186 Mr Love, On the Theory of Discontinuous [May 4, 



The £1 boundary is 



-1 



where c and — c are the values of u at the points in the base of 

 the vessel where the velocity is a maximum. 



7. This H region can be conformably represented upon the 

 half-plane w by means of the relation 



dO _ A w 2 — c 2 



du \/(u 2 — 1) . u (w 2 — a 2 ) 



•(H). 



Writing this in partial fractions 

 dO - Ai 



du V(l _ u *) 



VI a?-c 2 1 a 2 - c 2 1 

 a 2 w 2a 2 w — a 2a 2 u + a 



and integrating we have 



n=log\B 



i + va-^y 



W 



ylic 2 



1 - aw + V(l ~ a 2 ) V(l - u *) 1 + «w + V(l - a 2 ) V(l - w 2 ) 

 w — a tt + a 



so that 



2\-i-4i(a 2 -e 2 )- 



2oV(l-a s ) 



^ _ 1 R 



aw 2 



"i + v(i-u 2 y 



^ic 2 



l_ aM + ^/(l_ a 2 )V(l-^) l+aw+j/O-aVa-w 2 )! ^ ° 2) 



u — a 



u + a 



2a 2 V(i-« 2 ) 

 .(12). 



Now as in the preceding, since the corners of the z polygon 

 that correspond to 1 and — 1 are right angles 



-^(a 2 -c 2 ) _l 

 2aV(l-<> ~2' 



