338 



NATURE 



\Feb. 25, 1875 



all is a paradox more wonderful than ever, now that his method 

 gives the means of determining the conditions to be satisfied and 

 comparing their number with that of the disposable constants. 

 The orders for 3, 5, 7 bars are 0, 20, 72. Formerly the existence 

 of one was doubted ; now a finite number for ii-t-iy order of link- 

 work is rendered highly probable. — The .Secretary then read 

 portions of papers by Rev. W. H. Laverty, Mr. E. J. Routh, 

 F.R..S., and Mr. J. Griffiths. Mr. Laverty's paper discussed a 

 particular case of Peaucellier's problem. Mr. Routh discussed 

 Laplace's problem of three particles. Laplace showed that if 

 three particles be placed at the corners of an equilateral triangle 

 and be properly projected, they will move under their mutual 

 attractions so as always to remain at the angular points of an 

 equilateral triangle. On the supposition that the law of 

 attraction is the inverse k\\i power of the distance, Mr. 

 Routh .irrives at the following results : — i. The motion can- 

 not be stable unless k is less than 3. 2. The motion is 

 stable, whatever the masses may be, if the law of force be ex- 

 pressed by any positive power of the distance, or any negative 

 power less than unity. For other powers the stability will depend 

 on the relation between the masses. 3. Tl;e motion is stable 



to a first approximation if --,- — ; , >3 ( -1 



^^ Mm + Mm' -I- vim/ \3 - /.■/ 



where AF, in, m' are the masses. This agrees with a result given 

 by M. Gascheau (in a paper not seen by the author), if I; = 2^ 

 or the law of force be the law of nature. 4. When two of the 

 masses are much smaller than the third, the inequality in their 

 angular distances, as seen from the large body, has a much greater 

 coefficient than their linear distances from the same body. 5. On 

 proceeding to a second approximation it would seem that the form 

 of the triangle joining the three particles is very little altered by 

 any disturbance, but in certain cases, depending on the nature 

 of the disturbance, the size of the triangle may be subject to very 

 considerable variations. As a supplement, Mr. Routh generalises 

 the reasoning of the problem of the three bodies so as to obtain 

 the form of the determinantal equation to find the periods of 

 oscillation of any dynamical system about a state of steady motion 

 in which the vis viva is constant. Two limitations are made : 

 first, the system must be under a conservative system of forces ; 

 and, secondly, the vis viva can be expressed in terms of the co- 

 ordinates, so as not to contain the time expHcitly. The equation 

 is then shown to be always of an even order, and the condition of 

 stability is that all the roots should be real and negative, — The 

 results arrived at in Mr. Griffiths' note on some rel.itions between 

 certain elliptic and hyperbolic functions may be thus stated : — 

 Let E, F, J I stand for the integrals 



I Ji - e-sin.'^etie, I --= =^ 



J . J Ji -e^tm.-e 



I 



v't''-cosec."e - I cosec. OiiS 



respectively, the limits), in each case being Co to 9, and c/ = 1, 

 then 



and 



£-(i-'^)^ + (vT^ 



i"'- sin.- tf cot. 6 



)>' 



E + E' 



sin. e sin. ^ fin. Po sin. <^^ 



N + //' 



where the limits' in E', II' in <to to 1^, determined from the 

 equation — 



COS. cos. <;> — shi. sin 1^ \/i — /'' tin.-ju = cos. /» 

 = cos. e., cos. q>^ — sin. Bo sin. (p^ ^j - t-^ sinTV, 

 fi being a constanf. 



Geological Society, Feb. 10. — Mr. John Evans,' F'.R.S , 

 president, in the chair. — The following communications were 

 read : — 'Ihe phosphorite deposits of North Wales, by Mr. D. 

