OF ROTATING LIQUID CYLINDERS. 
81 
The Series of Circular Cylinders. 
§ 15. If it were possible to calculate the /‘’s and solve equations (53) in the most 
general case, Ave should arrive at a complete knoAvledge of the system of linear series 
of equilibrium configurations. This lieing impossible, we shall start from a known 
series, and calculate successive series by determining the various points of liifurcation. 
Noav AAm knoAV (§ 12) that for the small values of the a’s, is of the form 
fnfc Hence there is a solution of the system of equations (52) 
Th 
given by cq = cq = . . . = 0 . 
This is the series of circular configurations, and corresponds to the series of 
Maclaurin spheroids in the three-dimensional problem. When on > 'lirp the solution 
breaks down physically, since the pressure at every point of the liquid becomes 
negative. In fact, when reaches the value on ■= Unp the series gives place to 
a series of annular forms, foi' each of which ofi has the critical value. We can adjust 
the radius of the annulus so as to give any desired amount of angular momentum 
greater than the critical value which occurs in the circular configuration when 
0)^ = '2iTTp. 
Points of Bifurcation on Circidar Series. 
§ IG. To search for points of bifurcation on this linear series, we replace f, by ajn 
ill equation (^54). Every term in the determinant on the left hand now vanishes, 
except the terms of the leading diagonal, and the equation reduces to 
The dilferent roots correspond to the difierent integral Amines of n, and are 
given hy 
n = I, 2, 3, 4, 5, . . . oo, 
on/'Zirp = 0, '5, ‘ 666 , ’75, ‘ 8 , . . . I. 
The first point of hifurcation (n = I) may be rejected at once, the critical 
Aubration ” being merely a displacement of the entire cylinder as a rigid body. 
A displacement in AAdiich rq only occurs for the single value n = s Avill alter the 
potential energy only by a term proportional to the sijuare of cq. Hence the 
principal Aubrations correspond to the different Amines of n from 2 to go, and are such 
that a„ only occurs for a single Amiue of n in each. When oj^ = 0, all these vibrations 
are stable. When on reaches the point of bifurcation of order s, the Aubration of order 6‘ 
becomes unstable, and, since there is only one point of bifurcation of order s, this 
vibration remains unstable for ad values of oj^ greater than the value at this point of 
bifurcation. We therefore see that by the time that oT reaches the limiting value 
2Tip, eA’ery Aubration is unstable. 
AmL. cc.—A. 
AI 
