162 PROBLEMS IN THREE DIMENSIONS. [CHAP. 



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



|(o + X)* (6 2 + X)* (c 2 + X)* jj % = - (& 2 + X) (c 2 + X) & , 

 which may be written 



whence %=^ ..(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 



d<t>ld\ = -udxld\, for X = .................. (5). 



Let us write, for shortness, 



d\ 



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



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



It will be noticed that these quantities , /3 , 70 are pure 

 numerics. 



The conditions of our problem are now satisfied by 



pr vided 



that is C=H u ....... ..(9). 







