4o6 



NA TURE 



[February 24, 1898 



boulders are abundant, and the fine-grained matrix, which is 

 frequently argillaceous, is often well-laminated and false-bedded. 

 Some of these drifts are stratified, others unstratified, and con- 

 torted drifts occur. This type of moraine is remarkably like 

 some British boulder clay. The third class is sometimes formed 

 by land-ice, at other times beneath the sea ; the latter shows 

 stratification. The superglacial and intraglacial streams, so far 

 as seen, were usually clear of drift. Under the fourth head an 

 esker in a tributary of the Sassendal is described. The direct 

 geological action of the marine ice is of four kinds : transport 

 of material, contortion of shore-deposits, formation of small 

 ridges of boulder-terraces above sea level, and striation, round- 

 ing, and furrowing of rocks along the sea-shore. Traces of 

 former glaciation are described in the case of the Hecla Hook 

 beds, and of certain beds of late Mesozoic or early Cainozoic 

 age in Bunting Bluff. Under the head of general conclusions 

 the authors state that they have discovered no certain test to 

 distinguish between the action of land-ice and marine ice ; that 

 there is no evidence to prove that land-ice can advance far across 

 the sea ; and that there is evidence, which they regard as con- 

 clusive, of the uplift of materials by land ice. They note that 

 the mechanical processes connected with the advance of the 

 glaciers are of three kinds. All the material seen transported 

 by the glaciers was superglacial or intraglacial, and not sub- 

 glacial Some striation of intraglacial material is caused by 

 differential movement of different layers of ice. The advance 

 and retreat of the Spitsbergen glaciers is very irregular, and 

 apparently due to local changes. The observations of the authors 

 support the views of those who ascribe a limited erosive power 

 to glaciers. Lastly, the theory that glacial periods occurred as 

 a consequence of epeirogenic uplifts receives no support from 

 Spitsbergen. — An interesting discussion followed, in which Sir 

 Martin Conway, Prof. Bonney, the Rev. Edwin Hill, and Mr. 

 Marr took part. Mr. P. F. Kendall said that the paper would 

 mark a distinct epoch in British glacial geology. Hitherto, 

 one body of geologists had attributed the drift-deposits of 

 Britain to the agency of land-ice, while another had invoked 

 the agency of the sea. The latter had argued that glaciers 

 cannot move uphill, that they cannot transport materials from 

 lower to higher levels, that glaciers cannot gather up materials 

 over which they are moving, and that, even if they could pick 

 up shells they would grind them to powrler — "On a Quartz- 

 rock in the Carboniferous Limestone of Derbyshire," by H. H. 

 Arnold-Bemrose. The paper describes the occurrence in the 

 field and the microscopic structure of a rock consisting essen- 

 tially of quartz, which is found in the mountain limestone in 

 several localities. It occurs in irregularly-shaped bosses and 

 veins, and shows no signs of stratification. The author con- 

 siders that the quartz- rock is not a gritty limestone, altered by 

 the growth of crystalline quartz around the detrital grains, but 

 that it is a limestone replaced by quartz. 



Mathematical Society, February lo — Prof, Elliott, 

 F.R.S., President, in the chair.^Lieut. -Colonel Allan Cun- 

 ningham, R. E. , read a paper entitled ' ' On Aurifeuillians. " These 

 are defined as the algebraic ptime factors of two functions, viz. : 

 of (X2« -f Y'«) when 2«XY = D, and of (X« + /«+i. Y«) when 

 mXY = D, where n is odd, and i— \^ - i. Their salient 

 property (discovered by Mr. Aurifeuille, of Toulouse) is that 

 they are algebraually resolvible into two factors (say L, M). 

 Also L, M are expressible in the same 2'^ forms as their product- 

 function. The quotient of one Aurifeuillian by another of the 

 same order has the same properties. The properties of the two 

 Aurifeuillians of orders 2 and 3, viz. : 



{x^ + 2V) and {x^ -f- ff) ~ (x"' + Y)f 



were stated at some length. The application to Fermat's 

 numbers {2^"+^ + i) and (3^+' + i) was given, and a table of 

 the factorisation thereof into prime factors was given, extending 

 (with gaps) to (2^^" +1), 3^"" + l). In opening the discussion 

 on the paper, Mr. Bickmorc pointed out that Aurifeuille's 

 formulae, which were enunciated without complete proof by 

 Lucas, might be completely proved by the theory of complex 

 integers. Thus the formulae express the algebraical prime-factor 

 «— 1 



of a" - ( - 1 ) ^ b, of n as the difference of two squares, when n 



is an uneven integer, and — is a perfect square. But the alge- 

 n 



braical prime-factor is the norm of the binomial n'-' integer 



a - ( - i)^V. 

