Mav 25. 1S93] 



NA TURE 



95 



Mathematical Society, May n. — Mr. A. B. Ketnpe, 

 F.K.S., president, in the chair. — The followingjcommiinica- 

 lions were made : — On the collapse of boiler flues, by A. E. H. 

 Love. The problem consists in discovering the conditions of a 

 collapse of a thin cylindrical shell under external pres-ure, 

 when the ends are co: strained to occupy fixed positions. Since 

 all problems of collapse depend on the geometrical possibility 

 of finite displacements accompanied by only infinitesimal strains, 

 it appears at the outset that unless the shell can receive a dis- 

 placement of pure bending without stretching of the middle 

 surface collapse is impossible. The assumed condition of no 

 terminal displacement is equivalent to closing the ends of the 

 shell, and, since a closed surface cannot be bent without stretch- 

 ing, this condition apparently precludes the possibility of 

 collapse. On the other hand it is well known that, if the 

 external pressure exceeds a certain value, an infinitely long 

 cylindrical shell of given small thickness and given diameter 

 will collapse under the pressure. The critical pressure has been 

 determined by Bryan and Basset, who find the same result. It 

 is therefore to be expected that, if the cylinder is of sufficient 

 length, the extensional displacement which must be superposed 

 upon the displacement of pure bending in order to satisfy the 

 end conditions will be practically unimportant, except in the 

 neighbourhood of the ends. The problem is thus reduced to 

 <liscovering the order of magnitude of the length of the shell in 

 order that it may be treated as infinite when the thickness is 

 i-mall. For this purpose consider the case where the pressure is 

 just equal to the critical pressure, and the displacement of pure 

 bending in the infinite cylinder is consequently of the form 



o, 



= JA cos 2^), 



A sin 2i^, 



where A is a small arbitrary constant. The displacement u is 

 parallel to the generator, v is along the circular section, and 

 w along the radius outwards. By me.ins of displacements of 

 this form the equations of equilibrium can be satisfied, but the 

 boundary conditions at the ends cannot. Now take the case of 

 an infinite cylinder with an end x = o, at which v and w must 

 vanish, and seek a displacement involving both flexure and 

 extension of the middle surface to be superposed on the dis- 

 placement given by the above form, such displacement to satisfy 

 the equations of equilibrium ami the boundary conditions : — 

 (l) that the new v and tc are equal and opposite to tho5e above 

 given at A = o ; (2) that the new u, z\ -v vanish at j; = 00. The 

 lequired solution can be determined and is of the form 



« = e""" (Ai cos vix + Bi sin mx) sin 2cp, 



V = e'""' {A2 cos mx + Bo sin ntx) cos 2(p, 



,„,. nr d'^ ,„ A . , ■ 



- — - ( B„ cos titx — Aj sm mx) sm 2i^, 



4U + <r) 



in which Bo and Ao can be determined so as to satisfy the con- 

 ditions at r = 0, 0- is the Poissoo's ratio of the material of 

 the shell, and 



m = [12(1 -.r-^m^idi), 



t is the thickness and d the diameter of the shell. If a be 

 taken equal to \ the reciprocal of fit is about '546 of the mean 

 proportional between the thickness and the diameter, and it 

 follows that whenever x is great compared with this quantity 

 the influence of the end is unimportant, and the displacement 

 approximates to one of pure bending. To make the tendency 

 10 collapse occur in practice, it would be necessary that the half 

 length of the flue be great compared with m~^, and the prac- 

 tical conclusion would be that for a flue of length / stability 

 would be secured if 



ill < «//«, or / < N V (dl), 



where N is a considerable number. It is customary in sta- 

 tionary boilers to make the flues in detached pieces connected 

 by massive flanged joints, so that the effective length of the 

 flue is the distance between consecutive joints. If the number 

 N be taken equal to 12 we have the rule that the distance 

 between the joints must be not greater than twelve times the 

 mean proportional between the thickness and the diameter. 

 The value N = 12 accords well with what has been found safe 

 in practice, but the rule as to spacing the joints is new. — 

 On some formula: of Codazzi and VVeingarten in relation to the 

 application of surfaces to each other, by Prof. Cayley, F. R..S. — 

 On the expansion of some infinite products, by Prof. L. J. 

 Rogers, — On a theorem' for bicizcular'quarlics and for cyclides 

 corresponding to Ivory's theorem for conies and conicoids, by Mr. 

 A. L. Dixon. Using a form of the equation to these curves and 



NO. 1250, VOL. 48] 



surfaces (in quadricircular and penlaspherical co-ordinates) 

 already studied by Darboux and Casey, the writer deduced that 

 the ratio of the distance of any two points to the product of 

 the lengths of the tangents from them to a fixed focal circle or 

 sphere is the same as for the pair of corresponding points. 

