416 Sir William Thomson on the 



portion of Green's homogeneous medium left to itself in space 

 will have the same kind of stability or instability according 

 as A/B>4/3, or A/B<4/3. In fact A-jB, in Green's 

 notation, is what I have called the " bulk- modulus" of 

 elasticity*, and denoted by k (being infinitesimal change of 

 pressure divided by infinitesimal change from unit volume 

 produced by it : or the reciprocal of what is commonly called 

 " the compressibility"). B is what I have called the 

 "rigidity," as an abbreviation for " rigidity- modulus," and 

 which we may regard as essentially positive. Thus Green's 

 limit A/B>4/3 simply means positive compressibility, or 

 positive bulk-modulus : and the kind of instability that de- 

 terred him from admitting any supposition of A/B<4/3 is 

 the spontaneous shrinkage of a finite portion if left to itself 

 in a volume infinitesimally less, or spontaneous expansion if 

 left to itself in a volume infinitesimally greater, than the 

 volume for equilibrium. This instability is, in virtue of the 

 rigidity of the medium, converted into stability by attaching 

 the bounding surface of the medium to a rigid containing 

 vessel. How much smaller than 4/3 may A/B be we now 

 proceed to investigate, and we shall find, as we have anti- 

 cipated, that for stability it is only necessary that A be 

 positive. 



5. Taking Green's formula (C), but to make the energy 

 principle which it expresses clearer (he had not even the 

 words " energy/' or " work" !), let W denote the quantity of 

 work required per unit volume of the substance, to bring it 

 from its unstressed equilibrium to a condition of equilibrium 

 in which the matter which was at (x, y, z) is at (x + u, y + v, 

 z + w), u, v, w being functions of x, y, z such that each of the 

 nine differential coefficients dujdx, dujdy, . . . dv/dx, . . . &c. 

 is an infinitely small numeric, we have 



w -»{ A [(=+£+£)' 



tryfdv dw dw du du dv\ ") 



V dy dz dz dx dx dy) J * ' ' 



This, except difference of notation, is the same as the formula 



* ' Encyclopedia Britannica,' Article "Elasticity,": reproduced in 

 Vol. III. of my Collected Papers, soon to be published. 



