October 27. 19 10] 



NATURE 



plete elimination of oxygen caused was surprisingly great. 

 Thus the apparent specific velocity of the negative ion in 

 pure hydrogen was >230 cm. sec.-', volt cm.-', but it 

 rapidly dropped to 7-6 on the addition of oxygen up to 

 I per cent. The velocity of the positive ion was unaffected 

 by traces of oxygen impurity, and was 5-8 throughout. 



The change in the case of negative discharge might have 

 been caused in two ways : — (a) by a great diminution with 

 purity in the size of the negative ion in hydrogen ; (b) by 

 back discharge from the plate. If the latter were true the 

 wind-pressure method breaks down, and the above values 

 of velocity are not real. .'\s it was certain that some back 

 discharge was present, it seemed reasonable at the time 

 to attribute the whole effect to this cause, but some recent 

 work of Franck shows that it was probably not correct 

 to do so. 



Franck has shown (I'grJr. rf. D. Phys. Ges., xii., 291 

 and 613, loio) that in gases such as argon and nitrogen 

 the specific velocities of the negative ions obtained by 

 a. rays rapidly increased as the last traces of o.xygen were 

 removed. Thus in pure argon and nitrogen he obtained 

 values as high as 2064 and 1446 respectively. .\s in the 

 alx)ve, however, the positive ions were unaffected. 



In the light of these results it is probable that the 

 negative ions in point discharge in hydrogen, like those 

 in other o.\ygen-free gases, are either corpuscles or are 

 very small, although in our work the unknown amount 

 of back discharge present prevented the determination of 

 their true specific velocity. 



These results throw considerable light on various pheno- 

 mena occurring in point discharge, and I hope to publish 

 later a more complete discussion. 



A. M. TVNDALI.. 



Physical Department, University of Bristol. 



An Irish Pteridosperm. 



Readers of Nature familiar with the many valuable 

 additions to knowledge made by British palpeobolanists 

 within the last twenty years will be interested to know 

 that in the course of rearrangement of the fossil plants in 

 my charge in the botanical division of the National 

 Museum in Dublin I have found, while comparing the 

 "specimens of Sphenopteris in this collection with those in 

 the collection of the Geological Survey of Ireland (of which 

 my colleague. Prof. Grenville Cole, is director), that in 

 the latter collection there is a specimen of Sphenopteris 

 Hoeninghaiisi from the Coal-measures of Glengoole, co. 

 Tipperary, which shows all the characteristic features of 

 Lyginodendron Oldhammm (including its spines, sclerotic 

 network, venation, and conchoidal pinnule segments). 

 Moreover, the specimen shows, in direct continuity with 

 the vegetative part, the Calymmatotheca Stangeri condition 

 regarded first by Scott, and now by many others, as prob- 

 ably the seed-producing part of Lyginodendron, from which 

 the Lagenostoma Loinaxi seed has alreadv, as Oliver and 

 Scott have satisfactorily shown, in all probability fallen 

 out. The specimen in question, if my interpretation is 

 right, proves the correctness of the conjecture that 

 C. Stange'i is part of the true pteridosperm Lygino- 

 dendron. I hope to publish shortly an illustrated account 

 of the find. T. Johnson. - 



Royal College of Science, Dublin, October 24. 



Fermat's Theorem. 



The following proof of this theorem may be of some 

 interest. Take the scale of notation the radix of which 

 is X, and write down all the numbers of p digits, any or 

 all of which may be zero. The number of these numbers 

 is jr''. From one number we can, in general, derive p — i 

 others by cyclical permutation, the exceptions being those 

 numbers that are periodic with a period that is a sub- 

 multiple of p. Suppose /) to be a prime, so that its only 

 submultiple is unity. jhen all the numbers except the 

 X numbers that have their digits the same can be arranged 

 In sets of p each (which are easily seen to be mutually 

 exclusive). Hence the number of these numbers, which is 

 xi' — x, is divisible by p, and if x is prime to p we see 

 immediately that .v-' — i is divisible by p, which is 



NO. 21^9, VOL. 84I 



Fermat's theorem. It is clear that this proof depends on 

 permutations and combinations, not really on scales of 

 notation, which, indeed, we have only used because of the 

 clearness that they lend to its statement. 



H. C. POCKLINGTON. 

 1 1 Regent Park Terrace, Leeds. 



The Uganda-Congo Boundary. 



With reference to the note on this subject in Nature 

 of September i, has not the writer fallen into a slight 

 error in quoting (p. 268) the definition of limits in the 

 " Berlin Act " as applicable to the Congo State? Refer- 

 ence to the text of the .■\ct will show that the passage 

 quoted relates, not to the State, but to the Free Trade 

 area in the Congo basin and neighbouring territories con- 

 stituted at that time, with limits by no means coincident 

 with those of the State. In fact, the " Berlin Act " had 

 nothing to do with the State (as such), which was consti- 

 tuted, not by the conference, but by agreements with 

 individual Powers negotiated about the same time. 



The frontier originally claimed by the State in this 

 region, and definitely accepted by several of the Powers, 

 was the thirtieth meridian, for however unsatisfactory this 

 rhight be, there could hardly, in 1885, have been a ques- 

 tion of the adoption of the water-parting, which would at 

 that date have involved far more uncertainty than the 

 meridian. It was even doubtful to which of the two 

 basins Lake Edward belonged. The mistake in 1894 

 seems to have consisted, not in the gratuitous introduction 

 of the thirtieth meridian, but in its partial retention (viz. 

 in the Ruwenzori district), to the detriment of Uganda, 

 while replaced farther north by the water-parting, greatly 

 to the advantage of the Congo State. 



Edward Heawood. 



I Savile Row, September 16. 



It is quite true that the definition quoted in the note 

 was that of the Free Trade area, but as the recognition 

 of the Congo State, on our part, contained no definition 

 of frontiers, we w'ere entitled in 1894 to maintain that, 

 in default of any specific deviation agreed upon mutually, 

 the two frontier lines w^ere identical. So far as I am 

 aware, up to the date of the treaty of 1894 we had not 

 admitted, nor, indeed, seriously considered, any claim on 

 behalf of the Congo State to territories outside the Congo 

 basin. 



The history of the whole series of transactions is some- 

 what complicated and difficult, but it seems that, what- 

 ever were the intentions of the signatories to the Berlin 

 Act and of the framers of the original agreements with 

 the Congo State, any distinction between the State and the 

 Free Trade area disappeared at an early period of their 

 history, and had ceased to exist by 1804. 



The Writer of the .\rticle. 



An Agaiic with Sterile Gills. 



The occurrence of agarics with sterile gills is well 

 known in certain species ; it has been noticed most in 

 those with purple spores. A few days ago I met with 

 two specimens of Panaeoius campanulatits in Sutton Park, 

 near Birmingham, in which the gills were of a pinkish- 

 grey colour, somewhat closely resembling the tint of the 

 dry pileus. There was a total absence of the usual 

 variegated, grey and black, appearance. The pileus was 

 large and well developed, measuring I3 inches high and 

 broad, stipe quite 4 inches long, and presenting all the 

 characters of that of P. campanulatus. On examining 

 sections of the gills it was seen that numerous basidia 

 were present, projecting beyond the paraphyses ; very 

 many of them had the four sterigmata of full size, but 

 not one over the whole of the gills of both specimens had 

 produced a spore, nor were any produced afterwards, so 

 long as the fungi were preserved. Unfortunately, they had 

 been gathered before I saw them, so that it was impossible 

 to ascertain if there was any visible cause for the steriHty. 



W. B. Grove. 



The Botanical Laboratory, Birmingham Uni%-ersity. 