 He also showed how the theorem for the Cartesian oval could 

 be derived from its equation in terms of elliptic functions. — 

 A supplementary note on complex primes formed with the fifth 

 roots of unity, by Prof. Lloyd Tanner. The author investigates 

 a method of determining whether a complex number is 

 prime or composite. The process takes two distinct forms, 

 one of which was established, on different grounds, by 

 TchebichefT. The other appears to be new, and is 

 convenient in testing the sets of complex integers described in 

 the author's previous communication on the subject. The 

 discussion is based upon a certain classification of complex integers 

 according to the "orders" of their complexity, and this con- 

 ception facilitates the direct factorization of complex numbers. 

 The theory is restricted to the case of 5, but seems to be 

 quite general. — On the linear transformations between two 

 quadrics, by Mr. H. Faber. In Creile's Joitrtial, vol. v. (also 

 Phil. Trans,, 1S58), Cayley gave a representation of the auto- 

 morphic linear transformation of the uniparlite quadric func- 

 tion in the notation of the theory of matrices. In the present 

 paper the author extends Cayley's method to the determination 

 of the general linear transformation of a given quadric into 

 another given quadric, and applies the results to the determina- 

 tion of the general real linear transformation between two 

 equivalent quadrics and to the reduction of a quadric to a sum 

 of squares. The determination by this method of the general 

 linear transformation between two quadrics depends upon the 

 solution of an algebraic equation of the «"' degree, to which 

 the problem as it originally presents itself — viz., the solution of 

 a system of n" quadratic equations in «'- variables, is thus 

 reducible. — On maps and the problem of four colours, by 

 Prince C. de Polignac. — On Fermat's proof of the problem 

 that primes of the form 4« + I can be expressed as the sum of 

 two squares, by Mr. S. Roberts, F. R. S. 



Entomological Society, May 10, Mr. Henry John Elwes. 

 President in the chair. — Mr. R. McLachlan, F.R.S., exhibited 

 for Dr. Frilz-Miiller, of Blumenau, Santa Catarina, Brazil, 

 specimens of larvas and pupje of a dipterous insect, and read a 

 letter from Dr. Fritz-Miiller on the subject. The writer stated 

 that the larvae were similar to those exhibited by Mr. Gahan, at 

 a meeting of the society in October, 1890, and which were then 

 thought by Lord Walsingham, F. R.S., and Mr. McLachlan, 

 to be allied to the Myriapoda. — Mr. S. G. C. Russell exhibited 

 specimens of Hcspcria alveolus, including one of the variety 

 Taras. taken by him at Woking in April last. — Mr. J. M. 

 Adye exhibited a long series of Moma orion, Ettryineite dolo- 

 braria, Aiitphidasis helnlaria, Cloethora prasinana, and a few 

 specimens of Nolodonta dodonea, N. chaonia, and N. trefida, 

 Acroiiycta aliii, and Sclcnia illustraria, all bred by him in 

 March and April last, from larvoe obtained in the autumn of 

 1892 in the New Forest. — Mr. H. Goss read a copy of a letter 

 received by the Marquis of Ripon, at the Colonial Office, from 

 the Governor of the Gold Coast, reporting the occurrence of 

 vast swarms of locusts at Aburi and Accra, West Africa, about 

 the middle of February last. The writer stated that at Accia 

 the swarm extended from east to west as far as the eye could 

 see, and appeared to occupy a space about two miles wide and 

 from a quarter of a mile to a mile in height. — Colonel Swinhoe 

 stated that some years ago he had been requested by the Indian 

 Government to report on plagues of locusts. He said he had 

 witnessed swarms of these insects far larger than the one just 

 reported from the Gold Coast, and mentioned that many years 

 ago, when going up the Red Sea in one of the old V. and O. 

 paddle boats, the boat had frequently to stop to clear her 

 paddle-wheels from locusts, which had settled in such 

 swarms as to choke the wheels and stop their action. — 

 Mr. E. C. Reed, of Valp.araiso, Chili, communicated a papar 

 entitled " N oies on Acndiuiii paranensc, the migratory locust of 

 the Argentine Republic." Colonel Swinhoe, Mr. Champion, 

 Mr. Elwes, Mr. McLachlan, and Mr. Merrifield took part 

 in the discussion which ensued. — Prof. L. C. Miall, F. R. S., 

 communicated a paper entitled " Dicranota ; a Carnivorous 

 Tipulid Larva." — Dr. T. A. Chapman communicated a paper 

 entitled "On a Lepidoptercus pupa (Micropteryx furpurelld) 

 with functionally active mandibles." Mr. McLachlan said he 

 thought Dr. Chapman's observations were of great value, and 



