.) from Java. Scale = 5 mm

A

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The basal area of canines and the premolars are subequal in both toothrows; P4is wider

than long; mesostyles of M1and M2are less

developed but usually visible, the first up-per molar is with a more or less developed postcingular platform.

Description

The fur on the back is very long (9–10 mm in length, some guard hair can be as long as 14 mm) and woolly; the basal part

of the individual hair of underfur is dark brown with a bright yellow subterminal band and a dark brown tip. Guard hairs scattered all over the back (including the flanks and the nape) are dark brown in basal four-fifths and with shiny golden yellow tips. The general colouration of back is thus extremely similar to that of M. aurata being described as “rich mixture of gold and brown” (Bates and Harrison, 1997: 204). The dorsal side of the tail membrane, the tibia and the foot are all densely and evenly furred including the last caudal ver-tebra which is free from the uropatagium. The whole length of the forearm, the thumb and even the proximal part of the fifth metacarpal are covered with short golden coloured fur (Fig. 3). In ventral aspect the fur is shorter, dark brown at the base and light brown in the terminal one-third. Some individuals have greyish white hairs pre-dominated along the midline of chest and abdomen. The whole area of the tail mem-brane is also covered with dense, stiff, sil-very grey hairs. The ear is 12.5–13.0 mm in length, the ear conch is rounded with a very distinct emargination at the upper third of its posterior border; the tragus is medium long (6.5–8.0 mm) but wide at its base and gradually tapering to the backward-curved tip which just reaches the level of the notch; the base of tragus is with a small tooth-like projection at its outer margin.

The skull is delicately built, the brain-case is moderately bulbous, rostral profile is evenly ascending. There is no sagittal crest, the lambdoid crest is medium developed (Fig. 1A). The narial emargination is much longer than wide; there is no basioccipital fissure. Basioccipitale is with well-defined basial pits. The zygomatic arch is weak and slender, almost parallel-sided.

The inner upper incisor (I2) is anterior

to and only very little longer than the outer upper incisor (I3). Both upper incisors

are about two-third that of the C1in height;

FIG. 2. A — Occlusal view of left upper dentition;

B — lateral view of right mandible; and C — lingual view of left lower I3, C1and P2teeth of H. isodon sp. nov. (paratype, HNHM 2003.36.31.) from Taiwan.

Scales for A and B = 5 mm, and for C = 1 mm

A

B

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the basal area of second upper incisor is more than two-third that of C1 (Fig. 2A).

The basal area of C1, P2 and P4 are

sube-qual, their height are gradually decreasing. The metacones of M1and M2are distinctly

higher than the paracones. Second upper

premolar is wider than long, M1and M2are

usually with very small but recognizable mesostyle, the first upper molar is with a definite postcingular platform.

The mandible is slender, the corpus mandibulae is delicate; the symphysis is

Parameter Holotype n Paratypes min–max

Forearm length 31.40 9 31.00–35.60

Total length of skull 15.50 10 14.76–16.48

Condylobasal length 13.87 10 13.74–14.87

Upper canine width 3.65 10 3.65–4.02

Upper molar width 5.17 10 4.90–5.53

Zygomatic width 8.92 10 8.43–9.35

Mastoid width 7.36 10 7.29–7.96

Postorbital width 4.51 10 4.52–4.84

Maxillary toothrow length 4.98 10 4.97–5.63

Upper canine–premolar length 2.34 10 2.22–2.73

Length of mandible 10.28 10 10.15–11.32

Mandibular toothrow length 5.28 10 5.35–5.90

Lower canine–premolar length 2.14 10 2.02–2.44

Height of the coronoid process 3.48 10 3.36–3.98

TABLE1. Selected external and craniodental measurements of H. isodon sp. nov. (in mm)

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long and procumbent, protruding anterior-ly; there is a distinct indentation in the cor-pus in front of the angular process. The coronoid process is low (Fig. 2B). The low-er canine has a well developed additional cusp (Fig. 2C); C1 and premolar teeth are similar in both bulk and height among which C1is slightly less in height and P2 is smaller in basal area. The entoconid of M1 and M2is lower than the hypoconid but has a distinct cusp and is widely separated from the metaconid; therefore a clear posterior trigonid is present.

