OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 383 
Illustrations. 
10. Example I. Take the simplest case of the equation in \ given in Art. 4, viz., 
where c is a constant. 
dx^df) X ° 
x 2 (y _Y) 3 \ 
It is required to examine whether — -j- ' —— ) can represent vortex sheets 
v a 2 1 
in a motion, part of which is rotational and part irrotational ( f a , b being constants). 
Substituting in the equation 
Also 
( 2 C)( aS + ;,s) e 
«=/•—«— 
v=Y 
To find xfj it is necessary to solve the equation 
d± , j. 2(y—Y) # f2x\ df_ 
dt W dx dy ~ 
The auxiliary system of equations is 
dt dx 
dy dsjr 
1 2/(y — Y) y_2fx 
b 2 a? 
0 
One integral of which is 
)= const. = 
O 7 0 
a'- tr 
and the other 
x f f 
t — — sin 1 - a / A = const. = n 
Ij a V m 
where for to must be substituted its value. 
Hence 
/ o 
/ i 
0-A) 2 ' 
