PAPER BY PROF. OBERBECK. 161 



Therefore, the function W is known for the outer region and is 



In order to determine tliis function for the inner region also one must 

 go back to the equations (13) 



c\v ■ dy '\ dy :)x J 



?y ?J^ ' V ?'X' ^y 



First we make the assumption that C is constant throughout the 

 whole inner region : We can then write 



|(/,_C TT )- •^(;/,+C ^ )=0 



These equations are satisfied if we put 



/i — r Tr=constant; /o+T <^=constant. 



By considering equation (12) there follows from the last equation 

 especially 



A- Tl'-4-(A+^) <7>=Constant 



TT"= ^<7>+Constant. 



n 



From the first of these equations we also obtain, 



I- J ir+(A+r) J (p=0 

 or 



A- :=c (A+r) 



whence 



C=5 and TV — — = ffl+Constant 



But in general the values of IT thus found merge continuously into 

 €ach other at the borders of the two regious quite as little as do their 

 differential quotients. Hence it follows that the component velocities 

 also, and therefore both the velocity and also its direction, suffer sudden 

 changes of finite magnitude at the boundaries of the two regions. We 

 have therefore found only one special solution, and not one that obtains 

 in general. This special solution is that which Guldberg and Mohn 

 have used in the special case of a circular boundary for the inner 

 region. Corresponding to it they find that in the outer region the di- 

 80 A n 