Comparisons

Harpiola grisea, the only other species in the genus with only one specimen before Bhattacharyya (2002), is described by Hut-ton (1872: 712) as “colour above grey mouse-brown, beneath paler grey.” It is the same specimen, however, described by Dobson (1876: 154) as “fur, above, dark brown with yellowish brown extremities; beneath, similar, but the extreme points of the hairs are rather ashy”, which is closer to our observation. The external distinction between H. grisea and H. isodon sp. nov. in Taiwan seems to be the colour of extremi-ties of dorsal hairs, which is dark brown (underfur) or shining golden yellow (guard hairs) in the latter taxon. Interestingly, Bhattacharyya (2002) observed the Mizo-ram specimen as having hairs on the dor-sum with shining golden yellow tips, strik-ingly similar to those of H. isodon. Further materials of H. grisea, especially from sites that are close to the type locality, are still needed to make clear the range of variation in pelage colouration of this unique species, or that the Mizoram specimen actually rep-resents a separate taxon.

The condition of the H. grisea holotype skull makes any detailed morphological investigation and measuring impossible, and only the shape and size of teeth can be compared. Harpiola grisea differs dentally

FIG. 4. Occlusal view of the left P4, M1and M2teeth

of: A — H. isodon sp. nov. (paratype, HNHM 2003. 36.31.) from Taiwan and B — H. grisea (holotype, BM(NH) 79.11.21.117) from Uttar Pradesh, India.

Scale = 3 mm

A B

from H. isodon sp. nov. in having a rela-tively narrow second upper premolar which is almost as long as wide; M1 is with no

postcingular platform and M2 is without

any trace of mesostyle (Fig. 4) which is usually weak but present in H. isodon. Natural history

Individuals were sporadically captured in mountain areas with elevations between 1,000 and 2,400 m a.s.l. in Taiwan. They occurred mostly in coniferous plantations or mixed forests of coniferous and broad-leaf trees with more or less closed canopy. Two of them were found in a tunnel. One female (ESRI B0358), which was caught in May, 2002 had one fetus.

Zoogeography

Although Taiwan belongs to the Indo-malayan zoogeographical region, in the first compilation of the mammal fauna of the island Kuroda (1952: 285) emphasize the fact that “Formosa […] has rich ele-ments of the Palaearctic or the Himalayan on the mountainous parts”. The systematic and taxonomic composition of Taiwanese mammalian fauna has changed a lot in the light of the recent studies, but there are some examples among volant and non-volant mammal groups which occur only in Taiwan and in the high mountains of

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mainland Asia. This kind of connection is supported by the distribution patterns of shrews (Soriculus and Chodsigoa spp. — see Motokawa et al., 1997), mole-shrews (Anourosorex squamipes and A. yamashinai — see Motokawa and Lin, 2002; Motokawa et al., 2004), vesper bats (Arielulus aureo-collaris and A. torquatus — see Csorba and Lee, 1999) and voles (Volemys spp.). The species-pair of H. isodon and H. grisea is an important addition to this list.

ACKNOWLEDGEMENTS

We are very grateful to Hsi-Chi Cheng (Endem-ic Species Research Institute, Ch(Endem-ichi), Yen-Jean Chen (National Museum of Natural Science, Taichung), Liang-Kong Lin (Tunghai University, Taichung), Paula Jenkins (Natural History Museum, London), Jacques Cuisin (Muséum National d’Histoire Natur-elle, Paris) and Paul J. J. Bates (Harrison Institute, Sevenoaks) who kindly provided access to the speci-mens under their care. György Topál help us with his expert advice and by the preparation of some speci-mens. The work of HCK, YPF, and LLL was sup-ported by the Nature Conservation Grant of the Council of Agriculture, Taiwan, ROC. The work of GCS was supported by the European Commission’s Research Infrastructure Action via the SYNTHESYS Project.

LITERATURECITED

BATES, P. J. J., and D. L. HARRISON. 1997. Bats of the

Indian subcontinent. Harrisson Zoological Muse-um Publication, Sevenoaks, 258 pp.

BHATTACHARYYA, T. P. 2002. Taxonomic status of the

genus Harpiola Thomas, 1915 (Mammalia: Chi-roptera: Vespertilionidae), with a report of the oc-currence of Harpiola grisea (Peters, 1872) in Mi-zoram, India. Proceedings of the Zoological So-ciety, Calcutta, 55: 73–76.

