112 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



The functions X , &c. must of course separately satisfy (1), and 

 have their derivatives zero at infinity; the surface-conditions (3) 

 then render them completely determinate. 



101. When the motion is in two dimensions (xy} we have 

 only three functions to determine, viz. &amp;lt;j&amp;gt; lt &amp;lt; 2 , ^ 3 . In the last 

 chapter (Arts. 88, 89) general methods for discovering cases in 

 which one of these functions is known were given. In any case 

 of a liquid filling a cavity in a moving solid it is plain that the 

 conditions (3) are satisfied by &amp;lt;f 1? &amp;lt; 2 , &amp;lt; 3 = #, ?/, z t respectively, in 

 other words that if the solid have a motion of translation only, the 

 enclosed fluid moves as if it formed a rigid mass. We may there 

 fore regard the kinematical part of our problem as solved for the 

 cases where the cavity is in the form of an elliptic cylinder, or 

 a triangular prism on an equilateral base, for which ^ 3 has been 

 found*. 



In the more difficult problem of a cylindrical body moving 

 through an infinite mass of liquid, the complete solution has been 

 obtained for the case where the section of the cylinder is elliptic, 

 and for this case only. 



102. The number of cases in three dimensions for which the 

 functions &amp;lt; lf &c., % 1} &c. have been completely determined is very 

 small. We give here the chief of them. 



Example 1. An ellipsoidal cavity whose semiaxes are a, b, c. 

 If the principal axes of the ellipsoid be taken as axes of co-ordi 

 nates, and h be the perpendicular from the centre on the tangent 

 plane at (x, y, z\ we have then 



7 hx hy hz 



l,m,n=- a ,, -g,, ? -, 



respectively, so that the last three of conditions (3) give 



But also 



*i.&I + 4k + 46 a 



an ax ay dz 



* The student will find in Thomson and Tait, Natural Philosophy, Art. 708, 

 other forms of cylindrical cavity for which solutions can be obtained. 