 C. Davies. The deposit of phosphate of lime described by the 

 author is a bed varying from ten to fifteen inches in thickness, 

 which occurs at the top of the Bala limestone over a consider- 

 able district in North Wales, having been detected in various 

 localities Irom Llanfyllin to the hills north and west of Dinas 

 Mawddy. The bed is rendered black by the presence of 

 graphite, and appears to consist of concretions of various sizes 

 cemented together by a black matrix. The concretions are 

 richest in phosphate of lime, some of them containing 64 per 

 cent. ; the average amount in the bed, including the matrix, is 

 46 per cent. The deposit is underlain by a bed of crystalline 



limestone, and sometimes divided by thin beds of similar lime- 

 stone into two or three layers. The author noticed the principal 

 fossils occurring in the Bala limestone below the phosphorite 

 beds, and stated that many of those in the overlying shales, up to 

 a certain distance above the bed, are phosphatised. The author 

 referred to the presence of phosphate of lime in the inner layers 

 of Unio and Anodonia to the amount of as much as 15 percent., 

 and thought that the phosphate of lime in the deposit was pro- 

 bably of organic origin. It may have been an old sea-bottom 

 on which the phosphate of lime of MoUusca and Crustacea was 

 accumulated during a long period, and seaweeds may also have 

 contributed their share. It probably represented the remains of 

 an ancient Laminarian zone. The author suggested that the 

 phosphatic nodules of the so-called coprolite beds in other parts 

 of England might have been derived from tlie denudation of 

 similar deposits. — On the bone-caves in the neighbourhood of 

 Castleton, Derbyshire, by Rooke Pennington, LL.B. ; commu- 

 nicated by Prof W. Boyd Dawkins, F'.R.S. The author de- 

 scribed as a prehistoric cave the Cave Dale Cave, situated in 

 Cave Dale-, just below the keep of Peveril Castle. The uppei 

 earth in this cave contained fragments of late pottery mixed up 

 (by rabbits) with bits of rude prehistoric pottery, a tooled piece 

 of stag's horn, an iron spike, two worked flints, a piece of jet, 

 part of a bone comb, and a bronze celt of peculiar form, many 

 bones of B<>s longifrons and goat, broken to get out the marrow, 

 and remains of hogs ; chaicoal and human teelhalso attested the 

 occupation of the cave by man. Tliere were also remains of fox, 

 badger, cat, water-rat, dog, red deer, duck, fowl, and hare. 

 Lower down were remains of Bos longifrons, hog, red deer, 

 wolf, and horse ; and lower still, next the rock, more human 

 teelh, remains of animals, and a good flint. The cave seemed 

 to have been occupied Irom time to time during a lengthened 

 period, probably from the Neolithic age into those of bronze and 

 iron. A cave in .Gelly or Hartle Dale contained, in blackish 

 mould, bones (some broken) of goat, pig, fox, and rabbit, and 

 pieces of very rude prehistoric pottery. Of Pleistocene caves 

 and fissures the author described several. One in Hartle Dale 

 furni'hed remains of rhinoceros, aurochs {Bison pyiscus), and 

 mammoth, lying in yellow earth. The bones were probably 

 carried in by water. A fissure near the village of Waterhouses, 

 in Staffordshire, is six feet wide, and filled with the ordinary 

 loam. Bones of mammoths and the skeleton of a young bison 

 have been obtained fiom it, .and the author supposes the animals 

 to have fallen into the fissure while making for the river to drink. 

 The Windy Knoll fissure is situated near Castleton, in a quarry 

 near the top of th^ Winnetts, and close to the most northern 

 boundary of the mountain limestone of Derbyshire. The author 

 described particida-ly the situation of this fissure and drainage of 

 the district in which it is situated. The fissure itself is filled 

 with the ordinary loam, containing fragments of limestone, and 

 enclosing an astonishing quantity of bones of animals confusedly 

 mixed together, those lowest down near the rocks being coated 

 with and sometimes united by stalagmite. The author supposes 

 that this was a swampy place into which animals fell from time 

 to t me, and in rainy seasons their remains might be washed into 

 it from the neighbouring slopes. — The Mammalia found at 

 Windy Knoll, by Prof. W. Boyd Dawkins, F. R.S. This paper 

 contained an enumeration of the remains of Mammalia found in 

 the Windy Knoll fissure described by Mr. Pennington. They 

 were stated to belong to the following species : bison, reindeer, 

 grisly bear, wolf, fox, hare, rabbit, and water-rat. Great quan- 

 tities of bones and tee'.h were found, the number of individuals 

 represented by the remains being given roughly by the author as 

 follows ; — 



Bison 40-60 



Reindeer 20-30 



Grisly bear 4-5 



Wolf 7 



From the great excess of herbivorous forms, and the position of 

 the fissure, the author assumed that the latter lay in the line of 

 the annual migrations of the bison and reindeer, during which 

 some individuals might fall in ; and he explained the presence of 

 the carnivores by their having followed the migratory herds in 

 order to prey upon stragglers, as is now the case with the rein- 

 deer in Siberia and the bison in North America. He further 

 showed, from the examination of the young teeth of the bison 

 and the reindeer, that these animals must have passed this way 

 at different seasons of the year, and indicated that the deposit 

 must be regarded as of Pleistocene age, though whether pre- 

 or post-glacial is an open question. 



