170 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



by the envelope, must form an equilibrating system. The effect of 

 these latter pressures may be calculated from the formula 



A pressure uniform over the envelope has no resultant effect; 

 hence, since &amp;lt;f&amp;gt; is constant at the beginning and end, the only 

 effective part of the integral pressure fpdt is given by the term 



-tpffdt ........................... (2). 



Let us now revert to the original form of our problem, and 

 suppose the containing envelope to be infinitely large, and in 

 finitely distant in every direction from the moving solid. It is 

 easily seen by considering the arrangement of the tubes of flow 

 (Art. 37) that the fluid velocity q at a great distance r from an 

 origin in the neighbourhood of the solid will ultimately be, at 

 most*, of the order 1/r 2 , and the integral pressure (2) therefore of 

 the order 1/r 4 . Since the surface-elements of the envelope are of 

 the order r$-ar, where OT is an elementary solid angle, the force- 

 and couple-resultants of the integral pressure (2) will now both 

 be null. The same statement therefore holds with regard to the 

 time-integral of the forces applied to the solid. 



If we imagine the motion to have been started instantaneously 

 at time t Q , and to be arrested instantaneously at time t lt the result 

 at which we have arrived may be stated as follows : 



The impulse of the motion (in Lord Kelvin s sense) at time 

 j differs from the impulse at time t by the time-integral of the 

 extraneous forces acting on the solid during the interval ^ f. 



It will be noticed that the above reasoning is substantially 

 unaltered when the single solid is replaced by a group of solids, 

 which may moreover be flexible instead of rigid, and even 

 when these solids are replaced by portions of fluid moving 

 rotatiorially. 



117. To express the above result analytically, let f, 77, f, X, //-, v 

 be the components of the force- and couple-constituents of the 



* It is really of the order 1/r 3 when, as in the case considered, the total flux 

 outwards is zero. 



t Sir W. Thomson, I.e. ante p. 35. The form of the argument given above was 

 kindly suggested to the author by Mr Larmor. 



