446 Prof. Challis on the Principles of Hydrodynamics. 



neglected. Then we have the known approximate equations 

 a^da- du „ a^do- dv a^da ^"'_n 



'd^'^df' If "^ di ' IF "^ IF 



Hence by integration, 



n = C-a^-f^:^dt = C-a\'LM. 

 ^ dx dx 



^ dy dy 



rv = a'-a^r^-^dt=a^-a^.tf^, 

 J dz dz 



where C, C, C" are functions of x, y, and z not containing the 



time. For all cases of motion in which no part of the velocity 



is independent of the time, for instance, cases of vibratory 



motion, we shall have C = 0, C' = 0, C" = 0. Hence substituting 



6 for — a'^fadt, it follows that 



de dd dd 



dx' dy dz 



and consequently that udx + vdy + ivdz is an exact differential. 

 Since this inference has been drawn prior to the consideration of 

 any specified case of motion, it must, according to our principles, 

 be interpreted with reference to the motion resulting from the 

 mutual action of the parts of the fluid. And as the inference 

 depends on the assumption that no part of the motion is inde- 

 pendent of the time, the physical circumstance indicated by the 

 integrability of udx + vdy + wdz is, that the motion is vibratory. 

 In accordance with this conclusion, the foregoing exact investi- 

 gation of the motion along a rectilinear axis, to far as it is inde- 

 pendent of any arbitrary forms given to the function F(^), was 

 found to be vibratory motion. 



Again, the new general hydrodynamical equation^ viz. 



may be put under the form 



which, if the squares of the velocities be neglected, becomes 



This equation gives by integration, 



