394 
PROFESSOR M. J. M. HILL OR THE MOTION OF FLUID, PART 
When 
p=l, E=C\/a 2 +e4-D\/6 2 +e 
U= O' v/^+h+D V-& 2 -v 
1>=2, E=c( £ +^) +D y(a?+*j(V‘+ 7). 
U=C'(t>+~)+DVV+»)(-i J -») 
and so on. 
Now suppose it required to express A as the current function of an irrotational 
motion under the condition that 
. j%D CO / | /o\ I | !' -t 
A =/^+/^-2 wherever 
Assume if possible 
/$+/£-f (^+^) 
= A + Blog(v^'a 2 +e+ \/h J -f e) + 
c 
■^UDvV+ £ )(& 2 +e) 
a 2 +& 2 
The form of the expression on the right-hand side shows that it is the current 
function of an irrotational motion. 
But 
a a + 5 3 w a s + js\ a?-l\ . /a 3 -Z> 2N9 
£ +-2*jh + ~2~j=~( x -y-)~ 
cP + W a?-W-1 A 3 ?/ 2 
2 W 2 + e & 2 +1 
therefore the assumption is 
x '2 w 
=A+B log (vV+i+v^-H+C ^ y 2 
+ D~v/(a a +€)(^+ £ ) 
a 2 -Z> 2 
a 2 -6 2 . . / A 3 ?/ /2 
a 3 + e 6 2 + e 
#' 2 ?/' 3 
If this is to be satisfied when — + =1, and therefore when e=0, it is possible to 
i a 2 ~ 1 “ b n - 
l y n y'2 
to either side and then equate the coefficients of x' 2 , y' 2 . The con- 
