December 31, 1908] 



NATURE 



267 



lions, if any." The object is to prepare a bibliography 

 of contributions to science teaching in the last decade " that 

 will be a working basis for any teacher of science, and 

 especially for any in an institution with limited library 

 facilities." Since reviews of recent publications on science 

 tcachinfj are valuable in making up programmes of study, 

 tliis bibliography should be an aid in this way, and should 

 iiicourage the study of the literature of the subject. For 

 'onvenience and effectiveness in covering the whole field 

 of science teaching, specialists have been appointed to 

 undertake the work in six subdivisions. The federation 

 has already a membership of more than 1600, and is the 

 most representative body of teachers of science in America. 



SOCIETIES AND ACADEMIES. 



London. 



Royal Society, June 25. — "Note on the Instability of 

 Tubes subjected to End Pressure, and on the Folds in a 

 Flexible Material." By A. Mallock, F.R.S. 



When a straight rod is subjected to end compression 

 it is stable for small lateral displacements unless the com- 

 pressing force exceeds a definite limit, depending on the 

 elastic constants of the material of the rod and its length 

 and cross-section dimensions. 



If this limit is exceeded the rod is unstable, and the 

 least departure from straightness grows under the action 

 of the force, the axis of the rod then taking the form of 



assumed by the deformed tube depends on the ratio [h r) 

 of the thickness of the walls to the diameter, and will be 

 such that the potential energy of the combined bending 

 and shearing involved may be a maximum. 



If the crushing is continued until the tube is greatly 

 reduced in length the folds are seen to develop into the 

 symmetrical shapes shown in the photographs (Figs, i, 

 2, 3), for which n=i, 2, and 3 respectively. For )i = i 

 the folds are circular in plan and independent of 6 ; when 

 n = 2 the plan of the folds is a square, and when n = 3 the 

 plan is hexagonal. 



It may be noticed that the instability always shows itself 

 first at one end, and that since the reaction against end 

 pressure decreases as the deformation goes on, each fold 

 is completed in succession, the next not becoming marked 

 until the reaction is increased by the previous fold resting 

 against the last but one. 



Tidal Bores." By Lord 



one of the well-known elastic curves, and this is the only 

 form which a solid rod can take in the circumstances. 



With tubes and plates, however, the case is different, 

 for with the tube the ratio of the thickness of the walls 

 to the diameter of the tube has to be considered as well 

 as the ratio of the diameter to the length. Thus a tube 

 the length of which is insufficient to produce instability 

 involving a bending of the axis may become 

 unstable by the crumpling up of the walls, 

 the a.xis itself remaining straight. 



In the case of solid rods the governing ^__ 



condition is the constancy (to the first order) ^ 



of the length of the axis ; with tubes and > 



plates it is the constancy to the same order 

 nf the area of the mid-wall surface. Con- 



lering the case of tubes in rather more 



•.■lil, take the axis of the tube as z and let its unstrained 

 ; iflius be r,. 



Under end compression the surface may become unstable 

 by deformation into any of the cylindrical harmonics of 

 thr> type 



r = r,, + a cos /; 9 cos "-;, 



where 6 is the angle which r makes with a fixed diameter 

 of the tube and \ the length of the fold parallel to the 

 axis. The order of the harmonic which will naturallv be 



November 5. — " Note or 

 Rayleigrh, O.M., Pres.R.S. 



It was shown long ago by Airy that when waves advance 

 over shallow water of depth originally uniform, the crests 

 tend to gain upon the hollows (see also " Scientific 

 Papers," vol. i., p. 253, 1S99), so that the anterior slopes 

 become steeper and steeper. Ultimately, if the conditions 

 are favourable, there may be formed what is called a bore. 

 Ordinary breakers upon a shelving beach are of this 

 character, but the name is usually reserved for tidal bores 

 advancing up rivers or estuaries. Interesting descriptions 

 of some of these are 

 given in Sir G. Dar- 

 vi/in's " Tides " (Mur- 

 ray, 1898). 



Although the real 

 bore advances up the 

 channel, we may for 

 theoretical purposes 



" reduce it to rest " by 

 superposing an equal 

 and opposite motion 

 upon the whole water 

 system. We have then 

 merely to investigate 

 the transition from a 

 relatively rapid and 

 shallow stream of depth 

 I and velocity « to a 

 deeper and slower 

 stream of depth I' and 

 velocity n' (Fig. i). 

 The places where these 

 '^""- 5- velocities and depths 



are reckoned are sup- 

 posed to be situated on the two sides of the bore, and at 

 such distances from it that the motions are there sensibly 

 uniform. The problem being taken as in two dimensions, 

 two relations may at once be formulated connecting the 

 depths and velocities. By conservation of matter (" con- 

 tinuitv ") we have 



?» = ZV; (i) 



and since the mean pressures at the two sections are hgl. 

 hgl', the equation of momentum is 



lu{u-u') = igir--l-); 

 /hence 



u'- = ig{i+r).i'ii, u''=ig{i+i').iii'. 



The loss of energy per unit time at the bore is thus 



III (hi- + i -/) - /// (A//- + L"-'") = I't -sil' -I)' 



All' 



(4) 



That there should be a loss of energy constitutes no 



N"0. 2044, VOL. 79] 



