June I, 1876] 



NATURE 



123 



sequence of its having a fall of free-level in passing along that 

 stream-line.^ 



But the layer of water along the bottom, being by friction 

 much retarded, has much less centrifugal force in any bar of its 

 particles extending across the river ; and consequently it will 

 flow sidewise along the bottohi towards the inner bank, and will, 

 part of it at least, rise up between the stream-line and the inner 

 bank, and will protect the bank from the rapid scour of that 

 stream-line and of other adjacent parts of the rapidly flowing 

 current ; and as the sand and mud in motion at bottom are 

 cariied in that bottom layer, they will be in some degree brought 

 in to that inner bank, and may have a tendency to be deposited 

 there. 



On the other hand, along the outer bank there will be a general 

 tendency to descent of surface-water which will have a high 

 Velocity, not having been much impeded by friction ; and this 

 will wear away the bank and carry the worn substance in a 

 great degree down to the bottom, where, as explained before, 

 there will be a general prevailing tendency towards the inner 

 bank. 



Now further, it seems that even from the very beginning of the 

 curve forward there will thus be a considerable protection to the 

 inner bank. Because a surface stream-line c D, or one not close 

 to the l)ottom, flowing along the bank which in the bend becomes 

 the inner bank, will tend to depart from the inner bank at D, the 

 commencement of the bend, and to go forward along D E, or by 

 some such cours?, leaving the space G between it and the bank 

 to be supplied by slower moving water which has been moving 

 along the bottom of the river perhaps by some such oblique path 

 as the dotted line F G. 



It is further to be observed that ordinarily or very frequently 

 there will be detritus travelling down stream along the bottom 



» 



and seeking for resting places, because the cases here specially 

 under consideration are only such as occur in alluvial plains ; 

 and in regiors of that kind there is ordinarily- on the average 

 more deposition than erosion. This consideration explains 

 that we] need not have to seek for the material for deposi- 

 tion on the inner bank in the material worn away from the 

 outer bank of the same bend of the river. The material worn 

 from the outer bank may have to travel a long distance down 

 stream "before finding an inner bank of a bend on which to deposit 

 itself. And now it seems very clear that in the gravel, sand, and 

 mud carried down stream along the bottom of the river to the 

 place where the bend commences, there is an ample supply of 

 detritus for deposition on the inner bank of the river even at the 

 earliest points in the curve v/hich will offer any resting place. It 

 is especially worthy of notice that the oblique flow along the 

 bottom towards the inner hank begins even up stream from the 

 bend, as already explained, and as shown by the dotted line FG 

 in Fig. 3. The transverse movement comprised in this oblique 

 flow is instigated by the abatement of pressure, or lowering of 



'It must be here explained that, by the free- level for any particle, is to 

 be understood the level of an atmospheric end of a column, or of any bar, 

 straight or curved, of particles of statical water, having one end situated at 

 the levtl of the particle, and having at that end the same pressure as the 

 particle has, and having the other end, consisting of a level surface of 

 wattr, freely exposed to the atmosphere, ex else having otherwise atmo- 

 spheric pressure there ; or briefly we may say that the /ree-level for any 

 particle of water is the level of the atmospheric end of its pressure column, 

 or of an equivalent ideal pressure-column. 



" That is to say, except when by geological changes the causes which have 

 been producing the alluvial plane have become extinct, and erosion by the 

 river has come to predominate over deposition. 



free-level, in the water along the inner bank produced by centri- 

 fugal force in the way already explained. 



It may now be remarked that the considerations which have 

 in the present paper been adduced in respect to the mode of flow 

 of water round a bend of a river, by bringing under notice, con- 

 jointly, the lowering of free-level of the water at and near the 

 inner bank, and the raising of free-level of the water at and near 

 the outer bank relatively to the free-level of the water at 

 middle of the stream, and the effect of retardation of velocity in 

 the layer flowing along the bed of the channel in diminishing the 

 centrifugal force in the layer retarded, and so causing that re- 

 tarded water, and also frictionally retarded water, even in a 

 straight channel of approach to the bend, to flow obliquely 

 towards the inner bank, tends very materially to elucidate the 

 subject of the mode of flow of water round bends in pipes, and 

 the manner in which bends cause augmentation of frictional 

 resistance in pipes, a subject in regard to which I believe no 

 good exposition has hitherto been published in any printed books 

 or papers ; but about which various views, mostly crude and 

 misleading, have been published from time to time, and are now 

 often repeated, but which, almost entirely, ought to be at once 

 rejected. 



