Doctrine of Extraneous Force. 119 



surface equal per unit of area to three times the intrinsic 

 rigidity-modulus, would give quadrupled virtual rigidity, and 

 therefore doubled wave-velocity ! Positive normal pressure 

 inwards equal to the intrinsic rigidity-modulus would annul 

 the rigidity and the wave-velocity — that is to say, would 

 make a fluid of the solid. And, on the other hand, negative 

 pressure, or outward pull, on an incompressible liquid, would 

 give it virtual rigidity, and render it capable of transmitting 

 laminar waves ! It is obvious that abstract dynamics can 

 show for pressure or pull equal in all directions, no effect on 

 any physical property, of an incompressible solid or fluid. 



6. Again, pull or pressure unequal in different directions, on 

 an isotropic incompressible solid, would, according to Green's 

 formula (A) in p. 303 of his collected Mathematical Papers, 

 cause the velocity of a laminar wave to depend simply on the 

 wave-front, and to have maximum, minimax, and minimum 

 velocities for wave-fronts perpendicular respectively to the 

 directions of maximum pull, minimax pull, and minimum 

 pull; and would make the wave-surface a simple ellipsoid! 

 This, which would be precisely the case of foam stretched 

 unequally in different directions, seemed to me a very in- 

 teresting and important result, until (as shown in § 19 below) 

 I found it to be not true. 



7. To understand fully the stress-theory of double refraction, 

 we may help ourselves effectively by working out directly and 

 thoroughly (as is obviously to be done quite easily by abstract 

 dynamics) the problem of § 6, as follows : — Suppose the solid, 

 isotropic when unstrained, to become strained by pressure so 

 applied to its boundary as to produce, throughout the interior, 

 homogeneous strain according to the following specifica- 

 tion : — 



The coordinates of any point M of the mass which were 

 £, 7], £ when there was no strain, become in the strained 

 solid 



IV *, vV/3, fc/7 (i); 



V'a, V/5 5 \/7j or the " Principal Elongations " *, being the 

 same whatever point M of the solid we choose. Because of 

 incompressibility we have 



"07 = 1 (2). 



* See chap. iv. of " Mathematical Theory of Elasticity " (W. Thomson), 

 Trans. Roy. Soc. Lond. 1856, reprinted in vol. iii. of ' Mathematical and 

 Physical Payers/ now on the point of being published, or Thomson and 

 Tait's ' Natural Philosophy,' §§ 160, 164, or ' Elements,' §§ 141, 158. 