 NO. 1478, VOL. 57] 



ny'- 



«-i -k 



-)- ( - I) ^ 7P V 

 Gauss's results give 



(- ^{^ sln^i{p\ 

 a rational integral function of p ; hence, finally, 



N„{x2 .- (- i)^«jV} = N„{a: - jfiAp)} X N„{x ^ ypAp)\. 



The formulae also express as the difference of two squares the 

 algebraical prime-factor of d^" -f b''-", when n is an uneven 



integer, and — is a perfect square ; in this case the final result 



2» 



is 



N„(jr* -f 4«>V) = N«{x- - 2nyp/{p) -f- 2( - i)~^ nyy] 



X ^„{x^ + 2nyp/(p) + 2{- I)"2"„y2p2}. 



Kummer's tests show that if « be > 3, the absolute term in each 

 of the complex integers is correctly fixed ; hence, y being a 

 factor of every term except the absolute term, if either Auri- 

 feuillian factor be a prime, it has any prime factor of y as a 

 residue of order «, Avhen n > 3. The process also expresses 

 complex «"^ integers with more terms than two, which are 

 expressible in the form 



X-i - {- I) --i „f; 



(x and y being themselves complex integers of order n) as the 

 product of two complex «'<' integers. — The President ( Lieut. - 

 Colonel Cunningham, pro tern, in the chair) communicated a 

 paper by Mr. J. E. Campbell on the transformations which 

 leave the lengths of arcs on any surface unaltered. The object 

 of the paper was to obtain the infinitesimal transformations 

 which have the property of leaving unaltered the lengths of arcs 

 on any given surface in space of « -f i dimensions— that is, the 

 transformations which leave dx-^ -f . . . -t- dx,? -f dz^ invariant 

 where z = f{x^ . . . Xn). It is remarkable that this problem 

 can be solved completely whenw > 2, though not when n — 2. 

 At the conclusion of the paper it is proved that if H is the 

 Hessian of f{xy . . . x„), then 



n+2 



H-f- ji -f 



{A\ 



dxj 



+ 



(t.) } ' 



is an invariant for such substitutions ; this is a generalisation of 

 the well-known theorem that the measure of curvature (on a 

 surface in ordinary space) is unaltered by transformations which 

 leave the lengths of arcs invariant. — Mr. Hargreaves made a 

 short impromptu communication. 



Zoological Society, February 15. — Dr. Albert Giinther, 

 F.R.S., Vice-President, in the chair. — A letter was read from 

 Mr. Dudley Le Souef, of Melbourne, containing a summary of 

 some observations on the transfer by the mother of an embryo 

 kangaroo {Macropus gigatiteus) by her mouth into her pouch. 

 — A report was read, drawn up by Mr. A. Thomson, the 

 Society's head-keeper, on the insects exhibited in the insect- 

 house during the year 1897, and a series of the specimens was 

 exhibited. —The Secretary exhibited a series of specimens of 

 butterflies, which had formed part of a collection lately on view 

 at the Dunthorne Gallery, in illustration of the mode of mount- 

 ing employed in "Denton's Patent Butterfly Tablets." — Mr. 

 W. P. Pycraft read the first of a series of contributions to the 

 osteology of birds. The present part (of which the following is 

 an abstract) related to the Sleganopodes. The fact that in the 

 tropic-birds, cormorants, gannets, and frigate-birds, all the toes 

 are united by a common web, has led to the belief that these 

 forms are closely related ; they form the sub-order Steganopodes 

 or ToHpaimafce of authors. A comparison of the osteology of 

 the group confirms this opinion.— Dr. W. G. Ridewood read a 

 paper on the skeleton of regenerated linjbs of the midwife-toad 

 (Alytes obste(ricans). He demonstrated the possibility of the 

 development, in the regenerated hind limb of the larva, of 

 tarsal, metatarsal, and phalangeal cartilages identical in every 

 respect with those of the normal limb.— Mr. G. A. Boulenger, 



