Equations of Hydrodynamics, 439 



the requisite degree of generality. All this reasoning points to 

 the conclusion that a third fundamental equation is necessary 

 for eliminating the unknown function X, and obtaining a result- 

 ing general differential equation in which the principal variable 

 is yjrj and the other variables are x, y } z, and /. I proceed to 

 the investigation of this third equation. . : 



6. Preparatory to the investigation, it will be proper to take 

 account of the following general dynamical circumstance. The 

 accelerative forces which act on a given particle at any time are 

 the extraneous forces X, Y, Z, and the force due to the pressure 

 of the fluid, the components of which in the direction of the axes 



of coordinates are -4-% -~. -~, Now these forces are by hypo- 

 pdx pdy pdz J Jr 



thesis finite, and consequently the direction of the motion of a 

 given particle cannot alter per saltum, as it would require an 

 infinite accelerative force to produce this effect in an indefinitely 

 short time. Thus, although the course of a given particle cannot 

 generally be expressed by means of algebraic functions of con- 

 stant form, it must still be such that the tangents at any two 

 consecutive points do not make a finite angle with each other. 

 Hence also the directions of the surfaces of displacement which 

 cut at right angles the lines of motion in a given element at two 

 successive instants do not change per saltum. 



7. This being premised, since the function ■x/r is, by the fore- 

 going argument, applicable at all times to all parts of the fluid, 

 the equation 



/ 7 I \ U 7 V 7 W 7 ^ 



(fifyr) = - dx -f z-dy + -dz=0 



AAA 



is a general differential equation applicable to all the surfaces of 

 displacement at all times. If therefore (^r)=0 be taken to be 

 the differential equation of any one surface of displacement, the 

 coordinates of which are x, y, z at the time t, and if x + 8x, 

 y + Sy,z-{- §z, and t + St be substituted for these coordinates and 

 for t respectively, that equation will still be satisfied if the new 

 values of the coordinates apply to another surface of displace- 

 ment at the time t + St. But from what is argued above respect- 

 ing successive surfaces of displacement of a given element, this 

 will be the case if 



8x=uSt, Sy=vSt, hz = wht, 



that is if 8x, hj } <$z be the variations of the coordinates of any 

 given element in the indefinitely small time Bt. Now by the 

 substitution of the new values -ty is changed to 



f+ 'I± Bt+ d ± uSt+ f vSt+ '!± wSt> 



' at ax ay az 



