70 



NATURE 



[May 2 1, 1896 



investigated the conditions under which a group should have 

 no characteristic subgroup. This condition is that the group 

 should be generated by a number of holohedrically isomorphous 

 simple groups, such that every operation of any one of them is 

 permutable with every operation of all the rest ; or, in the 

 phraseology of Ilerr Holder, the group must be the direct 

 product of a number of holohedrically isomorphous simple 

 groups. The following theorem is then proved. If d is a 

 group which has no characteristic sub-group, and if R is the 

 group of greatest order that contains G self-conjugately, while 

 at the same time no operation contained in R, and not in G, is 

 permutable with every operation of G ; then the group R admits 

 of no contragredient isomorphisms, and contains no self-conjugate 

 operation except identity. A special case of this theorem is that 

 the group defined by the congruences 



.v,' = ai,.i;,-f ajj-i-j-t- . . . +ai„x„-l-;3j, 



.v.,' = a2,.ri-l-02jr2-f . . . -)- 02„x„ -I- (85, (Mod. A, prime) 



.v„^=o„i.«-i + o„2.«'.2-l- . . . -\-a„„Xn + &m 

 dmits only cogredient isomorphisms. In the third part I con- 

 sider the group of isomorphisms of certain simple groups, some 

 of whose properties I have already dealt with in vol. .\.kv. of the 

 Society's Proceedings. For the simple group of order 2"(2-" - I) 

 there defined, I show that the order of the group R of isomor- 

 phisms is 2"(2*'- l)", and that if H is the group of cogredient 

 isomorphisms, the factor group R/H is a cyclical group of order 

 It. For the simple groups of order JX"(\-" - I ), \ an odd prime, 

 it is .shown that the order of the group R of isomorphisms is 

 X>i(;j.On_ i)«^ the factor group R/H being the direct product of 

 cyclical groups of orders 2 and 11. The latter class includes 

 as a special case, (X" = 3-), the alternating group of six 

 symbols. In Herr Holder's paper, referred to above, the 

 isomorphisms of the alternating group are dealt with, and, 

 as compared with all other degrees, it is found that the 

 alternating group of degree 6 behaves exceptionally, and re- 

 quires rather elaborate treatment. There seems, however, to 

 be no reason for regarding the alternating groups of different 

 degrees as a set of groups which are characterised by common 

 group-properties in the same way, for instance, as the groups of 

 the modular equation for different prime transformations are ; 

 and this view is borne out by the fact that there is nothing 

 exceptional in the behaviour of the alternating group of degree 6 

 when regarded as one of the class of groups here considered. 



The President also read an abstract of a paper on division of 



the Lemniscate, by Prof. G. B. Mathews. — Dr. Hobson, F.R.S., 

 read a paper on some general formula; for the potentials of 

 ellipsoids, shells, and discs. — The President offered some remarks 

 on the compensation for difference of capital in gambling 

 h mitrance, being a contribution to the theory of the " Duration 

 of Play." — Mr. Basset, F.R.S., read a paper on the stabiUty of a 

 fi-ictionless liquid and the theory of critical planes. In the theory 

 of the stability of the steady motion of a frictionless liquid which 

 is bounded by the parallel planes j/ = o and j = a, the disturbed 

 motion depends upon the equation 



(»/* -f U)(cflvldy°- - k\<)^vdWldf- . . . . (i) 

 The usual process of solution fails whenever there is a plane, 

 called a critical plane, at which njk +17=0, and the object of 

 this paper is to examine the nature of the solution when such a 

 plane exists. In steady motion U = (p{}'), where (/> is a given 

 function ; and if a critical plane exists, - n/i = (p{c), which de- 

 termines the relation between the time-constant « and the wave- 

 constant i, provided a real value of c can be found which lies 

 between o and a. The integral of (l) is of the form 



V = kf,{y) + B/,(j)- 

 The boundary conditions require that v = o when _)' = o and j' = ir. 

 At a critical plane d-Uldy- = o orj' = o. If the first condition is 

 satisfied, and if neither of the functions/ become infinite between 

 _)'=o and y = a, the boundary conditions enable the constants A 

 and B to be eliminated, which leads to a relation of the form 

 F(«, /', i') = o, and the conditions for the existence of a critical 

 plane require that this equation should furnish at least one real 

 value of c lying between o and a. But if one of the functions — 

 say /, — becomes infinite between the limits, B = o, and the 

 boundary conditions cannot usually be satisfied, in which case a 

 critical plane cannot exist. When the form of U is such that 

 (PUIdyi- does not vanish when_)' = c-, a critical plane cannot exi.st 

 except in very special circumstances. The paper concludes by 

 showing that the particular solutions obtained by the hypothe.sis 



NO. 1386, VOL. 54] 



that X and / enter into the solution in the form of the factor 

 «"<-''+"" can always be generalised liy Fourier's theorem, so as to 

 include every possible disturbance which does not violate the 

 boundary conditions. The author and Mr. Love, F.R.S., joined 

 in a discussion on the subject of the communication. 



