June 2a. 1920] 



NATURE 



511 



Mathematics of Elasticity. 



A Treatise on the Mathematical Theory of Elas- 

 ticity. By Prof. A. E. H. Love. Third edition. 

 Pp. xviii + 624. (Cambridge : At the University 

 Press, 1920.) Price :i7s. 6d. net. 



THIS is the third edition of the classical 

 treatise in the English language upon the 

 theory of elasticity, and all students of the subject 

 will be grateful to Prof. Love for having brought 

 his masterly exposition of this difficult but fascin- 

 ating branch of mathematical science up to date. 

 The book is now thoroughly well established as 

 part of the education of such mathematicians as 

 have to deal with the engineering or physical 

 applications of elasticity; indeed, it needs some 

 hardihood, even for a specialist, to criticise it ; 

 every fresh perusal convinces the careful reader 

 of the inadequacy of his own knowledge. 



The changes made in this edition have been 

 slight in appearance, but attention to details will 

 reveal many improvements in both form and 

 matter. The numbering of the sections has not 

 been changed, which is of great help to those 

 students who have learnt' to use the second 

 edition. New sections have been interpolated here 

 and there, and these have been distinguished by 

 a letter — e.g. 79A. 



As previously, a great many references are 

 given to the experimental side of the subject, and 

 very rightly, for in many ways the mathematical 

 theory of elasticity is more closely verified by 

 experience (where verification has been seriously 

 attempted) than the cognate theory of hydro- 

 dynamics. \\'here discrepancies have occurred 

 they can usually be traced either to the inherent 

 difficulty of obtaining an exact mathematical solu- 

 tion of the problem, or to unintelligent experi- 

 menting. Far too much experimental work, for 

 example, has been done with systematic disregard 

 of the elastic limits, or without due precautions, 

 in anticipation of 5 per. cent, accuracy. For 

 various reasons, the engineer does not find it con- 

 venient to isolate effects, and he rarely carries 

 out experiments for the express purpose of testing 

 a mathematical theory. Thus what may be called 

 the physics of elasticity has advanced compara- 

 tively little. The methods of photo-elasticity, first 

 used by Clerk Maxwell, who applied the effect of 

 stress on polarised light (discovered by Brewster) 

 to the investigation of stress-distributions, and 

 recently developed as a working engineering 

 method by Prof. Coker, promise to do much to 

 remove this reproach and to get rid of the diffi- 

 culty mentioned by Prof. Love that "the com- 

 ponents of stress or of strain within a solid body 

 NO. 2643, VOL. 105] 



can never, from the nature of the case, be 

 measured directly " (p. 94). 



New sections have been added in chap. iv. on 

 the results of Hopkinson and Sears concerning 

 stresses maintained for a very short time, and 

 also on elastic hysteresis. The term "perfect elas- 

 ticity " to denote that condition in which the 

 stress-strain diagram is closed, althoup-h the load- 

 ing and unloading graphs do not coincide, seems 

 unfortunate, as elasticity can scarcely be called 

 perfect when elastic energy is being dissipated. 

 " Perfect recovery " might denote this case, " per- 

 fect elasticity " being reserved for the condition in 

 which loading and unloading graphs coincide. 

 "Linear elasticity" explains itself, but surely the 

 statement on p. 113, given on the authority of 

 Bauschinger, that the limits of linear elasticity 

 are higher than those of perfect recovery, can 

 scarcely be right, since the former condition should 

 imply the latter. An important appendix has been 

 added at the end of chap. ix. on Volterra's theory 

 of dislocations in the case of multiply connected 

 bodies. A simpler proof of Weingarten's theorem 

 that the discontinuities in the displacements on 

 crossing a "barrier" correspond to a rigid body 

 displacement can, however, be given. For if 

 Mq, i'o, W(, be one value of the displacement at 

 a point P, and Wj, v^, w^ the displacement at the 

 same point P after describing an irreducible 

 circuit, Mj — Wo, v^—Vq, u'j — u'^ are solutions of the 

 equations of elasticity which necessarily (since the 

 strains are supposed one-valued) correspond to 

 zero strain everywhere, and such displacements 

 must be rigid-body displacements. In this con- 

 nection it would make things clearer for the be- 

 ginner if in the proof of the uniqueness theorem 

 given in § 118 the limitations as to the nature of 

 the functions and the simply connected quality of 

 the space were stated. Todhunter and Pearson 

 have pointed out that the existence of more than 

 one solution for a multiply connected body is 

 immediately evident to anyone who turns a short 

 piece of indiarubber tubing inside out. The real- 

 isation of this fact is apt to shake the student's 

 faith if warning has not been given. 



In the chapter on the sphere a very valuable 

 new section gives the alternative method 

 developed by the author in his essay on " Some 

 Problems in Geodynamics," and another section 

 gives a number of new and important references 

 to work on geophysical problems, a branch of 

 elasticity which is assuming nowadays an increas- 

 ing importance. The work of Lamb and of G. W. 

 Walker in c6nnection with seismology is. noticed 

 on p. 314. 



§§ 226A and 226B deal with the torsion of a 



