OF ROTATING LIQUID CYLINDERS. 
85 
The remaining curves are the graphs of 
?/ = 
1 + 
n 
for the values n = 3, 4, 5, . . . 
The curve ( 66 ) cuts a — Q at the point y — ijn, and a = I at y 2 /n. The 
value of dylcloL is It is therefore obvious that the curves are convex to the 
axis of a, and since for any value of a the value of dyjda is greatest for that curve 
for which 7i is least, it is obvious that the curves can never intersect. 
We therefore see that the parabola (65) will meet each of the curves (66) once, 
and once only, for values of a between 0 and 1 . Moreover the smaller n is, the 
smaller the value of a at the intersection. 
As we move along the series of elliptic cylinders, the value of a increases Irom 
0 to I. Hence there will be an infinite number of points of bifurcation on this 
series, of orders 3, 4, 5, . . . . The point at which we arrive first is that of order 
91 = 3 ; those of orders 4, 5, . . . follow in succession. As before, we find that the 
configuration at the end of the series of elliptic cylinders (a = I, an infinitely long 
and thin ellipse) is unstable for every vibration. 
The linear series which we expect to be stable is that of order n = 3. To 
X 
find the point of bifurcation of this series ^ve require to solve the equation 
|-(1 — a~) = (I -|- a®) and the solution is found by inspection to be a — 
From equations (52) and (53) we find that at this point of bifurcation co^ = 
and a., = f. The elliptic cylinder at the })oint of bifurcation is therefore the 
cylinder 
-f + 7^3).(67) 
or, in Cartesian co-ordinates, 
X-+ 97/= 5r.d .( 68 ). 
If we reduce the linear scale of this until the area is cd, we find for its equation 
+ 9y^ = 3cr, 
and for its angular momentum, 1‘46 times the greatest angular momentum for which 
the circular form is stable. 
Poincare’s Series of Pear-shaped Cm'ves. 
§ 20 . The configuration of the new linear series of order n = 3 is, near the jioint 
of bifurcation, of the form 
= + + + . • • . ( 66 ). 
This new series is seen to be the series corresponding to Poincare’s series of pear- 
shaped figures."^' Instead of making a separate problem out of the determination 
of the constants and 63 , we shall, in order to avoid repetition at a later stage, pass 
* H. Poincare, ‘ Acta Math.,’ vol. 7, p. 347. 