Geological Society, April 29. — Dr. Henry Hicks, l-'.R.S., 

 President, in the chair. — Descriptions of new fossils from the 

 carboniferous limestone. (l) On Pcmmatiles constipatiis, sp. 

 nov., a lithistid sponge. (2) On Palaacis hitiiiilis, sp. nov., 

 a new perforate coral ; with remarks on the genus. (3) On the 

 jaw-apparatus of an Annelid, Eiiincilcs Reidii, sp. nov., 

 by Dr. G. Hinde. — (l) The Pcinniatilcs, belonging to genus 

 hitherto only known from the Permo-Carboniferous beds of 

 Spitzbergcn, was discovered in the Voredale beds of Yorkshire 

 by Mr. J. Rhodes, and is the only fairly complete sponge which 

 has hitherto been detected Jn the Yoredale beds of North-west 

 Yorkshire. The author gave a full description of the species. 

 (2) The Palmcis was found by the Rev. G. C. H. Pollen in the 

 carboniferous limestone and shale series, on the banks of the 

 Hodder, near Stonyhurst. The specific characters of the form 

 were given by the author, who, in the light of the new informa- 

 tion, gave a fresh definition of the genus Palcacis, which appears 

 to represent a distinct family of perforate corals, in some features 

 more nearly allied to the Favositidiie than to the Madreporid* 

 or Poritidre. (3) The third specimen was discovered by Miss 

 Margery A. Reid in the Lower Carboniferous beds of Ilalkin 

 Mountain, Flintshire, and is named in honour of its discoverer. 

 .\ description of it was given, and it was stated that, notwith- 

 standing certain peculiarities, the individual pieces correspond so 

 closely with those of the recent Eunice family that it may well be 

 included in the genus Eunicites. — The Eocene deposits of Dorset, 

 by Clement Reid. The new survey of the western end of the 

 Hampshire basin shows that the Reading beds become fluviatile 

 and gravelly in Dorset (as was already known), and contain, in 

 addition to chalk flints, many fragments of Greensand chert. 

 The London clay thins greatly and becomes more sandy, but is 

 apparently still marine. The Bagshot sands become coarser and 

 more fluviatile, changing rapidly west of Moreton Station, till 

 they consist mainly of coarse subangular gravel. These 

 gravels, formerly referred to the Reading series, are now shown 

 to be continuous with the Bagshot sands, which as they become 

 coarser cut through the London clay and Reading beds to rest 

 directly on the chalk. The Bagshot gravels contain, besides 

 chalk flints and Greensand chert, fragments of Purbeck marble 

 and numerous Palfeozoic grits and other stones probably derived 

 from the Permian breccias of Devon. — Discovery of mammalian 

 remains in the old river-gravels of the Dervvent near Derby, 

 Part i., by H. H. Arnold-Bemrose. A few mammalian bones 

 were found in sinking a well at AUenton. On April 8, 1895, the 

 authors commenced Further excavations, and were successful in 

 finding a number of Ijones of a Hippopotamus, an Elephas, and of 

 a Rhinoceros. They were found in a dark-coloured sand above 

 the river-gravel, at a depth of 9 feet 8 inches below the 

 surface, jlr. Clement Reid found some twenty or more species 

 of plant-remains in the sand. These plants " indicate a moist 

 meadow or swampy ground, and a temperate climate. The 

 species are all widely distributed." Part ii., by R. .M. Deeley. 

 The deposits in which the bones were found occupy a wide 

 trench which occurs on the inside edge of a gravel-terrace 

 stretching for several miles south of Derby, at a height of 15 

 or 20 feet above the modern alluvial plain. The gravels are 

 of later age than the great chalky boulder clay, and were formed 

 at a time when the rivers were removing from their preglacial 

 valleys the older boulder clays, with which tlrey had been 

 partially filled. Gravels of two ages are recognised : (a) recent 

 gravels well stratified, undisturbed, and covered in many places 

 by a thick layer of brick-earth ; and (h) high-level gravels 

 showing " trail " and contorted bedding. It is in these latter 

 gravels that the trench containing the mammalian remains 

 occurs. 



Zoological Society, May 5. — Dr. John Anderson, F.R.S., 

 \ice-President, in the chair.— Mr. W. E. Hoyle exhibited a 

 Rontgen-ray photograph of a snake in the act of swallowing a 

 mouse. — Mr. G. A. Boulenger, F.R.S., read a paper on some 

 little-known Batrachians from the Caucasus, based chiefly on 

 specimens recently transmitted to the British Museum by Dr. 

 Radde, of Tifiis. Among these was an example of the new 

 frog of the genus Pelodytes, for which he had proposed the 

 name P. caucasicus. Altogether ten species of Batrachians 



