IN HYDRODYNAMICS. 37 



Since V and — — are the same at all points of a given surface of displacement when the 

 as 



motion is rectilinear, it follows from equation (8) that in incompressible fluids - + — is the 



r r 



same at all points of the same surface of displacement. This is true also when the fluid is 



compressible. For since p is given for a given surface of displacement, and since the equation 



du , dv dw ,.1.. 11 (./.i-i 11 



— a* H dy + - — dx = 0, obtamed m Art. 7, proves that the surface ot displacement through 



dt dt dt 



a given point does not vary its position, it follows that -j- is the same at all points of the 



same surface of displacement. It has been already shewn that this is the case with ' and 



ds 



Vp. Hence equation (9) shews that - + — is the same at all points of a given surface of 



r r 



1 



displacement. Again, if Sr be an indefinitely small constant quantity, ^ + -; 5- 



"^ r + hr r + 67 



stant for the next contiguous surface of displacement. Hence if - + — = c and s — 1- , ^ 



r r r + cr r + dr 



11 ^ 



= c + Sc, we have — + --= — -^ = a constant. It follows that r and r must each be constant 



r r dr 



for a given surface, and consequently that not only is the curvature the same, but the principal 

 radii of curvature the same at all points of the surface. The only surfaces that possess this 

 property are the surface of a sphere and that of the common cylinder. Hence the only motions, 

 whether of incompressible or compressible fluids, that are possible when udx + vdy + wdz is 

 an exact differential, are in straight lines drawn from a fixed centre or perpendicular to a 

 fiaied axis. 



10. By reviewing the reasoning which has conducted to the above conclusion it will be 

 seen that after proving the dynamical equation to be integrable from any one point of the 

 fluid to any other whenever udx + vdy + wdis is an exact differential, the equations (8) and (9) 

 were assumed to be integrable in like manner. It is necessary therefore to inquire under what 

 circumstances the result obtained in the preceding Article is consistent with that assumption. 



Let udx + vdy + wdz be an exact differentia], be a function of r, and r' = x^ + y' + z". 

 Then the equation of continuity of a compressible fluid becomes, 



d(})\ dr(j> drcj) d^ d=(j) d^ /2A= Xx Yy Z%\ _ 

 dr^j dr' dt' dr'drdt dr \ r r r r ) 



which does not agree in giving tp a function of r unless the impressed force either be nothing 

 or a function of r. No such limitation is necessary with reference to incompressible fluids, 

 because the equation of continuity applicable to them becomes, 



d'(h d(b 

 -Ti + 2 X^ = 0, 

 dr^ dr 



which gives (p a function of r, whatever be the impressed force. It is, however, necessary 

 that JTdx + Vdy + Zdz be integrable. 



11. The investigation I have now gone through, shews that there are several defects in 

 the reasoning of my last pajjcr, which I will endeavour to point out as distinctly as possible. 

 The first occurs in Art. (i (p. 377), wlicre it is asserted that " f^dr is not an exact differential, 

 unless the variation of V from one point of space to anotiier at a given instant, depends only 



