OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 397 
V 
The difference of the values of +Y as given by the above expressions obtained 
from the motion inside and outside is 
2/ 2 ( j — 2/cj j i^-\~ |y) + arbitrary function of t. 
X '2 y '2 
As —■= 1, choosing the difference of the arbitrary functions of t to be constant 
and equal to 2fcb — the value of will be continuous at the surface. 
The current function of the irrotational motion is A. 
Therefore the equation to a surface always containing the same particles is 
(x' 2 + y’*) + A = const. 
That is, in elliptic coordinates :—- 
^(e+v+a 2 +b 2 )+ab lo g (v 7 a 2 +e+ \/6 3 +e) 
, a 2 J-6 2 \ 
v+—-—)= const. 
Now if e be chosen so as to make the coefficient of v vanish, it will always be 
possible to choose the constant so that the equation is satisfied. Therefore the 
elliptic cylinders corresponding to the values of e which make the coefficient of 
v vanish will be parts of surfaces \=const., and will therefore always contain the 
same particles. 
These values of e satisfy the equation 
ab 
( a 2 -& 2 ) 2 
. a 2 + 5 2 
ab ab 
0 , 7 c 
a~ + b‘ 
— 2 (/ — <u)v/(a'H -e)(b' + e) 
. a 2 + b~ 
- co~ - 
ab ab 
« 2 + & 2 \ a, (a 2 -6 2 ) 2 _/ ,Y + & 2 
2 / ' 2' ab 
co'j v 7 (cr -f-e) (b J -f-e) 
Solving this in the ordinary way, the roots obtained are 
It is necessary to examine whether these roots satisfy the above equation, or the 
equation obtained by changing the sign of the radical on the right-hand side. 
