796 



SCIENCE. 



[Vol.. II., No. 46. 



of finding the displacements in all parts of an 

 elastic solid of any figure sulijected to the action 

 of known forces applied to its exterior surfaces, 

 even wlien the solid is uniform in texture in all 

 directions (i.e., isotropic), transcends at present 

 the powers of analysis, tliough considerable 

 progress has been made toward a complete 

 theory. An important contribution to this 

 theory bj' Sir William Thomson is found on 

 pp. 461 to 408 in Appendix C, entitled 'Equa- 

 tions of equilibrium of an elastic solid deduced 

 from the principle, energy.' 



Bj- reason of the incompleteness of the gen- 

 eral theory, those simple cases are first treated 

 which are most completely amenable to analj*- 

 sis. The fortj^ pages succeeding p. 130 treat 

 the special case of the elastic wire, whose 

 fundamental equations were first thoroughlj' 

 investigated by Kirchhoff in 1859. This treat- 

 ment, which is of interest both to the mathe- 

 matician and engineer, investigates not only the 

 spirals which elastic wires of circular and of 

 rectangular cross-section assume under the 

 action of direct forces, and of couples produ- 

 cing bending and twisting, but also goes into 

 several important side-issues, one of which is 

 the so-called kinetic analogy. A simple case 

 of this, which is discussed at length, exists 

 between the plane curves assumed by a thin 

 flat spring, and the vibrations of a simple pen- 

 dulum which it graphicallj' represents. An- 

 other important side-issue is found in the 

 discussion of the common spiral spring, in 

 which the force resisting elongation is mostly 

 due to torsion of the wire. Verj- curiousl}', the 

 theorem of three moments of a straight beam 

 is omitted, although the principles to be em- 

 plo3'ed in establishing it are fully given. 



Another important elastic solid which is fully 

 amenable to analysis is the thin elastic plate. 

 The treatment of the thin plate, which occupies 

 thirt}' pages, discusses the flexure of a plane 

 plate under all combinations of forces tending 

 to produce either a state of synclastic stress 

 (i.e., a state in which the curvature at every 

 point is convex) or a state of anti-clastic stress 

 (i.e., one which tends to cause the surface to 

 become saddle-shaped) . Kirchhofl!"s boundarj' 

 conditions for a plate are also demonstrated at 

 length. These are of importance in most prac- 

 tical cases, — as, for example, that of the flat 

 steam-boiler head ; for evidently any plate 

 must have some kind of support or fastening at 

 its boundary-. 



The general subject of elastic solids is reached 

 at p. 204, and occupies a hundred pages, in 

 which, after the general equations of equilib- 

 rium between the applied stresses and the result- 



ing strains are established, several special cases 

 are treated at length. The first of these is the 

 celebrated torsion problem published by St. 

 Venant in 18;");') ; in which tlie distribution of 

 the stresses and strains throughout a right 

 prism of any cross-section wliatever, under the 

 action of forces applied to its ends, is com- 

 pletely determined. This is perhaps the most 

 complicated problem which has been entirel}' 

 worked out in the subject of elastic solids, and 

 twenty-four pages are devoted to it. The flex- 

 ure of beams having rectangular cross-sections 

 is discussed, especiall}' with reference to the 

 distortions which are suflTered by these cross- 

 sections. The distortions can be easily exhib- 

 ited b}' bending a thick rectangular piece of 

 rubber, when the upper and lower surfaces will 

 become saddle-shaped. 



The general problem is then further treated 

 by investigating the case of an infinite elastic 

 solid under various suppositions as to the force 

 applied through limited and through unlimited 

 portions of it. The spherical and cyliildrical 

 shells are then treated bj* the help of harmonic 

 analj'sis. 



The concluding hundred and sixty pages of 

 the work, beginning at p. .300, are devoted 

 ostensibly to hydrostatics ; but the first twentj-- 

 five pages finish those parts of the subject in- 

 cluded under that title in ordinary treatises, 

 and the remainder relates to the ph3-sics of the 

 earth as dependent upon its fluid condition, 

 past or present. The first great problem in this 

 department of inquirj' is to determine what fig- 

 ure will be assumed hj a rotating liquid mass 

 under the influence of centrifugal force and of 

 the mutual gravitation of its parts. That an 

 oblate spheroid is a figure of equilibrium for 

 such a mass is commonly known, having been 

 shown to be such by Newton ; but that an ellip- 

 soid with three unequal axes is also such a fig- 

 ure is not so commonly known, tliough this was 

 discovered to be the fact b}- Jacobi in 1834. 

 There are other possible figures, stable and 

 unstable ; but which of all these is the one 

 which will actually be assumed in any given 

 case ? In reply to this question, the authors 

 state, that "during the fifteen years whicli have 

 passed since the pubheation of the first edition 

 ■we have never abandoned the problem of the 

 equilibrium of a finite mass of rotating incom- 

 pressible fluid. Year after year, questions of 

 the multiplicity of possible figures of equilib- 

 rium have been almost incessantl}' before us ; 

 and 3'et it is only now, under the compulsion 

 of finishing this second edition of the second 

 part of our first volume, with the hope for a 

 second volume abandoned, that we have sue- 



