162 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



On substitution from Art. 109 (7) the equation (2) becomes 



x = - (Jf + X) (# + X) , 



which may be written 



whence y= - 5 ..(3), 



the arbitrary constant which presents itself in the second integra 

 tion being chosen as before so as to make % vanish at infinity. 



The solution contained in (1) and (3) enables us to find the 

 motion of a liquid, at rest at infinity, produced by the translation 

 of a solid ellipsoid through it, parallel to a principal axis. The 

 notation being as before, and the ellipsoid 



being supposed in motion parallel to x with velocity u, the surface- 

 condition is 



dct&amp;gt;ld\ = -udx/d\, for X = .................. (5). 



Let us write, for shortness, 



, r 00 d\ , r 00 d\ 7 r d\ 



a = abc / 2 . ^ s~ A &amp;gt; ^&amp;lt;&amp;gt; = abc fiA . -i\ A 7o = ooe -. . A 

 J o (a 2 + X) A J o (o + X) A J (c- -f X) A 



.............. (6), 



where A = }(a 2 + X) (6 2 + X) (c 2 + X)}* ............... (7). 



It will be noticed that these quantities , &&amp;gt; 70 are pure 

 numerics. 



The conditions of our problem are now satisfied by 



M~3 



P r Vlded 



that is C =- a6 -u ........................... (9). 