CORBET, G. B., and J. E. HILL. 1992. The mammals of

the Indomalayan region. Natural History Muse-um and Oxford University Press, Oxford, 488 pp. CSORBA, G., and L.-L. LEE. 1999. A new species of

vespertilionid bats from Taiwan and a revision of the taxonomic status of Arielulus and

Thainycte-ris (Chiroptera: Vespertilionidae). Journal of

Zo-ology (London), 248: 361–367.

DOBSON, G. E. 1876. Monograph of the Asiatic

Chiroptera, and catalogue of the species of bats

in the collection of the Indian Museum, Cal-cutta. Trustees of the Indian Museum, London, 228 pp.

ELLERMAN, J. R., and T. C. S. MORRISON-SCOTT. 1951.

Checklist of Palaearctic and Indian mammals 1758 to 1946. Trustees of the British Museum (Natural History), London, 810 pp.

HUTTON, T. 1872. On the bats of the north-western

Himalayas with notes and corrections in no-menclature by W. Peters, C.M.Z.S. Proceedings of the Zoological Society of London, 1872: 690–714.

KOOPMAN, K. F. 1993. Order Chiroptera. Pp. 137–241,

in Mammal species of the word: a taxonomic and

geographic reference. 2nd edition (D. E. WILSON

and D. M. REEDER, eds.). Smithsonian Institution

Press, Washington D.C., 1206 pp.

KOOPMAN, K. F. 1994. Chiroptera: systematics.

Hand-book of Zoology. Mammalia, part 60. Walter de Gruyter, Berlin, 217 pp.

KURODA, N. 1952. Mammalogical history of

Formo-sa, with zoogeography and bibliography. Quar-terly Journal of Taiwan Museum, 5: 267–304. MAEDA, K. 1980. Review on the classification of

lit-tle tube-nosed bats, Murina aurata, group. Mam-malia, 44: 531–551.

MOTOKAWA, M., and L.-K. LIN. 2002. Geographic

variation in the mole-shrew Anourosorex

squa-mipes. Mammal Study, 27: 113–120.

MOTOKAWA, M., H.-T. YU, Y.-P. FANG, H.-C. CHENG,

L.-K. LIN, and M. HARADA. 1997. Re-evaluation

of the status of Chodsigoa sodalis Thomas, 1913 (Mammalia: Insectivora: Soricidae). Zoological Studies, 36: 42–47.

MOTOKAWA, M., M. HARADA, L.-K. LINand Y. WU.

2004. Geographic differences in karyotypes of the mole-shrew Anourosorex squamipes (Inse-ctivora, Soricidae). Mammalian Biology, 69: 197–201.

OGNEV, S. I. 1928. Mammals of Eastern Europe and

Northern Asia. Volume 1. Insectivora and Chiro-ptera. Glavnauka, Moscow, 631 pp. [in Russian]. PAVLINOV, I. Y. 2003. Systematics of recent

mam-mals. Moscow University Publisher, Moscow, 296 pp. [in Russian].

PETERS, W. 1880. Mittheilung über die von Hrn. Dr.

F. Hilgendorf in Japan gesammelten Chiropteren. Monatsberichte der Königlich Preussischen Aka-demie der Wissenschaften, 1880: 23–25. SIMMONS, N. B. 2005. Order Chiroptera. Pp.

312–529, in Mammal species of the Word: a tax-onomic and geographic reference. 3rd edition (D. E. WILSON and D. M. REEDER, eds.). The

Johns Hopkins University Press, Baltimore, 2142 pp.

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TATE, G. H. H. 1941. Results of the Archbold

ex-peditions no. 40. Notes on Vespertilionid bats of the subfamilies Miniopterinae, Murininae, Ke-rivoulinae and Nyctophilinae. Bulletin of the American Museum of Natural History, 78: 567–597.

THOMAS, O. 1915. A special genus for the Himalayan

bat known as Murina grisea. Annals and Maga-zines of Natural History, 8, 16: 309–310. WALLIN, L. 1969. The Japanese bat fauna. Zoologiska

Bidrag frDn Uppsala, 37: 223–440.

YOSHIYUKI, M. 1989. A systematic study of the

Japa-nese Chiroptera. National Science Museum, Tokyo, 242 pp.

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