PART I. 



ANALYTICAL INVESTIGATION. 



Section I. 



Adaptation of the general equations to the case of the fluid surrounding a body which 

 oscillates as a pendulum. General laws which follow from the form of the equations. Solu- 

 tion of the equations in the case of an oscillating plane. 



1. In a paper " On the Theories of the Internal Friction of Fluids in Motion, eye," 

 which the Society did me the honour to publish in the 8th Volume of their Transactions, I 

 have arrived at the following equations for calculating the motion of a fluid when the internal 

 friction of the fluid itself is taken into account, and consequently the pressure not supposed 

 equal in all directions: 



dp I du du du du\ IcPu cPu d'u\ 



dm " \ dt doc dy dz) Xda? dy 2 dz") 



u d (du dv dw\ 



J. _ I u _j. 1 . . (l) 



3 dx \da> dy dW 



d (du dv dw\ 

 dy 



with two more equations which may be written down from symmetry. In these equations 

 u, v, w are the components of the velocity along the rectangular axes of oo, y, % ; X, Y, Z are 

 the components of the accelerating force ; p is the pressure, t the time, p the density, and n 

 a certain constant depending on the nature of the fluid. 



The three equations of which (l) is the type are not the general equations of motion which 

 apply to a heterogeneous fluid when internal friction is taken into account, which are those num- 

 bered 10 in my former paper, but are applicable to a homogeneous incompressible fluid, or to 

 a homogeneous elastic fluid subject to small variations of density, such as those which accom- 

 pany sonorous vibrations. It must be understood to be included in the term homogeneous 

 that the temperature is uniform throughout the mass, except so far as it may be raised or 

 lowered by sudden condensation or rarefaction in the case of an elastic fluid. The general 

 equations contain the differential coefficients of the quantity y. with respect to x, y, and % ; 

 but the equations of the form (l) are in their present shape even more general than is required 

 for the purposes of the present paper. 



These equations agree in the main with those which had been previously obtained, on 

 different principles, by Navier, by Poisson, and by M. de Saint-Venant, as I have elsewhere 

 observed*. The differences depend only on the coefficient of the last term, and this term 

 vanishes in the case of an incompressible fluid, to which Navier had confined his investiga- 

 tions. 



The equations such as (l) in their present shape are rather complicated, but in applying 



• Report on recent researches in Hydrodynamics. Report of the British Association for 1846, p. 16. 



