December 21, 1893] 



NA TURE 



189 



schedules to the four winds of heaven," and deal largely " with 

 some part of the present outlook as stated in Klein's ' Verj:;Iei- 

 chende Betrachtungen.' " Prof. Cleveland Ahbe gives (pp. 

 36-38) an analysis of papers on vortices in a compressible and 

 rotating fluid, by Mr. Charles Chree, and trusts that the strictly 

 meteori)Iogical work of his new position (at Kew) will tempt 

 him to apply mo-iern mathematical analysis to the winds, the 

 clouds, and the storms of the actual atmosphere. Dr. T. Craig's 

 high estimate of the Traite d' Analyse, by M. Kmile Picard, will 

 be seen from the space (pp. 39-66) he devotes to an account of 

 this first volume. The extraordinary developments in the theory 

 of functions, in differential equations, and in certain purely 

 algebraical theories, and the important applications of the results 

 of the-e developments to geometrical, physical, and astronomical 

 problems, have made such a treatise almost indispensable. 

 Notes and new publications occupy pages 65-72. 



The last volume of the Memoirs of the St. Petersburg Society of 

 Naturalists (vol. xxiii.), for the Section of Botany, contains, be- 

 sides the proceedings of the Society, a number of valuable papers. 

 N. V. Diakonoffgives, under the title " Typical Representatives 

 of the Life-substratum," a summary of his researches into the 

 behaviour of lower fungi in an atmosphere devoid of oxygen. 

 The Penicilliun glaucuin is a typical representative of organ- 

 isms in which no life is possible without the action of oxygen ; 

 while Mucor stolonifer may be taken as a representative of 

 organisms in which life is impossible without either the action 

 of oxygen or the presence of a substance capable of fermenting. 

 Prof N. Tsinger gives a description of the mosses of Tula, of 

 which 134 species are enumerated. M. L. P. Bowdin gives the 

 results of his very interesting experiments upon the breaking out 

 of buds upon cuttings of plants. S. G. Navashin describes and 

 figures a nevy parasite of the cupsulse of some mosses, Tilletia {?) 

 sphagni. N. Aiboff gives the results of his five years' explora- 

 tion of the flora of Abkhazia (on the north-east coast of the 

 Black Sea) ; his collection numbers no less than 1300 species, 

 and does not confirm the conclusion as to the flora of Caucasia, 

 arrived at by MM. Krasnoff" and Kuzeretsotf. S. Navashin 

 gives a very elaborate paper on the Discomycete, Sclerotinia 

 betiilcB, with several coloured plates. The veg-itation of i e 

 Zerafshun valley is shortly described by V. Komaroff; and N 

 Ponyatsky criticises Hugo de Vries' method for the analysis of 

 isotonic coefficients. 



Bulletin de la Societe des Naturalistes de Moscou, 1893 

 November i. — On some ecto- and ento-parasites of the Cyclo 

 pides, by Dr. W. Schewiakoff. One new species, Trichophrya 

 cordiformis, is described, as also the ento-parasitic slimes of 

 theCvclopides. — Note, by W. Zykoff, on the chorda of 6"zV^(/o« 

 pisciformis. — Note on a new skull of Amynodon, by Marie 

 Pavlow. — A catalogue of the Coleoptera of Kazan, by L. 

 Krulikovsky : the Nocture. — On the evolution of the ocean, by 

 H. Trautschold, being an attempt at tracing the gradual 

 development and modification of salinity in the ocean through- 

 out the earth's history. 



SOCIETIES AND ACADEMIES. 



London. 



Royal Society, November 23. — "A certain Class of 

 Generating Functions in the Theory of Numbers," by Major 

 P. A. MacMahon, R.A., F.R.S, 



The present investigation arose from my " Memoir on the 

 Compositions of Numbers," recently read before the Royal 

 Society and now in course of publication in the Philosophical 

 Transactions. The main theorem may be stated as folows : — 



IfXj, Xg, . . . . , Xi, be linear functions of quantities Xj, 

 -r^, . . . . , Xn given by the matricular relation 



(^1) ^2 X,i) = (otji ^12 • • «1h) (^1, x„ , x,^. 



that portion of the algebraic fraction 

 I 



(I - 5iXi) (I - 52X,) . . 



