402 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
The integrals are 
r=m , 
X 
-6 
t+ ¥(P) 0 - n ’ 
2 ' F'(r)J 1 Vn r 
= ( 1- 
therefore the integral of the original system is 
'=6 
■^ } +fy? F P»^+xK«) 
Take \=F(r), \p= y $'(m, n ) 
To reduce the components of the velocity to Clebsch’s forms, it is necessary to find 
X and so that 
i.e. 
Whence 
0 = 6 
F'(?•) rY\r) , (Y(r)-r¥"(r) 
{*»} 2 
yp>}41+“ 
M r w , bV n Y(r)-rY\r) 1 
+ bn {F^r)} 2 '\hmYbn {F'(r)} 2 J 
~F(r): 
F \r) 
F(r)J 
2 dr bX 1 tT' F(r) 
r im'F'(rW i»*F(r) 
bX , ,FF 
bm 
bX 
bn 
+F(r )^ = _ i{ F(r)}^-[lF'(r)}^ 
To satisfy these put ^F=?i$>(r) in each, then since m=r, it follows from the first of 
these equations that 
X= — wj'F(r)^> / (r)cZr+©(?i) 
Substitute in the second equation, therefore, 
. - fF(r)9.'(»-)*+e'W + F<r)®(r)= -i{F(r)}»-j {F(r)}*| r 
It is necessary that ®'(n) should be constant, and it will be found that the constant 
may be taken as zero. 
