NATURE 



505 



THURSDAY, APRIL 3, 1902. 



ELASTICITY FOR ENGINEERS. 

 Resistance des Maiei-iaux et Eldmenis de la Theorie 

 matltematique de PElasticiti!. Par Aug. Foppl. Traduit 

 par E. Hahn. Pp. 490. (Paris : Gauthier-Villars, 1901.) 

 Price fr. 15. 



DISCUSSIONS as to the amount of "learning of 

 mathematics " necessary for the engineer have not 

 been confined to this country, but have, as we learn from 

 the preface, been keenly carried on in Germany. There 

 appears to be a desire in some quarters to sacrifice mathe- 

 matical teaching to laboratory work, on the ground of 

 the great increase in the number of hours necessary for the 

 Latter in consequence of modern developments of electro- 

 technics ; and some writers have attempted to bring for- 

 ward as an argument the rarity of the occasions on 

 which mathematics is required by the engineer, and the 

 large amount of time that is required to obtain a know- 

 ledge of that subject. 



Prof Foppl takes an entirely different view. While 

 admitting that no advantage is gained by the study of 

 purely speculative branches of mathematics, he is 

 directly opposed to any tendency to lower the general 

 level of mathematical knowledge of our future engineers. 

 He lays stress on the value of mathematical training in 

 enabling the practical man to draw correct conclusions on 

 the numerous questions on which he is required to pro- 

 nounce judgment, even where actual calculations are not 

 required. 



This is a view which ought to be more keenly appre- 

 ciated among practical men than has been the case 

 hitherto, and we should like to see put forward an entirely 

 opposite suggestion to that which has apparently found 

 favour in some circles in Germany, namely that more 

 attention should be given to mathematics in the training 

 of engineers eveti at the expense of laboratory work. It is 

 sufficient to take up any volume of engineering trans- 

 actions to find the "practical" man "plunging" into 

 pages and pages of elaborate formulas involving " shines 

 and coshes," and in the end either getting no result of 

 interest, or obtaining a simple deduction of some principle 

 well known to every mathematician, but which the writer 

 of the paper puts forward as if it were a new physical 

 law. On the other hand, the engineer who has an inti- 

 mate knowledge of pure and applied mathematics will 

 know, for example, what partial differential equations are 

 involved in the solution of any problem placed before 

 him, and even if he knows that it is impossible to solve 

 these for the particular type of boundary with which he 

 has to deal, he will form a complete mental picture of 

 the machinery underlying the system he is investigating; 

 he will mentally classify the known and unknown 

 quantities involved in the problem, and will see at a 

 glance the right lines on which to determine the unknown 

 quantities experimentally, instead of spending hours in 

 floundering through formulre or wasting money over 

 superfluous experiments. 



The theory of elasticity probably enters into the work 

 of the engineer more intimately than any other branch of 

 applied mathematics, and, at the same time, there are 

 NO. 1692, VOL. 65] 



few subjects so difficult to present to the learner in a form 

 that will furnish him with a definite mental picture of 

 the phenomena concerned. So far as we are aware, no 

 one has, as yet, ever attempted to construct diagrams of 

 the deflections of a rectangular sheet of steel plating by 

 dotting it over with microscopists' cover-glasses, silvered 

 on the back, allowing these to reflect light from a source 

 on to a screen and observing or photographing the dis- 

 placements of the bright spots when one side of the plate 

 is submitted to pressure or a weight is applied at any 

 point of its surface ; yet this would not be difficult to do, 

 and would certainly give interesting results. 



The present work is the French translation of a book 

 written by Prof Foppl expressly for the purpose of giving 

 engineers an insight into the mathematical theory of 

 elasticity. It commences with a chapter on the analysis 

 of stresses, which we regret to see entitled " Forces in- 

 terieures ou actions moleculaires." When will writers 

 adopt some consistency in their use of the words molecule 

 and molecular ? In text-books on hydrodynamics, the 

 term molecular rotation is often found applied to a 

 quantity which by no means represents the rotation of the 

 actual molecules of the fluid, but is merely the curl of the 

 velocity. Thus it follows that the " molecular rota- 

 tion" of Basset and other text-book writers vanishes 

 when a fluid is moving irrotationally, while the kinetic 

 theory of gases tells us that in a mass of gas at rest the 

 molecules are in rapid rotation, the kinetic energy of the 

 true molecular rotation bearing a determinate ratio to 

 that of molecular translation, except probably in the case 

 of a monatomic gas. Similarly M. Foppl takes no 

 account of the molecular structure of the body, there is 

 no reference to Boscovich's or any other hypothesis, and 

 his "molecular" actions, so-called, are nothing more or 

 less than the ordinary stresses in the element dx dy dz. 

 Seeing that dx dy dz is called an element, why not con- 

 sistently use the term "elemental" or "elementary" for 

 such actions? In the next chapter, which deals with 

 analysis of strain as opposed to stress, this is done, 

 the nomenclature " deformations elementaires " being 

 adopted. This chapter contains discussions on the elastic 

 limits, Woehler's experiments and diagrams of the rela- 

 tions between stress and strain beyond the limits of 

 Hooke's law. The third chapter is devoted to flexure of 

 beams, and it includes digressions on moments of inertia 

 and the method of finding them with a planimeter. Of 

 graphical interest is the diagram showmg the enveloping 

 curves of the lines of stress (p. 120). The next chapter 

 deals with the potential energy of deformation, and in- 

 cludes Castigliano's interesting theorems and Maxwell's 

 reciprocal property. 



Chapter v. deals with curved beams or prisms, and 

 the next chapter with the problem (prisms resting on a 

 compressible base) presented by the yielding of railway 

 metals under the weight of a train. This is followed by 

 a chapter on the plane plate, after which come thin and 

 thick shells. The ninth chapter deals with the torsion 

 problem. The solution for a rectangular beam is, 

 however, wrong. The author finds stress components 

 represented by algebraic expressions of the third degree, 

 and shows that these satisfy the stress-equations, but he 

 omits to show that these expressions are compatible 

 with the strain-equations -professedly in order to obtain 



Z 