NO. 1260, VO . 49] 



(I - j„X„)' 



which is a function of the products, 



S^X^, Jo.Vo , SnXn, 



only, is i/\',„ where (putting s^ = s.^ — . . . . = s„ = i) 



V« = ( - )„X^X„ . . . . , Xn ao^, «22- l/x, •• a-'a 



«-a ^n2 •• a, 1,1- i/x,, 



The theorem is of considerable arithmetical importance, and 

 is also of interest in the algebraical theories of determinants and 

 matrices. 



The theory is developed at length in the paper, with illustra- 

 tive examples of arithmetical applications. 



Incidentally interesting results are obtained in the fields of 

 special and general determinant theory. The special deter- 

 minant, which presents itself for examination, provisionally 

 termed "inversely symmetric," is such that the constituents 

 symmetrically placed in respect to the principal axis have, 

 each pair, a product unity, whilst the constituents on the 

 principal axis itself are all of them equal to unity. The deter- 

 minant possesses many elegant properties which are of import- 

 ance to the principal investigation of the paper. The theorems 

 concerning the general determinant are connected entirely with 

 the co-axial minors. 



I find that the general determinant of even order, greater 

 than two, is expressible in precisely two ways as an irra- 

 tional function of its co-axial minors, whilst no determinant of 

 uneven order is so expressible at all. 



Of order superior to 3, it is not possible to assume arbitrary 

 values for the determinant itself and all of its co-axial minors. 

 In fact of order n the values assumed must satisfy 



2'» — ft' + n — 2 



conditions, but these conditions beings atisfied, the determinant 

 can be constructed so as to involve n - 1 undetermined 

 quantities. 



"On the Whirling and Vibration of Shafts." By Stanley 

 Dunkerley, Berkeley Fellow of the Owens College, Min- 

 chester 



December 7. — " Reptiles from the Elgin Sandstone : Descrip- 

 tio of Two New Genera." By E. T. Newton, F.R.S. 

 Com unicated by permission of the Director-General of the 

 Geolo:^ical Survey. 



Two new reptiles from the Elgin Sandstone are described in 

 detail. One of them is the property of Mr. James Grant, of 

 Lossiemouth. The bones themselves being absent, their forms 

 have been reproduced by gutta-percha casts taken from tlie cavi- 

 ties left in the stone. This reptile was evidently allied to Stago- 

 nolepis; it is represented by the skull, which is about three 

 inches long, and the anterior half of the body, with the pectoral 

 arch and both the fore limbs. The skull is depressed, has a pair 

 of large prelachrymal fossae ; the two nasal openings are small, 

 and placed near the end of the muzzle. The palate is narrow 

 and deeply grooved, with primitive posterior nares placed far 

 forwards. The vertebrae and limbs are Crocodilian in form. 

 Above the vertebrae there is a double row of pitted, and closely- 

 set scutes. This small Parasuchian is named Erpetosiuhus 

 Grant i. 



The second specimen was obtained by the Rev. Dr. Gordon, 

 from the quarry at Spynie. In this fossil the bones were pre- 

 sent, and the skull is still preserved, but many of the other bones 

 were too much crumbled to show their form, and the casting 

 process was again resorted to. The neck and fore limbs are 

 wanting. The skull, which closely resembles that of Cerato- 

 saurtts, is about 43 inches long, sharp anteriorly, and 

 bird like when seen from above, but deep when seen 

 from the side, and it has a large prelachrymal fossa. 

 The teeth are compressed and serrated anteriorly and pos- 

 teriorly. The palate is deep, and a median pair of apertures, 

 near the post-palatine vacuities, are believed to be primitive 

 posterior nares placed far back. Many oval scutes are scattered 

 above the neural spines. 



This reptile seems to be intermediate between the Dinosau- 

 rians and Crocodilians ; the skull and teeth are most like those 

 of Dinosaurs ; the pelvis and limbs might belong to either Dino- 

 saurs or Crocodiles ; while the free astragalus is certainly a 

 Crocodilian character ; provisionally it is referred to the Thero- 

 podous Dinosauria, and named Ornithosuchiis Woodivardi. 



