1891.J 



Fluid Motions in two dimensions. 



189 



When u > 1, we must write (14) in the form 



£-£n-.v<tf-i>] 



-|a+*) 



ft-tV(tt g -l) 

 V(w 2 -1 + F) 



1) 



V(w 2 -l+-t 2 ) + 2 



U 9 -l 



y (i* a - 1 + &■) VO 2 - 1 + & 2 )_ 



For the imaginary part of £ taking cosh = ujk' we have 



-■•|y(0 + sinh0cosh0)-0 



+ const., 



or writing z = %+ ty 

 1 



y=2 



(l + k)0- 5—^ (0 + sinh cosh 6) 



dy 



+ const. 



The greatest value is given at once by -j- = 0, so that 



l=Jc, 



Je + 1 1 



or 



u 



cosh 2 6 = 



l-F 1-k' 

 giving sinh 6 = *Jk (1 + ^k), 



0=y&(i + p). 



The value corresponding to u = 1 is given by 



COSll 2 = - To , 



giving sinh = k(l + |& 2 ), 



= jfe(l + p 2 ). 



Hence the height above the rim through which the free stream- 

 line rises before turning back is 



i(iH-*)v*a+i*)-iv*[(i+t*)+ a +i*)a+*)] 



- 1 (1 + k) k (1 + p 2 ) + ^ [(1 + iF) + (1 + p 2 ) (1 + ¥)}, 



and this is ultimately 



5 i$ 



rejecting higher powers of k. 



Hence if e be the ratio height of rim to breadth of disc the 

 greatest height above the rim to which the free stream-line rises 

 before turning back is 



^ (2e) f of the breadth (19), 



or about 



a e* of the breadth, 



