and the Laws of Plane Waves propagated through them. 133 



will be developed which will tend to increase the displacement. The func- 

 tion (36, a) will, therefore, represent solids, liquids, or gases, accordincr as 



I shall consider, first, the equations of homogeneous solids. The function 

 (36, a), substituted in equations (30), leads to the following equations of 

 motion. 



df ^ ^dx^-^\dx'^ df^ dz')' 



^^-(^-^>5^ + ^(cI? + ^+jij. (37) 



df ^ Uz^ W dy'^dz'-J- 



These equations of motion of solid bodies were first given by M. Cauchy •* 

 equations identical in form, but with the relation A = 3P between the coeffi- 

 cients, had been previously obtained by U. NAViER.f M. PoissoN deduced 

 equations identical with those of M. Navier. Mv. Green has used the equa- 

 tions (37), with two independent constants, in his theory of light ;J and Mr 

 Stokes has recently caUed attention to the importance of retaining the two coeffi- 

 cients (leaving their ratio to be determined by experiment for each soHd) in a 

 memoir published in the Cambridge Philosophical Society's Transactions.§ 



The relation ^ = 3P is a consequence of the use of definite hite<Tals for 

 the coefficients of the function V, and only represents a particular elast°c soHd • 

 Its introduction does not alter the form of equations (37), nor does it render 

 them more simple than they are in their present state. 



The additions necessary to be made to the equations of hydrodynamics in 

 order to take into account the friction of the fluid particles, have been given' by 

 many writers. M. Navier first stated the equations in their corrected form 

 for incompressible fluids. || M. PoissoN has treated of the subject in a me- 



* Exercices des Mathematiques, torn. iii. p. 180. 



t Memoires de I'lnstitut., torn. vii. p. 389. 



t Transactions of Cambridge Philosophical Society, torn. vii. p. 11. 



§ Vol. viii. part 3. || Memoires de i'lnstitut., torn. vi. p. 414. 



