Mr. R. Moon on Intei'nal Friction in Fluids. 529 



\dp _v_d{u) du ' 



p dx dt' — ^ d.v 



p di/ dt ~ chj I ' 



^ dp _,, _ d{w) j„ d w 



(2) 



p ds dt ~ dz \ 



where p denotes tlie density ; X, X> Z the iilipressed forces acting on 

 the element ; n, v, w the resolved parts of the velocity parallel to the 



coordinate axes ; -^ the total differential coefficient of u with 

 dt. 



respect to t, &c. ; and Ic replaces the li^ of the pre.ceding case. The 

 author considers that, for moderate ranges of density, the above equa- 

 tions accurately represent the whole- internal resistance. 



It is next shown, that when the fluid is inelastic, the same equa- 

 tions will represent the motion, provided that we obliterate p in the 

 terms involving F. 



The force of internal friction in an elastic fluid in which the 

 whole motion takes place parallel to the axis of x, and in which the 

 whole lateral variation of motion transverse to the axis of x occurs in 

 a direction parallel to the axis of z, is then ^hown to be properly 



represented by +H^p — , where n^ is a constant depending on the 



nature of the fluid ; the sign of the term to be introduced into the 

 equation of motion being determined by the consideration that fric- 

 tion must always be a retarding force. The author thence derives 

 the conclusion, that in order to represent the effect of internal fric- 

 tion in the motion of an elastic fluid, we must add to the first of 

 equations (2) a term of the form 



where 



V-=?<^H-«--|-2<;^ 

 and 



\ dtj dx ) \ dx dz J \ dz dy ) 



and similarly with regard to the other two equations. When the 

 fluid is inelastic, the terms in the equations of motion depending 

 upon friction will be identical with those in the preceding case if we 

 obliterate from the latter p. 



Reverting to the equation of sound, which (neglecting terms of the 

 second order) may be put under the form 



dt' dx^ dx dt 



the author next shows that if the initial disturbance consist of a con- 

 densation alone, it will be transmitted with the velocity «(1 — e) in 

 the direction in which its particles are moving ; and that if it consists 

 PIdL Ma(j. S. 4. No. 110. fiupid. Vol. 10. % M 



