1879.] liquid ellipsoid about its mean axis. 237 



a — b 



or ^iTTpab 



a + b 



= 2a, (a, + 0,') [a' - h') - jo)^ {^^^^ - <o'^) (a^ - h^) 



. . ,,„ . 4a^6^ ,2 4<7rpab 

 ^ ^ (a^ + bf (a+by 



As a particular case, we may have the elliptic cylinder stationary, 

 and then co-hco' = 0, 



and ^ (o =7rp \ . , ' . 



^ ab{a-\- b) 



In the particular case, considered by Kirchoff (Vorlesungen 

 ilber mathematische Physik, p, 263), 



CO -\- CO =%, co' = ^, 



(o + co' 2ab 



and therefore 



or 



co' (a + by 

 CO a^ + V 



co' {a + by 



Relatively to the cylinder, each particle of liquid describes an 

 ellipse ; for if xy refers to a particle of liquid, the axes being the 

 principal axes of the section, 



dx c^ — b^ 



^^-ya>^co-^^-^,y, 



d}J a'-b' 



-^ + XCO =6) o „ ^, 



dt a + 



da; 2a' dy 26' 



Therefore -5 -rr + ^ -7: = 0> 



a^ dt b^ dt 



0^ iy , , 



or -2 + T2 = constant, 



and if initially x = ma cos <^, y = mb sin ^, then at the time t, 

 x = ma cos ( -2 — 7-2 &>* — 9 ) , 

 2a& 



y = - mb sin f 2 ^ ^2 cot - (p 