Mathematical Society, May 11.— Prof. H. J. S. Smith, 

 F.R.S., president, in the chair. — Dr. Login was elected a mem- 

 ber of the Society. — Mr. Tucker communicated a paper by Mr. 

 S. A. Renshaw, on the inscription of a polygon in a conic 

 section, subject to the condition that each of its sides shall pass 

 through a given point by the aid of the generating circle of the 

 conic. The inscription of a polygon in a circle, subject to the 

 like condition, has been accomplished by several eminent geo- 

 meters, in a remarkably easy manner by the late Mr. Swale. 

 The object of Mr. Renshaw's paper is to show how, by an easy 

 transformation, effected by means of the generating circle, the 

 construction of the problem in the circle can be rendered avail- 

 able to the resolution of the same problem in the conic sections. 

 The author draws figures exhibiting the inscription of a pentagon 

 in an ellipse, and of a quadrilateral in a hyperbola. Mr. Renshaw 

 also extends some other properties (for the circle) given by Mr. 

 Swale in the Liverpool ApoUonius (p. 45) to the conic sections. — 

 Prof Cayley then spoke on the representation of imaginary 

 quantities by an («, «) correspondence. The Chairman and 

 Dr. Hirst spoke on the subject of this paper. Prof^ Cayley 

 having taken the chair, the President communicated two notes. 

 The first was on a theorem relating to the Pellian equation. Let 

 D be any integral number, let T and U be the least integral 

 numbers which satisfy the Pellian equation T--D U"^ — i ; 

 and let n^, fi^, flj, . . . fij„ be the period of complete quotients 



of the form '^L — g — ^ which is obtained in the development of 



the root of any quadratic equation of determinant Z? in a con- 

 tinued fraction. The equality 



fii X n.2 X . . . X n^n = r+ U ^D 



was established in the note, and an expression for the number of 

 non-equivalent quadratic forms of determinant D was deduced 

 from it. The second note was on the value of a certain arith- 

 metical determinant. Let (w, n) represent the greatest common 

 divisor of m and n ; and let >^ (///) represent the number of num- 

 bers prime to w, and not surpassing vi ; the equality 



2±(I, I) (2, 2) . . . (w, m) =;f,(l)i|/(2) . . . ^(tn) 

 was established in the note, and several consequences deduced 

 from it. 



Zoological Society, May 16.— Dr. A. Giinther, F.R.S., 

 vice-president, in the chair. — Dr. P. Comrie exhibited and 

 made remarks on the zoological specimens collected by him 

 during the survey of the south-eastern coast of New Guinea by 

 H.M. S. Basilisk, — Dr. Giinther exhibited and made remarks on 

 a collection of Mammals from the coast of Borneo, opposite to 

 Labuan. Among these were especially noticed a young example 

 of a Monkey [Alacacus melanotis) of which the exact habitat was 

 previously unknown, and a new species of Tupaia, proposed to 

 be called T. minor. — Dr. Giinther also read an extract from a 

 letter recently received from Commander Cookson, R.N., stating 

 that he was bringing home from the Galapagos Islands a living 

 pair of the large Land-tortoise, of Albemarle Island. Com- 

 mander Cookson stated that the male of this pair weighed 

 270 lbs., the female 1 17 lbs. — Mr. Sclater exhibited the skin of a 

 rare Pacific Parrot {Coriphilus ktikli), which had been obtained 

 by Dr. T. Hale Streets, U.S. Navy, at Washington Is'and, of 

 the Palmyra group, and had been sent to him for examinatiou 



