388 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
A particular integral is 
*=(/) 
v 2 cos” (6 — go) r 2 sin 2 (6 — a>) 
~o? •" 
if */(£+£) — 2&>=c(=—2£ of Kirchhofe’s “ Vorlesungen liber Mathematische 
Pbysik.” Zwanzigste Yorlesung). Therefore w is constant. 
Let x, y' be the coordinates of the point x, y if referred to the principal axes of the 
ellipse which rotates with uniform angular velocity do ; u' v' the components of the 
velocity parallel to these moving axes. 
Thus 
x=x cos oit-\-y sin dot ; y'= —x sin d>t-\-y cos dot 
The velocities along and perpendicular to the radius vector are 
= _Ux-^ = _y + |_n,,_//_/v. 
©= 
dr\" 2 J \a 2 1 b 2 \a 2 b 2 
r cos 2(6—a>) 
u — It cos (6—oj) — © sin (6 — co) = [^—cb^jy' 
A=B. sin (6— w)+© cos (6 —<y) = — — cb^jx' 
u—u cos dot — v sin dot 
v—u sin d>t-\-v cos dot 
lx' 2 v' 2 
x= wb + ? 
To find ip it is necessary to solve the equation 
The auxiliary system is 
g -±+M + M=o 
dt dx dy 
dt dx dy difr 
1 u v 0 
It will be more convenient to obtain xp in terms of x, yf. 
Since 
dx— cos cot.dx-\- sin dot.dy-\-doy .dt 
dy — — sin dot.dx-\- cos cbt.dy—cox ,dt, 
rV 
