NATURE 



529 



THURSDAY, APRIL 6, 1893. 



MATHEMATICAL ELASTICITY. 

 A Treatise on the Mathematical Theory of Elasticity. 

 By A. E. H. Love, M.A., Fellow and Lecturer of St. 

 John's College, Cambridge. Vol. I. (Cambridge: 

 University Press, 1892.) 



MR. LOVE'S treatise is the necessary complement 

 to Todhunter and Pearson's " History of the 

 Theory of Elasticity," in which an abstract is given of 

 all the most important original memoirs bearing on this 

 subject, arranged in historical order. 



But the student who wishes to make himself acquainted 

 with the works of these original authorities, by the 

 guidance of Todhunter and Pearson's History, will find 

 the necessity of an acquaintance with Mr. Love's work 

 as an introduction to the elements and to the notation 

 of the subject of elasticity. 



Mr. Love has prepared an elegant and modern 

 artillery of analysis ; and he is not afraid to fire off his 

 guns. To pursue the simile, there is no fear of the 

 subject being obscured in the smoke of his own guns — 

 in these days of smokeless gunpowder. 



The size of the book is kept within reasonable dimen- 

 sions, compared with the scale of a continental treatise, 

 by leaving the heaviest parts of the analysis as exercises 

 to be worked out by the trained mathematical student, 

 to whom the work is addressed. 



The author says in the preface, " I have not thought 

 it advisable to introduce collections of examples for 

 practice." But such collections do not exist, and the 

 author would find it as formidable a task as that he has 

 already carried out to attempt to construct the examples 

 himself. In the present state of his subject any really 

 novel example would be worthy to take rank as a new 

 and independent theorem. 



The examples which we see around us of the physical 

 and industrial applications of the Theory of Elasticity 

 are the best check in existence to keep the subject from 

 becoming a mere development of pure mathematics, with 

 such generalisations as to space of n dimensions, and 

 based upon physical laws adopted merely because of the 

 analytical elegance they confer, quite apart froni any 

 experimental verification. 



The first five chapters are occupied with the general 

 theory, including the analysis of strain and stress, stress- 

 strain relations, the strength of materials, and a number 

 of general theorems. In the analysis of strain the method 

 of Thomson and Tait's " Natural Philosophy " has been 

 followed, beginning with the geometrical and algebraical 

 theory of finite homogeneous strain, deducing thence the 

 physical state of infinitesimal strain. Hooke's law, made 

 such a mystery of by its inventor, now becomes a necessary 

 consequence of the expansion by Taylor's theorem of the 

 stresses as functions of the displacements and strains, 

 neglecting power above the first or second ; and the law 

 receives ample experimental justification in the observed 

 isochronism of the small vibrations of elastic bodies, as 

 exhibited by the musical notes they give out. 



In the treatment of the bending of a beam and the 

 torsion of acylinder in Chapter VI., Saint- Venant's method 

 NO. 1223, VOL. 47 j 



has been followed, and the warping and distortion of the 

 cross-section carefully investigated and illustrated in 

 fig. 10, p. 156. 



This warping effect is well known to engineers, though 

 hitherto generally ignored in the mathematical treatment, 

 as impairing the sweet simplicity of a bending moment 

 and consequent proportional curvature resulting only 

 from the extension and compression of the fibres, thus 

 ignoring the shearing stresses called into play. We 

 can now begin to perceive the reason why a beam is so 

 much stronger and stiffer than it ought to be according 

 to the ancient theory. 



In the investigation of the torsion of a cylinder, where 

 cross-section is a rectangle, the analysis of Thomson and 

 Tait has been closely adhered to. Dr. Ferrers, the 

 Master of Gonville and Caius College, has made this 

 analysis more complete and symmetrical, and has exhi- 

 bited the hydrodynamical analogies more clearly, by em- 

 ploying a pair of Fourier series, one proceeding by sines 

 and cosines of multiples of x, and the other of/ ; each 

 series representing separately the motion or displace- 

 ment corresponding to a simple shear of the rectangular 

 section. The elliptic function interpretation of this pair 

 of series, in which the corresponding moduli are 

 obviously complementary, is very interesting, but has not 

 been pursued by Mr. Love. 



Nowthat Prof. Karl Pearson has dedicated thefirst part 

 of the second volume of the History to the " Memory of 

 Saint- Venant," the political cloud, vaguely described in 

 M. Bertrand's recent //<?^^of Chasles, which overshadowed 

 Saint- Venant's official career, is clearing off, and full 

 tribute is beginning to be paid in France to the great 

 advances due to him. 



Lam^, too, like Saint-Venant, appears to have lived in 

 official neglect, although his method of Curvilinear Co- 

 ordinates, expounded in Chapter VII., has been a powerful 

 analytical engine for the solution of elastical problems, 

 and his " Theorie de I'lfclasticit^ " is a standard text-book 

 to the present day. 



The solution of the elastic deformation of a sphere, 

 tre ated in Chapter X., is also due to Lamd. Mr. Love 

 applies his analysis to the consideration of the effect ot 

 a flaw in the shape of a spherical cavity, and shows that 

 in this case the engineer's factor of safety of 2 is the 

 theoretically correct factor. 



The most important application on a large scale of the 

 analysis of the elastic deformation of a sphere is the in- 

 vestigation of the effective rigidity of the earth, con- 

 sidered as an elastic solid, under the action of its own 

 gravitation, and slightly disturbed by the rotation and the 

 tide-producing forces. Elaborate calculations and ob- 

 servations have been carried out by Prof, G. H. Darwin ; 

 if we could observe and measure the bodily tides in the 

 earth, an estimate of its rigidity could be obtained. Mr. 

 Love gives the numerical results corresponding to mean 

 rigidities equal to those of steel and glass. 



Mr. Chree's valuable investigations of the strain pro- 

 duced by rotation in an elastic circular disc, in a sphere 

 or an ellipsoid, are introduced here, and receive careful 

 analysis and interpretation. 



Chapter XI. treats of the vibrations of a sphere. The 

 free vibrations have been completely worked out by 

 Prof. Lamb. In the forced vibrations the lag or change 



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