the Theoi^ of Fluid Motion. 515 



will illustrate this remark. Suppose the fluid, ta be i^com<p 

 pressible: then )H| aioai ihiw yiini b^riuin oa vlltuii-ij rujiii .; . 



ds \r r^ )~ ' 



Hence, since ds=dr=dr^i we have by integrating, 



V= 



_ <pW 



rr 



1 * 



This expression for the velocity is general, having been ob- 

 tained prior to the consideration of any particular case of mo- 

 tion. It establishes the general law, that the quantity of fluid 

 which passes in a given small time a given small element of a 

 surface of displacement, being proportional to Yrr^, is given 

 for a given value oC (p{f), and consequently that the moment- 

 ary trajectory of the surfaces of displacement to which a single 

 disturbance gives rise is a straight line. 



If the motion take place in space of two dimensions, we 

 have 



r 



I formerly obtained this result for the case in which udx + 

 vdy is an exact differential, by a process which Professor 

 Tardy and M. Bertrand object to, and which I do not now 

 insist upon, because the above reasoning is inclusive of the 

 particular case, and the result is obtained in a more direct 

 manner. 



To apply the above general value of V to a given instance: 

 suppose a perfectly smooth sphere to move in any manner in 

 an incompressible fluid of unlimited extent, its centre remain- 

 ing on a given straight line. The general value of V applies 

 to the velocity impressed on the fluid by the surface of the 

 sphere. If V' be the velocity of its centre, then the motion 

 impressed at any point of the surface the radius to which 

 makes an angle Q with the direction of motion, is V^ cos Q. 

 Hence, R being the radius of the sphere, 



VI cos 9 = -^, or (p{t) = R2 V^ cos Q. 



Consequently, as there is no other arbitrary quantity to satisfy, 



R2 



v= % VI cose, 



where r^ is substituted for rr^, because as one surface of dt^-' 

 placement is spherical, all are spherical, their momentary tra- 

 jectory being rectilinear* This result does not agree with that 



2 L2 



